MICHIGAN

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This is to certify that the

dissertation entitled

THREE ESSAYS ON THE ELECTORAL MECHANISM

presented by

Sugato Dasgupta

has been accepted towards fulfillment
of the requirements for

Ph.D . degree in Political Science

fight

a Major professor

Date “1/”, q 7

MSU is an Affirmative Action/Equal Opportunity Institution

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THREE ESSAYS ON THE ELECTORAL MECHANISM
By

Sugato Dasgupta

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PI-HLOSOPHY

Department of Political Science

1997

 

 

 

 

ABSTRACT
THREE ESSAYS ON THE ELECTORAL MECHANISM
By

Sugato Dasgupta

This dissertation analyzes the efficiency of the electoral mechanism. Two distinct-
but-related topics are addressed in the three chapters.

Chapters one and two investigate the issue of electoral accountability. In the
presence of informational advantages for candidates, we investigate whether the threat of
electoral sanction provides sufficient incentive for candidates to comply with voter demands.
Using formal principal-agent models that combine adverse selection and moral hazard

elements, the theoretical possibility of candidate accountability is demonstrated. The

 

 

theoretical conclusions are buttressed with evidence from a series of laboratory experiments.

In chapters one and two, rules governing all social interactions are exogenous; that
is, they are taken as given. In contrast, chapter three constructs a formal model that
eIldogenizes rules. In brief, we posit that political parties in power establish rules so as to
confer benefits upon their constituents. We ask the following questions: What determines
the nature of the rules in place? When are they flexible and what makes them rigid? We
adOPL furthermore, a comparative perspective and contrast polities in a parliamentary system
With those in a separation-of-powers system. A surprising mathematical result is proved -—
rules generated in the parliamentary system are more flexible than those produced by the

SCparation-of-powers system.

L

 

 

1?!

ACKNOWLEDGMENTS

Writing a dissertation is a lonely and tedious task. Fortunately, several individuals,
in various capacities, helped lessen the burden. It is pleasant to acknowledge their assistance.

The members of my dissertation committee -- Scott Gates, Jim Granato, Gretchen
Hower and Mark Jones -- adequately monitored my progress. In return, they deserve
commensurate thanks. Rhonda Burns, Iris Dunn and Elaine Eschtruth were helpful beyond
the call of duty. They expertly ironed out numerous bureaucratic wrinkles without
complaint. Consequently, I was able to work in relative peace.

My friends -- Huimin Chung, John Uniack Davis, Erick Duchesne, Bryan Marshall,
Roger Moiles and Brandon Prins -- alleviated the monotony of graduate work. I remember
with fondness the many hours whittled away in idle banter. I owe an additional debt of
gratitude to John and Roger. John read much of the dissertation. His cogent comments
enhanced the quality of the exposition. Roger supplied me with much—needed
encouragement when work was progressing slowly. The completion of the dissertation is,
in part, due to his uncommon grace.

Finally, I want to thank my family -- Baba, Ma, Sona, Dola and Padmanabh —- for
years of love and support. Despite the physical separation, my mental world has always been

filled with their laughter and their tears.

L

 

 

 

 

TABLE OF CONTENTS

LIST OF TABLES ...................................................... vi
LIST OF FIGURES .................................................... viii
1. CHAPTER 1: EFFICIENCY IN A MODEL OF ELECTIONS WITH MORAL HAZARD
-- SOME EXPERIMENTAL EVIDENCE .................................... 1
1.1. Introduction ................................................... 1

1.2. The Experimental Model ........................................ 7

1.2.1. Phase One -- Incumbent Candidates' Effort Choice ............ 8

1.2.2. Phase Two -- Payoffs of Players ........................... 9

1.2.3. Phase Three -- Election Outcome ......................... 10

1.3. Experimental Predictions ....................................... 11

1.3.1. Definition of Sequential Equilibrium for the Model ........... 11

1.3.2. Refining the Sequential Equilibrium for the Model ........... 14

1.3.3. Predictions from the Model's Sequential Equilibria ........... 18

1.4. Experimental Design ........................................... 20

1.4.1. Repeated-Interactions Setup -- Experimental Procedures ....... 21

1.4.2. One-Shot Setup -- Experimental Procedures ................. 23

1.4.3. Experimental Parameter Values ........................... 24

1.5. Experimental Results .......................................... 26

1.5.1. Experimental Results for Repeated-Interactions Sessions ....... 26

1.5.2. Experimental Results for One-Shot Sessions ................ 54

1.6. Conclusion .................................................. 55

1.7 Formal Proofs of Propositions .................................... 57

2. CHAPTER 2: SIGNALING IN ONE-SHOT AND REPEATED ELECTIONS —- SOME

EXPERIMENTAL EVIDENCE ........................................... 62
2.1. Introduction .................................................. 62

2.2. The Experimental Model ....................................... 67

2.3. Solution to the Experimental Model ............................... 70
2.3.1. Model Solution: The One-Shot Case ....................... 70

2.3.2. Model Solution: The Repeated—Interactions Case ............. 75

2.4. Experimental Design ........................................... 81
2.4.1. One—Shot Setup -- Experimental Procedures ................. 82

2.4.2. Repeated-Interactions Setup -- Experimental Procedures ....... 83

iv

 

 

 

 

2.4.3. Experimental Parameter Values ........................... 85

2.5. Experimental Results .......................................... 89

2.5.1. Experimental Results for FI-OS and II-OS Sessions ........... 90

2.5.2. Experimental Results for FI-RI and II-RI Sessions ........... 102

2.5.3. Informational Efficiency in Experimental Elections .......... 107

2.6. Discussion .................................................. 109

2.7. Formal Proofs of Propositions .................................. 111

3. CHAPTER 3: RIGIDITY OF RULES IN ELECTORAL SYSTEMS ........... 114

3.1. Introduction ................................................. 1 14

3.2. A One-Party State ............................................ 121

3.3. A Two-Party Separation-of-Powers System ........................ 123

3.3.1. Description of the Model ............................... 123

3.3.2. Definition of a Markov Perfect Equilibrium ................ 128

3.3.3. The Model's Solution and Implications .................... 134

3.4. A Two-Party Parliamentary System .............................. 147

3.4.1. Description of the Model ............................... 147

3.4.2. Definition of a Markov Perfect Equilibrium ................ 151

3.4.3. The Model's Solution and Implications .................... 156

3.5. Comparison of the Two Electoral Systems ......................... 163

3.6. Conclusion ................................................. 170

3.7. Formal Proof of Result 1 ...................................... 172
APPENDIX A

TABLES FOR CHAPTER 1 ............................................. 191
APPENDIX B

TABLES FOR CHAPTER 2 ............................................. 196
APPENDIX C

FIGURES FOR CHAPTER 3 ............................................ 201

LIST OF REFERENCES ................................................ 206

v

 

 

 

Table 1:
Table 2:
Table 3:
Table 4:
Table 5:
Table 6:
Table 7:
Table 8:
Table 9:
Table 10:
Table 11:
Table 12:
Table 13:
Table 14:
Table 15:
Table 16:

Table 17:

LIST OF TABLES
Parameter Values (in francs) for the Experiment ...................... 191
Summary of Observations for Repeated-Interactions Sessions ........... 191
Regression-based Analysis of Candidates' Effort ...................... 192
Analysis of Voters' Behavior ..................................... 192
Analysis of Voter Heterogeneity .................................. 193
Predicted and Observed Candidates' Effort Levels ..................... 193
Regression-based Analysis of Effort Discrepancy ..................... 194
Relative Efficiency of Experimental Elections ........................ 194
Summary of Observations for One-Shot Sessions ..................... 195
Parameter Values (in francs) for the Experiment ..................... 196
Equilibrium Set for One-Shot Experimental Elections ................ 196
Equilibrium Set for Repeated-Interactions Experimental Elections ....... 197
Summary of Observations for FI-OS and II-OS Sessions .............. 197
Rationality of Candidates and Voters in FI—OS and II-OS Sessions ...... 198
Equilibrium Selection for One-Shot Elections (Consistent Responses) . . . . 198
Equilibrium Selection for One-Shot Elections (Likelihood Methods) ..... 199
Summary of Observations for the FI-RI and II-RI Sessions ............ 199

vi

 

 

 

Table 18: Equilibrium Selection for Repeated Elections (Consistent Responses) . . . . 200
Table 19: Informational Efficiency of Experimental Elections .................. 200
vi 3'.
L— 7

 

 

 

LIST OF FIGURES
Figure 1: Solution for the Separation-of-Powers System ....................... 201
Figure 2: Welfare Loss Triangles for Party PL ................................ 202
Figure 3: Solution for the Parliamentary System .............................. 203
Figure 4: Comparison of the Two Electoral Systems .......................... 204
Figure 5: g-(wiggle) Mapping for Lambda-space ............................. 205

viii

 

 

 

iii

i'vfi

1. CHAPTER 1: EFFICIENCY IN A MODEL OF ELEC-
TIONS WITH MORAL HAZARD — SOME EXPERI-

MENTAL EVIDENCE

1.1. Introduction

In a well-fimctioning representative democracy, elected public officials serve to im—

plement some notion of the “collective will.” Periodic elections are the mechanism

 

by which the electorate disciplines these public ofiicials. In essence, irrespective of
the extent of divergence between the innate interests of the public and the elected
officials, the threat of electoral defeat provides sufficient incentive for the latter
to comply with popular demands. Nevertheless, the degree to which electoral
incentives constrain the behavior of public officials is debatable.

As Austen—Smith and Banks (1989) observe, in an ideal world of complete
information, effective accountability can be easily achieved. The presence of in—
complete information on the part of voters substantially weakens the power of
electoral control. Informational asymmetries arise for at least two reasons. First,

since public officials Specialize in the day-to—day tasks of government, they are

 

 

(m

 

 

privy to more information than the voters. Second, even if all pertinent informa-
tion is readily available, rational voters acquire information only if the incremental
benefit exceeds the resource cost of its acquisition. Since each voter is a small
percentage of the population, the probability of casting a decisive vote is minis—
cule. Thus, the value of information is close to zero and voters choose to remain
rationally ignorant (e.g., Downs (1957) and Ferejohn (1990)).1 If informational
asymmetries characterize all real-world democracies, then the appropriate policy
question becomes: how does the quality of available information affect a repre-
sentative democracy?

Researchers have examined this question by constructing abstract election
models that acknowledge, in varying detail, the presence of informational fric—
tions between voters and potential candidates. Theoretical discussions subdivide
into two camps: Barro (1973), Ferejohn (1986), and Austen—Smith and Banks
(1989) consider informational problems from a moral hazard (hidden action) per—
spective; Rogoff and Sibert (1988), Alesina and Cukierman (1990), Reed (1990),
Rogoff (1990), Harrington (1993), and Banks and Sundaram (1993) consider in-

formational problems from an adverse selection (hidden types) point of view. By

 

1We should note, however, that within a rational choice framework, Ledyard (1986) and
Palfrey and Rosenthal (1985) demonstrate that a single vote may be more relevant than is
generally believed.

 

 

 

computing a model’s equilibrium, the theoretical exercises investigate the effect
of information in an election environment.

The theoretical literature provides intuition regarding the performance of demo-
cratic systems when information is of poor quality but the predictive validity of
such models is difficult to ascertain. In fact, electoral outcomes are without excep—
tion dependent on variables such as information structure, utility fimction char-
acteristics, and the values of rewards and punishments. Yet, data seldom provide

2 Therefore, we adOpt a different

even error-laden measures of these variables.
stance. First, we construct a model of elections that incorporates informational
frictions between the elected public officials and the electorate. Second, we eval-
uate the theoretical model using experimental techniques. Within the structured
environment of a laboratory, it is relatively simple to both induce and systemati—
cally control the values of the parameters of interest. This capacity for variation

allows for a thorough consistency-check of the theory.3

The theoretical setup involves an infinite horizon model of elections comprised

 

2A large empirical literature tests the reduced form comparative static predictions 0f theoret—
ical political-agency models. Examples include Kalt and Zup an (1990), L0“ and Davis (1992),
Lott and Bronars (1993), and Besley and Case (1995a, and 1995b).

3We are not arguing that experimental methods are intrinsically superior to non-experimental
methods. Rather, they are a valid tool for testing theory when naturally occuring data is of
dubious quality. The close relationship between experimental evidence and innovative field
studies is explored in Roth (1991).

 

 

 

 

 

of two candidates and an electorate of homogeneous voters. In each period, the
incumbent candidate for that period selects an effort level from a choice set.
Effort, for example, can take the form of time spent drafting legislation, casework
solicitation and accessibility to constituents. One feature of effort is that only
a fraction of the electorate will be cognizant of how much efiort the incumbent
expends (e.g., Stokes and Miller (1962) and Abramowitz (1980)). In sum, there
is a significant moral hazard component to effort.4
In our model, voters estimate the effort level of the incumbent by examining
the realized output (i.e., what projects or policies the incumbent has actually
produced). We model output as being probabilistically dependent on the effort
expended. Consistent with the principal—agent literature, larger effort levels pro-
duce better output realizations on average.5 We emphasize that when interpreting
the performance of the incumbent, the electorate never directly observes the ef—
fort level. Rather, since eifort level and realized output are correlated, ex post
performance conveys some information to voters about the incumbent’s diligence
in oflice.

The root of the principal-agent problem is as follows. Effort undertaken by

 

4For an exposition of moral hazard models, the reader is referred to Holmstrom (1979) and

Grossman and Hart (1983).
5The need to model “good” output realizations as likely signals of “high” effort is explicitly

reCOgHized by Milgrom (1981).

 

 

 

 

the incumbent iS privately costly in terms of resources expended (e.g., time).
Therefore, output levels preferrable to the voter impose higher costs on the in—
cumbent candidate. How, then, can the incumbent be induced to expend costly
and unobservable effort? Electoral control is partially accomplished by using the
office-holder’s ambition to discipline her. Specifically, since the incumbent derives
private benefit from holding office, expending effort becomes rational should vot—
ers reward “good” performance by reelecting the incumbent and punish “poor”
performance by electing the opposing candidate. In sum, the institutional struc-
ture of repeated elections provides the electorate with an incentive device with
which to mitigate the informational advantages of the elected public official.6

For the model sketched above, we provide some theoretical guidance regarding
likely electoral outcomes. Given our multi—player, repeated-interactions environ-
ment, there exists an abundance of sequential equilibria. For the purposes of
sharper predictions, we focus on the set of symmetric and stationary sequential
equilibria. Our theoretical predictions fall into two categories. First, we prove the
existence of a continuum of symmetric, stationary sequential equilibria and we ex-

plicitly compute an upper bound on the amount of effort that can be elicited from

 

6Our formal model is similar to that of Ferejohn (1986). Voters in Ferejohn’s model confront a
moral hazard problem: they observe incumbent candidates’ actions with some noise. However,
unlike our setup, incumbent candidates in Ferejohn’s model know the consequences of their
actions with certainty: they observe the realizations of noise prior to action choices.

 

 

 
 
 
 

 

 

 

candidates. We interpret the “efiort upper bound” as representing the maximal
feasible efficiency of the elections model. Second, holding constant the behavior
of voters, we find that a candidate’s effort level is an increasing function of 1)
the private benefit of holding office, 2) the rate of time discounting, and 3) the
productivity of candidate effort.7

To test the predictions of the model, we conducted a series of laboratory ex-
periments. The experimental sessions were of two sorts. In one-shot sessions, with
the composition of the electorate held fixed, each pair of candidates participated
in only a single election. By contrast, in the repeated-interactions sessions, the
same two candidates participated in a series of structurally identical elections.
Across both session-types, we varied two parameters of interest: 1) the private
benefit of holding office; and 2) the productivity of candidate effort.

The data are consistent with the theoretical predictions of the baseline model.
Our principal findings are twofold. First, given the absence of reelection pressure,
candidates in one-shot sessions were unwilling to expend nonnegligible levels of
effort. Second, in repeated—interactions sessions, average candidate efiort was

increasing with respect to both the private benefit of office and the productivity

 

7Productivity of candidate effort is defined as follows. We exogenously increase candidate
eifort at the margin and we let productivity be the associated increase in the probability with
which “better” electoral outcomes obtain.

 

 

 

 

 

of effort. In sum, candidates exhibited behavioral patterns that were sharply
responsive to incentives implicit in the theoretical setup.8

The remainder of this chapter is organized as follows. In section 1.2 we de—
scribe the experimental model. Section 1.3 provides the analytical solution to the
experimental model. The experimental design for our empirical tests is in section
1.4 and the results are described in section 1.5. Section 1.6 concludes the chapter
from a substantive perspective. All theoretical proofs are relegated to section 1.7

while tables presenting the empirical results are gathered in Appendix A.

1.2. The Experimental Model

The model explores the interaction between two candidates, denoted by K E
{A, B}, and an electorate of n voters, denoted by N E {1,2, ...,n}. The inter—
action spans an infinite sequence of structurally identical periods. Each period
consists of three phases. In phase one, the incumbent candidate for that period se-
lects an effort level from a choice set. In phase two, voters and the two candidates
receive payoffs. In phase three, an election determines whether the incumbent

candidate is reelected. A description of the three phases follows.

 

8Several researchers in experimental economics have investigated the impact of repeated
interactions in ameliorating various enforcement problems implicit in one-shot games. Examples
include Palfrey and Rosenthal (1992), Feinberg and Husted (1993), and Davis and Holt (1994).

 

 

\- gr]

 

 

1.2.1. Phase One — Incumbent Candidate’s Effort Choice

Each period, t E Z+, begins with a candidate already in office. We refer to the
current office-holder as the period-t incumbent candidate and we term the remain-
ing candidate as the period-t challenger. Unobserved by either the electorate or
her challenger, the period—t incumbent candidate selects an effort level, denoted
at, from the set E '5 [0,1).

Once et is chosen, an output, denoted yt, is stochastically generated from the
two-point set Y _=_ {yL, yH}, where yH > 3);, > 0. The following three assumptions
are imposed on the output generation process.

[Al] Probability{yt = yH | 6t} = 7rL + 6,; x (7TH — 7rL).

[A2] Probability{yt = yL I ct} = (1 — 7rL) — at x (7rH — 7rL).

[A3] 7r” 2 7rL and 7r, E [0, 1],z'E {L,H}.

In the above notation, “Probability{yt = yilet}” represents the probability that
the period-t output is y,- (y,- E {yL, yH}) when the period-t effort is at. 7TH, m, are
two arbitrarily chosen numbers from the unit interval such that 717; weakly exceeds
FL, Assumptions [A1] and [A2] jointly ensure that: Probability{yt :2 3m | et}+
Probability{yt = yL | ct} = 1. Thus, subsequent to the choice of 6,, the ensuing
output must equal either y L or yH.

The three assumptions assert that, as the effort level rises, there is a corre-

 

 

 

 

sponding increase in the probability that the realized output is the superior out-
come (yH). Specifically, observe that: 6 [Probability{yt = my I et}I/8 e, = (7TH—
7rL) Z 0.

In unambiguous terms, [Al] - [A3] define the productivity of candidate ef-
fort. We measure productivity by the increase in the probability of generat-
ing 3);, corresponding to an exogenous unit increase in the effort level (i.e., 6
[Probability{yt = m; I et}I/8 et). Since 6 [Probability{yt = 3);; I et}I/6 6, equals

(7m — 7rL), note that candidate effort is more productive as (7rH —— m) is raised.

1.2.2. Phase Two —— Payoffs of Players

We now describe the period-t payoffs of the players in the model. The period-
tpayoff of each voter i E N is set to yt. By endowing voters with identical
preferences, all aspects of preference heterogeneity are abstracted away.9 The
period-t payoff of the period—t challenger is set to 0. Since the period-t challenger
undertakes no task, the normalization is without loss of generality. Finally, the
period-t payoff of the period-t incumbent candidate is given by (B — k x [523]),
where B > 0 and k > 0. The incumbent candidate’s payoff is comprised of two

components: the private benefit of office, summarized by B, and the private cost

 

 

9For a general discussion of the effects of preference heterogeneity, the reader is referred to
Ferejohn (1986).

 

 

 

 

 

 

 

 

 

 

   

”P1“

 

10

82

of choosing effort 6; (i.e., k X [4]).

The chosen payoff functions highlight the principal-agent problem. As the
effort level, et, is raised, both the probability of generating yH and the private cost
borne by the period-t incumbent candidate increase. Therefore, the electorate and
the period—t incumbent candidate have diametrically opposed preferences over the
set E. The electorate’s (incumbent candidate’s) current payoff is maximized when

e, is chosen to be the largest (smallest) value in E.

1.2.3. Phase Three — Election Outcome

Once voters receive their period-t payoffs, they cast their ballots and decide
whether to retain the period—t incumbent candidate. Following the vote, period t
concludes.

The majority Winner of the period-t election is assigned to be the period-(15+ 1)
incumbent candidate. The sequence of events in period (t + 1) replicates that of

period t.

 

~
:.

 

 

 

 

 

 

11

1.3. Experimental Predictions

In this section, we develop the predictions which logically follow from the model.10
The section is organized as follows: First, we formally define a sequential equi—
librium of the model. Second, we refine the sequential equilibrium by imposing
stationarity and symmetry conditions. Third, we state and briefly discuss the

principal predictions that follow from the model’s solution.

1.3.1. Definition of Sequential Equilibrium for the Model

To define a sequential equilibrium, we introduce the following notation.11 A his-
tory of length t, denoted ht, specifies all public events through period t: namely,
the identity of the incumbent candidate in each period, the output generated in
each period and the distribution of votes in the end-of-period election. H t denotes
the set of all possible histories of length if and H 0 E 45. For each h‘ 6 H t, a partial
history, denoted 12;, is a. specification that includes all the component elements of
ht except the distribution of votes realized in the period-t election. H; denotes

the set of all possible partial histories of length If and HE E (b.

M

10For readers uninterested in the details of computing equilibria of the model, much of section
1.3 can be skimmed. The principal implications of the model’s solution are presented in section
1.3.3.

”A general description of sequential equilibrium for finite games is given in Kreps and Wilson
(1982). In defining a sequential equilibrium for the model, much of the notation is adapted from
Banks and Sundaram ( 1993).

 

 

 

 

 

 

 

 

12

Voter i’s strategy is represented as a sequence of functions a, = (115),,‘21 where,
for each t, of : H; —> [0,1] is a measurable map that specifies the probability with
which voter 'i casts her ballot in favor of the period-t incumbent candidate as a
function of the partial history h; V; denotes the set of all possible strategies for
voterz' and V E HI; V,.

For each ('01, ..., an) E v 6 V, r(v) E (r(v)‘)f:1 is a sequence of fimctions where,
for each t, r(u)‘: Hf, —> [0,1] is a measurable map that specifies the probability
with which the period—t incumbent candidate is retained in office as a function of
the partial history h; and voting strategy 1).

Given the effort space of candidates (E E [0,1)), is}; denotes the Borel sets
of E and A(E) denotes the space of probability measures on the measurable
space (E, ‘33). The strategy for candidate 1' E K is represented as a sequence of
fimctions c,- E (oflglwhere, for each t, cf: H “1 —> A(E) is a measurable map that
specifies the probability measure in A(E) chosen by candidate 2' if after history
h“1 she is the period-t incumbent candidate. C,- denotes the set of all possible
strategies for candidate 2' and C E CAX CB-

For partial history hf, E H; and fixed strategy profile (cm) 6 C x V, Av(h,‘,;
c,v) denotes each voter’s expected discounted sum of payoffs over the infinite

horizon, conditional on being at 11%,. Similarly, for history h‘ E H t and fixed

 

 

 

 

13

strategy profile (0,21) 6 C X V, {A,(h‘;c,v)},eK denotes candidate i’s expected
discounted sum of payoffs over the infinite horizon, conditional on being at h‘.
(an) is called a sequential equilibrium if and only if after every h], E H; (ht 6
Hi), the voting (effort) strategy of each voter (candidate) is privately Optimal.
Specifically, we require the following two inequalities to hold:

IC.1IAV(h;;c,v) Z Av(h;,; c, v_,,17,-);‘v’i7,- E V; and Vi 6 N.

[0.2] /\.;(ht;c,v) 2 A,(ht; c_,-, 5,,0); V5,; E C,- and W E K.

Consider condition [0.1]. Av(h;,; c, v_,;, 17,) denotes each voter’s expected dis-
counted sum of payoffs over the infinite horizon conditional on being at h; when:
1) candidates adept strategy 0; 2) voters, except for voter 7L, adopt strategy n_,;
and 3) voter 2' deviates from strategy 2),- by selecting 27, instead. If strategy pro-
file (0,1)) is a sequential equilibrium, condition [0.1] requires that a unilateral
deviation by voter i be unprofitable.

Consider condition [C2]. A,(h“; c_,-, 5,, 1)) denotes candidate t’s expected dis-
counted sum of payoffs over the infinite horizon conditional on being at ht when:
1) the other candidate adOpts strategy c_,-; 2) candidate 2' deviates from strategy
Ci by selecting E,- instead; and 3) voters adept strategy 1). If strategy profile (0, v)

is a sequential equilibrium, condition [C2] requires that a unilateral deviation by

candidate 2' be unprofitable.

 

 

 

 

 

 

14
1.3.2. Refining the Sequential Equilibrium for the Model

Given the multi-player, repeated-interactions design, there exists multiple sequen-
tial equilibria. For the purposes of sharper predictions, we focus on symmetric
and stationary sequential equilibria in which both candidates play pure strategies.
We impose the following four restrictions on admissible strategy profiles (c, 2)):

[A4] For i E K, t E Z+ and H4 E Ht_1, cflht—l) assigns a probability mass
of one to some point in E.

[A5] Let ht and h? be any two histories following which candidate 2' E K is in
office. Then, 0,-(ht) = c,-(h?). Hereafter, E,- 6 E denotes the effort level chosen by
candidate i when she is the incumbent.

[Ac] ‘6}; = a3 (2 5).

[A7] For each t and h; 6 Hg, 12 is such that r(v)t(h;,) depends only on the
realized output in period t, 3),. Hereafter, rL(rH) denotes the probability with
Which the incumbent candidate is reelected when output equals yL(yH).

We explain the four restrictions ([A4] — [A.7]). Assumption [A4] forbids the
l3W0 candidates from randomizing over the effort space, E. If after histories ht and
hf, candidatez' E K is reelected, the subgames following h‘ and h? are structurally
identical, Assumption [A.5] requires candidate t’ to expend equal effort in the two

Situations. Assumption [A.6] is a symmetry statement. Since the two candidates

 

 

 

 

 

 

 

 

rm

 

 

15

possess identical preferences, we consider equilibria in which they expend equal
effort. Finally, assumption [A.7] is a necessary imposition if candidate strategies
are to obey assumptions [A.4] — [A.6]. Note that if reelection of the period—
tincumbent candidate involves considerations other than the current realized
output, yt, then optimal period-t effort will depend on the details of history ht’l,
thereby violating assumption [A.5].

Given assumptions [A.4] - [A.7], we can define an equilibrium in a more com-
Pact manner. 8 E {3A, 83} is the state space of the model and 3,4(53) is a subgame
in which candidate A(B) is the incumbent. Q : S X E —> S is a transition map
that specifies, as a function of the current state and effort expended, the proba—
bility of the two states in the next period. The following two conditions regarding

Q can be readily established:12

Q(3iI3i,€) = Probability{y = yHIe} x 7‘}; + Probability{y = yLIe} X TL

CNS—43¢, e) = Probability{y = yHIe} x (1— TH) + Probability{y = yLIe} x (1— n)

12 .
In our notatlon, 3-, is a shorthand for the element of 3 not equal to 3r

 

 

16

Starting from a node of the game tree in which candidate 1' is the incumbent,
Q(s,Is,-,e) denotes the probability of reelection when effort expended is equal to
e. Given assumption [A.7], candidate 2' is reelected with probability 'rH(rL) when

realized output is yH(yL). Furthermore, since effort is fixed at e, the probabil—

 

ity of yH(yL) is Probability{y = yHIe} (Probability{y = yLIe}). The expression

for Q(s,-Is,~,e) now follows from a standard conditioning argument. Analogous

 

interpretations apply for Q(s_,-Is,-, e).13

{Id-(5,; 6)},‘J6K is the discounted sum of payoffs (value function) for candidate
is K when: 1) the current state is Sj E S and 2) each candidate, when in office,
expends effort equal to e. Standard recursion arguments yield the following two

conditions:

Vitae) = (B — k x [652]) + 6 X [Q(sz-l8.-,e) >< V4856) + Q(8—¢|8ae) >< W's—i; 62)]

V,(s_,-;e) = 6 x [Q(s_,-Is_,-, e) X Wis—176) + Q<StI8-ia 9) X 146136)]

 

13Starting from a node of the game tree in which candidate 1' E K is the incumbent, Q(s_,-
I 8;, e) denotes the probability of electoral defeat when effort expended is equal to e. Thus,
Q(8¢l8ae) + Q(3—iI3i; 8) = 1.

 

 

 

 

17

Starting from a node of the game tree in which candidate 2' is the incumbent,
V,(s,;e) denotes her total discounted sum of payoffs. When candidate 2' expends
effort e, her current period payoff is (B — k X [923]). Thereafter, with probability
Q(s,-Is,-, e)(Q(s_,Isi, 6)) she is reelected (rejected) and obtains, over the remainder
of the game, a discounted sum of payoffs equal to 6 X V,(s,~; e) (6 x V,(s_,-; (2)).14
V,(s,'; e) is computed as the sum of her current and expected future payoffs. Anal-
ogous interpretations apply for V,(s_,-; 6).

Using the unimprovability criterion from dynamic programming, it follows that
(EA, 53) E (eye) can be supported as the effort level of a sequential equilibrium
satisfying assumptions [A4] — [A.7] if and only if for i E K and E E E the following

inequality is satisfied:
(“3’2
V,(s,-;é) 2 [B—kx [3]]+6 X lQ(8.-|8¢; E) X l/z-(sz-;E)+Q(s_:lsi, 5) >< Wan—6)] (1-1)
The unimprovability criterion posits that EA (EB) E e is an equilibrium if a one-

time deviation from E is unprofitable for candidate 2' e K. Consider a node of

the game tree in which candidate 2' is the incumbent. If she does not deviate from

 

14We assume that both candidates discount future payoffs at rate 6 6 (0,1).

 

 

 

 

 

 

18

E, her total discounted sum of payoffs is given by V,(s,-; e). On the other hand,
should she deviate from E and choose E : 1) her current payoff is (B — k X [%]);
and 2) her future discounted sum of payoffs is 6 x V,(s,-;E)(6 x l€(s_,-;E)) if she
is reelected (rejected). Furthermore, given E, reelection (rejection) occurs with
probability Q(s,- I 3,,E) (Q(s_,- I s,,E)). By a standard conditioning argument,
it is immediate that the right—hand side of equation (1.1) is the expected total
discounted sum of payoffs to candidate 2' following deviation E. The inequality of

equation (1.1) ensures that all contemplated deviations from E are unprofitable.

1.3.3. Predictions from the Model’s Sequential Equilibria

While details of the model’s solution are provided in section 1.7, we summarize
the two main predictions here. First, we introduce additional notation.

The model has five exogenous parameters: B, Is, (5, 7rL and 717;. For fixed voter
behavior (TL, ’I‘H held constant) and parameter configuration, E(B, k,6,7rL,7rH;
mm) denotes the effort level in a sequential equilibrium satisfying assumptions
[A4] - [A.7]. Propositions 1 and 2 provide detailed characterizations of this “effort

function.”

Proposition 1: 'é(B, k,6, 7rL,7rH; 11,771) is increasing in B, 6, 7TH and decreasing in

 

 

 

19

(6,711,. Furthermore, 5(B, k,6,7TL,7TH;TL,’I‘H) = 0 if 6 = 0 or 7rH :: 71-1].

Proposition 2: As voter behavior, (77,, TH), is varied, the model generates a con—

tinuum of equilibrium effort levels. Any number in the interval [0, [336$] can be

 

 

supported as an equilibrium effort level.

We now draw out the substantive implications of the above propositions.
Proposition 1 formalizes several standard intuitions. Consider, first, the com-

parative statics of effort with respect to B and k. When the private benefit from

 

holding office, B, is increased, there is a corresponding increase in the marginal
benefit of effort. When the effort cost parameter, k, is increased, there is a cor-
responding increase in the marginal cost of effort. Since candidates equate the
marginal benefit and marginal cost of effort, effort levels increase (decrease) as
B(k) is raised. Second, if the discount factor, 6, is increased, future consider—
ations begin to carry more weight. The added desire for reelection engenders
greater electoral discipline and elicits larger effort levels from candidates. Third,
When 5 is equal to zero, future payoffs become irrelevant. Candidates maximize
current payoffs and expend no effort. Forirth, as nH(7rL) is increased, the produc-

iiVity of candidate effort increases (decreases). Effort expended is an increasing

 

 

lie

Elli
‘ifilon

if A ll

 

 

 

20

function of effort productivity; hence, a rise in 7TH(7TL) produces an increase (de—
crease) in effort levels. Finally, when nH is equal to 7r L, the productivity of effort
becomes zero. Consequently, candidates are unwilling to exert any effort.
Proposition 2 is unsurprising. Since the setup is a repeated game, the abun-
dance of equilibria is a consequence of Folk theorem-type results.15 If electoral
efficiency is measured by the amount of effort extracted from candidates, then
proposition 2 derives a theoretical upper bound. The effort level actually elicited

remains primarily an empirical issue.

1.4. Experimental Design

To test the predictions of the electoral model, we performed a series of experi-
ments using undergraduates from a large public university. In the recruitment
phase, care was taken to ensure that our subjects were unexposed to formal de-
cision/game theory. The experiments were conducted on a computer network
system and, except for reading the instructions (available upon request), all com-
munication took place over the network.

The experimental sessions were of two sorts: repeated-interactions and one-

shot sessions. The repeated-interactions sessions approximated the conditions of

 

 

15For an illumm' ating introduction to the literature on Folk theorems, the reader iS referred
to Phdenberg and Maskin (1986).

 

 

 

 

. f,

‘1
z}?!

 

 

”1"!"

 

21

the electoral model by allowing two candidates to interact in a sequence of periods.
By contrast, in one-shot sessions candidate interactions were restricted to a single
period. The possibility of repeated-game strategies was thereby eliminated.

The motivation for two session types is twofold. First, by mimicing the base-
line model, the repeated-interactions sessions enable us to evaluate its predictive
validity. Second, differences in experimental findings across session types allow
estimation of the degree to which reelection pressure, present only in repeated—
interactions sessions, elicits larger effort levels from candidates. The experimental

procedures followed in each session type are detailed below.

1.4.1. Repeated-Interactions Setup —— Experimental Procedures

An experimental session consisted of a cohort of nine subjects divided into an
electorate of five voters and a pool of four potential candidates. A trial, with a
fixed configuration of parameter values, lasted for a variable number of periods
and involved the electorate facing the same pair of candidates. The uncertain
termination date, coupled with unchanging candidates, was deemed sufficient to
induce repeated-game considerations. A trial proceeded as follows.

[Step 1] At the start of the trial, two subjects from the candidate pool were

randomly chosen to participate. Only the two “active” candidates were aware of

 

 

 

 

.L‘r

 

 

22

their role assignments. Thereafter, by random choice, one of the two candidates
was designated to be the incumbent for the first period.
[Step 2] At the start of period t, the incumbent candidate, denoted 2'(t), selected
her effort level, denoted 6,, from the interval [0, 95].16
[Step 3] Given 8,, the computer program generated a number, denoted yt,
from the two—point set Y E {yL, yH}. y, equalled yH(yL) with probability given
by (KL + (1566) X (7rH — nL)) ((1— 7:7,) — (TEE) X (7TH — 7173)). The yt-value was
transmitted to the seven subjects participating in the trial.17
[Step 4] Given 3),, voters cast their ballots and decided, by majority rule,
whether to reelect or reject candidate i(t). Once votes were tallied, the final
outcome of the election as well as the vote margins realized were transmitted to
the seven subjects participating in the trial.
[Step 5] Candidate 2'(t) received a period—t payoff of (B — k X [323]). Each voter
received a period-t payoff of 31,.
[Step 6] The computer program switched to period (t + 1). The winner of

the period-t election was designated to be the incumbent candidate for period

 

161,, the theoretical model (see section 1.2), candidates’ effort space is the half-open unit
interval. To allow for greater variation in observed behavior, the experimental setup expands
the effort space to be [0, 95].

”The chosen effort, 6, 6 [0,95], is mapped into the unit interval by the transformation
(ct/100). Thereafter, assumptions [A.l] - [A.3] (see section 1.2) are used to determine the prob-
ability of generating 31” and 111,.

 

 

 

 

 

 

23

(HI). At the end of a stochastic number of periods, the trial terminated.18 Each

subject’s payoff for the trial was the sum of her period payoffs.

1.4.2. One-Shot Setup — Experimental Procedures

An experimental session consisted of a cohort of eleven subjects divided into an
electorate of five voters and a pool of six potential candidates. A trial, with a
fixed configuration of parameter values, lasted for six periods. Each candidate
was randomly chosen to participate in two of the periods. A trial proceeded as
follows.

[Step 1] At the start of period t, two subjects from the candidate pool were
selected to be the period-t candidates. Only the two chosen candidates were
aware of their role assignments. By random choice, one of the two candidates was
designated to be the period-t incumbent.

[Step 2] The incumbent candidate, denoted z'(t), selected her effort level, de-
noted et, from the interval [0, 95].

[Step 3] Given 6,, the computer program generated a number, denoted y,,

g

18Each trial was terminated as follows. Beginning in period twenty-five, the computer program
generated a number from one to ten where each number was drawn with equal probability.
The experiment was terminated if the number one was drawn; otherwise another period was
conducted. This process was continued until, eventually, a one was drawn. All subjects were
informed of the termination procedure adopted.

 

 

 

 

 

 

24

from the two-point set Y E {yL, pH}. 3;, equalled yL(yH) with probability given
by (7rL +(1—‘2,L0)X(7rH — 111)) ((1 — wL) — (14%) X (7rH — 7rL)). The yt-value was
transmitted to all eleven subjects.

[Step 4] Candidate i(t) received a period—t payoff of (B —— k x [522]). Each voter
received a period—t payoff of 3),.

[Step 5] The computer program switched to period (t + 1). The sequence of
events in period (t + l) replicated that of period t. At the end of six periods, the
trial terminated. Each subject’s payoff for the trial was the sum of her period

payoffs.19

1.4.3. Experimental Parameter Values

An experimental session consisted of four trials. Individual trials differed in the
values assumed by the four exogenous parameters: B, It, 7rL, and 7rH. Table 1 in
Appendix A displays the parameter value configurations that were considered.
For example, in Treatment 1, we set the private benefit of office, B, to be 7425;
the scale parameter of effort cost, It, to be 1; and the parameter vector of the

output generation process, (7rL, 7rH), to be (0, 1). The parameter values in Table

 

19In contrast to the repeated—interactions case, there was no “voting phase” in one—shot ses-
sions. Therefore, it was impossible for the electorate to express approval (disapproval) of the
incumbent’s performance by reelecting (rejecting) her.

 

3' to l

.I,

4 MW

 

 

25

1 correspond to subjects’ payoffs denominated in a laboratory currency called the
franc. At the conclusion of the experimental session, cumulative earnings in francs
were converted into dollars using a preassigned exchange rate.

For repeated-interactions sessions, the primary goal of our experimental work
was to sort out the factors that affect candidates’ effort choice. Specifically, we
considered the influence of two potentially important factors. First, by a pairwise
comparisons of Treatments 1 and 3 and Treatments 2 and 4, we investigated the
effect of the benefits of office. In short, we asked: if B is increased (decreased),
is there a systematic effect on the level of candidates’ effort? Second, by a pair-
wise comparisons of Treatments 1 and 2 and Treatments 3 and 4, we considered
the impact of effort productivity. In brief, we asked: if (7m — 717,) is increased
(decreased), is there a systematic effect on the level of candidates’ effort? For
Treatments 5 and 6, the productivity of effort is equal to zero. Thus, we asked:
if effort is entirely unproductive, are candidates nonetheless willing to expend
effort?

By contrasting experimental findings across session types, we evaluated the
extent to which reelection pressure elicits effort from candidates. Note, first,
that reelection pressure is present only in repeated-interactions sessions. Hence,

if candidates’ effort in one-shot trials is significantly less than that in repeated—

 

 

 

26

interactions trials, the disciplining role of reelection pressure will have been es-

tablished.

1.5. Experimental Results

Our experiments consisted of two session types: repeated-interactions and one-
shot. In this section we present the experimental results for each session type.

We consider the repeated-interactions sessions first.

1.5.1. Experimental Results for Repeated—Interactions Sessions

The analysis of the data is comprised of four parts. First, we provide results
that characterize the aggregate behavior of candidates and voters. Second, we
estimate two alternative models that explain the electoral choices of individual
voters. Third, we estimate a model that accounts for the effort decisions of
individual candidates. Fourth, we compute the relative efficiency of experimental
elections.20
Aggregate Behavior in Repeated-Interactions Sessions

The repeated~interactions sessions consisted of six treatment conditions (see

Table 1). For each treatment condition, we conducted several trials where each

 

20The definition of “relative efficiency” is given later.

 

 

 

 

27

trial, in turn, consisted of a variable number of election periods. Corresponding
to each treatment condition, we pooled all the observations. Table 2 in Appendix
A provides a summary of the data.

In Table 2, for each treatment condition, “# of trials” indicates the number
of trials that were conducted. Corresponding to each treatment condition, we
pooled observations across trials and periods. “Average effort” (“std. dev. of
effort”) computes the average (standard deviation) of the effort levels chosen by
candidates. “Realized output” is a two—element vector where the first and second
elements represent, respectively, the number of instances that the observed output
was yL(yH). “Reelection probability” is a two-element vector where the first and
second elements represent, respectively, the empirical probability of reelection
conditional on outputs M and 3119.

Table 2 is read as follows. In the experiment, we conducted six trials with
rIreatment 1-parameter values. These trials yielded a total of 201 observations:
the average of candidates’ effort levels was 67.43 while the standard deviation
Was 18.66. The vector “realized output” indicates that the observed output was
i/LfyH) on 67(134) occasions. The “reelection probability” vector indicates that
an output of yL(yH) resulted in reelection 8%(100%) of the time.

Consider, first, aggregate behavior in Treatments 5 and 6. In both treatments,

 

 

 

 

* ale

F“? .,....

28

the productivity of candidate effort is equal to zero. Hence, proposition 1 main—
tains that candidates will be unwilling to expend effort. Table 2 provides striking
support for this prediction. Specifically, in Treatments 5 and 6, average candi—

date effort was, respectively, only 5.24 and 2.51.21 Conclusion 1 summarizes our

 

findings.

 

Conclusion 1: As predicted by proposition 1, when the productivity of candidate

effort is equal to zero, candidates expend negligible effort.

Consider, now, aggregate behavior in Treatments 1, 2, 3 and 4. Table 2 al-
lows us to make the following three observations. First, for each of the treatment
conditions, reelection probabilities are increasing in output levels.22 Since the
electorate rewards (punishes) “good” (“poor”) candidate performance, an incen-
tive to undertake costly effort emerges. Second, candidates’ effort levels averaged

over Treatments 1 and 2 (B = 7425) is 55.93 while that averaged over Treatments

 

3and 4 (B = 5000) is 40.50.23 Thus, in accord with proposition 1, average effort

 

21For comparison, recall that the feasible upper bound for candidate effort is 95.

22Consider Treatment 1. The “reelection probability” vector indicates that the probability of
reelection when 34;, occurs (0.08) is less than that when yH occurs (1.0).

2355.93 is the average of two numbers: 67.43 and 44.42. 40.50 is the average of two numbers:
48.72 and 32.27.

 

is}.

4'1

r—_______i

29

increases when the private benefit of office is raised. Third, candidates’ effort
levels averaged over Treatments 1 and 3 ((7rH — 7rL) = 1) is 58.08 while that aver-
aged over Treatments 2 and 4 ((7rH — m) = 0.6) is 38.35.24 Thus, in accord with
proposition 1, average effort increases when the productivity of candidate effort

is raised. Conclusion 2 summarizes the above observations.

Conclusion 2: The data provide support for the comparative statics predictions of
proposition 1. Specifically, the average of candidates’ effort levels is an increasing

function of 1) the private benefit of office and 2) the productivity of candidate

 

effort.

We supplement conclusion 2 with regression-based analyses of candidates’ ef-

fort choices in Treatments 1, 2, 3 and 4. In the first regression, the dependent

 

variable is the effort expended by candidates while the independent variables con-
trol for treatment conditions. The base group, represented by a constant, (:2, refers
to the treatment in which the private benefit of office, B, is equal to 5000 and

the productivity of candidate effort, (77H — 7rL), is equal to 0.6.25 Additionally,

 

2458.08 is the average of two numbers: 67.43 and 48.72. 38.35 is the average of two numbers:

44.42 and 32.27.
25The base group represents Treatment 4 (see Table 1).

 

 

30

two dummy variables are included. Specifically, Highb is a dummy variable that
equals 0(1) if the B-value for the treatment equals 5000(7425).26 The coefficient
of Highb, denoted ,81, represents the difference in candidates’ average effort level
when, ceteris paribus, B is raised from 5000 to 7425. The second dummy vari—
able, called Highprod, equals 0(1) if the (7r H — 7rL)-value of the treatment equals
06(1).27 The coefficient of Highprod, denoted ,62, represents the difference in can-
didates’ average effort level when, ceteris paribus, (7rH — 7rL) is raised from 0.6 to

1.0. In sum, the first regression estimates the following model:

6,, = a + filHighb,t + figHighprodit + 23,-, (1.2)

where: 1) ea is the effort expended by candidate i in period t; 2) 6’8 are coeffi—
cients; 3) Highb” is a dummy variable that equals 0(1) if candidate 7; in period t
is in a treatment with B equal to 5000(7425); 4) Highprod,-t is a dummy variable
that equals 0( 1) if candidate 2' in period t is in a treatment with (7rH — 7rL) equal
to 0.6(1); and 5) 5,, is an i.i.d. error term.

The results of the estimation are detailed in column 1 of Table 3 (refer to
Appendix A). The point estimates of Bl and [32 are positive and statistically

26Highb is equal to 0(1) for Treatments 3 and 4 (1 and 2) (see Table 1).
27Highprod is equal to 0(1) for Treatments 2 and 4 (1 and 3) (see Table 1).

 

 

 

 

 

31

significant. Therefore, there is a positive relationship between candidates’ effort
levels and the B — and (7TH — 7rL)—values of the electoral environment.

Another method of estimating the impact of treatment conditions on candi-
dates’ effort levels explicitly recognizes heterogeneity in the pool of candidates.

The errors-components approach estimates the following model:

eit = fllHZghblt + figHtghprodit + 01 + 5it (1.3)

Note that the constant, 04, in equation (1.2) is replaced by the fixed effects, 0,, in
equation (1.3). Heterogeneity in the candidate pool is modeled by allowing the
(ifs to vary across candidates.28

We estimate equation (1.3) using two approaches. First, we include the fixed
effects directly as regressors. The coefficients (fi’s) are measured by using the
within-subject variation in treatment conditions.29 The results are detailed in
column 2 of Table 3. As in the OLS case, the point estimates of the B’s are posi—
tive and statistically significant, thereby confirming conclusion 2. Since the fixed
effects are treated as regressors, they have no distribution. However, a measure of

candidate heterogeneity can be obtained by computing the sample standard de—

 

28For an introduction to panel data models, the reader is referred to Chamberlain (1984).
29Each candidate subject participated in more than one treatment condition. Thus, treating
the (95’s as regressors becomes a feasible way of estimating equation (1.3).

 

 

.1'

 

 

 

 

32

viation of the estimated 6,93 (61’s). In the data, the sample standard deviation of
the 03s is 10.73. Since the average of candidates’ effort levels across Treatments 1

through 4 is 48.21, the sample standard deviation of the 61’s represents significant

 

heterogeneity in the candidate pool.30

Given the random assignment of candidates to the various treatments, the
fixed effects are uncorrelated with the two treatment variables on the right—hand
side of equation (1.3). Consequently, a random-effects estimator is consistent and
potentially more efficient. This is a generalized—least—squares (GLS) estimator of
equation (1.3).31 The results of the random-effects estimator are detailed in col—
umn 3 of Table 3. They are similar to the results of the fixed—effects estimator.32
As before, the point estimates of the ,B’s are positive and statistically significant.
Since the fixed effects now have a distribution, we can formally check for het-
erogeneity in the candidate pool by determining whether the standard deviation

of the distribution of the fixed effects is zero. The Breusch and Pagan Lagrange

 

3”Another measure of heterogeneity is the ratio: (sample standard deviation of the 02’s) +
(estimated standard deviation of the en’s). In the data, this ratio is equal to 0.70. Hence, the
conclusion of “significant heterogeneity” is validated.

31For details, the reader is referred to Chamberlain (1984).

32If the random-effects specification is correct, then fixed— and random—effects estimators
should yield comparable point estimates of the fi’s. We performed a Hausman test (see Haus—
man (1978)) to check whether the point estimates of the 6’s in columns two and three of Table
3are statistically indistinguishable. The null hypothesis of “equality of the 3’s” could not be
rejected at the 0.1 level of significance.

 

 

 

 

33

Multiplier test detects heterogeneity in the candidate pool.33 The results of the

three regressions are summarized in the following conclusion.

Conclusion 3: The regression-based analysis supports conclusion 2. Specifically,
the average of candidates’ effort levels is an increasing function of 1) the pri-
vate benefit of office and 2) the productivity of candidate effort. Also, there is

heterogeneity in the candidate pool.

Voter Behavior in Repeated-Interactions Sessions

To analyze voter behavior, we introduce additional notation. Let t(t) E {A, B}
be the identity of the incumbent in period t and let gt 6 Y E {yL,yH} be the
realized output in period t. In the experiment, voters are imaware of candidate
effort or the mechanism by which expended effort stochastically generates output
values. Thus, in period t, a voter’s history consists of two parts: 1) the identity
of the incumbent for periods 1 through t (i.e., {i(j)}§=1); and 2) the stream of
realized outputs for periods 1 through t (i.e., {mg-=1). We wish to construct

plausible models that account for how a voter in period t uses her information in

 

33Under the null hypothesis of “no heterogeneity,” the test statistic is distributed as Xfll' At
the 0.1 significance level, the critical value is 2.71. In our dataset, the test statistic assumes a
Value of 373.03.

 

 

 

 

 

f—w

34

deciding whether to reelect or reject candidate i(t).

We build two distinct micro-models of voter behavior. In both models, a voter
utilizes the available retrospective information and rewards (punishes) candidate
i(t) with reelection (rejection) when her performance is deemed to be satisfactory
(unsatisfactory). The models differ in the specification of the process by which
candidate i(t)’s performance is rated.

In model 1 of voter behavior, called the Average Payoff Model (hereafter,
APM), a voter in period 13 first computes the average payoff received over periods
lthrough (t — 1). Thereafter, if the observed yt—value is above (below) the payoff
average, candidate 2'(t) is reelected (rejected). For the two-point output set, Y
5 Ln, 3111}, the prediction of APM is as follows: If the sequence of outputs upto
period (t — 1) comprises both 3)], and yH, then candidate 2'(t) is reelected with
probability equal to 0(1) if y; equals yL(yH).

To test the explanatory power of APM we proceed as follows. First, for each
of the six treatment conditions, we pool the vote decisions across trials and peri—
ods. Thereafter, we compute the percentage of vote decisions that violate APM’s
predictions. The results are detailed in Table 4 of Appendix A.

In model 2 of voter behavior, called the Discriminating Average Payoff Model

 

 

 

 

 

 

 

 

35

(hereafter, DAPM), a voter discriminates between the two candidates, A and B34
Therefore, in period t, a voter computes two average (perhaps, discounted) payoffs
received over periods 1 through t —— one for each candidate. In period 75, let EV,-z (t)
denote the discounted average payoff attached to candidate j E {A, B} by voter
i. EVj’(t) is calculated by considering only those periods for which the incumbent
candidate is j. More formally, suppose that A is candidate i(t). In addition, let
p.- 6 [0,1] denote the rate at which voter 7L discounts past observations. Then,

{EV/((15), EV§(t)} satisfies the following recursion:

EVE“) = (1 - 1a)]c >< EVN - 1) +11 - (1 - pdkl >< yr

rwya=E@a—n

where k is the number of periods since A was the incumbent last.35 Given

{El/Ht), EV§(t)}, voter 2' casts her ballot for candidate A(B) if EVA (t) > EV1§(t)

 

34The idea behind DAPM, though developed independently, is identical to that in Collier et
al (1987). For a lengthier discussion of DAPM, the reader is referred to that paper.

35MB is candidate i(t), {EV}; (t), EVé(t)} is computed as follows:

E%0=c—merec—n+u—o—mmXa

EVg(t) = EVA“ _ 1)

Where It is the number of periods since B was the incumbent last.

 

 

 

 

36

 

(EV§(t) > EVZ(t)). In the case of a tie (EVflt) = EV§(t)), voter t’ randomly
selects one of the two candidates.

DAPM is empirically implemented as follows. Note, first, that the model
involves the following free parameters: 1) voter 13’s discount rate, p,; 2) voter i’s
initial expectation about candidate A, EVj](0); and 3) voter i’s initial expectation
about candidate B, EV1§(0). For each of the six treatment conditions, we consider
one trial at a time. We set EVflO) = EV§(0) = EV(O) and, using a grid search,
find the trial-specific EV(0)— and p,—values that minimize the number of “errors”
in voting behavior. The results are detailed in Table 4.

Table 4 is read as follows. In our experiment, the six trials with Treatment
l-parameter values yielded observations for two hundred and one periods and,
hence, one thousand and five vote decisions.36 The “error rate, APM” (“error
rate, DAPM”) entry indicates that 10.75% (14.23%) of the vote decisions violated
the predictions of APM (DAPM).

For the entire experiment, APM and DAPM explain, respectively, 81.90% and

77.89% of the vote decisions.37 If, on the other hand, voters cast their ballots with—

 

36Since the electorate consists of five voters, the number of vote decisions is five times the
number of periods.

3781.90 is computed as the difference between 100 and the average of the six error rates: 10.75,
13.06, 18.77, 26.37, 23.95 and 15.73 (see Table 4). 77.89 is computed as the difference between
30 and the average of the six error rates: 14.23, 22.59, 23.09, 27.67, 23.42 and 21.64 (see Table

 

 

 

 

37

out considering candidates’ performance, a model of “random choice” accounts
for 50% of the vote decisions. Relative to a “random choice” model, both APM
and DAPM possess superior predictive power. Voters, therefore, use candidates’
performance in determining electoral outcomes.

While APM has a slightly higher prediction rate than DAPM, the difference
is not substantively significant. For both models, approximately 20% of the vote
decisions remain unexplained. Voter errors, for the most part, occur when the
realized output is yL.38 Despite the unsatisfactory performance, voters reelect the
incumbent with a probability that exceeds the predictions of either model. A
more complete theory of voter behavior remains to be developed. Conclusion 4

summarizes the above discussion.

Conclusion 4: Voters use candidates’ performance in determining electoral out—

 

comes. Specifically, the reelection probability of the incumbent candidate is higher
when the realized output is 31;] rather than yL. However, about twenty percent of

the vote decisions cannot be accounted for by the two theoretical models.

A more detailed analysis of voter behavior detects significant heterogeneity

 

3”The details of the analysis are available upon request.

 

 

 

 

38

in the subject pool. We consider heterogeneity of two sorts. First, we consider
heterogeneity of voter error rates. We proceed as follows. Since APM has higher
predictive power than DAPM, we measure an individual voter’s error rate relative
to the APM predictions. Corresponding to each of the six treatment conditions,
we equate heterogenity of voter error rates with the standard deviation of the
error rates of voters assigned to that treatment. Second, we consider heterogene-
ity of voter preferences. The empirical implementation of DAPM generates an
estimate of each voter’s discount rate, [3,. Corresponding to each of the six treat—
ment conditions, we equate heterogeneity of voter preferences with the standard
deviation of the discount rates of voters assigned to that treatment. The results
are detailed in Table 5 of Appendix A.

To read Table 5, consider the Treatment 1 sessions. The entry for the “avg,
error rates” (“std dev., error rates”) column indicates that the average (standard
deviation) of the error rates of voters participating in Treatment 1 sessions was
10.75(6.70). The benchmark for voter behavior was APM. The entry for the “avg,
W8” (“std dev., fifs”) column indicates that the average (standard deviation)
0f the discount rates of voters participating in Treatment 1 sessions was .35(.28).
The benchmark for voter behavior was DAPM.

Table 5 reveals that for each of the six treatment conditions, both types of

 

 

iii

 

J

iidat

‘ l
he]

.2] M
351121]

 

 

 

 

 

39

voter heterogeneity - measured by the two standard deviations - exist. We now
ask: which type of voter heterogeneity is more substantial? Corresponding to each
treatment condition, we compute two summary measures. Normalized heterogene-

ity of voter error rates is defined to be the ratio: (“std dev., error rates”) / ( “avg,

 

error rates”). Normalized heterogeneity of voter preferences is defined to be the
ratio: ( “std. dev., fi,’s”)/(“avg., [0,-’3”). Averaged over the six treatments, the

normalized heterogeneity of voter error rates (preferences) assumes the value of

045(091).39 Since 0.91 exceeds 0.45, heterogeneity of preferences is more sub-

 

stantial than heterogeneity with respect to error rate. Conclusion 5 summarizes

the above discussion.

Conclusion 5: In the pool of voters, there is heterogeneity with reSpect to error
rate and discount rate. Heterogeneity of discount rates is more substantial than

that of error rates.

Candidate Behavior in Repeated-Interactions Sessions

Table 2 indicates that for Treatments 1 through 4, the average of the candidate

 

' 390.45 is the average of the renewing six numbers: 0.62, 0.43, 0.39, 0.44, 0.33 and 0.54. 0.91
18 the average of the following six numbers: 0.80, 1.05, 0.91, 1.38, 0.91 and 0.41.

 

 

 

- I'll?"

40

effort levels was significantly less than the feasible upper bound of the effort choice
set (95). In this subsection we first construct a theoretical model of how the
incumbent in period t uses her information in selecting an effort level. Once the
theoretical model is empirically estimated, we determine whether the observed
effort levels can be rationalized.

To describe the theoretical model, we introduce additional notation. Let 1)
i(t) E {A, B} be the identity of the incumbent candidate in period t; 2) y, E

Y E {MM/H} be the observed output in period t; and 3) 0(t) E {1,2,3,4, 5}

 

be the number votes received by candidate 2(t) in the period-t election. In the
experiment, besides knowing her own sequence of past efforts, candidate 73(t) is
aware of the public history {2( j), y,, 'u( j) ];11 . We now construct a model that
specifies, as a function of history, the process by which candidate z'(t) picks her
period-t effort, denoted emp(t).

Before selecting em” (t), candidate t(t) must form an opinion on two issues: 1)
the efiort level of her Opponent should she be placed in power and 2) the voting
behavior of the electorate.We posit that candidate z’(t) views her environment as
being stationary and that she employs likelihood techniques in estimating all the
unobserved parameters pertaining to the above two issues.

Consider, first, how candidate i(t) estimates her opponent’s effort level, de-

 

 

 

 

 

, .
35m

5 at

 

41

noted 8"”(t), from the available information. Between periods 1 through (t — 1),
let l(t) be the number of times that candidate i(t)’s opponent, denoted —t'(t), is
in office and let s(t) be the number of times that voters’ payoffs under candidate
—z'(t)’s administration is yH. ems (t) is the maximum likelihood estimate of can-
didate —z'(t)’s effort level conditional on the information {l(t), s(t)}. Specifically,

e"’e(t) solves the following program:

me e s e —s
e (t)6argeg[1dagc5] [7TL+(l—(i)')X(7TH—7TL)] (t)x[(1—7rL)_(fié)X(7rH_7rL)]l(t) (i)

 

(1.4)

The interpretation of the program is as follows. Under the administration

of candidate —i(t), there are s(t)(l(t) — s(t)) draws 0f yH(yL)- The probability

of generating output equal to yH(yL), conditional on effort equal to e, is (717, +

(1:70) x (7TH _ 7%)) ((1.. 77L) _ (T56) x (7TH — 7rL)). Therefore, the probability of

generating a sample of s(t) yH—values and (W) — 3(0) yL-values i35 l7TL + (fi) X

(«H _ ”LN-90) X [(1 __ 7m) _ (fi) >< (7TH — 7rL)]’(")‘:“(‘). The maximum likelihood

estimate of candidate —i(t)’s effort choice, eme(t), is the e-value that maximizes
the probability of observing the sample.40

m\_________
40If lft) ecluals zero, we set eme(t) to be fiftY- The empirical results are insensitive to the
ch01ce of initial value.

 

 

 

 

{Fl

in

 

 

42

Consider, now, how candidate i(t) estimates voter behavior from her avail-
able information. Candidate z'(t) assumes that voters are behaviorally identical.
Specifically, each voter is characterized by two parameters, m and pH, where
pL(pH) is the probability with which a voter reelects the incumbent candidate
when the observed output is yL(yH). Given her information, let (p'Lne(t), p’fie(t))
be candidate i(t)’s maximum likelihood estimate of (p L, p H). Specifically, (pine (t),

0590)) solves the following programs:

pZ’e(t) E arg max 1:1 1[yj = 311,] X {03], X (pL)”J' X (1 —PL)5_vj} (1.5)

pLG[0,1] J=1

t—l
10306 are 1033be H llyj = ya] x {03, X um)“ >< (1 -pH)5‘”J'} (1.6)
i j=1

Where: 1L1},- = yL] (1[yj = yH]) is the indicator function which equals one when
the Period-j output, 3),, equals yL(yH) and is zero otherwise.
The interpretation of the programs are as follows. Consider equation (1.5).

Given period t, we first look at all the past periods for which the realized output is

 

 

flat)

rpm
”ll- Ti,

 

 

43

3”,. For one such period, say period j , the probability of generating 12,- votes for the
period-j incumbent candidate is C; X (pL)“J‘ X (1 — pL)5‘”J'. Consider the observed
subhistory: {(yj, 1),-My,- = yL, j S (t — 1)}. For a fixed value of pL, the probability
of generating this subhistory is: H]: 1[y,- = yL] X {ij X (pL)”J' X (1—pL)5‘”J'} The
maximum likelihood estimate of 131, (i.e., p$e(t)) is the pL—value that maximizes
the above expression. A similar interpretation applies to equation (1.6).41

Having estimated {eme(t), p'L”8(t), p’fie(t)}, the optimal effort level of candidate

i(t) (i.e., em”(t)) solves the following program:

one = [m(fi)xers-arx[fax(psaorxo-pmar-ki (1.7)
k=3

0i? X (p?e(t))k >< (1-p's’“(t))5‘kl

Mei

+104.) — (a x («s-m] x ’.

II
on

Welt) = (B — k x [923]) + a x [0cm x vI(elt)+(1— Q(elt)) >< V0(elt>] (1.8)

V0(elt) = 6 X [Q(eme(t)|t) x vow) + (1 — Q<em€(t)lt)) >< V’(elt)l (1.9)

4 - . . .
1H ”1 Period t, there is no previous occurence of yL(yH), we arbltrarlly set 1771:1805) (1971711th
to be 0(1)- The empirical results are insensitive to the choice of initial values.

 

 

 

 

44

mp t I
e ( ) E arg 631%] V (elt) (1.10)

The basic idea behind the program is simple. Given voter behavior, (192“ (t),
p”H’e(t)), Q(e]t) is candidate i(t)’s estimate of an incumbent’s reelection probability
when her efiort level is equal to e.“2 Suppose, now, that candidate z'(t) always
chooses an effort level of 6 when placed in office. Then, VI(e|t) (VO(e]t)) is
candidate i(t)’s estimate of her discounted sum of payoffs over the infinite horizon
Starting at any node of the game tree where she is in (out of) office. Consider the
expression for V1(e]t). When candidate i(t) expends effort 6, her current period
payoff iS (B — k X [%]). Thereafter, she is reelected (rejected) with probability
equal ’00 Qfelt) (1 — Q(e|t)) and her discounted sum of future payoffs is 6 X VI (eIt)
(6 X V0040). Equation (1.8) indicates that V1 (elt) comprises of candidate i(t)’s
Current and estimated expected discounted sum of future payoffs. Consider the
expressmn for Vo(e]t). When candidate i(t) is at a node where she is out of office,
her Current period payoff is zero. Thereafter, when candidate —z'(t) expends her

42Note, first, that an incumbent requires at least three votes to be reelected. Given voter
behavior: Manages», Prob{incumbent reelected I 311,} is equal to [232:3 02 x (202160))“ ><
(i‘P’i’e(t))5‘k] While Prob{incumbent reelected I 1111} is equal to [212:3 Ci: X (Pffegnk x (1 _
Pte(t))5‘k]. From [A.l] and [A.2] (see section 1.2), Prob{yLle} is equal to [(1—7rt)-(m)><(7m—
Ml] while Prob{yH]e} is equal to [7,, 3, (fl) X (7”, _ m]. Finally, Q(e|t) = Prob{incumbent

reelected i yL}XPIOb{yLIe} + Prob{incumbent is reelected I yH}><Prob{yH]e}- Substituting
for each of the four terms in the above expression for Q(€|t), W6 Obtaln equation (1'7)'

 

 

 

 

 

 

ff fi"—-l_

45

estimated effort, eme(t), candidate i(t) is subsequently placed in (out of) oflice
with probability equal to (l — Q(eme(t)|t)) (Q(eme(t)[t)) and her discounted sum
of future payoffs is 6 X VI(e|t) (6 X VO(e|t)). A standard conditioning argument
immediately yields equation (1.9). Finally, equation (1.10) posits that candidate
i(t)’s period-t effort choice, 6’”? (t), maximizes her estimated expected discounted
sum of future payoffs starting from period t (i.e., VI(e|t)).

For each treatment condition and past history, the theoretical model of can—
didate behavior generates an “optimal” effort choice.43 We now compare the
theoretically predicted effort levels with observed candidate behavior. We pro—
ceed as follows. For each treatment condition, we pool observations across trials
and periods. Thereafter, we compare the average and the standard deviation of
the observed candidate effort levels (en’s) with the corresponding summary statis-
tics of the predicted candidate effort levels (emp(t)’s). The results are detailed in
Table 6 of Appendix A.

To read Table 6, consider the Treatment 1 sessions. Across all trials and

periods With Treatment l-parameter values, the average (standard deviation) of

 

Observed candidates’ effort levels was 67.43(18.66). The average (standard devia—

¥ . '
43T° Obtain emp(t), equations (1.4) — (1.10) need to be solved Simultaneously. Slnce closed
form solutions are unavailable, numerical techniques were employed.

 

7'1

 

 

it?“ ‘

 

 

 

46

tion) of predicted candidates’ effort levels was 66.53( 18.65).

A striking pattern appears in Table 6. Except for the Treatment 1 sessions,
observed effort is larger on average and more volatile than the theoretical predic-
tions. What accounts for these anomalies?

Consider, first, the anomaly in the level of observed effort. In Treatments 5
and 6, average observed effort was larger than zero because for some subjects
it took a few periods before they realized that it was unprofitable to expend
effort when its productivity is zero. For Treatments 2, 3 and 4, we conjecture
that the main reason for the discrepancy between theory and data is the risk
aversion of candidates. In the theoretical model of effort choice, the risk neutrality
assumption is implicitly invoked: candidates maximized the expected discounted
sum of period payoffs. If, on the other hand, candidates are risk averse, the model’s
solution will be misleading. In brief, a larger effort level smoothens the benefit
stream of a candidate by increasing her probability of retaining power and thereby
avoiding the drastic payoff of zero when she is out of office. For a risk—averse
candidate, there is a utility gain from this “benefit smoothing”. Consequently,
expended effort will be larger than the optimum under risk neutrality. Our risk

aversion explanation is, at best, partially correct. We cannot account for the

close match between theory and data for Treatment 1 sessions. Conclusion 6

 

 

 

47

summarizes the above discussion.

Conclusion 6: Except for Treatment 1 sessions, the average of candidates’ effort
levels exceeds that predicted by the theoretical model. The discrepancy may

reflect the risk aversion of candidates.

Consider, now, the anomaly in the volatility of observed effort. Why in Treat—
ments 2, 3, 4, 5 and 6 is the standard deviation of observed candidate effort larger
than the theoretically predicted magnitude? We conjecture that two forces ac-
count for this discrepancy. First, as candidates learn the structure of the game,
their actions change. Thus, learning induces volatility in the data. Second, as
indicated in conclusion 3, there is heterogeneity in the candidate pool. Aggregat-
ing all treatment specific periods generates an additional volatility that reflects
(merely) the inherent heterogeneity of candidates. Our explanations are, at best,
partially correct. They cannot account for the close match between theory and
data for Treatment 1 sessions. Conclusion 7 summarizes the above discussion.
QM Except for Treatment 1 sessions, the standard deviation of can—

didates’ effort levels exceeds that predicted by the theoretical model. The dis-

1
u

 

 

 

48

crepancy may reflect individual learning and/ or heterogeneity in the candidate

pool.

We now ask whether discrepancies between observed effort levels and theoret-
ical predictions decrease as candidates gather experience. Since non—trivial effort
choice occurs only in Treatments 1, 2, 3 and 4, we restrict our analysis of “effort
discrepancy” to these treatments.44

The regression-based analysis of “effort discrepancy” is as follows. In the first
regression, the dependent variable is the absolute value of the difference between
the effort expended by candidates and the level predicted by the theoretical model.
The independent variables control for treatment conditions and the experience of
candidates. The treatment conditions are represented with two dummy variables:
Highb and Highprod. Highb equals 0(1) if the B-value of the treatment equals
5000(7425). The coefficient of Highb, denoted 61, represents the difference in error
discrepancy when, ceteris paribus, B is raised from 5000 to 7425. Highprod equals
0(1) if the (7m — 7rL)-value of the treatment equals 06(1). The coefficient of

Highprod, denoted ,32, represents the difference in error discrepancy when, ceteris
paribus, (TI'H — m) is raised from 0.6 to 1.0. Candidate experience is represented

\_____
44 . . . . . .
The substantive concluSlons are unchanged if, instead, we consrder all srx treatments.

 

 

 

49

with the variable Exper. Exper specifies the number of times in the past that the
current incumbent was in office.45 The coefficient of Exper, denoted [33, represents
the difference in error discrepancy when, ceteris paribus, a candidate is in ofiice

for one more period. In sum, the first regression estimates the following model.

dig = a + ,BlH’ighbit + figHighp’f'Odit + ,63E37p67‘it + Ez’t (1.11)

where: 1) d“ is the absolute value of the difference between effort expended by
candidate 2‘ in period if (6a) and the theoretically predicted effort level (emp(t)); 2)
6’5 are coefficients; 3) Highb“ is a dummy variable that equals 0(1) if candidate
iin period t is in a treatment with B equal to 5000(7425); 4) Highprod,-t is a
dummy variable that equals 0(1) if candidate 2' in period t is in a treatment with
(Try — 7rL) equal to 06(1); 5) Exper,-t computes the number of times, between
periods 1 and (t — 1), that candidate z' is in power; and 6) a“ is an i.i.d. error
term.

The results of the estimation are detailed in column 1 of Table 7 (refer to
Appendix A). The point estimate of 63 is negative and statistically significant.
Thus, as candidates gather experience, the discrepancy between data and theory

w
45 . , . . . .
We experlrnented With several ways of measuring candldate experience. The substantive
Conclusrons are robust to the various approaches.

 

50

diminishes. The absolute value of 63 (0.25) is not large. Thus, the rate at which
effort discrepancy diminishes is slow.

Another method of estimating the effects of treatment conditions and expe-
rience on effort discrepancy explicitly recognizes heterogeneity in the candidate

pool. The errors-components approach estimates the following model.

dit = 61H ighbit + flgHighprodit + flgEmperit + 0i + Sit (1.12)

Note that the constant, or, in equation (1.11) is replaced by the fixed effects, 6,,
in equation (1.12). Heterogeneity in the candidate pool is modeled by allowing
the 0,’s to vary across candidates.

We estimate equation (1.12) using two approaches. First, we include the fixed
effects directly as regressors. The coefficients (6’s) are measured by using the
Within-subject variation in treatment conditions and experience. The results are
detailed in column 2 of Table 7. As in the OLS case, the point estimate of 63
is negative and statistically significant, thereby confirming that as candidates’
experience accumulates effort discrepancies decline. Since the fixed effects are
treated as regressors, they have no distribution. Nonetheless, a measure of subject

heterogeneity can be obtained by computing the sample standard deviation of the

 

 

51

estimated 0,33 (61’s). In the data, the sample standard deviation of the 61’s is
5.08. Since the average of candidates’ effort discrepancies across Treatments 1
through 4 is 10.67, the sample standard deviation of the 675s represents significant
heterogeneity in the candidate pool.46

Given the random assignment of candidates to the various treatments, a
random-effects (GLS) estimator is consistent and potentially more efficient. The
results of the random-effects estimator of equation (1.12) are detailed in column
30f Table 7. As in the OLS and fixed—effects case, the point estimate of ,83 is
negative and statistically significant.47 Since the fixed effects now have a distribu~
tion, we can formally check for heterogeneity in the candidate pool by determining
whether the variance of the distribution of the fixed effects is zero. The Breusch
and Pagan Lagrange Multiplier test detects heterogeneity in the candidate pool.48

The results of the three regressions are summarized in the following conclusion.

 

46Another measure of heterogeneity is the ratio: (sample standard deviation 0f the 558) +
(estimated standard deviation of the (fl-L’s). In the data, this ratio is equal to 0.44. Hence, the
conclusion of “significant heterogeneity” is validated-

47If the random-effects specification is correct, then fixed- and random—effects estimators
should yield comparable point estimates of the 6’s. We Performed a Hausman teS’t (see Haus-
man (1978)) to check Whether the point estimates of the 6’s in columns two and three of Table
7are statistically indistinguishable. The null hypothesis of “equality of the 13’s” could not be
rejgcted at the 0.1 level of significance.

Under the null hypothesis of “no heterogeneity,” the test statistic is distributed as Xfll' At
the 0.1 significance level, the critical value is 2.71. In our dataset, the test statistic assumes a
value of 144.56.

 

 

 

its

an.

n

#.

1w

{7

it

 

 

52

Conclusion 8: As candidates gather experience, the discrepancies between candi—
dates’ effort levels and the corresponding theoretical predictions decrease. There
is heterogeneity in the candidate pool with respect to effort discrepancy. Thus,

the model fits the behavior of some candidates better than others.

Relative Efficiency of Experimental Elections

We conclude by computing the relative efficiency of experimental elections.
Our procedure is as follows. For each of the six treatment conditions, proposition 2
places a theoretical upper bound on the effort that can be elicited from candidates
in any symmetric and stationary sequential equilibrium. Corresponding to each
treatment condition, we pool all the observations and determine the average effort
level of candidates. We define relative efficiency to be the ratio of the average
candidate effort to its theoretical ripper bound. Table 8 in Appendix A presents
the results.

To read Table 8, consider the Treatment 1 sessions. The “effort upper bound”
of 70.36 is computed by plugging the B— and k—values of Treatment 1 (B = 7425

and k = 1) into the formula, (356$, derived in proposition 2. On average,

 

Candidates assigned to Treatment 1 expend effort equal to 67.43. Therefore, the

 

 

 

li.‘

li.‘

/‘
/,=

 

r

53

resulting relative efficiency is (%) X 100% = 95.84%.49

Two conclusions Show up in Table 8. First, relative efficiency averaged over
rlfeatments 1 and 2 (B = 7425) is 79.49% while relative efficiency averaged over
Treatments 3 and 4 (B = 5000) is 70.14%. Thus, relative efficiency increases
when the private benefit of office is raised. Second, relative efficiency averaged
over Treatments 1 and 3 ((7rH - m) = 1) is 90.11% while relative efficiency
averaged over Treatments 2 and 4 ((7rH — m) = 0.6) is 59.51%. Thus, relative
efficiency increases when the productivity of candidate effort is raised. We do not

possess a theory that accounts for relative efficiency. We simply summarize the

above observations in the following conclusion.

Conclusion 9: The relative efficiency of experimental elections is an increasing

 

function of 1) the private benefit of office and 2) the productivity of candidate

effort.

 

49For Treatments 5 and 6, since (7TH — 7rL) is equal to zero, proposition 1 maintains that
candidates will be unwilling to expend effort. Hence, the upper bounds on effort are zero. To
avoid dividing by zero, the relative efficiency numbers are not computed.

 

 

 

Lg].

 

 

 

54
1.5.2. Experimental Results for One-Shot Sessions

The one-shot sessions consisted of Treatments 1, 2, 3 and 4. For each treatment
condition, we conducted two trials where each trial, in turn, consisted of six
election periods. Corresponding to each treatment condition, we pooled all the
observations.50 Table 9 in Appendix A provides a summary of the data.

To read Table 9, consider the Treatment 1 sessions. In the experiment, we
conducted two trials with Treatment l-parameter values. In these trials, the
average of candidates’ effort levels was 8.58 while the standard deviation was
16.39.51

In one-shot sessions, since the reelection pressure is absent, proposition 1 main-
tains that candidates will be unwilling to expend costly effort. In each of the four
treatment conditions, while the theoretically predicted effort level of zero was
exceeded, the magnitude of the discrepancy was small. Therefore, we draw the

following conclusion.

 

Conclusion 10: As predicted by proposition 1, when the reelection pressure is

 

50With two trials and six periods per trial, each treatment condition had twelve observations.

51In one-shot sessions, voters have no task to undertake. They only receive the period pay—
offs that are stochastically generated from candidates’ effort choices. Thus, effort choices of
candidates are the relevant observations for these sessions.

 

 

 

 

 

 

55

absent, candidates are willing to expend negligible effort levels.

1.6. Conclusion

Using a principal-agent model of elections with moral hazard, we experimentally
investigated the extent to which the desire for reelection elicits costly and unob-
served effort from candidates.

The experimental data supports the theoretical predictions. Specifically, the
average of candidates’ effort levels is an increasing function of the private benefit
of office and the productivity of candidate effort. Furthermore, candidates expend
negligible effort when the productivity of candidate effort is zero or the reelection
pressure is absent.

The individual decisions of voters and candidates reveal two anomalies. First,
While voters use candidate performance in determining electoral choices, poor
candidate performance is punished less harshly than theoretical models predict.
Second, candidates’ effort choices respond to the incentives implicit in the elections
game, but the average effort level exceeds that predicted by the theoretical model.
The reasons for these anomalies are as yet largely unexplained.

Many theoretically-interesting questions remain to be explored. The experi—

ment considered the case of a homogenous electorate —— the realized period output

 

 

 

 

 

 

56

was equally valued by all voters. An extension of the experimental setup could
allow the incumbent candidate to divide the realized output in any manner among
the voters. The data generated from this modified experiment allow estimation
of the loss of electoral accountability from two factors: 1) unobserved candidate
effort, and 2) the ability of the incumbent to endogenously select her own “voting
constituency.”

In addition, our experiment bases candidate evaluation on the sequence of out-
puts generated. A future extension could graft a spatial structure onto the current
model. Specifically, policy-motivated candidates could simultaneously choose un—
observed effort levels and locations on a policy space. The data generated from
this modified experiment allow estimation of the loss of electoral accountability
from two factors: 1) unobserved candidate effort, and 2) differences in policy
preferences of the median voter and candidates.

Drawing broad conclusions from our analysis would be a risky undertaking.
While the experiments themselves are extensive, the challenge of verisimilitude
is a threat to the validity of any experimental research. Our study nonetheless
Provides a useful look at voters’ capacity to overcome the moral hazard problem.
Given the institutional structure of repeated elections, voters do have the requi-

site ability to elicit effort from candidates. However, candidates’ effort choices

 

 

 

17’

57

fall short of the feasible upper bound. Thus, the electorate’s ability to sanction

candidates is not sufficient to eliminate the rents of office.

1.7. Formal Proofs of Propositions

In this section, we formally prove propositions 1 and 2. Throughout, the notation
employed is that of section 1.3.

In section 1.3, we defined the state space of the model — S E {rs/1,33} — and
a transition map, Q : S X E —> S. From the formulas supplied in the text, the

following equalities are immediate:

Q(3A l 8.4.6) = 62(3); I 53.6) (1.13)

Q<3A l 83,6) = Q(SB I SAae) = (1‘ Q(3A I 3.4)) (1.14)

Q(3A l 3,4,6) = (7rL+e>< (WH—WL» XTH+((1—TFL)—6X(7TH—7I‘L))X1‘L (115)

 

In section 1.3, we defined the value function {V,-(sj;e)},-,jeK and supplied the

 

 

 

58

following relevant formulas:

V,(s,,;e) = (B—kx[e§])+6x(Q(sA I 3,, e)XVA(sA;e)+Q(sB | 8A,e)><VA(sB;e))

(1.16)

VA(sB;e) = 6 X (Q(sA | 83,6) x VA(sA;e) + Q(sB | 83,6) x VA(sB;e)) (1.17)

For fixed voters’ strategy — i.e., TL, T‘H held constant —— 5(B, k, 6, 7rL,7rH; TL, TH)
denotes the symmetric equilibrium effort level of both candidates. For notational
convenience, we suppress the dependence on B, k, 6, 7rL and 71'}; and write, instead,
Emmi). From equation (1.1) of section 1.3, it is clear that for all 5 e E, 501, TH)

satisfies the following inequality:

VA(sA;E(rL,rH)) 2 (B — (k x 752) + 2) + 6 X (CNS/dang) X (1-18)

VA(3A'>E(7‘L77'H)) + Q($B|3Aa5) X VA(SB;'é(7"L,TH)))

In Other words, the right-hand side of equation (1.18) attains its maximum value

 

 

 

 

f 2 —fi

59

when E is set equal to €01,771). Therefore, the derivative of the right-hand side
of equation (1.18) with respect to E equals zero when evaluated at e(rL,rH).

Performing the above manipulations, we obtain the following condition:

501,771) = (g) X (TH — TL) X (7rH — 7rL) X (VA(sA;E(rL, TH)) —— VA(SB;E(7‘L,7‘H)))

(1.19)

Subtracting equation (1.17) from equation (1.16) and rearranging terms, we obtain

the following condition:

(VA(3A;€) —VA(sB;e)) X (1 — 6 X Q(sA|sA; e) — (5 X Q(SB|3A;€)) = (B _ k x [63])
(1.20)

 

Combining equation (1.19) and equation (1.20) and rearranging terms, we obtain

the following condition:

E(I’nLi'r'H) X (1— 5 X Q(SAlSA;—é(7’L,T'H)) — (S X Q(SB|sAi—é(TL1TH))) =

 

 

 

 

 

60

(B — k X C(T‘L,T'H)2 + 2) X (g) X (TH — TL) X (7TH — 7(1) (1.21)

We use equation (1.14) and equation (1.15) to plug in the formulas for Q(sA|sA;
E(rL,rH)) and Q(sB| 3A;E(7'L,TH)) in equation (1.21). Rearranging terms, we

obtain the following quadratic equation:

a X E(mm)2 — (3 x 501.171) + 7 = 0 (1.22)

where:
oz = 6 X (7m — 7TL) (1.23)
18=(2X(1+6—2X6X?))+(3X(TH—TL)) (1.24)

7=(§)X(%)XBX(7TH—7TL) (1.25)

Consider 6 = 0 or (7TH _ 7“) = 0_ Then, a = 7 = 0 (see equations (1.23) and
(M5))- It follows from equation (1.22) that 601,771) = 0. Consider, now, 6 > 0

and (7rH — 7rL) > 0. The solution to equation ( 1.22) is as follows:

 

 

 

 

61

E(rL,rH) = {b — b2 — 2:5} + 2; where b E B + (7rH — 7rL) (1.26)

 

Direct computations reveal that com, TH) is increasing in B, 6 and 7rH and de-
creasing in 77L and k. Hence, proposition 1 is proved.
When 6 > 0 and (7rH — 71,) > O, we obtain the upper bound on candidate
effort by maximizing 601,771) in equation (1.26) with respect to TL and TH. The
st

upper bound, denoted é“, obtains when (TL, TH) is such that 62 = (m). The

corresponding Eu—value is [33]? Hence, proposition 2 is proved. (Note: The

 

formula for 6" reveals that E“ E E iff lfzfifclf < 1. We ensure that this condition

 

is satisfied in all experimental setups.)

 

l

 

 

 

 

2. CHAPTER 2: SIGNALING IN ONE-SHOT AND RE-
PEATED ELECTIONS — SOlVLE EXPERIMENTAL EV-

IDENCE

2.1. Introduction

A key factor in the social contract is the voter’s ability to sanction public offi—
cials. This presupposes that citizens can distinguish between “good” and “bad”
outcomes and that they also estimate, with some degree of accuracy, the extent
to which realized outcomes reflect candidate characteristics. In a complete infor»
mation environment, electoral accountability can be readily achieved. However, if
informational asymmetries characterize all real—world democracies, then the rele-
vant policy question becomes: how does the quality of available information affect
a representative democracy?

Researchers have examined this question by constructing abstract election

models that acknowledge, in varying detail, the presence of informational fric-

62

 

[v

63

tions between voters and potential candidates. Theoretical discussions subdivide
into two categories: Barro (1973), Ferejohn (1986), and Austen-Smith and Banks
(1989) consider informational problems from a moral hazard (hidden action) per—
spective; Rogoff and Sibert (1988), Alesina and Cukierman (1990), Reed (1990),
Rogoff (1990), Harrington (1993), and Banks and Sundaram (1993) consider in-
formational problems from an adverse selection (hidden types) perspective. By
computing a model’s equilibrium, the theoretical exercises investigate the effect
of information in an election environment.

The theoretical literature provides intuition regarding the performance of demo-
cratic systems when information is of poor quality. Yet, predictive validity is
difliwlt to ascertain. Without exception, electoral outcomes depend on variables
such as information structure, characteristics of utility functions, values of re-
wards and punishments. Data seldom provide even error-laden measures of the

aforementioned variables.52 Therefore, we adopt a different stance. First, we con-

 

struct a model of elections that incorporates informational asymmetries between
the elected public officials and the electorate. Second, we evaluate the theoretical
model using experimental techniques.53 Within the structured environment of a

k“—
52A large empirical literature tests the reduced form comparative static predictions of theoret-
ical political-agency models. Examples include Kalt and Zupan (1990), Lott and Davis (1992),
Lott and Bronars (1993), and Besley and Case (1995a, 1995b).
53We are not arguing that experimental methods are intrinsically superior to non—experimental

 

 

.1,“
‘t. N

64

laboratory, it is relatively simple to both induce and systematically control the
values of the parameters of interest. This allows for a thorough consistency-check
of the theory.

Our theoretical model is based on the idea that although the quest for reelec—
tion is dependent on various exogenous factors (health of the economy, redistrict—
ing, public mood, etc.), incumbent candidates, in addition to revealing preferences
on underlying issues, also desire to stress that they are of high competency. We
treat competency as a shorthand for attributes (administrative efficiency, policy
expertise, etc.) that all voters find desirable. Therefore, when an incumbent
candidate possesses an abundance of such attributes, she will wish to emphasize
them. However, the electorate never directly observes the competency parameter.
It follows that the electorate encounters an adverse selection environment.

The baseline model is a one—shot election with one incumbent candidate and
an electorate of identical voters. The sequence of events is as follows. The incum—
bent candidate, cognizant of her competency level, implements a privately costly
but publicly observable policy (casework solicitation, sponsoring of legislation,

etc). Consistent with the principal-agent literature, the marginal and average

 

methods. Rather, they are a valid tool for testing theory when naturally occuring data is of
dubious quality. The close relationship between experimental evidence and innovative field
studies is explored in Roth (1991).

 

 

 

 

 

 

 

 

 

65

cost in policy-space is presumed to be decreasing in the incumbent candidate’s
competency level. Once the incumbent candidate’s policy choice is witnessed,
the electorate attempts to evaluate her implicit competency. Thereafter, if the
estimated competency is satisfactory, reelection follows.

We compute the set of rational expectations equilibria. The presence of private
candidate-specific information leads to multiple equilibria that are qualitatively
distinct. The equilibrium set can be divided into two cases. In the “pooling” case,
the incumbent candidate’s choice of policy does not depend upon her competency
level. The consequent lack of transmission of competency information leads to
electoral outcomes that are inefficient. By contrast, in the “separating” case, the
incumbent candidate’s choice of pOIiCy increases with her competency level. Since
there is complete transmission of competency information, electoral outcomes are
efficient.

To test the predictions of the model, we conducted a series of laboratory ex
periments. A primary goal of the experimental work was to sort out the various
factors that could affect equilibrium selection. To this end, we varied three back—
ground conditions of interest: 1) the incumbent candidate’s private benefit of

1101(1ng office; 2) the incumbent candidate’s private cost of implementing policy;

and 3) the amount of information available to voters.

 

 

 

c
the

 

 

66

We also conducted experimental sessions involving repeated interactions. In
a repeated-interactions session, with the composition of the electorate held fixed,
the same candidate subject participated in a series of structurally identical election
periods. This allows the incumbent candidate to develop reputations of various
kinds. The experiments were designed to examine the robustness of one-shot
experimental outcomes to reputational considerations.

In both one-shot and repeated-interactions sessions, experimental outcomes
are, for the most part, consistent with equilibrium signaling. Since candidate
specific information is transmitted in a signaling equilibrium, experimental elec-
tions are informationally efficient. Somewhat surprisingly, background conditions
(electorate’s information level and parameters of the utility functions) do not sub-
stantively affect the probability with which signaling emerges in the experimental
setups. In sum, extant theoretical models of elections with adverse selection
demonstrate the coesxistence of pooling and signaling equilibria. Using experi—
mental techniques, our study establishes that signaling is likely to be observed in
practice.

The remainder of this chapter is organized as follows. In section 2.2 we de-
scribe the experimental model. Section 2.3 provides the analytical solution(s) to

the experimental model for both the one—shot and repeated interactions cases.

 

 

 

 

f

67

The experimental design for our empirical tests is in section 2.4 and the results
are described in section 2.5. Section 2.6 concludes the chapter from a substan—
tive perspective. All analytical proofs are relegated to section 2.7 while tables

presenting the empirical results are gathered in Appendix B.

2.2. The Experimental Model

The model consists of two periods. The basic setup is as follows. There is one
incumbent candidate and one challenger. The incumbent exhibits either high
competence, I H, or low competence, I L. I H and I L are represented as elements
(numbers) in §R+ with I H > I L. Similarly, a challenger possesses either high com—
petence, OH, or low competence, CL — and just as with I H and I L ~— CH and
CL are represented as elements in 3%,. with OH > CL. A candidate’s type is not

known to voters. The commonly known and shared prior belief is that there is

 

a probability 7r 6 (0,1) that a candidate is highly competent and a probability
1— 7r that she possesses low competence.

At the beginning of the period, the incumbent candidate chooses a policy
outcome, denoted y; 6 82+, at a privately borne cost given by k X [Ti—I54 Two

assumptions are implicit in the cost function. First, the incurred cost is a convex

 

54Recall that policy outcomes refer to the tangible benefits conferred upon the incumbent
candidate’s constituents. Examples include pork barrel projects and casework.

_;/

. £1

‘2.
"W

 

68

and increasing fimction of the policy outcome level. Second, total and marginal
cost in policy-space is lower for the high competency incumbent candidate than
for the low competency incumbent candidate.55

All voters observe the choice of y I. Once y; is observed, the electorate votes and
decides, by majority rule, whether to reelect the incumbent candidate. Following
the vote, period one concludes.

Before proceeding, we introduce additional notation for later use. Both V
and B are mappings from the positive real numbers, §R+, to [0,1]. V(yI) is the
proportion of votes cast in favor of the incumbent candidate when the observed
policy outcome is y). Conditional on having observed the policy outcome y], B (y I)
is the electorate’s posterior probability that the incumbent has high competence
(In).56 When the policy outcome is y; and the electorate uses vote function V,
P(V,y1) is the probability with which the incumbent candidate is reelected.

In period two, the incumbent candidate and voters receive their payoffs. Should
the incumbent candidate be reelected, she receives a gross reward of W, where

W represents the value of holding office for a single additional term. When the

 

55The “total” and “marginal cost” conditions are equivalent to the singlecrossing property of
Spence (1973). For a detailed theoretical treatment of one-period signaling games, see Mailath
(1987). For a detailed survey of signaling games in political science, see Banks (1991).

“Voters have identical information and preferences. In equilibrium, the beliefs and vote
decisions of voters should be indistinguishable. Thus, it is legitimate to let B and V represent,
respectively, “the” electorate’s beliefs and vote function.

 

 

 

  

    

69

electorate adopts the vote fimction V and the incumbent candidate of type, I X,
chooses policy outcome, 31;, her net expected payoff is given by [W X P(V,y1) —
kx 1:311.

Consider, now, a voter’s payoff. Should the incumbent candidate be reelected,
each voter receives a payoff of R(IX). Otherwise, each voter’s payoff is R(CX)
since the challenger is elected. We maintain, furthermore, that: 1) R(IH) =
R(CH); 2) R(IL) = R(CL); and 3) R(_,H) > R(_,L). Two observations clarify
our interpretation of voters’ payoffs. First, a voter’s interest in the incumbent
candidate is restricted to her competency. Policy outcome is relevant only because
it provides a noisy signal of the incumbent’s competency. Second, when voters
reject the incumbent candidate, their subsequent payoffs are also random due to
the ex ante uncertainty concerning the competency of the new office holder.57 In
words, if voters are convinced that the incumbent is of low competence, they are
to take a chance on the challenger being highly competent.

&_____

57A more elaborate model would allow voters’ payoffs to depend on period one policy outcome,
in, the incumbent candidate’s type, I X, as well as the policy outcome and type of the period
two incumbent candidate. However, it can readily be established that in any subgame perfect
eqllilibrium of our “elaborate” model, the policy outcome selected by the period two incumbent
Candidate is 0. Thus, electoral decisions in period one reduce to a choice between the incumbent
candidate and her challenger on the basis of their type characteristics. The simple model
Presented in section 2.2 fully captures the asymmetric information aspects of the vote decision.

 

 

 

70

2.3. Solution to the Experimental Model

While our experimental model is a one-period game, the experimental sessions
were of two sorts. In one-shot sessions, we approximate the single period interac-
tion between the incumbent candidate and the electorate at large. By contrast,
in repeated-interactions sessions, we allow the incumbent candidate to partici-
pate in a sequence of structurally identical election periods. Consequently, the
repeated-interactions sessions allow for the possibility of supergame strategies.

The motivation for introducing two session—types is twofold. First, by mimicing
our baseline experimental model, the one—shot sessions enables us to evaluate
its predictive validity. Second, by contrasting the experimental findings across
session-types, we evaluate the extent to which reputational considerations displace
the predictions of one-period signaling games.

In this section, we provide theoretical solutions for our experimental model for
both the oneshot and repeated-interactions case. We consider first the one-shot

version.

2.3.1. Model Solution: The One-Shot Case

In this subsection, we define and characterize rational expectations equilibria

(hereafter, r.e.e.) of the one-shot model. An r.e.e., denoted by the triple <

 

 

 

~70!

‘,‘ 0

71

Y",B*,V* >, is comprised of three parts: Y*(IL)(Y*(IH)) is the policy outcome
chosen by the incumbent candidate when her competency turns out to be low
(high). Conditional on the realized policy outcome y], B*(yI) is the electorate’s
belief function about candidate competency while V*(y1) is the electorate’s vote

flmction.

Definition 1: We call <Y*, B*, V*> a one-shot r.e.e. if and only if the following
three conditions are satisfied:

(i) Y*(IX) E arg max [W X P(V*,y1) — k X [fill

yrEiR+
1 If y] = Y*(IH) and Y*(IH) 75 Y*(IL)
(ii) B*(yx) = 0 if y, = Y*(IL) and Y*(IH) 7e Y*(IL)
7r if y, = Y*(IX) and Y*(IH) = Y*(IL)
1 If B*(y1)> 71'
(m) V’Iyrl =
0 If B *(yl) < 71'
Since the incumbent candidate is a rational actor, condition (i) maintains that the
policy outcome choice must maximize her expected ex ante utility. We also require
that the electorate’s belief function be consistent with the incumbent candidate’s

p01icy outcome choice. Specifically, condition (ii) stipulates that for y; on the

 

 

LA

 

 

72

equilibrium path (i.e., y] E {Y*(IL), Y*(IH)}), beliefs are pinned down by Bayes’
Rule. Suppose that the electorate observes a policy outcome of y1. B*(yI) is the
posterior probability that the incumbent candidate is of type 1H. Therefore, if the
incumbent gets reelected, each voter receives a payoff of R(IH) with probability
B‘(y1) and R(IL) with probability (1 — B*(y1)). If the challenger is elected, each
voter receives a payoff of R(CH) with a probability 7r and R(CL) with probability
(1— 77). Since voters maximize their expected payoff, condition (iii) requires that
the challenger (incumbent) be selected when B*(yj) exceeds (is less than) 71.
While details of the r.e.e. are given in section 2.7, we summarize the main
findings. The model generates multiple equilibria. The set of equilibria can be
divided into two cases. In case one, the “pooling case,” Y*(IL) = Y*(IH). In
case two, the “separating case,” Y*(IL) 74 Y*(IH). We consider the pooling

(separating) case in Proposition 1 (Proposition 2).

KW There is a continuum of pooling equilibria. By definition, in a

Specific pooling equilibrium, Y*(IL) = Y‘le) E 36- Any 9* 6 l0) II‘VICKI X ILlfl can

be supported as an equilibrium outcome.

In a pooling equilibrium, the incumbent candidate’s choice of policy outcome,

 

 

M

y‘, is independent of her competency. The electorate realizes that the observed y*
is a thoroughly uninformative signal. Given that the incumbent candidate and her
challenger are ex ante identical in terms of competency, the incumbent candidate
is reelected with some prespecified probability. As a result, the electoral system
is informationally inefficient. Ex post informational efficiency requires that the
incumbent candidate be reelected if and only if she possesses high competency.
Since votes cast are not conditioned on the incumbent’s type, informational effi-
ciency occurs only by chance.

The continuum of pooling equilibria can be ranked in terms of aggregate wel—

 

fare. Recall that the policy outcome does not directly affect a voter’s payoff: the
policy outcome is relevant only because it potentially contains information about
the incumbent candidate’s competency. In every pooling equilibrium, y* trans—
mits no competency information. Thus, a voter’s expected payoff in every pooling
equilibrium is the same. However, since higher values of y* impose larger costs on
the incumbent candidate, her net payoff is maximized in the pooling equilibrium
with the smallest policy outcome level (i.e., y* = 0). Aggregate welfare, measured
as the sum of the payoffs of all agents (the incumbent candidate and voters) in

the model, is maximized when y" = 0. Aggregate welfare declines as y* is raised.

 

 

74

Proposition 2: There is a continuum of separating equilibria. By definition, in
aspecific separating equilibrium, Y*(IL) 7e Y*(IH). Any pair {Y*(IL),Y*(IH)}
such that: 1) Y*(IL) = 0 and 2) Y*(IH) 6 us] x Ids, [[g] x 1.4%] can be

supported as an equilibrium outcome.

In a separating equilibrium, the incumbent candidate’s choice of policy out-
come varies with her competency. The electorate recognizes that the observed
y; is a fully informative signal of the incumbent’s competency. Reelection of the

incumbent occurs if and only if y1 = Y*(IH). Equivalently, the incumbent can—

 

didate is reelected if and only if she possesses high competency. Therefore, the
electoral system is ex post informationally efficient.

The continuum of separating equilibria can also be ranked in terms of aggre-
gate welfare. In every separating equilibrium, the policy pair {Y*(I L), Y*(I 11)}
completely transmits the incumbent candidate’s competency information to the
electorate. Thus, a voter’s expected payoff in every separating equilibrium is
the same. However, since higher values of Y*(IH) impose larger costs on the
incumbent candidate, her net payoff is maximized in the separating equilibrium
with minimal separation ~ i.e., {Y*(IL), Y*(IH)} = {0, (H?) X Ali}. Aggregate

welfare, measured as the sum of incumbent candidate’s and voter’s payoffs, is

 

 

75

maximized at this “minimal separation” equilibrium. Aggregate welfare declines
as Y*(IH) is raised.

The principal characteristics of the equilibria generated by our experimental
model are: 1) for a fixed set of exogenous parameters, there is a continuum of
equilibria. The equilibrium set can be divided into a “pooling set” and a “sepa—
rating set”; 2) in a pooling equilibrium, no information is transmitted regarding
the competency of the incumbent candidate and the electoral system is ex post in—
formationally inefficient. In a separating equilibrium, the incumbent candidate’s
competency is fully revealed to the electorate. Consequently, the electoral system

is ex post informationally efficient.

2.3.2. Model Solution: The Repeated-Interactions Case

Since our subjects participated in a sequence of structurally identical games with
uncertain termination date, we employ the theory of repeated games to provide
theoretical guidance regarding likely experimental outcomes. Given our multi-
player, repeated-interactions environment, Folk theorem—type results guarantee
the existence of an abundance of equilibria. For the purposes of sharper predic—
tions, we focus on solutions that satisfy an additional condition of stationarity.58

mg“
58Stationarity is satisfied if along the equilibrium path: 1) the incumbent candidate of compe-
tency I x selects a time- and history—independent policy Y*(IX); and 2) the electorate reelects

 

 

’fiej

tion,

with

3%

76

The emphasis on stationarity yields an extra dividend: solutions for one-shot and
repeated-interactions setups become directly comparable.

As in the one-shot case, the equilibrium set can be divided into two cases. In
the “pooling case,” the incumbent candidate chooses the same policy outcome
for both levels of competency. In the “separating case,” the incumbent candidate
chooses policy outcomes that vary with her competency. We consider the pooling
(separating) case in Definition 2 and Proposition 3 (Definition 3 and Proposition
4).

In a pooling equilibrium — along the equilibrium path — Y*(IL) = Y*(IH) E
y" is the policy outcome chosen by the incumbent candidate. Also, 1) V*(y*)
is the portion of votes cast in favor of the incumbent candidate; 2) P(V*, y*) is
the resulting probability with which the incumbent candidate is reelected; and
3) B*(y*) is the electorate’s posterior probability that I X is equal to I H. Finally,
Q*(IX) denotes the incumbent candidate’s expected discounted sum of payoffs
over the infinite horizon when her current competency is I X. Given the above
notation, it is immediate that: Q*(Ix) = [W X P(V*,y*)— k X [31*[2 + Ix) + 6 X
[(1— it) x Q*( I L) + 7r x Q*( I H)], where 6 is the probability with which the game

is continued from one period to the next.

 

the incumbent candidate with a time- and history—independent probability P(V*,Y* (I X))

 

 

 

 

Definition 2: We call <y*, B*(y*), V*(y*)> a stationary path of a repeated— inter—

actions pooling equilibrium if and only if the following two conditions are satisfied:
(1) Q*(IL) Z 0 and Q*(IH) Z 0
(ii) B*(y*) = W and V*(y*) E (0.1)

,

To support “repeated play of y*’ as an equilibrium path, we need to specify
voter behavior subsequent to a possible deviation by the incumbent candidate.
Given our experimental setup, any candidate deviation from the pooling equilib~
riurn in period—t is immediately detected by the electorate. Without loss of gen—
erality, we invoke the harshest possible punishment following a deviation: from
period-t onwards, the incumbent candidate is never selected by the electorate.59
Subsequent to a deviation, the incumbent candidate’s expected discounted sum
of payoffs over the infinite horizon is 0. Thus, condition (i) ensures that it is
unprofitable for the incumbent candidate to deviate from the putative pooling

equilibrium. Consider, now, condition (ii) of the above definition. Since the equj_

‘—

59Such drastic punishments can easily be supported as part of a repeated—game equilibrium.
Specifically, following a deviation, let the electorate harbor the belief that B(y1) = 7r;\7’ y, E Y.
Since the electorate is now indifferent between the incumbent candidate and her challenger, it
becomes rational to always reject the incumbent candidate. Finally, given voter behavior, the
incumbent candidate, for both levels of competency, chooses a policy outcome of 0.

 

 

 

78

librium choice of policy outcome, 31*, does not reveal the incumbent candidate’s

current competency level, the electorate’s posterior assessment about candidate
competency equals its prior assessment. Hence, B*(y*) = 7r. Finally, given the
electorate’s beliefs, the incumbent candidate and her challenger are identical op-
tions. As a result, the electorate’s vote behavior, summarized by V*(y*), remains

unconstrained.

Proposition 3: There is a continuum of stationary pooling equilibria. By def-
inition, in a specific pooling equilibrium, Y*(IL) = Y*(IH) E y“. Any 34* 6

[0,A X [[121] X I Llfl can be supported as an equilibrium outcome, where A :

[13 + [IH — 6 X 71' X [IH — IL]]]i is greater than 1.

As in the one—shot setup, stationary pooling equilibria in the repeated— interac-
tions setup are both informationally inefficient and Pareto ordered. Furthermore,
a comparison of propositions 1 and 3 shows that the set of pooling equilibrium
outcomes expands relative to the one—shot setup when reputational considerations
are introduced through the repeated-interactions setup.

In a separating equilibrium — along the equilibrium path — Y*(I L)(Y*(I 11))

is the policy outcome chosen by the incumbent candidate when her competency

 

 

 

 

 

79

is IL (IH). Also, for Ix E {ILJH}: 1) V*(Y*(IX)) is the proportion of votes
cast in favor of the incumbent candidate when the chosen policy outcome is
Y*(IX); 2) P(V*,Y*(IX)) is the resulting probability with which the incum-
bent candidate of type I x is reelected; and 3) B*(Y*(Ix)) is the electorate’s
posterior probability that I X is equal to I H conditional on observing the pol-
icy outcome Y*(IX). Finally, Q*(IX) denotes the incumbent candidate’s ex-
pected discounted sum of payoffs over the infinite horizon when her current
competency is I X. Given the above notation, it is immediate that: Q*(IX) =
[W X P(V*.Y*(Ix)) - k X [Y‘flxh2 + Ix] + <5 X [(1— 7T) X Q*(IL) + 7? X Q*(IH)l.
where 6 is the probability with which the game is continued from one period to

the next.

 

Definition 3: We call <Y*(IX),B*(Y*(IX)), V*(Y*(Ix)) I IX 6 {IL.IH}> a sta-
tionary path of a repeated—interactions separating equilibrium if and only if the
following four conditions are satisfied:

(i) Q*(IL) Z [W X P(V*.y1) — k X [311]2 + [LINE/I E Y\{Y*(IL)}

(11) can.) 2 [w x P(V*,y1) — k x [11,]2 + Islam 6 Y\{Y*(IH)}

(iii) B*(Y*(IL)) = 0 and B*(Y*(IH)) = 1

(ii!) V*(Y*(IL)) = 0 and v*(y*(1H)) = 1

1M

 

80

To support “repeated play of {Y*(IL),Y*(IH)}” as an equilibrium path, we
need to specify voter behavior subsequent to a possible deviation by the incumbent
candidate. If a low competency incumbent deviates from the separating equilib-
rium in period t, her period-t payoff is the right-hand—side expression of condition
(i). However, the period—t candidate deviation is detected by the electorate prior
to period-(15+ 1) play. Therefore, from period (t + 1) onwards the incumbent
candidate is never selected by the electorate. Subsequent to a deviation, the in-
cumbent candidate’s expected discounted sum of payoffs over the infinite horizon
is the right-hand—side expression of condition (i). Thus, condition (i) ensures that
it is, at all times, unprofitable for a low competency incumbent candidate to devi-
ate from the putative separating equilibrium. Condition (ii) is the corresponding
“110 profitable deviation” criterion for a high competency incumbent candidate.
Condition (iii) demands that voters’ beliefs be consistent with the fact that the
e(Illilibrium choice of policy outcome reveals the incumbent candidate’s current
competency level. Condition (iv) requires that the electorate maximize its own
payoff by reelecting the incumbent candidate if and only if the observed POHCY

outcome is Y*(IH),

 

 

81

Mon—4; In the repeated-interactions experimental model, there is a con-
tinuum of stationary separating equilibria. The stationary separating equilibria
are both informationally efficient and Pareto ordered.60 The set of separating
equilibrium outcomes for the repeated-interactions setup strictly includes that

corresponding to the one-shot setup.

2.4. Experimental Design

To test the predictions of the electoral model, we performed a series of experiments
using undergraduates from a large public university. In the recruitment stage, care
was taken to ensure that the subjects were unexposed to formal decision/ game
theory. The experiments were conducted on a computer network system and,
except for reading the instructions (available upon request), all communication
took place over the network.

The experimental sessions were of two sorts: one-shot and repeated—interactions
sessions. The amount of information available to voters varied. In full-information
sessions, only the incumbent candidate’s realized competency, I x, was left undis-
CIosed. By contrast, in incompleteinformation sessions, voters were unaware of
the parameters in the incumbent candidate’s utility function. The experimental

M
The reason is identical to the one given for the one-shot case.

 

82

procedures followed in each session type are detailed below.

2.4.1. One-Shot Setup - Experimental Procedures

An experimental session consisted of a cohort of fifteen subjects divided into an
electorate of five voters and a pool of ten potential candidates. A trial, with a
fixed configuration of parameter values, lasted for either ten or twenty structurally
identical election periods. In a ten-period (twenty~period) trial, each candidate
was randomly chosen to participate in exactly one (two) of the election periods.
The fixed termination date, coupled with randomly chosen candidate subjects,
was deemed sufficient to eliminate supergame considerations. In full—information
situations, all subjects were aware of the trial-specific parameter values for W, k,
7T,R(_,L) and R(_,H). In incomplete—information situations, only candidates were
aware of all trial—specific parameter values. Voters were apprised of their own
payoff-specific parameter values (7r,R(_,L),R(_,H))- A detailed account 0f the
experimental procedures adopted in a ten-period (twenty—period) trial follows.
[Step 1] At the start of election period 15, one candidate, denoted i(t), was
designated to be the period-t incumbent candidate. Only candidate i(t) was made
aware of her role assignment.

[Step 2] The computer program generated a random variable, denoted 1,, where

 

 

 

 

 

if? "WT

83

It assumed the value of I L (I H) with probability (1 — 7r)(7r). Only candidate i(t)
was made aware of her realized competency, 1,.

[Step 3] Candidate z'(t) selected her policy outcome, yt, from the four—point
set, Y E {0, 3, 6, 9}. The recorded yt—value was transmitted to all subjects.61

[Step 4] Given yt, voters cast their ballots and decided, by majority rule, to
reelect or reject candidate 2'(t). Once votes were tallied, the final outcome of the
election, as well as the vote margins realized were transmitted to all subjects.

[Step 5] Candidate i(t) and the five voters received their period-t payoffs ac—
cording to the rules of the model in section 2.2.

[Step 6] The computer program switched to election period (t + 1). At the end
of ten (twenty) election periods, the trial terminated. Each subject’s payoff for

the trial was the sum of her period payoffs.

2.4.2. Repeated-Interactions Setup - Experimental Procedures

An experimental session consisted of a cohort of subjects divided into an elec~
torate of five voters and a pool of ten potential candidates. A trial, with a fixed
configuration of parameter values, lasted for a variable number of structurally

\________

61While the theoretical model in section 2.2 allows the policy outcome set to be 92+, we
have, for tractability, restricted the experimental policy outcome set, Y, to be {0, 3, 6, 9}. The
distinction between pooling and separating outcomes, as well as equilibrium multiplicity, applies
to Y.

 

 

 

84

identical election periods and involved the electorate facing the same candidate
each time. The uncertain termination date, coupled with an unchanging candi-
date, was deemed suflicient to induce the possibility of reputational equilibria.
As in the one—shot case, repeated-interactions sessions were of two types: full—
information and incomplete-information. The information available to subjects
in each session mirrored that of the one-shot setup. A trial proceeded as follows.

[Step 1] At the start of the trial, one candidate was designated to be the
incumbent candidate. Only the chosen candidate knew her role assignment.

[Step 2] At the start of election period t, the computer program generated a
random variable, denoted It, where It assumed the value of I L (by) with proba-
bility (1 — 7r)(7r). Only the incumbent candidate knew her competency, It.

[Step 3] The incumbent candidate selected her period—t policy outcome, ya,
from the four-point set, Y E {0, 3, 6, 9}. The recorded yt-value was transmitted
to all subjects.

[Step 4] Given yt, voters cast their ballots and decided, by majority rule,
to reelect or reject the incumbent candidate. Once votes were tallied, the final
outcome of the election, as well as the vote margins realized were transmitted to

all subjects.

[Step 5] The incumbent candidate and the five voters received their period—t

 

 

 

85

payoffs according to the rules of the model in section 2.2.
[Step 6] The computer program switched to election period (t +1). At the end
of a stochastic number of election periods, the trial terminated.62 Each subject’s

payofir for the trial was the sum of her period payoffs.

2.4.3. Experimental Parameter Values

An experimental session consisted of a number of trials. Individual trials differed
in the values assumed by the five exogenous parameters: W, k, I L, I H, and 7r. Table
10in Appendix B displays the parameter value combinations that were considered.
For example, in Treatment 1, we set the benefit of reelection, W, to be 600; the
scale parameter of policy cost, k, to be 100; the “low” competency level, I L, to be
1; the “high” competency level, I H, to be 10; and the prior probability of a high
competency incumbent candidate, 7r, to be 0.5. The parameter values in Table
10 correspond to subjects’ payoffs denominated in a laboratory currency called
the franc. At the conclusion of the experimental session, cumulative earnings in
francs were converted into dollars using a preassigned exchange rate.

The primary goal of our experimental work was to sort out the factors that

 

62Each trial lasted for at least fifteen election periods. Thereafter, at the conclusion of each
election period, the trial was terminated with a probability of 1—10. Subjects were aware that the
termination date of a trial was stochastic. They were not apprised of the process by which trial

length was determined.

 

rm

86

affect equilibrium selection. We considered the influence of two potentially im—
portant factors. First, by a pairwise comparison of Treatments 1 and 2, we inves—
tigated the influence of policy cost. Specifically, if policy cost (It) is increased (or
decreased), is there a systematic effect on the equilibrium selected? Second, by
a pairwise comparison of Treatments 2 and 3, we considered the effect of reelec—

tion benefits. If the reelection benefit (W) is increased (or decreased), is there a

 

systematic effect on the equilibrium selected? The equilibrium predictions of the
one-shot and repeated—interactions models are detailed, respectively, in Tables 11

and 12 of Appendix B.

 

A pooling equilibrium is a two-element vector. The first element is the policy
outcome chosen by the incumbent candidate; the second element is the probability
of reelection. A separating equilibrium is a four—element vector. The first element
is the policy outcome chosen by the incumbent candidate when her competency is
low; the second element is the policy outcome chosen by an incumbent with high
competency; the third element is the probability of reelection when the observed
POIicy outcome is Y*(IL); the fourth element is the probability of reelection if the
observed policy oucome is Y*(IH).

For each treatment condition, the one-shot model generates multiple pooling

equilibria (column two of Table 11). The pooling equilibria are obtained as follows.

 

87

We plug the values of the exogenous parameters from Table 10 into proposition
1 to recover the y*-values that can be supported as outcomes of some pooling
equilibrium. It can be demonstrated (see section 2.7) that if y* 7e 0, the reelection
probability must be one.
The cost of implementing policy, k, is smaller in Treatment 2 than in Treatment
1; the benefit of reelection, W, is larger in Treatment 3 than in Treatment 2. A
decrease in policy cost or an increase in reelection benefit makes pooling relatively
attractive. Thus, there is an increase in the number of pooling equilibria as we
move from Treatment 1 to Treatment 3. Since no payoff-relevant information
is transmitted in a pooling equilibrium, efficient pooling requires that the chosen
policy outcome be the cost minimizing one, Y* (I X) = 0. This intuition is reflected
in the entries in column three of Table 11.
The set of separating equilibria for the one-shot model (column four of Table
11) is obtained as follows. We plug the parameter values from Table 10 into propo—
sition 2 to obtain the pairs (Y*(IL), Y*(IH)) that can be supported as outcomes of
some separating equilibrium. The reelection probability corresponding to Y*(IL)
(Y*(IH)) is 0(1). Thus, a particular separating equilibrium, in the notation of
Table 11, becomes (Y*(IL),Y*(IH),0,1).

For Treatments 1 and 2, there are multiple separating equilibria. As we move

 

.....

r_______

88

from Treatment 1 to Treatment 3, separation becomes increasingly difficult and
the set of separating equilibria shrinks to a singleton. Since all payoff-relevant
information is transmitted in a separating equilibrium, efficient separation requires
that policy choices corresponding to candidate competencies I L and I H be distinct
and as small as possible. For each treatment condition, the entry in column five
of Table 11 selects the separating equilibrium (from column four of Table 11)

satisfying the two conditions noted above.

 

Table 12 indicates that for each treatment condition, the repeated—interactions
model generates multiple pooling and separating equilibria. Furthermore, a com-
parison of Tables 11 and 12 reveals that for each treatment condition, the equilib—
rium set of the repeated-interactions model weakly includes that of the one-shot
model. Thus, reputational considerations expand the set of equilibrium outcomes.

The pooling equilibria of the repeated-interactions model (column two of Ta—
ble 12) are obtained as follows. We plug the parameter values from Table 10 into
proposition 3 to recover the y*-values that can be supported as outcomes of a
Poolng equilibrium. It can be demonstrated (see section 2.7) that if Y*(IX) 7e 0,

the reelection probability must equal one.63 As in the one-shot case, the pooling

 

63h repeated-interactions sessions, each trial lasted for at least fifteen election periods. There-
after, at the conclusion of each election period, the trial was terminated with a probability of— 10
Wheil computing the set of pooling equilibria, we let 6— — — .The equilibrium set is invariant
'00 Slight perturbations 1n the chosen 6—value

 

 

3(‘0

file

 

89

equilibrium set expands as we move from Treatment 1 to Treatment 3. Eflicient
pooling requires that the incumbent candidate’s policy choice be the cost mini—
mizing one — Y*(IX) = 0, IX 6 {IL,IH}. The entries in column three of Table
12 reflect this intuition.

The separating equilibria of the repeated—interactions model (column four of
Table 12) are obtained as follows. For each treatment condition, we note the
parameter values detailed in Table 10. Thereafter, we check for policy pairs
(Y*(IL),Y*(IH)) that satisfy conditions A and B of section 2.7. The reelection
probability corresponding to Y*(IL) (Y*(IH)) is 0(1). Thus, a particular sepa-
rating equilibrium becomes (Y*(IL),Y*(IH),0, 1).64 Efficient separation requires
that policy choices impose minimum cost on the incumbent candidate. The entry
in column five of Table 12 selects the separating equilibrium (from column four

of Table 12) that is least expensive for the incumbent candidate.

2.5. Experimental Results

Our experiment consists of four session types: full—information one-shot (here-
after, FI—OS), incomplete-information one-shot (hereafter, II-OS), full-information

repeated-interactions (hereafter, FI—RI), and incomplete—information repeated in-

 

1

64When computing the set of separating equilibria, we set 6 = 10. The equilibrium set is
1nvariant to slight perturbations in the chosen 6—value.

 

 

 

 

 

 

90

teractions (hereafter, II-RI). In this section we present the experimental results

for each session type. We consider the one-shot sessions first.

2.5.1. Experimental Results for F I-OS and II-OS Sessions

The FI—OS and II—OS sessions consisted of three treatment conditions (see Table
10). We conducted four trials for each treatment condition. Each trial consisted
of a single ten-period or twenty-period election game. Table 13 in Appendix B
summarizes the data.

3

In Table 13, “policy outcome, I X’ is a four-element vector. For each session
type and treament condition, we pooled all cases for which the realized incum-
bent candidate competency is I X. The first, second, third, and fourth elements
represent, respectively, the number of instances that policy outcomes 0,3,6 and 9
were chosen. “Reelection probability” is a four—element vector where the first, sec—
ond, third, and fourth elements represent, respectively, the empirical probability
of reelection conditional on policy outcomes 0,3,6 and 9.

Table 13 is read as follows. For the FI—OS Treatment 1 session there were

40 observations and incumbent candidates’ realized competency was low on 27

occasions and high on 13 occasions. The vector “policy outcome, I L” indicates

that for the 27 observations involving low competency, the policy outcome of 0

 

 

 

 

 

91

was always selected. The vector “policy outcome, I H” indicates that for the 13
observations involving high competency, the policy outcome of 0 was selected
once and the policy outcome of 3 was selected 12 times. The electorate observed
a policy outcome of 0 on 28 occasions; the observed policy outcome was 3 on 12
occasions. The “reelection probability” vector indicates that a policy outcome
of 0(3) resulted in reelection 14%( 100%) of the time. Since policy outcomes of
6 and 9 were not observed, the corresponding reelection probabilities were not
computed.

Two conditions must be satisfied for each session type if signaling character—
izes the aggregate data of Table 13. Since larger policy outcomes signal higher
incumbent candidate competency, reelection probabilities are required to be non-
decreasing in policy outcome levels. Second, the distribution of policy outcomes
when incumbent candidates’ competency is high should stochastically dominate in
a first-order sense the distribution of policy outcomes when incumbent candidates’
competency is low.

Consider, first, whether reelection probabilities are (weakly) increasing in pol-
icy outcome levels. Note that for each of the six session types, reelection prob-
abilities satisfy the required weak monotonicity property (column four of Table

13). To check whether the second condition for signaling applies, observe the

 

 

 

 

 

 

 

92

distribution of policy outcomes chosen by incumbent candidates when realized
competency is low (high) (columns two and three of Table 13). For each of the
six session types, application of the Median Test and Kolmogorov-Smirnov Two
Sample Test (see Conover 1980) rejects the null hypothesis of equality between
the two empirical distributions (.01 significance level).65 Conclusion 1 summarizes

our results.66

Conclusion 1: For each of the six session types, aggregate data reveals separation

in both policy outcomes and reelection probabilities.

Having ascertained that signaling characterizes the aggregate data, we now
determine whether the individual decisions of incumbent candidates and voters
are consistent with the private incentives implicit in the model. We evaluate the

Optimality of incumbent candidates’ behavior as follows.

 

65As an example, for the FI—OS Treatment 1 session, we test whether the sample distribution
of policy outcomes contingent on low competency - i.e., (27, 0, 0, 0) - is stochastically dominated
by the sample distributon of policy outcomes contingent on high competency — i.e., ( 1, 12, 0, 0).

66For the Median Test, the test statistic is distributed as X21 . The critical value, at the .01
Significance level, is 6.64. The minimum realized value of the test statistic, across the six session
types, is 7.86. For the Kolmogorov-Smirnov Two-Sample Test, the critical value of the test
statistic, at the .01 significance level, is 9.21. The minimum realized value of the test statistic,
across the six session types, is 9.96. For both tests, the minimum value occurs in the II—OS
Treatment 3 session.

 

 

 

—1.

:1

 

93

For each of the six session types, we observe the vector of reelection probabil-
ities (column four of Table 13). Given the empirical reelection probabilities, for
each level of candidate competency, we rank order the four policy outcomes in
terms of their average payoff for the incumbent candidate. Thereafter, for each
competency level, we compare the computed optimal policy choice with the ob-
served distribution of policy outcomes (colurrms two and three of Table 13). Our
results are presented in Table 14 of Appendix B.

In Table 14, “optimal policy choice” is a two-element vector, where the first
(second) element equals the policy outcome that yields the highest average payoff
when the incumbent candidate’s competency is low (high). “Modal policy choice”
is a two-element vector, where the first (second) element equals the modal policy
choice when the incumbent candidate’s competency is low (high). Consider, for
example, the FI—OS Treatment 1 session. Given the vector of reelection prob-
abilities ((0.14,1.0,_,_)) and the parameter configurations detailed in Table 10
(k = 100 and W = 600) computations indicate that the policy outcome of 0(3)
maximizes the expected payoff to the incumbent candidate when her competency

is low (high).67 Columns two and three of Table 13 show that the modal policy

 

67Let 7r(y1) be the empirical probability of reelection when policy outcome is yr 6 Y. For
an Ix-type incumbent candidate, the expected payoff from y; is: [W X 7r(y1) - k x [y1]2 + IX].
The theoretically optimal policy for an I X-type incumbent candidate is the ypvalue yielding

 

 

94

chosen by the incumbent candidate is also 0(3) when realized competency is low
(high).

In nine out of twelve cases (two competency levels X six session types), in-
cumbent candidates’ modal policy choice is the privately optimal one. For the
three anomalous cases, the modal policy choice of incumbent candidates turns
out to be the second best alternative. We have conclusive evidence that incurn—
bent candidates’ policy choices are consistent with the maximization of private
utility.

Next, we evaluate the optimality of voters’ behavior. For each session type,
we pool the data and compute, for each policy level, the empirical conditional
probability that the incumbent candidate’s competency is high. We classify the
electorate’s aggregate behavior to be privately rational if, for each policy level,
the reelection probability is greater (less) than % when the empirical conditional
probability of high competency is greater (less) than 568 The results are presented
in Table 14.

In Table 14, “reelection probability” is a four-element vector identical to that

 

the highest expected payoff.

8Since rejection of the incumbent candidate results in the electorate’s payoff being a 50-50
gamble between R(CL) and R(CH), a more stringent test of the electorate’s rationality would
require that the reelection probability be 1(0) when the conditional probability of I H is greater
(less) than %. However, in experimental setups, subjects estimate the conditional probability of
I” only as evidence accumulates. Our proposed test is a reasonable compromise.

 

 

 

95

shown in column four of Table 13. “Conditional probability of I H” is a four—
element vector where the first, second, third, and fourth elements represent, re-
spectively, the empirical probability that the incumbent candidate’s competency
is high conditional on policy outcomes 0,3,6 and 9. Consider the FI-OS Treat—
ment 1 session. The vector “conditional probability of I H” is computed as follows.
During the session, a policy outcome of 0 was observed 28 times and a policy out—
come of 3 was observed 12 times. Out of the 28 observations of policy outcome

0, incumbent candidates’ competency was high on 1 occasion; of the 12 obser—

 

vations of policy outcome 3, incumbent candidates’ competency was high on 12

 

occasions. Thus, corresponding to the policy outcome of 0, the conditional prob—
ability of high competency equals 2—18; for the policy outcome of 3, the conditional
probability of high competency equals ]—§. Since policy outcomes of 6 and 9 were
unobserved, the corresponding conditional probabilities were not computed.
Note that in the FI—OS Treatment 1 session the conditional probability of high
competency exceeds % when the observed policy outcome is 3 and is less than %
when the observed policy outcome is 0. The electorate’s aggregate behavior is
rational in this case: the reelection probability corresponding to policy outcome
of 3 exceeds % and the reelection probability corresponding to policy outcome of

0 is less than %. Columns four and five of Table 14 demonstrate that in twenty

 

 

m

96

out of twenty-one cases, reelection probabilities and conditional probabilities of
high competency satisfy our condition. The electorate’s vote decision is consistent
with the maximization of private utility.

Conclusion 2 summarizes our findings regarding candidate and voter behavior.

Conclusion 2: For each of the six session types, the behavior of incumbent candi—

dates and voters is uniformly consistent with the maximization of private utility.

 

Recall fiom section 2.3.1 that the one-shot model generates a multitude of
equilibria. We now determine which of the equilibria in Table 11 is most consistent
with experimental observations.

For each of the six session types, we pool the data. An individual observation,
1', is viewed as a triple (yr, 1,, 6,) where: 1) yr 6 Y iS the choice of policy outcome;
2) Ir E {I L, I H} is the realized competency of the incumbent candidate; and 3)
er 6 {0,1} is a binary choice variable that assumes the value 0 if the incumbent
candidate is rejected and 1 if she is reelected. A specific equilibrium predicts
the occurrence of certain kinds of experimental observations. For example, when
incumbent candidates separate such that {Y*(IL) = 0,Y*(IH) = 3} (i.e., the

equilibrium is (0, 3, 0, 1) in the notation of Table 11) experimental observations

 

 

 

r—_—____

97

are predicted to be either (0, I L, O) or (3, I H, 1). For each session type and each
possible equilibrium we compute the percentage of observations consistent with
its prediction. Table 15 in Appendix B presents the results.

In Table 15, each entry is a three-element vector (a, fi, 7). a depicts a partic—
ular equilibrium. [3 E {3, p} is a binary variable equal to 3 if the corresponding
equilibrium, a, involves separation and p if it involves pooling. 7 E [0, 100] denotes
the percentage of the observations consistent with the equilibrium predictions of

69

a.

Table 15 reads as follows. For the FI-OS Treatment 1 session the equilibrium

 

 

with the best empirical fit is the separating equilibrium (0, 3, 0, 1). In fact, 90%
of the observations for the session matched equilibrium predictions (i.e., were of
the form (0, I L, 0) or (3, I H, 1)). The second-best empirical fit is achieved by the
separating equilibrium (0,6,0, 1); the pooling equilibrium (0,0) is the third-best
fit.

Table 15 provides additional support for the preponderance of signaling in the
data. Except for the II—OS Treatment 3 session, the best-fitting equilibrium model

involves signaling (column two). On comparing column five of Table 11 with col-

 

. ”Recall that if 6 = s, then a is a four-element vector, (Y*(IL),Y*(IH),O,1). Alternatively,
lffi =1), then a is a two-element vector (Y*(IX), P(V*, Y*(IX))).

 

 

98

umn 2 of Table 15, a striking fact emerges. Except for the II—OS Treatment 3
session, the best—fitting signaling model is also the most efficient one. Experi-
mental elections are not simply informationally efficient. Informational efficiency
obtains at a minimum cost to the incumbent candidate.

Why does signaling cease in the II—OS Treatment 3 session? We conjecture
that two factors are at work. First, since the benefit of reelection is large, the
separating equilibrium set shrinks to a singleton and entails extreme separation;
that is, the policy choice of the incumbent candidate for low (high) competency
is 0(9). Second, since payoff information is unavailable to voters, observation of
policy outcome equal to 9 does not immediately establish that the incumbent
candidate’s competency is high. On the other hand, after observing a policy
outcome equal to 9, should the electorate choose the challenger, the incumbent
candidate incurs a substantial loss. In sum, in an incomplete information setting,
extreme and risky separation is difficult to induce. We summarize these results

in conclusion 3.

Conclusion 3: Except for the II-OS Treatment 3 session, the data is best explained

 

by the efficient separating equilibrium.

 

[j

99

 

By ranking the various equilibria by consistent observations, we have discarded
information present in unpredicted outcomes. Therefore, we suggest a simple
theory of errors and reanalyze the data using maximum likelihood procedures.70
For each session type, we pool the data and obtain a sample of size N where
observation 2' is a triple (yin,, 6,). Thereafter, for each of the equilibria in Table
11, we compute the probability of generating the observed sample {(y,,I,-, e,)}i1:1.
The computation of the probability requires estimation of two parameters, the
error rates of incumbent candidates and voters. Finally, Akaike’s Information
Criterion (see Amemiya 1985) enables us to rank order the various equilibria in
terms of the log of the computed probabilities.

For the sake of brevity, we only show how to compute the probability of gen-
erating the sample {(y,,I,-, e,)},.1:1 when the putative separating equilibrium is as
follows: 1) Y*(IL) = 0; 2) Y*(IH) = 3; 3) P(V*,0) =1; and 4) P(V*,3) = 1.71
If some of the observed data violates the equilibrium predictions above, we en-
counter the zero probability problem. To avoid the problem, we assume that,
with probability 51(5E), the incumbent candidate (electorate) randomly selects

an outcome diflerent from the equilibrium prediction.

 

700m estimation procedure is based on Harless and Camerer (1995) and Hey (1995).
71The probability computations for other equilibria are available upon request.

 

 

 

l1]

 

100

 

Before proceeding, we introduce additional notation. Let 7r1(yI|IX) denote the
probability that the incumbent candidate chooses policy outcome y; when her
competency is I X. Let 77E(y1) denote the probability that the electorate reelects
the incumbent candidate when the observed policy outcome is yr. A “noisy”
version of the putative separating equilibrium specifies the following conditional
probabilities: 1) 7r;(0|IL) = 7rI(3|IH) = 1 — 351; 2) 7r;(y1|IX) = 51 if (y1,IX)
¢ {(0:1L)a(3,IH)}; 3) «19(0) = 513; and 4) 7IE(yI) = 1 — €19in 7é 0-72 It is
now immediate that the probability of generating the sample {(11/,-,I,-,e,-)},1:1 is
Il£17r1(y,|I,) X 1rE(y,-)“’i x (1 — 77E(y,))1‘ei. The probability is a function of two
unknown parameters, 51 and 5E. However, 51 and E}; can be estimated by standard
likelihood methods and the probability of generating the sample evaluated. The
results of ranking the various equilibria in terms of log likelihoods are detailed in
Table 16 of Appendix B.

For Table 16, consider the FI-OS Treatment 1 session.73 The equilibrium with

 

72Conditions 2 and 3 represent the random deviations from the putative “noiseless” equilib—
rium by, respectively, the incumbent candidate and the electorate. Conditions 1 and 4 represent,
respectively, the probability with which the incumbent candidate and the electorate implements
the predicted actions of the putative “noiseless” equilibrium. We have implicitly assumed that
the electorate’s voting rule has the monotonicity property (i.e., in the “noiseless” equilibrium,
P(V*,y.) 2 P(V*,yj)if1,~ > y,).

73An entry in columns two, three, and four is a two-element vector (01,6). Using the notation
of Table 11, a depicts a particular equilibrium. 5 E {3, p} is a binary variable that is set equal
to 3 if the corresponding equilibrium, or, involves separation and is equal to p if the equilibrium
involves pooling. For the equilibrium with the best empirical fit, indicated in column two, glis
the MLE point estimate for the incumbent’s error rate and EE is the MLE point estimate for

 

 

For

en

lo

101

 

the best empirical fit (largest log likelihood value) is the separating equilibrium
(0,3,0, 1). The second-best empirical fit is achieved by the separating equilibrium
(0,6,0, 1); the third—best empirical fit is achieved by the pooling equilibrium (0, 1).
For the best—fitting equilibrium, the maximum likelihood point estimates of the
error rates for incumbent candidates and voters are, respectively, .04 and .14.
Conclusion 3, based on the crude counting of consistent outcomes, is robust to
the introduction of a simple error structure and maximum likelihood techniques.
Except for the II-OS Treatment 3 session, the efficient signaling equilibrium best
characterizes the data (see column 2 of Tables 15 and 16). The data from the II-
OS Treatment 3 session is difficult to characterize; the best-fitting pooling model,
(6,1), estimates substantial errors for incumbent candidates and voters. We sum—

marize these results in conclusion 4.

Conclusion 4: Except for the II—OS Treatment 3 session, maximum likelihood
estimation indicates that the data is best explained by a noisy version of the

efficient separating equilibrium.

 

the electorate’s error rate.

 

 

102

 

2.5.2. Experimental Results for FI-RI and II-RI Sessions

The FI-RI and II—RI sessions consisted of three treatment conditions (see Table
10). For each treatment condition we conducted an average of eleven trials. Each
trial consisted of a single game with a variable number of election periods. Table
17 in Appendix B provides a summary of the data.

In Table 17, “Policy outcome, I X” is a four—element vector. For each session
type and treatment condition, we pooled all cases for which the realized incum-
bent candidate competency is I X. The first, second, third, and fourth elements
represent, respectively, the number of instances that policy outcomes 0,3,6 and 9
were chosen. “Reelection probability” is a four-element vector where the first, sec—
ond, third, and fourth elements represent, respectively, the empirical probability
of reelection conditional on policy outcomes 0,3,6 and 9.

Table 17 is read as follows. For the FI—RI Treatment 1 session, there were
209 observations and incumbent candidates’ realized competency was low on 105
occasions and high on 104 occasions. The vector “policy outcome I L” indicates
that for the 105 observations involving low competency, policy outcomes of 0, 3 and
6 were chosen, respectively, 100, 4 and 1 times. The vector “policy outcome, I H”
indicates that for the 104 observations involving high competency, policy outcomes

of 0 and 3 were chosen, respectively, 6 and 98 times. The “reelection probability”

 

 

 

 

[it

im

 

103

vector indicates that policy outcomes of 0, 3 and 6 resulted in reelection 14%, 99%
and 100% of the time. Since the policy outcome of 9 was not observed, the
corresponding reelection probability was not computed.

Two conditions must be satisfied for each session type if signaling characterizes
the aggregate data of Table 17. First, the reelection probability must be (weakly)
monotonically increasing in policy outcome levels. Second, the distribution of
policy outcomes when incumbent candidates’ competency is high should stochas—
tically dominate in a first—order sense the distribution of policy outcomes when
incumbent candidates’ competency is low.

Consider, first, whether reelection probability is (weakly) monotonically in-
creasing in policy outcome levels. Observe that across the six session types, the
only minor violation of weak monotonicity occurs in the FI—RI Treatment 1 ses-
sion: the policy outcome of 3 produces a reelection probability of 0.99 while the
POIiCy outcome of 6 results in a reelection probability of 0. This anomaly may be
dismissed since the policy outcome of 6 was observed only once.

To check whether the second condition for signaling applies, note the distri-
bution of policy outcomes chosen by incumbent candidates when realized com-
petency is low (high) (columns two and three of Table 17). For each of the six

session types, application of the Median Test and Kolmogorov—Smirnov Two Sam-

 

 

 

m

 

104

ple Test (see Conover 1980) rejects the null hypothesis of equality between the
two empirical distributions (.01 significance level).74 Conclusion 5 summarizes

our results.75

Conclusion 5: For each of the six session types, aggregate data reveals separation

in both policy outcome and reelection probabilities.

Recall from section 3.3.2 that the repeated—interactions model generates a
multitude of equilibria. We now determine which of the equilibria in Table 12 is
most consistent with experimental observations.

For each of the six session types, we conducted several trials. For each trial,
We pooled the data and obtained a sample {(y,, I,, (2)}:1 where (31,-, 1,, 6,) denotes
the observation corresponding to the i’th election period.76 Corresponding to each
possible equilibrium, detailed in columns two and four of Table 12, we computed

M

74“As an example, for the FI-OS Treatment 1 session, we test whether the sample distribution
of policy outcomes contingent on low competency — i.e., (100, 4, 1, 0) — is stochastically dominated
by the sample distributon of policy outcomes contingent on high competency — i.e., (6, 98, 0, 0).

7:’For the Median Test, the test statistic is distributed as M21]. The critical value, at the .01
significance level, is 6.64. The minimum value of the test statistic, across the six session types, is
66.68. For the Kolmogorov—Smirnov Two Sample Test, the critical value, at the .01 significance
level, is 9.21. The minimum value of the test statistic, across the six session types, is 82.93, For
both tests, the minimum value occurs in the II—RI Treatment 3 session.

76Recall (yi, I,-, 6i) is a three-element vector where: 1) 11,- E Y is the choice of policy outcome;
2) Ii 6 {I L,I H} is the realized competency of the incumbent candidate; and 3) e,‘ 6 {0,1} is a
binary variable that assumes the value 0(1) if the incumbent candidate is rejected (reelected).

 

 

 

105

the percentage of observations consistent with its predictions. Finally, each trial
was categorized according to the equilibrium with the best empirical fit.77 Table
18 in Appendix B presents the results.

Table 18 is read as follows. For the FI—Rl Treatment 2 session, twelve trials

 

were conducted. A separating (pooling) equilibrium had the best empirical fit

in eleven (one) of the trials. The efficient separating equilibrium (0,3, 0, 1) best

accounted for the data in six of the eleven trials with separation.

The results in Table 18 are similar to those for one-shot elections: except
in the II-RI Treatment 3 session, signaling explains a substantial portion of the
experimental data. Note that excluding the Il—Rl Treatment 3 session, we con-
ducted a total of fifty-three trials (column two of Table 18). Remarkably, data for
forty-eight of the trials (colurrm three of Table 18) are characterized by signaling.

A distinction, however, emerges between one-shot and repeated—interactions
elections. In repeated-interactions elections, efficient separation is not guaranteed
to emerge. Consider, for example, the II-RI Treatment 2 session. While ten
of the eleven conducted trials produced separation, efiicient separation —— i.e.,
(0,3,0,1) — had maximal predictive power in only two of the trials. Excluding

the II—RI Treatment 3 session, separation was observed in a total of forty—eight

 

M
77Detailed results for each trial are available on request.

 

 

 

 

106

trials (column three of Table 18). Of these trials, efficient separation occurred
only twenty—seven times (column four of Table 18).78
Table 18 also indicates that information levels affect the probability of observ-
ing efficient separation. We consider, first, only trials involving separation (col-
umn three of Table 18). Thereafter, for each of the three treatment conditions,
we construct a two—element vector (pF1,pH), where pF1(pH) denotes the proba-
bility that efficient separation emerges when the session involves full-information
(incomplete—information). From columns three and four of Table 18, for Treat-
ments 1, 2 and 3, the vectors are, respectively, (§,%),(1—61,%) and (g, g).79 Since
pp; exceeds pH for each treatment condition, efficient separation survives more
readily in complete information environments. Conclusions 6 and 7 summarize

the analysis of Table 18.

Conclusion 6: Except for the II—RI rTreatment 3 session, a signaling model best

 

accounts for the data in almost all trials.

 

78By contrast, in one-shot elections, efficient separation survived in each of the corresponding
five session types.

7‘9The (p131, p11) vector corresponding to Treatment 1 is derived as follows. In the FI—RI
Treatment 1 session there were 9 trials with separation; efficient separation occurred in all 9 of
the trials. Hence, pp] 2 g. In the II—RI Treatment 1 session there were 10 trials with separation;
efficient separation occurred in 4 of the 10 trials. Hence, p1; = 130-.

 

 

 

 

107

Conclusion 7: Efficient separation is more likely to emerge in complete rather than

incomplete information environments.

Before conducting the experiments, we believed that in repeated-interactions
environments, efficient separation would be trivial to induce. The data provides
a sharp contradiction. Why does efficient separation not occur more frequently?
Our conjecture is as follows. In repeated-interactions elections, the strategy spaces
for incumbent candidates and voters are “large.” As a result, no single equilibrium
(e.g. efficient signaling) emerges as a natural focal point.

A slight variation of the above argument may also explain why complete in—
formation environments are relatively conducive to the occurrence of efficient sep—
aration. When incumbent candidates’ payoffs are known to voters, it is possible
to evaluate the cost associated with each policy pair (Y*(IL), Y*(IH)). Perhaps,
the computation of equilibrium cost enables the least costly signaling equilibrium

to emerge as a focal point.

2.5.3. Informational Efficiency in Experimental Elections

We conclude by comparing the informational efficiency of our experimental elec—

tions. Recall that informational efficiency obtains if and only if the incumbent

 

 

rm

 

108

candidate is reelected when her realized competency is high. For each session
type, we pooled the data and determined the proportion of cases for which elec-
toral outcomes are informationally efficient. The results are detailed in Table 19
of Appendix B.

As a baseline, note that an electoral equilibrium of complete pooling (separa-
tion) generates informational efficiency of 50% (100%). Except for the incomplete-
information Treatment 3 sessions, informational efficiency of our experimental
elections is substantial and reflects the preponderance of signaling in the aggre-
gate data.

Two surprising conclusions show up in Table 19. First, informational effi-
ciency averaged over one-shot sessions is 83.92% while informational efficiency
averaged over repeated-interactions sessions is an indistinguishable 82.78%.80 In—
formational efficiency does not depend on the frequency of interactions (one-shot
or repeated). Second, excluding Treatment 3, informational efficiency averaged
over full-information sessions is 86.21% while informational efficiency averaged
over incomplete-information sessions is an indistinguishable 86.23%.81 Informa-

tional efficiency does not depend on voter information. We therefore draw the

 

8083.92 is the average of the following six numbers: 92.50, 80.00, 90.00, 92.22, 85.00, and 63.75.
82.78 is the average of the following six numbers: 87.56, 8476,8198, 79.35, 88.44, and 74.69.

8186.21 is the average of the following four numbers: 92.50,80.00,86.56, and 84.76. 86.23 is
the average of the following four numbers: 92.22, 85.00, 79.25, and 88.44.

 

 

 

109

following conclusion.

Conclusion 8: Full information of payoff parameters or repeated interactions are
unnecessary for informational efficiency. Electoral accountability obtains even

when background conditions appear, a priori, to be unpromising.

2.6. Discussion

We derive a model of elections that incorporates informational asymmetries be-
tween elected public officials and voters and compute the set of rational expec-
tations equilibria. The baseline model is a one—shot election with one incumbent
candidate and an electorate of homogenous voters. We compute the set of rational
expectations equilibria. The presence of private candidate-specific information
leads to multiple equilibria. In the “pooling” case, the incumbent candidate’s
choice of policy does not depend upon her competency level. The consequent
lack of transmission of competency information leads to electoral outcomes that
are inefficient. By contrast, in the “separating” case, the incumbent candidate’s
choice of policy increases with her competency level. Since there is complete
transmission of competency information, electoral outcomes are efficient.

We conducted a series of lab experiments. A primary goal of the experimental

 

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validity 0

110

  

work was to sort out the various factors influencing equilibrium selection. These
factors included: 1) the incumbent candidate’s private benefit of holding office;
2) the incumbent candidate’s private cost of implementing policy; 3) the amount
of information available to voters, and 4) the frequency of interactions (one-shot
versus repeated-interactions elections). These factors were adjusted individually
in order to isolate their respective influence on equilibrium selection.

In regards to experimental outcomes, the principal similarity between one-shot
and repeated-interactions sessions is the preponderance of signaling in the data.
Since candidate specific information is transmitted in a signaling equilibrium,
experimental elections are informationally efficient. While signaling explains most
of the outcomes for both one-shot and repeated—interactions elections, there is one
distinction: in one-shot elections, efficient signaling is more likely to occur.

These findings aid in resolving the ambiguities raised in theoretical models
of elections with adverse selection that previously demonstrate the coexistence
0f Pooling and separating equilibria. Using experimental techniques, this chap-
ter establishes that signaling is likely to be observed in practice. Drawing broad
generalizations from our analysis would be a risky undertaking. While the exper-
iments themselves are extensive, the question of verisimilitude is a threat to the

validity of any experimental research. Our study, nonetheless, provides a useful

 

 

 

 

of: at rotors (

‘2 requisite a

2.7. formal

Ti section 1
t: notation .
Proof of ‘
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we set B‘(
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B3180 triv
Sif] x
be “le0!
outshotv

[MSW

 

111

look at voters capacity to overcome the adverse selection problem. Voters do have

the requisite ability to make the social contract function.

2.7. Formal Proofs of Propositions

This section presents formal proofs for propositions 1, 2, 3 and 4. Throughout,
the notation employed is that of section 2.3.

Proof of Proposition 1.— Suppose Y*(IL) = Y*(IH) E 31*. It follows from
definition 1 (see conditions (ii) and (iii)) that B*(y*) = 71' and V*(y*) is any
number in the unit interval [0,1]. To ensure that y* is indeed an equilibrium
outcome, we need to specify beliefs, B*(yI), and vote decisions, V*(y1), for out—
Of-equilibrium policy outcomes —— i.e., y; 7E y“. Without loss of generality, we
Specify the “harshest punishments” for all contemplated deviations. Specifically,
we set B*(yI) = 0 and V*(y1) = 0 if yl aé 31*. (Note, for all 311 e §R+, the
specified B" and V* functions satisfy conditions (ii) and (iii) of definition 1.) It
is also trivial to check that condition (i) of definition 1 is satisfied as well iff: y*
S [[%] x I L x V*(y*)]i. Finally, to obtain the entire set of y*-values that can
be supported as outcomes of pooling equilibria, set V*(y*) = 1. In sum, in the
one-shot electoral model, there is a continuum of pooling equilibrium outcomes.

In a specific pooling equilibrium, we have: 1) Y*(IL) = Y*(IH) = y* and 2) y*

 

 

 

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112

e [0, H] x mt].

Proof of Proposition 2.— Suppose Y*(IL) aé Y*(IH). It follows from defi—
nition 1 (see conditions (ii) and (iii)) that B*(Y*(IL)) = V*(Y*(IL)) = 0 and
B*(Y*(IH)) = V*(Y*(IH)) = 1. To ensure that {Y*(IL),Y*(IH)} is indeed an
equilibrium policy outcome pair, we need to specify beliefs, B*(yl), and vote deci—
sions, V*(y1), for out-of-equilibrium policy outcomes — i.e., 111 t? {Y* (I L), Y*(I H)}
Without loss of generality, we specify the “harshest punishments” for all contem—
plated deviations. Specifically, we set B*(y1) = V*(y1) = 0 for 91 ¢ {Y*(IL): Y*(IH)}-
(Note, for all y; E §R+, the specified 3* and V* functions satisfy conditions (ii)
and (iii) of definition 1.) It is also trivial to check that condition (i) of definition
1 is satisfied as well iff: 1) Y*(IL) = 0 and 2) Y*(IH) E [[%] >< IL]i, l%l X IHlfl-
In sum, in the one-shot electoral model, there is a continuum of separating equi-
librium outcomes. In a specific separating equilibrium, we have: 1) Y*(IL) = 0
and 2) Y*(IH) e [[g] X he, [g] x not].

Proof of Proposition 3.—— Suppose Y*(IL) = Y*(I H) —=- y*. It is then immediate

that Q*( I H) 2 Q*( I L) Thus, condition (i) Of definition 2 is satisfied iff: Q*(IL) Z

0. It is also clear that Q*(IL) is increasing in V*(y*). Hence, to obtain the

entire set of y*-values that can be supported as outcomes of stationary pooling

equilibria, set V*(y*) = 1. The correSponding Q*(IL) Z 0 condition reduces to:

 

 

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There is a (-(
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and B. Fm,
Satisfies 001
for the re]

OHEShOt r

 

113

y* 5 A X [[3,3] X IL]%, where A E [IH + [IH — 6 X 7r X (1H — IL)]]% In sum, in the
repeated-interactions electoral model, there is a continuum of stationary pooling
equilibrium outcomes. In a Specific stationary pooling equilibrium, we have: 1)
Y*(IL) = Y*(IH) a y* and 2) y* e [0, A x W] x mt].

Proof of Proposition 4.— Suppose Y*(IL) at Y*(IH). Given conditions (iii)
and (iv) of definition 3, we obtain: 1) Q*(IL) = [[—k X Y*(IL)2] + IL] + 6 X [(1 —
7r) X Q*(IL) + 7r X Q*(IH)] and 2) Q*(IH) = [W — [k X Y*([H)2] + 1H] + 6 X
[(1— 7r) X Q*(IL) + 7r X Q*(IH)]. The above two equations can be solved to obtain
expressions for Q*(IL) and Q*(IH). It can then be shown that conditions (i)-(iv)
of definition 3 reduce to: A) [W— [[k X Y*(IH)2] —:—IH]] 2 0 and B) Q*(IL) 2 Max
{0, [W— [[k X Y* (I H)2] +1 L]]} In sum, in the repeated—interactions electoral model,
there is a continuum of stationary separating equilibrium outcomes. In a specific
stationary separating equilibrium: {Y*(IL), Y*(I 11)} must satisfy conditions A
and B. Finally, we observe that every separating equilibrium of the one-shot model
satisfies conditions A and B. Therefore, the set of separating equilibrium outcomes
for the repeated-interactions model strictly includes that corresponding to the

one-shot model.

 

 

3. CHAPT

SYSTE

3.1. introd‘

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3. CHAPTER 3: RIGIDITY OF RULES IN ELECTORAL

SYSTEMS

3.1. Introduction

In a well—fimctioning representative democracy, it is commonplace to view public
policies as outcomes of a principal-agent game. Legislators, as agents of the elec—
torate, condition their behavior on some notion of the “common will.” Periodic
elections are the mechanism by which the electorate disciplines these legislators.
In sum, irrespective of the extent of divergence between the innate interests of
the public and elected officials, the threat of electoral defeat provides sufficient
incentive for the latter to comply with popular demands. Despite its elegance,
the principal-agent paradigm overlooks an important consideration. In a modern
state, legislators rarely implement public policy. Rather, the task of policy im-
plementation is delegated to administrative agencies. A question naturally arises:
What prevents unelected bureaucrats from engaging in bureaucratic drift _ i.e.,
flouting legislative intent?

The scholarly literature investigating bureaucratic drift makes two substan—

114

 

 

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111111 bill

 

115

tive contributions. First, the various types of bureaucratic drift are elucidated.
Calvert, McCubbins and Weingast (1989), Epstein and O’Halloran (1994, 1996),
Ferejohn and Shipan (1990) and Hill (1985), for example, assume that bureau-
crats have policy preferences distinct from that of legislators. Given discretion,
bureaucrats may override democratic values and implement public policies in ac—
cord with their own desires. Among others, Banks (1989), Banks and Weingast
(1992), Bendor, Taylor and Van Gaalen (1985), Miller and Moe (1983) and Niska—
nen (1971, 1975) posit that bureaucrats seek to maximize an agency’s budget.
Left unmonitored, bureaucrats provide their political overseers with inflated cost
estimates for projects assigned to them. Finally, Laffont and Tirole (1990, 1991),
Peltzman (1976), Posner (1974), Stigler (1971) and Tirole (1986) have pioneered
the economic approach to agency capture: bureaucrats entrusted with the task
of regulating an industry engage in graft and begin, instead, to champion the
industry’s concerns.

Given the possibility of bureaucratic drift, several authors have explored the
Ways by which politicians can design the structure and procedures (i.e., rules and
regulations) of an agency so as to control bureaucrats. Arnold (1987) emphasizes
the myriad oversight techniques available to politicians: extensive hearings may

ruin bureaucratic careers; the annual appropriations process can reward (punish)

 

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116

 

compliant (deviant) agencies; and the original legislation authorizing a program
can be made suitably specific. McCubbins and Schwartz (1984) indicate how
legislators can institute procedures that enable individual citizens and interest
groups to examine an agency’s decisions and sound a ‘fire alarm’ when legislative
intent is violated. Finally, McCubbins, Nell and Weingast (1987, 1989) elaborate
on the arguments in McCubbins and Schwartz. Specifically, they demonstrate
that administrative procedures affect an ‘agency’s range of feasible actions’: the
constraints of due process imposed by the Administrative Procedures Act make
it relatively cheap for any interested party to gather information about agency
behavior; and the ease with which agency decisions can be challenged in court
determines the level of agency discretion.

As Horn and Shepsle (1989), Moe (1990a, 1990b), Moe and Wilson (1994) and
Shepsle (1992) rightly point out, the extensive preoccupation with the ramifica—
tions of bureaucratic drift has glossed over other factors affecting agency design.
In particular, the aforementioned papers suggest that an agency’s structure is also
fundamentally affected by the possibility of ‘coalitional drift.’ The basic idea is as
follows: Consider an enacting coalition (of legislators and interest groups) engaged

in the task of setting up an agency. By making the agency’s structure sufficiently

Pel‘meable, the enacting coalition can exercise strict control over public policy in

 

 

 

 

5: present.
ration 103
:3' Stone pc
l’i’t agency
.. ‘ihe ca;
ie‘ldesign
éiiIi-'
ln cont
in. this (
exception.
Bolate th.
Presence
agency (it
Our 1
Sll'ie par
employs
tile two
Daily is

Party is

 

117

  

the present. Political power, however, is ephemeral. Thus, when the enacting
coalition loses authority, the deal struck at enactment is subjected to predation
by future political coalitions — a permeable agency structure serves only to facil—
itate agency capture. In sum, an enacting coalition must create an agency that
has ‘the capacity to survive and prosper in an uncertain political environment.’ A
well-designed agency exhibits resilience to the ‘authority that its opponents might
gain.’

In contrast to extant theoretical models exploring the politics of agency de-
sign, this chapter simply assumes away bureaucratic drift —— bureaucrats, without
exception, Obey legislative intent. Instead, the principal goal of the chapter is to
isolate the design implications of coalitional drift. Specifically, we ask: in the
presence of legislative turnover —— and, hence, coalitional drift — what sort of
agency does an enacting coalition construct?

Our theoretical setup is as follows. We examine a polity in a Westminster-
Style parliamentary system with a unicameral legislature. There is an agency that
employs public funds to provide a service valued by citizens. Furthermore, there
are two political parties that assign different shadow prices to public funds —- one
Party is more fiscally conservative than the other. When a particular political

Party is in power (i.e., controls the legislature), it establishes the budgetary rules

 

 

 

it 'he ageni
that de
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if; 10 1an
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the riskj
8'2\
The

0'iiallon
which C(
aphotos

118

 
    
    
 

for the agency.

What determines the nature of the budgetary rules that emerge in equilibrium?
Suppose that the political party currently in power writes flexible budgetary rules
that cede discretion to future legislative coalitions. Should the incumbent party be
removed from office, its replacement can subsequently exploit the granted flexib-
lity to implement any public policy it desires. In short, when political uncertainty
is substantial and the two political parties are ideologically separated, flexible
rules are risky structures. If the risk associated with flexible rules is deemed un-
acceptable, the incumbent party institutes rigid budgetary rules for the agency.
Rigid rules, by definition, are riskless structures: public policy implemented in the
future is determined in advance and is independent of the electoral fortunes of the
two political parties. However, rigid rules impose a cost as well —— because public
policy cannot respond to changing circumstances, inefficient outcomes are fre—
quently generated. In sum, equilibrium budgetary rules reflect a tradeoff between

the riskiness of flexibility and the inefficiency of rigidity.82

 

8"’The idea behind our paper is similar to that developed independently in Epstein and
iO’Halloran (1994, 1996), They construct a one-period model of an administrative state in
which Congress determines the amount of leeway granted to bureaucrats. Since the president
appoints agency heads, who in turn implement public policies, Congress places tighter (looser)
constraints on bureaucrats when congressional and presidential preferences diverge (converge)

Despite the underlying similarity between our paper and their, the formal structures are quite
distinct. Furthermore, our paper contrasts two distinct electoral arrangements: the parliamen—
tary system and the separation-of-powers system. Epstein and O’Halloran only consider the

 

 

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119

Our chapter also has a comparative perspective. Besides the aforementioned
parliamentary system, we analyze budgetary rules for polities in a separation-of-
powers system. Such polities consist of one agency and two political parties, with
differing preferences, that jointly oversee the agency. However, the legislature is
now presumed to be bicameral. Therefore, in periods of divided government, bud-
getary rules for the agency involve a compromise between the two political parties.
By contrast, in periods of unified government, the party in control unilaterally
determines the agency’s budgetary rules.

The comparative perspective yields an unexpected dividend. We directly com-
pare the two electoral systems (parliamentary and separation-of-powers) in terms
of the rigidity of the budgetary rules that emerge in equilibrium. Surprisingly,
for an enacting coalition, flexible budgetary rules are always riskier structures
in a separation-of-powers system than in a parliamentary system. Since an en-
acting coalition is presumed to maximize its own welfare, equilibrium budgetary
rules in a separation-of-powers system are uniformly more rigid than that in a
parliamentary system. In sum, the two electoral arrangements can theoretically
be rank—ordered in terms of rigidity (or flexibility). Our theoretical result has

an empirical counterpart. Moe (1990a) and Moe and Caldwell (1994) examine

 

separation—of-powers case.

%m

 

‘36 striictn
and ('01
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diagrm

120

the structural design of regulatory agencies. The evidence they summarize spans
several countries (though, the United States and Britain are emphasized) and
principally concerns environmental regulation. Without exception, they discover
that agencies in a parliamentary system are less encumbered by rigid rules than
those in a separation—of-powers system.

The rest of the chapter is structured as follows. As a baseline case, in section
3.2 we consider a polity in which a single political party has jurisdiction over an
agency. The goal of this section is twofold: we 1) establish basic notation and 2)
demonstrate that, absent electoral uncertainty, the equilibrium budgetary rule for
the agency is both flexible and efficient. In section 3.3, we extend our analysis of
budgetary rules to polities in a separation—of-powers system. Budgetary rules in
a parliamentary system are examined in section 3.4. In section 3.5, we compare
the two electoral systems in terms of the rigidity (or flexibility) of the budgetary
rules that are generated. Section 3.6 concludes the chapter from a substantive
perspective. All theoretical proofs are relegated to section 3.7 while pertinent

diagrams are gathered in Appendix C.

 

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121
3.2. A One-Party State

As a baseline case, we construct a model of a one-party state. The model consists
of two players: an agency, denoted by A, that supplies a service valued by citizens
and a political party, denoted by P, that oversees the agency and undertakes
budgetary decisions. The interaction spans an infinite sequence of structurally
identical periods.

Each period, t E Z+, proceeds as follows. At the start of period t, Nature
makes a draw from the distribution of a random variable, 9. 0 is uniformly dis-
tributed on the interval [0,9] and 9t denotes its period-t realization. Without
exception, agencies provide services (e.g., unemployment compensation) that are
subject to demand shocks of a transitory character. We interpret 0, to be the
period-t benefit obtained by citizens in aggregate for each dollar spent by the
agency.

Once the agency is apprised of the Qt—value, it truthfully transmits the Qt-value
to its overseer, party P. Since the agency does not engage in information manip-
ulation, standard principal-agent problems are assumed away. Thus, delegation
of authority engenders no loss of control.

Given 6,, party P fixes the size of the agency’s budget as follows. Party P can

raise a maximum of B dollars through taxation. However, distortionary taxation

 

 

 

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State is

 

122

causes inefficiencies in the system: specifically, A 6 [0,6] is the deadweight loss
inflicted on citizens in order to levy one dollar for the party. The party behaves
benevolently and selects a budget level that maximizes citizens’ net surplus. Thus,

the budget, denoted by b(6,, A), solves the following program:

b(6,,A) E arg max {b X (6, — A)} (3.1)
be[0,§]

Since the objective function in equation (3.1) is linear, the solution is immediate:

{0} if 6, < A
{B} if 6, > A

The budget rule, b(. , .), has two properties. First, since b(6,,A) is weakly
increasing in 6,, the budget rule is flexible and responds to demand shocks. Second,
the budget rule is efficient; i.e., the budget size is at a maximum (minimum) when
benefits, 6,, are greater (less) than costs, A. In sum, an agency in a one-party

state is an optimal organization.

 

 

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123

3.3. A Two-Party Separation-of-Powers System

In this section, we examine a stylized model that approximates a separation-of-
powers system. The section is divided into three parts. First, we provide a verbal
description of the model. We then formally define an equilibrium for the model.

Finally, we discuss the principal predictions that follow from the model’s solution.

3.3.1. Description of the Model

To emphasize the dispersion of political power in a separation—of-powers system,
we consider a polity with a bicameral legislature; the two chambers being labelled
C and P83 There are two political parties, denoted PL and PR, that stochastically
occupy the two chambers and jointly oversee an agency, denoted .484

Like the model in section 3.2, the agency employs public funds to provide a
service valued by citizens. Raising public funds, however, induces welfare—reducing
distortions in the economy. The two political parties have different estimates
of the magnitude of these distortions: specifically, parties PL and PR assign,
respectively, a deadweight loss to society of A L and A R in order to levy one dollar

for the agency. Party PR is (weakly) more fiscally conservative than party PL;

 

83It is also reasonable to interpret chamber C (P) as Congress (Presidency). In fact, the labels
are chosen to encourage this association.
“P L and PR are mnemonics for “left party” and “right party” respectively.

 

 

 

hence. /\

The i

\Hriab
Period

in agg
\

85111
W

C0Hstit
“5G1

 

 

124

hence, AR 2 AL > O.85

 

The interaction between the two political parties spans an infinite sequence of
structurally identical periods. Each period, in turn, consists of three phases. In

phase one, the parties (or party) in power determine(s) the agency’s budget for

 

,that period. In phase two, budgetary rules for the next period are written. In
phase three, elections in both chambers indicate whether incumbents are reelected.
‘A description of the three phases follows.
‘Phase One: Determining the agency’s budget

Each period, t E Z+, begins with a particular configuration of political power.
We let PC“) and Pp“) denote, respectively, the party controlling chambers C’ and
P in period t.86 When Pea) and Pp“) are both party PL(PR), we say that period
t exhibits “unified party PL(PR) Control.” Instead, if P00) and Pp“) represent
different parties, we say that period t exhibits “divided government.”

In each period t, Nature makes a draw from the distribution of a random
variable, 9. 0 is uniformly distributed on the interval [0,5] and 6, denotes its

period-t realization. 0t is interpreted to be the period-t benefit obtained by citizens

in aggregate for each dollar spent by the agency.

 

85In an alternative interpretation, the two political parties have distinct constituencies. AR
5 larger than AL because a dollar raised in taxes inflicts greater deadweight loss on party PR’s
:onstituency than on party PL’s constituency.

6Given our notation, C(t) and P(t) are elements of the set {L, R}.

 

 

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to its p0?
mod-t l
Nix-rem

DISC?
‘91 P3”.
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level 3C

The re
Henge
Be

125

Once the agency is apprised of the Ot-value, it truthfully transmits the Ht-value
to its political overseers, parties P00) and Ppm. Given Hi, how is the agency’s
period-t budget determined? We now need to distinguish between two cases: the
“discretion regime” and the “no discretion regime.”

Discretion regime— Case one considers the possibility wherein the parties
(or party) in power in period (t — 1) established budgetary rules that yielded
discretion to the future coalition. Under this “discretion regime,” the agency’s
period-t budget involves the following compromise between the two chambers,
C and P. The political parties (or party) in power in period t — i.e., P00) and
P P(t) — can raise a maximum of F dollars through taxation. Given this exogenous
constraint on the availability of public funds, note that the “ideal” period-t budget

level according to party P0(t)(PP(t)) is b(0t, A00») (b(0t, /\ 13(0)) where:

b(Ht,)\j) = arg max [b x (0t — Aj)]; j E {C(t),P(t)} (3.3)

be[0,§]

The realized period—t budget simply averages the desires of the two chambers.

Hence, it assumes the value: [(%) >< (b(6t, Acm) + b(0t, Ap(t)))].

Before proceeding further, we observe that the two chambers have conflicting

“divided government” (C(25) % P(t))

budget demands only when period t exhibits

 

 

 

rd dema
:enrollet
iv part?
:equal 1

Na d;
:01 have
Case m-
rigidly i
no discr
Shock 9.

Phase

a

126

and demand shock 6,: lies in the interval (AL, AR). In this situation, one chamber,
controlled by party PL, seeks a budget of F while the other chamber, controlled
by party PR, hopes to shut down the agency. The resulting compromise budget
is equal to g

No discretion regime— The parties (or party) in power in period (t -— 1) may
not have written budgetary rules that ceded discretion to the period-t coalition.
Case two considers the situation wherein the period-(t — 1) enacting coalition
rigidly fixed the period-t agency budget at some level b 6 [0,3]. Under this “b
no discretion regime,” the parties in period t have their hands tied. The demand
shock 0t is ignored and the period-t funding of b is implemented.
Phase Two: Writing budgetary rules

Once the agency’s period-t budget is determined, the budgetary regime for
period (t + 1) needs to be established. Who writes the rules? A separation—

Of'130wers system forces the two chambers to bargain over rules. We model this

bargain starkly: specifically, with probability oz(1—oz), the party controlling cham-

ber C (P) is invested with the rights to stipulate the budgetary regime for period

(t+1).

The party choosing the budgetary regime for period (t + 1) selects an element

ation of the

from the set [0,F] U {“DR”} where: 1) b 6 [0,3] implies implement

 

 

 

both eha
erogeno

ii perio

idiom

W

127

“b no discretion regime” in period (t + 1); and 2) “DR” implies implementation
of the “discretion regime” in period (t + 1).
Phase Three: Elections

After the budgetary regime for period (t + 1) is decided, elections are held in
both chambers of the legislature. We model elections crudely and posit that with
exogenous probability 7rc(7rp), the party controlling chamber 0 (P) retains power
in period (t + 1). Without loss of generality, we let no weakly exceed 7rp (i.e.,
71'0 2 WP). We assume, furthermore, that incumbency in both chambers is strictly
advantageous. Thus:
(3.4)

1 1
7r0>§and7rp>§

Following the two elections, period t concludes. The sequence of events in period

(t + 1) replicates that in period t.

AS mentioned previously, our model has an infinite horizon. Both parties

discount future payoffs at rate 5 6 (0,1) and have preferences that are separable

over time. In sum, the parties’ objective functions are as follows.

(3.5)

w. = Edie x it x (0. — at}
t=0

 

 

 

rise: b,
team’s

salable

3.3.2. I

5 deta
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128

WR = Ed; at x [1). x (a. — AR)]} (3.6)

where: bt is the agency’s budget in period t; 6, is the period-t demand shock for the
agency’s services; E0 is the expectation operator conditional on the information

available at time 0; and WL(WR) is the objective function of party PL(PR).

3.3.2. Definition of a Markov Perfect Equilibrium

As detailed in section 3.3.1, the agency’s budget in generic period t depends on
three factors: 1) the period-t demand shock, 9t; 2) the identity of the parties
(or party) controlling chambers C and P in period t (P003) and PP(t))§ and 3)
the budgetary regime currently in place (“discretion regime” or “b no discretion

regime”), While factors one and two are outcomes of an exogenously specified

stochastic process, factor three is determined endogenously — i.e., as part of the

model’s equilibrium. This section defines an equilibrium for the model.

Our model is a dynamic game, for which the solution concept typically em—

PlOyed is perfect equilibrium. Unfortunately, given the repeated interactions be-

tween the two political parties, folk theorem type results guarantee the existence

of a plethora of perfect equilibria. Most of the perfect equilibria, however, are

g elaborate history—

implausible: they depend critically on each party implementin

 

 

 

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129

dependent punishment strategies that deter deviation from the putative equilib-
rium. For purposes of credibility, we therefore consider a suitable refinement of
perfect equilibrium: namely, Markov perfect equilibrium.87

Defining a Markov perfect equilibrium is notationally cumbersome. For clarity,
we divide our exposition into three parts. First, we define the state space of the
model and present the transition function specifying how states evolve over time.
Second, we describe the strategies and value functions of the two political parties,
P L and PR. Third, we provide the necessary and sufficient conditions for strategies
to constitute a Markov perfect equilibrium.
State Space and Transition Function

Fix a generic period t and consider the phase in which budgetary rules for
period (t + 1) are written (i.e., phase two). This phase is described by a three—
element vector — s E (so,sp,a) —— where: 1) 30 equals 3L(5R) when party
PL(PR) controls chamber 0; 2) SP equals 8L(SR) when party PLipR) controls
chamber P; and 3) a equals 0(1) when chamber 0 (P) has the authority to specify
the eight possible

the budgetary regime for the next period. S, the collection 0f

states 3, is the state space of the model.

87For an interpretation of Markov perfection, the reader is referred to Baron (1996), Fudenberg
and Tirole (1991) and Maskin and Tirole (1988, 1994).

 

 

 

 

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safer: ,5
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leSl [Se
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lllt f0]

130

We let go : S X S —> [0,1] denote the transition function for the model; thus,
go(s,s’) is the probability that the state next period is s’ E S when the current
state is s E S. The construction of (p is as follows. For .9, s’ E S, we first build
a three-element vector p(s, s’) .=_ (p1(s,s’),p2(s,s’),p3(s, s’)) where: 1) p1(s, s’) is
a binary variable that equals 0(1) if s’C = so (s’C sé so); 2) p2(s,s’) is a binary
variable that equals 0(1) if s’P = 31: (3’1, 74 3p); and 3) p3(s, s’) = a’. In words, the
first (second) component of p(s, 3’) assumes the value of 1 only when movement
from State 3 to state 3’ involves a different party controlling chamber 0 (P) The
third component of p(s, 3’) simply identifies the chamber writing the budgetary
rule in state 3’. Note, now, that electoral outcomes in the two chambers and the
granting Of rule writing authority are independent events. Therefore, we obtain

the following expression for <p(s,8')1

as, s’) = agree» x (1 — mam} >< {wt—”2‘3”” x <1 — we“ >}
x{a(1-pa(878’)) X (1 _ a)p3(8.8’)} (3.7)

Strategy Spaces and Value Functions

We let 73 be a mapping from S into {PLi PR} where 73(5) identifies the party

 

 

 

rang ll
repre
Sateen:

lie 1

131

writing the budgetary rules in state .9 E S. Generic strategies of the two parties
are represented by a mapping 9 : S —> [0,3] U {“DR”} where g(s) specifies the
budgetary regime chosen by party ’P(s) in state 3 E S.

We let UL(s,7‘,9) (UR(s,r,0)) denote the period payoff received by party
PL(PR) when: 1) the current state is s E S; 2) the current budgetary regime
is 7‘ 6 [0,3] U {“DR”}; and 3) the current demand shock is 6 6 [0,9]. From the
model’s description in section 3.3.1, the following expressions for U L(s, r, 6) and

UR(s,r, 0) can be derived:

Uj(s, “DR”, 9) = b(s, “DR”,0) x (a — A,); j e {L, R} (3.8)

Uj(s,r,0) =7'X (e—Aj); je {L,R} andre [0,5] (3.9)

In equation (3.8), b(s, “DR”, 0) denotes the agency’s budget when: 1) the current

state is s E S; 2) the current regime is the “discretion regime;” and 3) the current

demand shock is 6.88 Since (6 — M) is the per dollar net benefit to party P], the

total period payoff is the product of b(s, “DR,”0) and ((9 —— Aj). Consider now

88When state 3 exhibits “unified party PL control”. (i.e., s E {(SL,5L,0)1;,(5(IE::La:)}:
b(S,“DR” 0) = b(0 AL). When state s exhibits “unlfied party .PRucontro . ., t”
{(3111 3R 0,) (33 83 f)}) b(s “DR”,0) = b(6,)\R). When stateriexhibits d1v1dzd6giv¢3r)r]1men
(i-e-use{<sL.sR.a),(sR,sL,a)Iae{0,1}}).b(s.“DR”.6)=I(§)x(b<0,m+ <. R .

 

 

 

raw l

Give
tinned
l) the (
have a1

follow

132

equation (3.9). Under the “b no discretion regime,” the agency’s budget is fixed
at b. Once again, the total period payoff to party P]- is the product of the current
agency budget, b, and the current per dollar net benefit, (6 — Aj).

Given generic strategies 9, VL(s) (VR(s)) denotes the expected present dis-
counted sum of period payoffs for party PL(PR) over the infinite horizon when:
1) the current state is s E S; and 2) the agency’s services for the current period
have already been paid for and consumed. By a standard recursion argument, the

following two equations can be established:

VL(S) = 6 X 2943,33 x {VL(3') + /0§UL(S',g(s),6) X %d6} (3.10)
VR(S) = 6 x Z cp(s,s’) X {VR(s') + /: UR(s’,g(s),6) >< %d6} (3.11)
s’ES

Consider equation (3.10). In state 8 E 5, party 79(5) selects budgetary regime

' ' ’ . t t
9(8) Thereafter, with probability <p(s, 3’) the state next period is s E S In s a e

_ 1 '
3,, party PL’S expected period payoff is [f59 UL(8’,g(8),9) X 5d6] Whllea OVGI‘ the

 

 

fr par

Condit

Ween!
when:
period
budge
and 4

equal.

133

remainder of the game, its expected discounted sum of period payoffs is VL(s’).
Thus, at state 3’ , the total expected payoff for party PL is [[05 U L(s’ , 9(3), 6) x %d6+
VL(s’)]. Equation (3.10) indicates that from an ex ante perspective, conditional
on starting at s e S, VL(s) is simply: 25165943, 3’) X 6 X {total expected payoff
for party PL starting at s’}. Identical interpretation applies for equation (3.11).
Conditions for Markov Perfect Equilibrium

We divide the state space S into two disjoint subsets: specifically, we let
S L(SR) denote the set of states in which party PL(PR) writes the budgetary rules
for the next period.89 For .9 E SL(SR), VL(s,a) (VR(s,a)) denotes the expected
present discounted sum of period payoffs for party PL(PR) over the infinite horizon
when: 1) the current state is s E SL(SR); 2) the agency’s services for the current
period have already been paid for and consumed; 3) party PL’s (PR’s) choice of
budgetary regime for next period is a 6 [0,?] U {“DR”} (not necessarily 9(3));
and 4) that henceforth both parties play according to 9. Following the logic of

equations (3.10) and (3.11), the next two equations can be derived:

VL(s,a,) = 6 x E (00373,) X {VL(8,) + /09 UL(s’,a,6) X %d0}; 8 E 8;, (3.12)

s’ES

 

89Hence, SL E {3]P(s) = PL} and SR E {sI’P(s) 2 PR}.

 

H.511] :6)

lie now provide
{inert equilibri
ing (see Belln
shor deviations

conditions ensu

3.3.3. The 1)

The construe
System is (let

the SOllition,

A Polity,

134

VR(s,a) = 6 X Z <p(s,s’) X {VR(s’) + /0§UL(s’,a, 6) X %d6}; 3 6 83 (3.13)

s’ES

7e now provide the necessary and sufficient conditions for g to be a Markov
arfect equilibrium. Using the unimprovability criterion of dynamic program—
ing (see Bellman (1957)), g is a Markov perfect equilibrium if and only if one
rot deviations by parties PL and PR are improfitable. Hence, the following two

)nditions ensure the optimality of g:

VL(s) 2 VL(s,a); Vs 6 SL and Va E [0,B] U {“DR”} (3.14)

VR(s) Z VR(s,a); V8 6 SR and Va 6 [0,B] U {“DR”} (3.15)

.3.3. The Model’s Solution and Implications

‘he construction of the Markov perfect equilibrium for the separation-of-powers
(stem is detailed in section 3.7. In this section, we present the main features of
re solution, provide certain intuitions, and emphasize the principal implications.

A polity, in our model, is characterized by a four-tuple (AL,)\R,7rC,7rp). A

 

futiathIl-inov
(WAR. rc
ti 7,; held fixe
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it turns out
identify two qt:
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135

aparation—of-powers system is defined to be the set of admissible polities —
e., {(AL,/\R,7rc,7rp)|)\,- E [O,6],7rj E (%,l],)\R 2 AL and no 2 ma}. With 71:0
nd 7rp held fixed, the cross-section of admissible polities becomes the A-space,
(AL,/\R)|)\,- 6 [0,6] and AR 2 AL}. In Figure 1 of Appendix C, triangle XYZ
epicts this A—space.

It turns out that the Markov perfect equilibrium of the model allows us to
ientify two qualitatively distinct regions of the A—space: 1) A polity lying in
agion RBR (see Figure 1) exhibits rigid budgets. In such a polity, for each state
6 S, when party 73(3) selects next period’s budgetary regime it refuses to yield
iscretion. Instead, the budget is some fixed element of the set [0,B]. 2) A polity
ring in region RBF (see Figure 1) exhibits flexible budgets. In such a polity, for
ach state 3 E S, when party P(s) selects next period’s budgetary regime it yields
iscretion (“DR”). The precise locations of regions BBB and RBF are detailed in

esult 1.90

 

{esult 1: For a separation—of-powers system, fix the reelection parameters, («0,
'P), and consider the A—space, {(AL, AR)|)\,- 6 [0,6] and AR 2 AL}. Then:

H

90Note that there are polities in the A-space that do not belong in either RBR or RBF. These
rolities exhibit hybrid behavior: for certain states, the party setting next period’s budget yields
.iscretion; for other states, the party setting next period’s budget induces rigidity. In the interest
if brevity, we do not consider such polities.

 

i,,={ii,.
where: j' E \
Hg; = (M.
where: j" E

Prof: See secti

figure 1 illn
generates mult
depends on its

Result 1 er
g9] flexibility l
the reelection
of admissible
lSltS defined
’mallon C11]
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the ”ODOlari
“”0 isopolar

m

136

BBB = remain 2 (lg—21> x A. and AR 2 (E...) + (E...) x AL}

where: j’ E (1 — ”f — 123

V

RBF = {(ALaARMAR 3(1—j—hfi)x )‘L and )‘R S (1717) + (11%) X AL}

where: j” E (% + 12‘; — If)

’roof: See section 3.7.

Figure 1 illustrates an important observation: the separation—of-powers system
enerates multiple behavioral patterns; the budgetary regime of a specific polity
.epends on its location in the A—space.

Result 1 enables us to derive predictions about the relationship between bud—
et flexibility (or rigidity) and the level of polarization. Specifically, we hold fixed
he reelection parameters, (7T0,71'P), and consider the corresponding cross-section
f admissible polities (i.e., the A-space). Associated with a generic polity (AL, AR)
5 its defined polarization level, MAL, A3) = AR ~ )‘L- In the A—space, the isopo-
arization curves are the level sets of 1,6. In Figure 1, the isopolarization curves
.re straight lines parallel to the diagonal XY; the polarization level increases as
he isopolarization curves move farther away from the diagonal. Consider, now,
wo isopolarization curves, the A—line and the B-line.91 Polities located on line A

9’ The A—line refers to the line through points A1 and A4. The B—line refers to the line through

 

 

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randomly 52mph
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Conclusion 1.1 i

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Result 1 a
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137

possess a higher polarization level than those located on line B. Observe that a
randomly sampled polity from line A(B) exhibits budget flexibility with probabil—
ity equal to 0((length of line segment BgB3) + (length of line segment BIB4)).92

Conclusion 1.1 generalizes the foregoing discussion.

Conclusion 1.1: In a separation—of—powers system, as the level of polarization in-
creases, the probability of observing budget rigidity (flexibility) increases (de—

creases) .

Result 1 also predicts how the regions of budget rigidity (R312) and budget
flexibility (REF) change as the reelection rates, (no, mo), are varied. Holding fixed
7T0 and the A—space, two changes occur as 7rp is increased: 1) Line 1(2) rotates
counter-clockwise (clockwise) about the point X(Y) (see Figure 1). Hence, region
R33 shrinks.93 2) Furthermore, line 3(4) rotates counter-clockwise (clockwise)
about the point X(Y) (see Figure 1).94 Hence, region REF expands. Conclusion

Points 31 and B4.

92Line segment B233 refers to the line segment with endpoints 32 and B3. Line segment
3134 refers to the line segment with endpoints B1 and B4 (see Figure 1).

93The equation of line 1 is: A3 = (149%) x AL. The equation of line 2 is: AR 2 (F027)+(T4%) x
)‘L- To determine the rotation of the two lines, note that the derivative of j’ with respect to 7rp
is negative.

-H

94‘The equation of line 3 is: AR = (311.9%) X AL. The equation of line 4 is: AR E (%) +

Zionisu

incisor

EEK}?

The e
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Ittion it
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138

1.2 sums up the comparative statics of budgetary regimes with respect to ma.

Conclusion 1.2: In a separation-of—powers system, as the rate of reelection in cham—

ber P, inn, is increased, budget rigidity is unambiguously reduced.

The effect on budget rigidity of raising no is, however, ambiguous. Holding
fixed 71'}? and the A-space, two changes occur as no is increased: 1) Line 1(2) ro-
tates counter-clockwise (clockwise) about the point X(Y) (see Figure 1).95 Hence,
ref-{ion RBR shrinks. 2) However, line 3(4) rotates clockwise (counter-clockwise)
about the point X(Y) (see Figure 1).96 Hence, region REF shrinks as well. Con-
clusion 1.3 summarizes the above discussion.

Conclusion 1.3: In a separation-of—powers system, as the rate of reelection in cham-

 

ber C, 7rC, is increased, the impact on budget rigidity is ambiguous; the regions
0f budget rigidity (R33) and budget flexibility (RBF) Shrink.

(Ii-j") X AL. To determine the rotation of the two lines, note that the derivative of j” with

respect to m: is negative.

5Given the equations of lines 1 and 2, the rotation is
0f 9" with respect to no is negative-

96Given the equations of lines 3 and 4, the rotation is
Of j” with respect to we is positiVe-

determined by noting that the derivative

determined by noting that the derivative

 

139

We now provide the intuition behind conclusions 1.1 - 1.3 and the mechanics
of the model’s solution. The construction of a Markov perfect equilibrium (see
equations (3.10) — (3.15)) demands that at each node 3 E S, the budgetary regime
selected by party ’P(s) maximizes its expected next period payoff. Hence, even in
an infinite-horizon setup, optimality of party P(s)’s action is determined in terms
of rewards in the immediate future.

To fix ideas, consider a polity in which AL is less than 3 while AR exceeds $.97
SUppose, furthermore, that in generic period t, the polity is in state 3 6 SL—
i.e., party PL sets the budgetary regime for period (t + 1).98 In an ideal world for
party PL, the period—(t + 1) budget follows the rule, b(6t+1, AL); specifically, the
budget is set at D(0) when the period-(t + 1) demand shock, 6,4,1, exceeds (is less
than) )‘L- Given the structure of the model, however, party PL cannot enforce
the b(9t+1, AL)-rule. Recall that party PL has only two budgetary options: 1) By

selecting “DR”, party PL can cede discretion to the coalition in power in period

(t ‘l' 1). 2) Alternatively, party PL can rigidly fix the period (t +1) budget at some

level between 0 and B.

. We only require

MIQI

97The intuitions do not depend on the locations of AL and AR relative to

that AR weakly exceeds AL- -
98The intuitions are unchanged if, instead, we consrder state 5 6 SR.

 

the bi

140

 

Consider, now, what happens when party PL opts to give discretion. With
some probability, denoted p2(s), the polity in period (t + 1) exhibits “unified
party PR government.” 99 Since party PR now has the right to set the agency’s
period-(t+ 1) budget, it therefore follows the individually rational b(6t+1, A 12)-rule;
the budget is set at 3(0) when 6t+1 exceeds (is less than) AB. The behavior of
party PR differs from the ideal of party PL: specifically, when 6H1 6 (AL, AR),
Party PR shuts down the agency while party PL prefers that the budget be B.
This incongruence in behavior inflicts on party PL an expected welfare equal to:
lp2(s) x H x [35(6 -— AL) >< %d6].100

Furthermore, with some probability, denoted p3(s), the polity in period (t+ 1)
exhibits “divided government.” 101 Given discretion, the agency’s budget satisfies
the [(é) x (b(6t+1, AL)+ b(6t+1, ARM—rule; the compromise budget is set at: 1) B if
6H1 exceeds AR; 2) g if 6H1 6 (AL, AR); and 3) 0 if 6t+1 is less than AL. Once again,
the above decision rule is suboptimal for party PL in the region 6H1 6 (AL, AR):
Specifically, party PL desires a budget of B but obtains only half that amount.
This incongruence in behavior inflicts on party PL an expected welfare loss equal

to: [p3(s) x a x ff(6 — AL) x %d6].1°2

 

99The reader should note that p2(s) varies across the states in S L.
100The expected welfare loss is relative to the first-best payoff of party PL.
1(“The reader should note that p3(3) varies across the states in S L.
102The expected welfare loss is relative to the first—best payoff of party PL.

 

141

 

The discussion in the above two paragraphs makes the following argument —
when party PL chooses to give discretion, it encounters two sources of welfare loss:
1) that under “unified party PR government” and 2) that under “divided govern-
ment.” The total expected welfare loss, denoted AW5(“DR”, s), combines losses

from both sources. Hence, AWS(“DR”, s) is given by the following expression:

 

_ A 1
AWS(“DR”,s) = [P2(s) + ”35”] x B x A ”(a — AL) >< gdé’l; s e 8L (316)

AWs(“DR” , s), is partially depicted in Figure 2 of Appendix C. Note that AWS(
“DR”, 3) is proportional to the product of two terms: 1) the “total probability
factor,” [p2(s) + 95232]; and 2) the “total preference factor,” f,(\L3(6 — AL)d6, equal
to the area of triangle ABC.103

Two observations regarding AW5(“DR”, 8) turn out to be crucial. First, for
a generic polity, (AL, AR), the “total preference factor” (area of triangle ABC
in Figure 2) is increasing in the level of its polarization, 160%, A3) 2 AR —— AL.
In sum, AW5(“DR”, s) is an increasing function of a polity’s polarization level.

Second, the “total probability factor” ([p2(s) + Egg-1]) varies across states 5 E S L.

 

103The constant of proportionality is %.

 

142

 

When party PL controls both chambers — the polity is under “unified party
PL control” — the advantages of incumbency make the total probability factor
relatively small in value. Therefore, AW5(“DR”,s) is of smallest magnitude
when 8 E {(sL,sL,0),(sL,sL,1)}. On the other hand, the status of party PL is
most precarious when the polity is currently under “divided government” and
it, furthermore, controls the chamber for which incumbency is less advantageous
(i.e., chamber P). Hence, the total probability factor and AW5(“DR”,3) assume
their largest value when 5 = (SR, 5);, 1).

We have exhaustively discussed the consequences for party PL, when in state
3 6 SL, it yields discretion. However, party PL has another option: it can set the
budget for the next period (period (t + 1)) at some level between 0 and B. By
assumMion, we have fixed AL to be less than g. Hence, on average, the agency’s
services in period (t + 1) are worth more than party PL’s estimate of the cost.
The optimal rigid budget therefore involves precommitting all available funds, B.

The fixed budget of B leads to an expected welfare loss for party PL: specif~
ically, when the demand shock for period (t + 1), 6,4,1, is less than AL, party PL
prefers ex post to shut down the agency but is instead committed to a funding level

of B. The welfare loss from rigid budgets, denoted AWS(B, s), is as follows:104

 

10“The welfare loss AWS (B, s) is relative to the first-best payoff of party PL.

 

143

 

_ _ AL
AW3(B, s) = [B x /0 (AL — 6) x $39], .9 6 3L (3.17)

Observe that welfare loss AWS(B, s) is proportional to the area of triangle CAD
in Figure 2.105

In sum, party PL has essentially two budgetary options in state 5 E S L: “DR”

 

and B. Relative to the first-best, AWS(“DR, ”s) (AW5(B, 8)) measures the ex-
pected welfare loss to party PL when it chooses discretion (a fixed budget of B).
Therefore, the optimal strategy for party PL is to simply select the budgetary
option with the smaller welfare loss. Given this intuition, we now provide expla—
nations for conclusions 1.1 — 1.3.

Consider, first, conclusion 1.1. As a polity becomes more polarized, (AR —
AL) increases. Figure 2 indicates that the area of triangle ABC (and, hence,
AWS(“DR” , 3)) increases as well. On the other hand, the area of triangle CAD
(and, hence, AWS(B, 3)) does not depend on (AR — AL). Thus, as the level of
polarization rises, there is an enhanced incentive for party PL to induce rigid
rules. In sum, polarization level and budget flexibility (rigidity) are negatively

(positively) correlated.106

 

<0 Ital

105The constant of proportionality is _ .
1°6For ease of exposition, we only consider the behavior of party PL (i.e., s E S1,). Identical

 

144

Consider, now, how the region of budget rigidity, R B R, varies with the reelec-
tion rates, (no, np). From party PL’s perspective, R B R is the subset of the A—space
in which, for all states 5 6 SL, AWS(“DR”,3) weakly exceeds AWS(B, s).1°7 Re-
call that AWS(“DR” , s) varies across states in S L and attains its minimum value
when the polity exhibits “unified party PL control” — i.e., s E {(sL,sL,0), (3L,
sL,1)}.108 On the other hand, AWS(B, s) does not vary across states in SL.109
Given the foregoing observations, region RBR can be characterized simply as fol—
lows: it is the subset of the A—space in which party PL prefers to set a rigid budget
even though it currently controls both chambers of power, 0 and P. Suppose, now,
that the incumbency advantage is exogenously increased; i.e., no or np is raised.
Since party PL controls both chambers, such a change enhances its current status
and makes it less likely that there is “divided government” or “unified party PR
control” next period. Consequently, ceding discretion becomes less costly and

AW5(“DR” , 3) decreases in magnitude.110 In sum, when no or np is raised, there

 

a«lguments apply for party PR (i.e., s 6 53,) as well.

107For ease of exposition, we only consider the perspective of party PL. With two political
Partiesfio consider, R B R is the subset of the A—space in which: 1) AW5(“DR” , s) weakly exceeds
AWS(B,s) for all s 6 SL; and 2) AW5(“DR”,s) weakly exceeds AW5(0,3) for all s 6 SR.

108Note that AWS(“DR”,5) is proportional to the product of two terms: “total preference
faCtOI” and “total probability factor.” While “total preference factor” does not vary across
states in S L, we have argued that “total probability factor” attains its minimum value when
3 E {(SLisLjo),(3L,-SL,1)}

MAE/fig, s) is independent of electoral factors and, hence, does not vary across states 3 E S L.
AWS(B, s) is proportional to the area of triangle OAD in Figure 2.

IIOFOT S E {(sL,sL,0), (3L,sL,1)}, we can show that (p2(s) + 3&9) is equal to (1 — 12¢ — 1211).

 

145

is less incentive for party PL to induce budget rigidity; i.e., region RBR shrinks in
size.

Consider, now, how the region of budget flexibility, R B F, varies with the reelec-
tion rates, (7T0, 7rp). From party PL’s perspective, R BF is the subset of the A—space
in which, for all states 3 6 8L, AWS(§, s) weakly exceeds AWS(“DR”, 3).111 Re—
call that AW5(“DR” , s) varies across states in SL and attains its maximum value
when .3 = (s R, s L, 1) — i.e., the polity exhibits “divided government” with cham-
bers C and P controlled, respectively, by parties PR and PL.“2 On the other
hand, note that AWS(§, s) does not vary across states in S L. Given the foregoing
observations, region RBF can be characterized simply as follows: it is the subset
of the A-space in which party PL prefers to give discretion even though it currently
controls only the weakest institution of power (chamber P).

Suppose, now, that the incumbency advantage of chamber P (i.e., ms) is ex-

ogenously increased. Since, in state 3 = (33, 3L, 1), party PL controls chamber P,

 

Thus, the partial derivatives of (p2(3) + 3&9) with respect to 7rc and m: are negative. Since
the “total probability factor” is a decreasing function of 7rC and ma, so too is AWS(“DR” , s).

lllFor ease of expOSition, we only consider the perspective of party PL_._With two political
parties to consider, RBF is the subset of the A—space in which: 1) AWS(B, s) weakly exceeds
AWS(“DR”,s) for all s 6 5L; and 2) AWS(0,3) weakly exceeds AWS(“DR”,s) for all s 6 33.

112Note that AWs(“ D R”,s) is proportional to the product of two terms: “total preference
factor” and “total probability factor.” While “total preference factor” does not vary across
states in S L, we have argued that “total probability factor” attains its maximum value when
3 = (3R,3L,1)-

 

 

 

 

146

an increase in 70: enhances its current status: specifically, the “total probability
factor,” (p2(s)+p—32(32), associated with AWS(“DR” , .9) decreases.113 Consequently,
because AWS( “DR”, .9) is of smaller value, ceding discretion becomes less costly
for party PL. In sum, when 7rp is raised, there is less incentive for party PL to
induce budget rigidity; i.e., region RBF expands in size.

By contrast, an increase in the reelection rate of chamber C (i.e., W0) weakens
the status of party PL by making it more likely that party PR, currently control—
ling chamber C’, will retain power next period. Therefore, the “total probability
factor” and AWS(“DR”,s) increase in magnitude. Since ceding discretion be-
comes more costly, there is more incentive for party PL to induce budget rigidity;
i.e., region RBF shrinks in size.

Our discussion of the comparative statics of regions BER and R B F with respect
to reelection rates, (7T0, 7rp), provides an explanation for conclusions 1.2 and 1.3.
The two conclusions highlight an interesting asymmetry: An increase in 71’]: shrinks
region BER and expands region RBF, thereby reducing the total rigidity in the
A-Space. By contrast, an increase in to has an ambiguous impact on the total

rigidity in the A-space; both the regions BBB and RBF shrink in size.

 

INFO! 3 = (33,314, 1), we can show that (p2(s) + &$31) is equal to (% + 1,} — £211) Thus, the
partial derivative of (p2(s) + P—aéfl) with respect to 7rp(7ro) is negative (positive).

 

 

 

 

147

3.4. A Two-Party Parliamentary System

In this section, we examine a stylized model that approximates a Westminster-
style parliamentary system. The section is divided into three parts. First, we
provide a verbal description of the model. We then formally define an equilibrium
for the model. Finally, we discuss the principal predictions that follow from the

model’s solution.

3.4.1. Description of the Model

To emphasize the concentration of political power in a parliamentary system, we
consider a polity with a unicameral legislature. There are two political parties,
denoted PL and PR, and an agency, denoted A.

As before, the agency employs public funds to provide a service valued by
citizens. Raising public funds, however, induces welfare-reducing distortions in the
economy. The two political parties have different estimates of the magnitude of
these distortions: specifically, parties PL and PR assign, respectively, a deadweight
loss to society of AL and AR in order to levy one dollar for the agency. Party PR
is (weakly) more fiscally conservative than party PL; hence, AR 2 AL > 0.

The interaction between the two political parties spans an infinite sequence of

structurally identical periods. Each period, in turn, consists of three phases. In

 

 

 

148

phase one, the party in power (i.e., the party controlling the unicameral legisla-

ture) determines the agency’s budget for that period. In phase two, budgetary

rules for the next period are written. In phase three, an election indicates whether

 

the incumbent party is reelected. A description of the three phases follows
Phase One: Determining the agency’s budget

Each period, t E Z+, begins with one of the two political parties, PL or PR,
controlling the unicameral legislature. The party in power in period t is denoted
Pw(t).114 When PW“) equals PL(PR), we say that period it exhibits “party PL(PR)
control.”

In each period t, Nature makes a draw from the distribution of a random
variable, 0. 6 is uniformly distributed on the interval [0,5] and (9, denotes its
period-t realization. 6, is interpreted to be the period-t benefit obtained by citizens
in aggregate for each dollar spent by the agency.

Once the agency is apprised of the 0t-value, it truthfully transmits the 0t-value
to its political overseer, party PW“). Given 6,, how is the agency’s period—t budget
determined? We now need to distinguish between two cases: the “discretion
regime” and the “no discretion regime.”

Discretion regime.—— Case one considers the possibility wherein the party in

 

114Given our notation, W(t) is an element of the set (L, R}.

 

 

 

149

power in period (t — 1) (i.e., party PW(t_1)) established budgetary rules that
ceded discretion to party PW“). Under this “discretion regime,” party Pwm can
unilaterally raise a maximum of F dollars through taxation. Subject to this
exogenous constraint on the availability of public funds, party PW“) selects a
period-t budget level that maximizes its version of citizens’ net surplus. In sum,

the period-t budget level equals b(6t, )‘W(t)) where:

b(0t, Ara/(0) = arg max [b X (0, — Aw(t))] (3.18)
be[o,’1§]

N0 discretion regime.— The party in power in period (t — 1) may not have
written budgetary rules that ceded discretion to party Pym). Case two considers
the situation wherein party Pw(t_1) rigidly fixed the period-t agency budget at
some level b 6 [0,5]. Under this “b no discretion regime,” party PW“) is ren—
dered powerless. The demand shock 0,; is ignored and the period-t funding of b is
implemented.

Phase Two: Writing budgetary rules

Once the agency’s period—t budget is determined, the budgetary regime for

period (t + 1) needs to be established. Who writes the rules? In a Westminster-

Style parliamentary system, the party in power — to a first approximation -— 18 all-

 

150

powerful. We assume, therefore, that it is invested with the rights to unilaterally
stipulate the budgetary regime for period (t + 1).

Party Pwm determines the budgetary regime for period (t + 1) by selecting
an element from the set [0,3] U {“DR”} where: 1) b E [0,E] implies implemen-
tation of the “b no discretion regime” in period (t + 1); and 2) “DR” implies
implementation of the “discretion regime” in period (t + 1).

Phase Three: Election

Once the budgetary regime for period (t + 1) is decided, an election takes
place. We model electoral outcomes crudely and posit that with exogenous prob—
ability 7rW, the incumbent party retains control in period (t + 1). Incumbency,
furthermore, is assumed to be strictly advantageous; thus: WW > %.

Following the election, period t concludes. The sequence of events in period
(15+ 1) replicates that in period t. As mentioned previously, our model has an

infinite horizon. Both parties discount future payoffs at rate 6 6 (0,1) and have

. . , . . .
Preferences that are separable over time. In sum, the parties objective functions

are as follows:

(3.19)

WL = E0{i6t X [bt X (at ’ AL)”

 

 

151

wR = Ed: at x [a x (a, — AR)]} (3.20)

where: bt is the agency’s budget in period t; 9, is the period-t demand shock for the
agency’s services; E0 is the expectation operator conditional on the information

available at time 0; and WL(WR) is the objective function of party PL(PR).

3.4.2. Definition of a Markov Perfect Equilibrium

As detailed in section 3.4.1, the agency’s budget in generic period it depends
on three factors: 1) the period-t demand shock, 0t; 2) the identity of the party
controlling the legislature in period t (Pu/(o); and 3) the budgetary regime in place
in period t (“discretion regime” or “b no discretion regime”). While factors one
and two are outcomes of an exogenously specified stochastic process, factor three
is determined endogenously * i.e., as part of the model’s equilibrium.

The motivation for defining equilibrium in terms of Markov perfection is given

in section 3.3.2 and need not be reprised here. Our exposition of the conditions

for Markov perfect equilibrium is given in two parts. FiI‘Sta W9 SPeCify the state

space of the model and the strategy spaces and value functions of the two political

d sufficient conditions for

parties, PL and PR. We then provide the necessary an

strategies to constitute a Markov perfect equilibrium.

 

 

 

152

State Space, Strategies and Value Frmctions

Fix a generic period t. We say that the polity, in period t, is in state §L(§R)
if party PL(PR) is in power. We define S, the collection of possible states, to be
the state space of the model; thus, S E {§L, s3}.

Generic strategies of the two parties are represented by a mapping g 2 S —>
[0,D] U {“DR”}, where g(§L) (g(§3)) specifies the budgetary regime chosen by
party PL(PR) in state §L(§R). UL(§,7:,9) (UR(§,T,0)) denotes the period payoff
received by party PL(PR) when: 1) the current state is a E S; 2) the current
budgetary regime is 1" E [0, E U { “DR” }; and 3) the current demand shock is 6 6
[0,5]. From section 3.4.1, the following expressions for UL(§, 7“, 0) and UR(§,7',9)

can be derived:

U,(§, mags) = b(g, «1913”,0) x (0 — A,); je {L, R} and 3' e S: (3.21)

U,(§,r,6) = 7‘ x (a — A,); j e (L, R}, 376 3 and r e [0.3] (3.22)

In equation (3.21), b(s, “DR” , 6) denotes the agency’s budget when: 1) the current

state is g E S; 2) the current regime is the “discretion regime;” and 3) the current

 

 

153

demand shock is 6.115 Since (6 — A,) is the per dollar benefit to party P], the
total period benefit is the product of b(s, “DR, ”6) and (9 —— Aj). Consider now
equation (3.22). Under the “b no discretion regime,” the agency’s budget is fixed
at b. Once again, the total period payoff to party P,- is the product of the current
agency budget, b, and the current per dollar net benefit, (0 — Aj).

Given generic strategies 9, VL(§) (VR(§)) denotes the expected present dis-
counted sum of period payoffs to party PL(PR) over the infinite horizon when:
1) the current state is '5” E S; and 2) the agency’s services for the current period
have already been paid for and consumed. By a standard recursion argument, the

following two equations can be derived:

5 1
VL(§L) = 5 X 7rW X [/0 UL(§L19(§L)a0) X 5039 + VL(§L)l + (3-23)

6x (1W) x [AQULemgeaa x %d6 + View]

me.) = 3m. x [Agassi/(3330) >< gde + VR<§R>J+ (3.24)

\—
“5In state :91, b(IEL, mama) = b(6, AL). In state 33,, b(ER, “DR”,9) = b(0, AR).

 

154

6x (1—7FW)X [/0§U13(§L.9(§R),9)><%d9 + as,»

Consider equation (3.23). In state EL, party PL selects budgetary regime g(§L) for
the next period. Two possibilities now arise. With probability 7rw, party PL is
reelected (i.e., the state next period remains Q). In state EL, party PL’s expected
period payoff is [ f? U L(§L, g(§L), 6) x %d6] while, over the remainder of the game,
its expected discounted sum of period payoffs is VL(§L). Hence, conditional on
reelection, the total discounted sum of period payoffs is: U05 U L(§L, g(§ L), 6) X %d6
+ VL(§L)]. On the other hand, with probability (1 —7rW), party PL is not re—elected
(i.e., state next period is ER). An analogous argment establishes that conditional
on not being re—elected, party PL’s total discounted sum of period payoffs is:
[foL(§R,g(§L),6) x %d6 + VL(§R)]. Equation (3.23) indicates that from an ex
ante perspective, VL(§L) is simply: 6 x [probability{state next period is Q] current
state is EL} x {total discounted sum of period payoffs for party PL starting at 3}}
Plus probability{state next period is s3] current state is §L} >< {total discounted
sum of period payoffs for party PL starting at ER}]. Identical interpretation applies
for equation (3.24).

Conditions for Markov Perfect Equilibrium

We let VL(§L, a) (VR(§R, a)) denote the expected present discounted sum of

 

 

 

155

period payoffs for party PL(PR) over the infinite horizon when: 1) the current
state is §L(§R); 2) the agency’s services for the current period have already been
paid for and consumed; 3) party PL’s (PR’S) choice of budgetary regime for next
period is a E [0,B] U {“DR”} (not necessarily that specified by g); and 4) that
henceforth both parties play according to g. Following the logic of equations (3.23)

and (3.24), the next two equations can be immediately derived:

5 1
VL(§L,a) = 6 X7rW x [/0 UL(§L,a,6) x §d6 + VL(§L)]+ (3.25)

3 x (1—7rw) >< [AEUL(§R,G,9) x %d6 + was]

.. 5 ~ 1 -
VR(sR,a) = (3wa x [/0 UR(sR,a,6) >< §d6 + VR(sR)]+ (3.26)

5 1
6x (1-7rw) x [/0 UR(5L,0,9) x Ede + new

We now provide the necessary and sufficient conditions for g to be a Markov
Perfect equilibrium. Using the unimprovability criterion of dynamic program-

111ng (see Bellman (1957)), g is a Markov perfect equilibrium if and only if one

 

 

156

shot deviations by parties PL and P3 are unprofitable. Hence, the following two

conditions ensure the optimality of g:

VL(§L) 2 VL(§L,a); Va 6 [0,157] o {“DR”} (3.27)

VR(§R) Z VR(§R3 (1); Va 6 [0,3] U {“DR”} (3.28)

3.4.3. The Model’s Solution and Implications

The construction of the Markov perfect equilibrium for the parliamentary system
is analogous to that for the separation-of—powers system and is, hence, omitted. In
this section, we simply present the main features of the model’s solution, provide
certain intuitions, and emphasize the principal implications.

A polity, in our model, is characterized by a triple (AL, AR, 7rw). A parliamen—
tary system is defined to be the set of admissible polities — i.e., {(AL, AR, 77W) |A,~ E
[0,6],7rw E (%, 1] and AR 2 AL}. With reelection rate my held fixed, the cross-
section of admissible polities becomes the A-space, {(AL, have 6 [0,6] and AR 2
AL}. In Figure 3 of Appendix C, triangle XYZ depicts this A-space.

It turns out that the Markov perfect equilibrium of the model allows us to

identify two qualitatively distinct regions of the A-space: 1) A polity lying in

 

 

157

region RBB (see Figure 3) exhibits rigid budgets. In such a polity, for each state
3 E S, when party 73(3) selects next period’s budgetary regime it refuses to yield
discretion. Instead, the budget is some fixed element of the set [0,D]. 2) A polity
lying in region R B F (see Figure 3) exhibits flexible budgets. In such a polity, for
each state E E S, when party ”P(§) selects next period’s budgetary regime it yields
discretion (“DR”). The precise locations of regions RBR and RBF are detailed in

result 2 1167117

M For a parliamentary system, fix the reelection parameter 7rW and con-
sider the A—space, {(AL, AR)|)\, 6 [0,6] and AR 2 AL}. Then:
RBR = {(AL,AR)|AR 2 (1—ji) x AL and AR 2 (Ti—j) + (1&7) x AL}

REF = {(AL.AR)|AR g(l—;1)x AL and AR g (1%) + (T5,) x AL}

where: j E ,/1— 7TW

Figure 3 illustrates an important observation: the parliamentary system gen—

erates multiple behavioral patterns; the budgetary regime of a specific polity

 

11"The complete proofs are available upon request.

117Note that there are polities in the A—space that do not . .
polities exhibit hybrid behavior: one of the tWO parties yields discretlo
budget rigidity. For brevity, we do not consider such polltles-

belong in either RBR or REF. These
n while the other induces

 

 

 

158

depends on its location in the A—space.

We employ result 2 to derive predictions regarding the relationship between
budget flexibility (or rigidity) and the level of polarization. Specifically, we hold
fixed the reelection parameter, ’ITw, and consider the corresponding A—space. As
in the separation-of-powers system, the isopolarization curves in this A-space are
straight lines parallel to diagonal XY; the polarization level increases as the isopo—
larization curves move farther away from the diagonal. In Figure 3, we now con-
sider two isopolarization curves, the A—line and the B-line.118 Polities located on
line A possess a higher polarization level than those located on line B. Observe also
that a randomly sampled polity from line A(B) exhibits budget flexibility with
probability equal to 0((length of line segment B2B3) + (length of line segment

3134)).119 Conclusion 2.1 summarizes the foregoing discussion.

Conclusion 2.1: In a parliamentary system, as the level of polarization increases,

 

the probability of observing budget flexibility (rigidity) decreases (increases).

 

118The A-line refers to the line through points A; and A4. The B—line refers to the line through
Points 31 and B4.

119Line segment 3233 refers to the line segment with endpoints 32 and B3. Line segment
3134 refers to the line segment with endpoints B, and B4 (see Figure 3).

159

Result 2 also predicts how the regions of budget flexibility (REF) and budget
rigidity (RBR) change as the reelection parameter, 77W, is varied. Holding fixed
the A-space, two changes occur as 7TW is increased: 1) line 1 rotates counter-
clockwise about the point X; and 2) line 2 rotates clockwise about the point Y
(see Figure 3).120 Hence, region RBR shrinks in size while region REF expands.
The comparative statics of budgetary regimes with respect to 7m; is summarized

in conclusion 2.2.

Conclusion 2.2: In a parliamentary system, as the reelection rate of the chamber,

7rw, is raised, budgetary rigidities are unambiguously reduced.

We now provide the intuition behind conclusions 2.1 and 2.2. The construction
of a Markov perfect equilibrium (see equations (3.23) — (3.28)) demands that at
each node 3 E S, the budgetary regime selected by party 73(37) maximizes its
expected next period payoff. Thus, despite the infinite-horizon structure of the
model, optimality of party 79(19):; action is measured in terms of rewards in the

immediate future.

 

120The equation ofline 1 is, /\R 2 (L331) >< AL. The equation of line 2 is: AR = (%)+(T§) >< AL.
To determine the rotation of the two lines, note that the derivative of j with respect to 7rw is

negative.

 

160

To fix ideas, consider a polity in which /\L is less than g while AR exceeds 9121
Suppose, furthermore, that in generic period t, party PL sets the budgetary regime
for period (t + 1) — i.e., the polity is currently in state §L_122 In an ideal world
for party PL, the period-(t + 1) budget follows the b(6t+1, AL)—rule. Specifically,
the budget is set at B(0) when the period—(t + 1) demand shock, 6t“, exceeds
(is less than) AL. Given the structure of the model, however, the b(6t+1, AL)—rule
cannot be implemented. Recall that party PL has only two budgetary options: 1)
By selecting “DR”, party PL can cede discretion to the party in power in period
(t-l- 1). 2) Alternatively, party PL can rigidly fix the period (t+ 1) budget at some
level between 0 and B.

Consider, now, what happens when party PL opts to yield discretion. With
probability (1 — 7rW), the polity in period (t + 1) has party PR in power. Since
party PR has the right to set the agency’s period (t+1) budget, it therefore follows
the individually rational b(6,+1, AR)-rule; i.e., the budget is set at B(0) when 6t+1
exceeds (is less than) AB. The behavior of party PR differs from the ideal of party
PL: specifically, when 6H1 E OMAR), party PR shuts down the agency while

party PL prefers that the budget be B. This incongruence in behavior inflicts on

 

121Our intuitions do not depend on the locations of AL and AR relative to %. We only require
that AR weakly exceed AL.
122Our intuitions are unchanged if, instead, we consider state 33.

 

161

party PL an expected welfare loss, denoted AWp(“DR”,sL). The expression for
AWP(“DR” , n) is as follows:123
AR

AWP(“DR”,§L) = (1 — 3W) x (s x (L (a — AL) x gag) (3.29)

The welfare loss, AWP(“DR”,§L), is partially depicted in Figure 2. Note that
AWP(“DR” , EL) is proportional to the product of two terms: 1) the “total proba-
bility factor,” (1 — 7rW); and 2) the “total preference factor,” f (:(6 — AL)d6, equal
to the area of triangle ABC.124

Two observations regarding AWp(“DR” , (S'L) turn out to be important. First,
as 7rW is increased, the “total probability factor,” (1 — 7rw), decreases. Therefore,
AWP (“DR”,§L) is a decreasing function of my. Second, as the polarization
level of a polity, (AR ~ AL), rises, the area of triangle ABC (and, hence, the
“total preference factor”) increases. Therefore, AWp(“DR”, EL) is an increasing
fimction of a polity’s polarization level.

We have discussed the consequences for party PL, when in state 3, it chooses
to yield discretion. However, party PL has another option: it can set the budget for

the next period (period (t+1)) at some level between 0 and D. By assumption, AL

 

’23The welfare loss AWP( “DR”, 3;) is relative to the first-best payoff of party PL.
124The constant of proportionality is g.

 

162

is less than g. Hence, on average, the agency’s services in period (t + 1) are worth
more than party PL’S estimate of the cost. The optimal rigid budget therefore
involves precommitting the entire available fimds, B.

The fixed budget of B leads to an expected welfare loss for party PL: specif-
ically, when the demand shock for period (t + 1), 6H1, is less than AL, party
PL prefers ex post to shut down the agency but is instead committed to a fund—
ing level of D. The welfare loss from rigid budgets, denoted AWP(B, EL), is as
follows:125

/\L

AWPG, 3,) = [s x /0 (AL — 0) x %d6] (3.30)

Observe, now, that welfare loss AWP(B, EL) is proportional to the area of triangle
OAD in Figure 2.126

In sum, party PL has essentially two budgetary options in state EL: “DR” and
D. Relative to its first-best, AWp(“DR” , gL) (AWP(D, EL)) measures the welfare
loss to party PL when it chooses discretion (a fixed budget of B). The optimal
strategy for party PL is to select the budgetary option with the smaller welfare

loss. Given this intuition, we now provide explanations for conclusions 2.1 and

 

125The welfare loss AWP (D, EL) is relative to the first~best payoff of party PL.
126The constant of proportionality is 45—.

163

2.2.

Consider conclusion 2.1. Figure 2 illustrates that as the polarization level,
(AR — AL), rises, the area of triangle ABC (and, hence, AWp( “DR”, §L)) increases
as well. On the other hand, the area of triangle OAD (and, hence, AWP(B, E,» is
independent of (AR —— AL). Thus, as the level of polarization rises, there is greater
incentive to introduce rigid budgetary rules. In sum, polarization level and budget
flexibility (rigidity) are negatively (positively) correlated.

Consider conclusion 2.2. As the reelection rate of the chamber, 7rw, is raised,
the “total probability factor” (1 — Fry) (and, hence, AWp(“DR”, §L)) decreases.
On the other hand, the area of triangle OAD (and, hence, AWP(B, 31)) is inde-
pendent of my. Thus, an increase in 7rW produces a greater incentive for party PL
to yield discretion. In sum, the reelection rate and budget flexibility (rigidity) are

positively (negatively) correlated.

3.5. Comparison of the Two Electoral Systems

In sections 3.3 and 3.4, we analyzed in isolation the separation-of—powers system
and the Westminster-style parliamentary arrangement. We now directly compare

the two electoral systems in terms of the rigidity of the rules that emerge in

equilibrium.

 

 

 

 

164

Our method of comparison is as follows. Associated with the separation—
of-powers system are two reelection parameters, we and ma, corresponding to
chambers C and P. By contrast, associated with the parliamentary system is
a single chamber and, hence, one reelection parameter, 7rW. We let the three
parameters — no, mo and my — assume some fixed common value, denoted
7r 6 (%,1]. Then, we compare the A—spaces corresponding to the two electoral
systems to determine whether one cross-section generates more rigid rules than
the other.

Recall that result 1(2) in section 3.3(3.4) permits us to divide the A—space of
the separation-of-powers (parliamentary) system into three subsets: 1) a region
of rigid budgets, R33; 2) a region of flexible budgets, REF; and 3) a region of
hybrid budgets. Result 3, which follows directly from results 1 and 2, compares

the A-spaces of the two electoral systems.

Result 3: Let the reelection parameters 7ro, 7r p and 7rw assume some fixed common

 

value, denoted 71' E G, 1]. For the two electoral systems, now consider the A-spaces,
{(AL, MIA. 6 [0,5] and AR 2 AL}. Then:

1) The region of rigid budgets, R B R, is identical across the two electoral systems.

2) The region of flexible budgets, RBF, in the separation-of-powers system is a

 

 

165

strict subset of the corresponding region in the parliamentary system.

Figure 4 in Appendix C illustrates the above result. For both electoral systems,
the A-space is represented by triangle XYZ and the region of rigid budgets, R33,
is depicted by area ZDEF. The region of flexible budgets, RBF, is triangle XGY
(XEY) in the separation-of-powers (parliamentary) system. Since triangle XGY is
contained in triangle XEY, we derive a surprising conclusion: for every parameter
value 7r 6 (%,1], the separation-of-powers system generates uniformly more rigid
budgetary rules than does the parliamentary system.

What is the intuition for result 3? Consider, first, why the region of rigid
budgets, R33, is the same across the two electoral systems. To fix ideas, we focus
on polities for which AL is less than g while AR exceeds 3. Suppose, furthermore,
that in generic period t, party PL is assigned the right to set the budgetary regime
for period (t + 1).

In a separation-of-powers system, region RBR is the subset of the A-space in
Which party PL prefers to set a rigid budget for period (t + 1) even though it cur-
rently controls both chambers, C and P127 Thus, for s E {(sL,sL,0), (5L,8L,1)},

R33 is the region of the A—space in which the welfare loss to party PL from setting a

 

127This observation has been discussed at length in section 3.3.3.

 

 

166

rigid budget (AWS(B, 3)) is less than that from ceding discretion (AWS( “DR” , 3)).
From equations (3.16) and (3.17), we obtain, respectively, expressions for AW5(
“DR”, s) and AWS(B, 3). Using the two expressions, RBR becomes the subset of

the A—space satisfying the following condition:128

[3 >< [L6, — 0) x gde] S 012(8) + ”7””) X [F X AT“? _ A”) X bdg] (3'31)

Furthermore, when 8 E {(sL,sL,0), (3L,sL,1)} (i.e., the polity exhibits “unified
party PL control”) and the two reelection rates, no and 7r p, are each equal to 7r, the
total probability factor, (p2(s) + (432(9), associated with AWS(“DR”,3) assumes

the value (1 — 70.129 In sum:

.(p2(s) + $31?) = (1 —— 7r); for s E {(sL,sL,0), (3L,3L,1)} (3.32)

Now consider the parliamentary system. Region BBB is the region of the A—space

 

128For ease of exposition, we only consider the perspective of party PL. With two political
Parties_to consider, R B R is the region of the A—space in which: 1) AW5(“DR” , s) weakly exceeds
AWs(B,s) for 3 E {(sL,sL,0), (5L,3L,1)}; and 2) AWS(“DR”,s) weakly exceeds AWS(0,s)
f01r293 €{(3R13R10)7(5R13Ra1)}-
When 3 E {(sL,sL,0), (5L,sL,1)}, we have: 1) p2(s) (the probability of “unified party PR
control” in period (t + 1)) equal to [(1-r0).(1-7rp)]; and 2) [13(3) (the probability of “divided
government” in period (t + 1)) equal to [7T0.(1-7FP) + (1—7ro).7rp]. Thus, (p2(s) + $531) is equal

to; (Ilia-122). When 7rc = 7n: = 7r, (p2(s) + L’s—2(2) assumes the value: (1—7r).

 

 

 

 

 

167

in which party PL, when establishing budgetary rules, prefers to yield discretion to
the party assuming oflice in period (t + 1). Thus in region R312, the welfare loss to
party PL from setting a rigid budget (AWP(R, 314)) is less than that from yielding
discretion (AWp( “DR”, 31)). Equations (3.29) and (3.30) provide expressions
for, respectively, AWp( “DR”, EL) and AWP(B, 3L).130 Using the two expressions,

RBR becomes the subset of the A-space satisfying the following condition:

[B x foALO‘L — 6) x %d6] 3 (1 —- 7r) X [B x A:R(6 — AL) x %d6] (3.33)
Given condition (3.32), constraints (3.31) and (3.33) are identical. Therefore, for
the two electoral systems, the regions of rigid budgets coincide.

Why, then, is the region of budget flexibility, R B F, smaller in the separation-of-
powers system than in the parliamentary system? Recall that in a separation-of—
powers system, R B p is the subset of the A-space in which party PL prefers to cede
discretion even though it currently controls only the weakest institution of power,

i.e., chamber P_131 Thus, for 3 = (312,812, 1), RBF is the region of the A~space

 

130For ease of exposition, we only consider the perspective of party PL. With two political
Parties to consider, RBR is the subset of the A-space in which: 1) AWp(“DR”,I§L) weakly
engeds AWp (3,31,); and 2) AWp(“DR”,§R) weakly exceeds AWp(0,'§R).
This observation has been discussed at length in section 3.3.3.

 

 

 

 

 

 

 

168

in which the welfare loss to party PL from setting a rigid budget (AW5(B, 5)) is
more than that from ceding discretion (AWS( “DR” , 3)). Using the expressions for
AW5(“DR”,5) and AWS(D,S) (see equations (3.16) and (3.17)), RBF becomes

the subset of the A-space satisfying the following condition:132

(p2(s) + y) x [B x [ATM — AL) x %d6] 3 [B x fthL — 6) x %d6] (3.34)

Furthermore, when 3 = (SR, 31,, 1) (i.e., the polity currently exhibits “divided
government” with party PR(PL) controlling chamber C(P)) and the two reelection
rates, 7ro and mo, are each equal to 7r, the total probability factor, (p2(8) + 33$),

associated with AWS(“DR, ”3) assumes the value 315.133 In sum:

 

(P2(8) + p358” = g; for 8 = (3R73L, 1) (3.35)

Now consider region R B F in the parliamentary system. In region R B F, the welfare

132For ease of exposition, we only consider the perspective of party PL. With two political
parties to consider, REF is the subset of the A—space in which: 1) AWS (B, s) weakly exceeds
€W3(“D)R”,3) for s 2 (33,3L, 1); and 2) AWS(O,s) weakly exceeds AWS(“DR”,3) for s =
3L,SR,1 .

133When .9 = (8R,8L,1), we have: 1) p2(3) (the probability of “unified party PR control” in
period (t + 1)) equal to [7T0.(1-7rp)]; and 2) p3(3) (the probability of “divided government” in
Period (25+ 1)) equal to [fig/Ir}? + (1-71'0).(1-7Tp)]. Thus, (p2(s) + flag-‘31) is equal to: (% + if — 2.3:)
When 71'0 = 7n: = 7r, (p2(s) + $592) assumes the value: %.

 

 

 

.-.lllli‘

 

169

loss to party PL from setting a rigid budget (AWP(B, 7.51)) is more than that from
ceding discretion (AWp(“DR”, 3L)).134 Using the expressions for AWp( “DR” , ’31)
and AWP(B,I§L) (see equations (3.29) and (3.30)), RBF becomes the subset of

the A-space satisfying the following condition:

(1 — 7r) x [s x C‘s — AL) x %d6] 5 [s x (”(AL — 0) x %d6] (3.36)

By assumption, 7r exceeds %. Given condition (3.35), we now note that constraint
(3.34) is more stringent than constraint (3.36). Therefore, the region of budget
flexibility in the separation—of—powers system is a proper subset of the correspond-
ing region in the parliamentary system.

In brief, the message of this section is as follows. In a. separation-of-powers
system, when a political party —— say, party PL — cedes discretion in generic
period t, there are two potential sources of welfare loss: 1) that under “divided
government” in period (t + 1); and 2) that under “unified party PR government”

in period (t + 1). By contrast, in a parliamentary system, when party PL cedes

 

 

134“For ease of exposition, we only consider the perspective of party PL. With two political
parties to consider, RBF is the subset of the A—space in which: 1) AW3(D,§L) weakly exceeds
AWS(“DR”, EL); and 2) AW3(0, ER) weakly exceeds AWS(“DR”,§R).

 

 

 

 

170

discretion in generic period t, there is only one potential source of welfare loss:
that under “party PR control” in period (t + 1). Somewhat surprisingly, for every
parameter value 7r 6 (%,1], the probability weighted welfare loss from granting
discretion is weakly larger in a separation-of-powers system than in a parliamen-
tary system. Given the relative costliness of discretion, budget rules tend to be

more flexible (rigid) in a parliamentary (separation-of—powers) system.

3.6. Conclusion

In this paper, a polity has the following basic structure. There is an agency
that employs public funds to provide a service valued by citizens. There are two
political parties, with different preferences, that jointly establish the budgetary
rules to which the agency is subjected. Polities, however, are differentiated by
the electoral arrangement in place. Certain polities have a bicameral legislature
and, hence, are considered members of a separation-of-powers system. Other
polities possess a unicameral legislature and, hence, are considered members of a
parliamentary system. Throughout, we ask one question: what determines the
nature of the budgetary rules — flexible or rigid ~ that emerge in equilibrium?

Our models generate four predictions regarding the nature of budgetary rules.

First, in both electoral systems, as the level of polarization increases ~ i.e., as

 

 

 

171

the two political parties move farther apart in preference — budgetary rules be-
come rigid. Second, in the parliamentary system, as incumbency becomes more
profitable — i.e., as the reelection rate of the unicameral chamber increases —
budgetary rules become flexible. Third, in the separation-of-powers system, the
relationship between the reelection rates of the two chambers and the nature of
the budgetary rules is asymmetrical (conclusions 1.2 and 1.3 make this precise).
Fourth, budgetary rules in the separation-of-powers system are, on average, more
rigid than that in the parliamentary system. While all of our four predictions
generate precise empirical hypotheses, we view the comparative result (prediction
4) to be the most surprising.
Our analysis of equilibrium budget rules is far from exhaustive. Several exten-
sions to this paper seem especially worthwhile. In an attempt to obtain closed-
form solutions that can be readily interpreted, we have consistently employed sim—
ple functional forms (linear utility for political parties, demand shocks that are
i.i.d. and uniformly distributed, etc). It is now desirable to ascertain whether
the principal predictions of this paper are robust to more realistic functional form
specifications.
This paper only addresses the implications of coalitional drift for equilibrium

budget rules — for analytical tractability, we have simply assumed away bu—

 

 

 

 

 

172

reaucratic drift. Specifically, note that the agency never rnisreports the demand-
related information to its political overseer(s). This explicit assumption of agency
honesty is at odds with the traditional notion (see, e.g., Niskanen 1971, 1975)
that agency executives strategically manipulate private information so as to re-
ceive larger funds from politicians. Our paper will be considerably strengthened
should the agency be modeled as a strategic player engaged in budget maximiza-
tion. Results from this modified three-player game will shed light on how the two
drift forms (coalitional and bureaucratic) interact in the formation of equilibrium
budget rules. Preliminary work on this topic is in progress.

Finally, our paper considers polities with only two political parties. Most
democracies, however, possess multiple parties. Hence, extending our results to

multi-party polities should prove challenging but worthwhile.

3.7. Formal Proof of Result 1

In this section, we formally prove result 1. To reduce inessential details, we
consider polities for which both AL and AR are different from g. The restrictions
are without loss of generality; all proofs can be extended to include these “knife-

edge cases.” Throughout, the notation used is that of section 3.3.

 

 

 

 

 

173

[Step 1] Consider equations (3.10) — (3.15). The equations indicate that g is a
Markov perfect equilibrium if and only if the following two conditions are satisfied.

First, for all a 6 [0,3] U {“DR”} and for all 3 6 51,, we require:

79- 6
6X 2 <,0(3, 3') X/O UL(3’,g(s), 6)d6 2 6X 2 90(3, 3’) X/O UL(3',a,6)d6. (3.37)
3'65 3’63

Second, for all a E [0,D] U {“DR”} and for all 3 6 SR, we require:

6X 2 cp(3,3’) X [05U3(3’,g(3),6)d6 2 6X 2 go(3,s’) X [06 UR(3',a,6)d6. (3.38)

3,63 3’68

Thus, g is a. Markov perfect equilibrium if at node 3 E S, budgetary regime 9(3)

maximizes the expected next period payoff of party 79(3).

[Step 2] Suppose that in generic period t, a polity is in state 3 E S; i.e., party
P(s) is in office. In the phase where party 79(3) selects the budgetary regime for
period (t + 1), assume that the “discretion regime” (“DR”) option is exogenously

excluded. Let 3(3) 6 [0, D] now be the fixed budget level that maximizes party

 

 

 

 

 

174

’P(s)’s ex ante expected payoff in period (t + 1).
The average demand shock in period (t + 1) is the expectation of the random

. Given the risk neutral preferences of the two parties, it is

NIQI

variable, 9; i.e.,

immediate that §(s) is as follows:

0 if < A-value of party 73(3)

NIQI

as) = (3.39)

tolml

E if > A-value of party ’P(s)

We describe the g-mapping graphically. Corresponding to a polity is a specific
pair (AL, AR). Since AL 3 AR and A,- 6 [0,3], the collection of possible polities is
represented by the A-space: {(AL,)\R)[ AL 3 AR and A,- E [0,3]} In Figure 5 of
Appendix C, the A-space is the triangle ACF. Triangle ACF is subdivided into
three regions: A polity is placed in region 1 (triangle ABD) if its (AL, AR)—va1ue
satisfies: 1) AL < g and 2) AR < g. A polity is placed in region 2 (square DBEF)
if its (AL, Ala-value satisfies: 1) AL < g and 2) AR > g. A polity is placed in region
3 (triangle BCE) if its (AL, AR)-value satisfies: 1) AL > ~2- and 2) AR > g.

The fi-mapping in each of the three regions is as follows: 1) In region 1,

W3) = §,V8 E S. 2) In region 2, §(s) = 0 for .3 6 SR and 9(3) 2 F for s 6 SL. 3)

In region 3, §(s) = 0,Vs E S.

 

 

 

 

175

[Step 3] Suppose that in generic period t, a polity is in state 3 E S. By the
construction in step 2, it is clear that when party 73(3) selects a budgetary rule for
period (t+1), the choice is between §(3) and “DR”. Thus, 9(3) 6 {§(3), “DR”}.
Furthermore, step 1 indicates that in deciding between §(3) and “DR”, party

’P(3) picks the Option that maximizes its expected payoff in period (t + 1).

[Step 4] Suppose that in generic period t, a polity is in state 3 E S and that party
73(3) chooses budgetary regime §(3). Given 9(3), we now compute the expected
payoff to party 70(3) in period (t + 1), denoted V(§(3), 3). To compute l7('g'(3), 3),
we shall consider two subcases: 1) the A-value of party 73(3) is less than g (i.e.,
5(3) = F) and 2) the A-value of party 77(3) exceeds g (i.e., §(3) = 0).

Subcase 1: A-value of party P(3) less than g.—— In an ideal world, party 79(3)
desires a period (t + 1) budget of 73—(0) when 0H1 exceeds (is less than) its A-value.

Thus, for 3 6 SL (3 6 SR) the first-best expected payoff for party PL(PR) is:

[D x ffL (6 — AL) >< %d6] ([1??- x fwa — AR) >< %d6]). This first-best expected payoff

for party PL(PR) is denoted as WWI/'2“).

However, given the rigid §(3)—regime, the flexible budgets of the ideal world

cannot be enforced, When the A—value of party 73(3) is less than 2, step 2 indicates

 

 

 

 

 

176

that §(3) = D — i.e., the budget for period (t + 1) is set at F. The fixed budget
induces a welfare loss for party 79(3): specifically, when 9H1 is less than the A-

value of party 73(3), party P(3) prefers ex post to shut down the agency but is

instead committed to a funding level of D.

In sum, when 3 E S L (3 6 SR), the budget rigidity inflicts on party PL(PR)
an expected loss of: [F x [OALO‘L — 0) X %d6] ([13 x fOARQR — «9) x %d6]). Given
this expected welfare loss relative to the first-best, the following two conditions

are immediate:

[\DIQI

"_ —maX — AL 1
V(§(3), 3) = VL — B x /0 (AL — 3) x Ede; 3 e 3;, and AL < (3.40)

(3.41)

[\qubl

— —max — AR 1
V(§(s),s) = vR - B x/ (AR — 9) x Ede; 3 6 SR and AR <
O

Subcase 2: A-value of party 79(3) exceeds 9— Step 2 indicates that §(3) = 0
~ i.e., the budget for period (t + 1) is set at 0. The fixed budget induces a

welfare loss for party 79(3): specifically, when 6H1 is more than the A—value of

 

 

 

 

 

 

177

party 73(3), party P(3) prefers ex post to let the agency’s budget be D but is
instead committed to a funding level of 0.

In sum, when 3 6 SL (3 E S R), the budget rigidity inflicts on party PL(PR)
an expected loss of: [D X Kim — AL) X %d0] ([5 X 5109 — AR) X %d9]). Given
this expected welfare loss relative to the first-best, the following two conditions

are immediate:

— _maX _ a 1
V(§(3),3) = VL — B x/ (3 — AL) x =d6; 3 6 3L and AL >

3.42
AL 6 ( )

[01%|

5
R

_ _max _ 1 3
V(§(s), 3) = vR — B x A (9 — AR) x .533; 3 3 SR and AR > 5 (3.43)

[Step 5] Suppose that in generic period t, a polity is in state 3 E S and that

party 73(3) chooses the “discretion” regime, “DR”. Given the “DR”-regime,

we now compute the expected payoff to party 79(3) in period (t + 1), denoted

~

V(“DR”, 3). First, we introduce additional notation.

 

 

 

 

 

 

 

178

We divide the state space S into four subsets. SLL(SRR) denotes the set of
states in which party PL(PR) controls both chambers. SLR(SRL) denotes the set
of states in which party PL(PR) controls chamber C while party PR(PL) con-
trols chamber P. Given our theory of elections, characterized by reelection rates
(30, ms), the transition probabilities between the (above) four subsets of S are

given by the matrix H.

 

 

 

 

 

SLL SLR SRL SRR
SLL 7T0.7TP 7T0.(1-’/Tp) (1-7rC).7rp (1-7T0).(1-7Tp)
H= SLR mam.) «amp (1-7r0).(1-7rp) (1-7r0).7rp
SRL (1-71'0).7Tp (1-7rC).(1-7rp) Wc-WP 7TC-(1‘7TP)
sRR (1-71'0).(1-7rp) (1-7r0).7rp mum.) mm.

 

 

 

 

 

 

The (i, j)’th element of II denotes the probability of transiting (in one step) from
a state represented by the i’th row of II to a state represented by the j’th column
0f H. For example, H(1,1) denotes the probability that the state next period is an
element of SLL when the current state is an element of SLL. Since this requires
the incumbent party PL to win elections in both chambers, 110,1) = 770 X 77 P.
Given H, for each state 3 E S we construct a three-element vector p(3) where:
1) the first element (p1(8)) represents the probability of “unified PL government”

- - u - (1
next period; 2) the second element (p2(3)) represents the probability of umfie

 

 

 

 

179

PR governmen ” next period; and 3) the third element (p3(3)) represents the
probability of “divided government” next period. The following four conditions

can be readily derived from the elements of II.

p(3) = («0.7172, (1-7Tc).(1-7TP), 7TC.(1-7l'p) + (1-7rc).7rp); 3 E SLL (3.44)

p(s) = (71'0.(l—7TP), (1—770).7rp, 7T0.7f'_p + (1-71'0).(1-7l'p)); S E SLR (3.45)

p<s> = ((1-«mp, «an-m»), «W + (Mom-7w»; s e Sm (346)

(”(5) = ((1-7T0).(1-7Tp), Wcfl'p, 71'0.(1—1Fp) -[- (1—7rc).7l'p); 3 E SRR (3.47)

Consider now V(“DR”,S)9 S 6 SL- Three cases arise: 1) With probability

 

 

180

p1(3), party PL controls both chambers in period (t + 1) and fixes the agency’s
budget at a level that is first-best for party PL — i.e., the budget is _B_(0) if
6H1 exceeds (is less than) AL. 2) With probability p2(3), party PR controls both
chambers in period (t + 1) and fixes the agency’s budget at 5(0) if 0H1 exceeds
(is less than) AR. The behavior of party PR diflers from the ideal rule of party PL;
specifically, when 6H1 6 (AL, AR), party PR shuts down the agency while party
PL prefers that the budget be S. This incongruence in behavior inflicts on party
PL an expected welfare loss of [D X f (A: (6’ — AL) X %d6]. 3) With probability p3(3),
there is “divided government” in period (t + 1). Under divided government, the
agency’s budget is set at: a) 0 if 6H1 S AL, b) g if 6H1 6 (MAR), and c) R if
6t+1 2 AR. The decision rule under divided government is suboptimal for party
PL in the region 0t+1 6 (AL, AR); specifically, party PL prefers a budget of P but
obtains only half that amount. This incongruence in behavior inflicts on party PL
an expected welfare loss of [g X §LR(0 — AL) X %d6]. By a standard conditioning

argument, the following expression for {V( “DR”, 3)]3 E S L} can be derived:

AR

 

WM», 3) = V?“ — (Ms) + ”5“”) x ('3’ X/. (9 ‘ W X 3‘”); 3 6 51’ (3'48)

2

L

 

 

 

 

 

 

181

By an argument identical to that given above, the following expression for {7

(“DR”, 3)]3 6 SR} can also be established:

V3012», 3) = V3” — (91(3) + ”5“”) x (B x AARQR — a) x gag); 3 6 SR (3.49)

L

 

[Step 6] For brevity, we only provide the necessary and sufficient conditions for
the Markov perfect equilibrium to exhibit complete budget rigidity or complete
budget flexibility. Thus, we compute: 1) the subset of the A-space, denoted RBR,
for which 9(3) = 9(3),\7’3 E S (i.e., the region of budget rigidity); and 2) the
subset of the A-space, denoted RBF, for which 9(3) = “DR”,\7’3 E S (i.e., the
region of budget flexibility).

[Step 7] Consider, now, RBR. By step 1, RBR is the region: {(AL, AR)|V(§(3), 8) Z

V(“DR”7 S);VS E S}
Consider, first, 3 6 SL. Given the transition matrix H and equations (3.44)

3 h
‘ (3-46), Simple algebra reveals that (92(3) + 435—2) has the smallest value w en

3 6 SL C SLL. (The comparison is across three subsets of SL : SL (1 S LL, S L C SLR

 

 

 

 

 

 

182

and SL (1 SRL. Note also that SL 0 SLL = SLL.) Horn equation (3.48), it therefore
follows that V( “DR, ”3) has the largest value when 3 E SLL. Finally, observe that
equations (3.40) and (3.42) jointly imply that the value of V(9(3), 3) does not
vary across states 3 6 SL.

Consider, now, 3 6 SR. Given the transition matrix 11 and equations (3.45)
- (3.47), simple algebra reveals that (p1(3) + 53232) has the smallest value when
3 6 SR (1 SRR. (The comparison is across three subsets of SR : SR 0833, S30 SLR
and SRHSRL. Note also that SR (1 S RR = SRR.) From equation (3.49), it therefore
follows that 7( “DR”, 3) has the largest value when 3 6 S33. Finally, observe that
equations (3.41) and (3.42) jointly imply that the value of V(9(3), 3) does not
vary across states 3 E S R.

Given the observations in the above two paragraphs, RBR is as follows:

RBR = RBR(SLL) fl RBR(SRR) (350}

where : RBR(SLL) = {(AL, AR)|V(§(3), .9) 2 V(“DR”, 3);Vs e SLL} (3.51)

 

 

 

 

 

 

183

RBR(SRR) = {(AL, AR)]V(§(3), S) 2 V(“DR”, S);VS E SRR} (3.52)

We begin by characterizing RBR(SLL). To characterize RBR(SLL), we shall

N|<b|

consider two subcases: 1) AL < g (regions 1 and 2 of Figure 5) and 2) AL >
(region 3 of Figure 5).
Subcase 1: AL less than 9— Figure 5 indicates that 9(3) = B. From equations

(3.40), (3.44) and (3.48), 7(B, 3) _>_ V( “DR”, 3) if and only if:

71-0 71'}: _ AR 1 _ A
(————)XBX/A (B—AL)X§d62BX/OL(AL—6)X%d6’ (3.53)
L

Simplifying the above expression, V(9(3), 3) Z 7(“DR”, 3) if and only if:

 

 

1+ "
AR 2( ‘7’] ) X AL; where j, E W— 11.22 " 3213‘) (3.54)

Subcase 2: AL exceeds ga— Figure 5 indicates that 9(3) = 0. From equations

(342), (3.44) and (3.48), V(0, 3) 2 7( “DR”, 3) if and only if:

 

 

 

 

 

 

184

AR 1 _ 9 l
x/ (e—AL)X=dasz/ (H—AL)X=d0 (3.55)
,\ 6 AL 9

L

l
:1
Q
3
ml

Condition (3.55) is violated for all parameter configurations.

Combining equations (3.54) and (3.55), RBR(SLL) is as follows:

1 + " 3
RBR(SLL) = {(AL, ARMAR 2 ( 3,7 ) x AL and AL < 5} (3.56)

 

We now characterize region RBR(SRR). To characterize RBR(SRR), we shall
consider two subcases: 1) AR < g (region 1 of Figure 5) and 2) AR > 3 (regions 2
and 3 of Figure 5).

Subcase 1: AR less than 9— Figure 5 indicates that 9(3) = B. From equations

(3.41), (3.47) and (3.49), 7(B, 3) 2 V( “DR”, 3) if and only if:

A _ A 1
(—W—C—“—P)><Bx/“(AR—0)x%desz/OBorax-3533 (3.57)
/\

L

Condition (3.57) is violated for all parameter configurations.

Subcase 2: AR exceeds 9— Figure 5 indicates that 9(3) = 0. Mom equations

 

 

 

 

 

 

185

(3.43), (3.47) and (3.49), 7(0, 3) 2 7( “DR”, 3) if and only if:

770 7TB — AR 1 _ 5
(1————)XBX[\ (AR—9)X§d6’ZBX/(9—AR)X%d9 (3.58)

L AR

Simplifying the above expression, V(9(3), 3) Z V( “DR”, 3) if and only if:

3 j’
.,) +( .,
1+] 1+]

 

 

AR 2 ( ) x AL (3.59)

Combining equations (3.57) and (3.59), RBR(SRR) is as follows:

'I 6
) +(1-Jl—j’)X)\L and AR>§} (3.60)

 

RBR(SRR) = {(AL, ARMAR 2 (m

Finally, equations (3.50), (3.56) and (3.60) yield, after simple manipulations, a
characterization of RBR. RBR is the subset of the A-space satisfying the following

two requirements:

 

 

 

 

 

 

186

 

) X )‘L (3.62)

[Step 8] We now compute RB F (the region of budget flexibility). By step 1, RBF
is the region: {(AL, AR)|V( “DR”, 3) 2 V(9(3), 3);V3 E S}.

Consider, first, 3 E S L. Given the transition matrix H and equations (3.44)
- (3.46), simple algebra reveals that (p2(3) + 53232) has the largest value when
8 6 SL 0 SRL. (The comparison is across three subsets of SL : SL 0 SLL, SL 0 SLR
and SL 0 S R L.) From equation (3.48), it therefore follows that V( “DR”, 3) has the
smallest value when 3 E S L F1 331,. Observe, also, that equations (3.40) and (3.42)
jointly imply that the value of V(9(3), 3) does not vary across states 3 E S L.

Consider, now, 3 E S R. Given the transition matrix H and equations (3.45)
- (3.47), simple algebra reveals that (p1(3) + 39432) has the largest value when
3 6 SR 0 SLR. (The comparison is across three subsets of SR : S R D S 33, SR fl SLR
and SR fl S'RL.) From equation (3.49), it therefore follows that V( “DR”, 3) has the
Smallest value when 3 6 SR 0 S LR. Observe, also, that equations (3.41) and (3.43)
jOintly imply that the value of V(9(3), 3) does not vary across states 3 6 SR.

Given the observations in the above two paragraphs, R B F can be expressed as

 

 

 

 

 

 

187

follows:

where I RBF(SL fl SRL) = {(AL, AR)[V(“DR”,3) Z V(§(3), 3);‘v’3 E 5L (1 SRL}

(3.64)

RBF(SR fl SLR) = {(AL, AR)|V(“DR”, 3) 2 V(9(3), 3);‘v’3 6 SR 0 SLR} (3.65)

We begin by characterizing RBF(SL fl SRL). Two subcases need to be consid-
ered: 1) AL < g (regions 1 and 2 of Figure 5) and 2) AL > g (region 3 of Figure
5).

Subcase 1: AL less than 9— Figure 5 indicates that 9(3) = B. Ftom equations

(3.40), (3.46) and (3.48), 7( “DR”, 3) Z [7(B, 3) if and only if:

1
1 no WP _ AR 1 — f“) _9 x=d6 (3.66)
(5+?——)XBXA (Q—AL)X9-d0§BX 0 (L ) 6)

 

 

 

 

 

 

188

Simplifying the above expression, 7( “DR”, 3) 2 V(9’(3), 3) if and only if:

 

1+j/I

)‘R S( j”

 

l
) X AL; where j” E (/(E + 322 — g) (3.67)

Subcase 2: AL exceeds 9— Figure 5 indicates that 9(3) = 0. From equations

(3.42), (3.46) and (3.48), V(“DR”,3) Z V(0, 3) if and only if:

3

_ A _ 1
(1+7;_C_7T_P)XBX/R(6—AL)X%dHSBX/(Q—AL)X§d9 (3.68)
A

L AL

Condition (3.68) is satisfied for all parameter configurations.

Combining equations (3.67) and (3.68), RBF(SL fl SRL) is as follows:

1+jl/

j”

 

3 3
RBF(SLCSRL) = {(ALyARHAR S ( )XAL and )‘L < ‘2'} UfkaaARH ’\L > 5}

(3.69)
We now characterize RBF(SR C S L R). Two subcases need to be considered: 1)

AR < 3 (region 1 of Figure 5) and 2) AR > 3 (regions 2 and 3 of Figure 5).
Subcase 1: AR less than 33— Figure 5 indicates that 9(3) = B. From equations

(3.41), (3.45) and (3.49), V(“DR”, 3) 2 T703, 3) if and only if:

 

 

 

 

 

 

189

1 7T 71' _ AR 1 _ A
(_+_2£__P)XBXA (AR—6)X§d63BX/OR(AR—6)X%d6 (3.70)

Condition (3.70) is satisfied for all parameter configurations.
Subcase 2: A R exceeds 9— Figure 5 indicates that 9(3) = 0. From equations

(3.43), (3.45) and (3.49), V(“DR”,3) 2 7(0, 3) if and only if:

1 _ A _ 3 1
(—+E—W—P)XBX/R(AR—0)X%d93BX/(B—AR)><§d9 (3-71)
A

L AR

Simplifying the above expression, l7( “DR”, 3) Z V(0, 3) if and only if:

'9' j”
AR 2 (m) +(1—+—J7) X )‘L (3-72)

Combining equations (3.70) and (3.72), RBF(SR fl SLR) is as follows:

RBF(SR f7 SLR) = {(AL, ARMAR < '3'} U {(ALaARM

 

g -/I 6
)‘R (1’7) +( J I ) X ALand AR > 5} (3.73)

1+j’

 

 

 

 

190

Finally, equations (3.63), (3.69) and (3.73) yield, after simple manipulations, a

characterization of R B F. R B F is the subset of the A-space satisfying the following

two requirements:

 

 

 

1 ‘I/
AR_( 4]” )er (3.74)
A <( 3 )+(j")x1 (375
R— 1+j” 1+j” L . )

[Step 9] The locations of the two regions (RBR and RBF) in A-Space are depicted

in Figure 1 of Appendix C. The location of RBR(RBF) is derived from equations

(3.61) - (3.62) ((3.74) _ (3.75)).

 

 

 

 

APPENDICES

 

 

 

 

 

APPENDIX A

TABLES FOR CHAPTER 1

Table 1: Parameter Values (in francs) for the Experiment

Treatment # B k 11'.L an

 

 

 

 

 

 

 

Treatment 1 7425 1 0.0 1.0
Treatment 2 7425 1 0.2 0.8 ][
Treatment 3 5000 1 0.0 1.0 ]]
Treatment 4 5000 1 0.2 0.8 I
Treatment 5 7425 1 0.5 0.5
Treatment 6 5000 1 0.5 0.5

 

Table 2: Summary of Observations for Repeated-Interactions Sessions

_reatment # # .f .verage .td. Dev. .ealized _eelection
_rials _ffort -f Effort _utput _robability

Treatment 1 (67,134) (.,08 1.0)
Treatment 2 (93, 77) (.10, 1.0) ]]

(86, 76) (.21, 1.0)
(100, 54) (.57, .98)
(79, 73) (.40, 1.0)
(25, 36) (.08, 1.0)

Treatment 3

Treatment 4

 

Treatment 5

NKJILIIQJIKII

Treatment 6

 

 

 

 

 

192

Table 3: Regression-based Analysis of Candidates' Effort

Variable (1) (2) (3)

OLS OLS-FE OLS-RE
Constant 30.46 - 27.31

(1 . 19) (3.15)
Highb 15.60 16.88 16.82

(1.33) (1.78) (1.72)
Highprod 19.98 19.59 19.65

(1.33) (1.27) (1.26)
Fixed Effects No Yes No
R-squared 0.35 0.33 0.33
11 = 687

 

Notes: OLS-FE refers to the fixed effects estimates of equation (1 .3). OLS-RE
refers to the random-effects estimates of equation (1 .3). The numbers in
parentheses are standard errors.

Table 4: Analysis of Voters' Behavior

-reatment # # _f # _f Error —APate, Error .Apate,
_rials _lection Pds.

 

 

 

 

 

 

 

Treatment 1 6 10 75 14.23
Treatment 2 5 170 13.06 22.59
Treatment 3 5 162 18.77 23.09
Treatment 4 5 154 26.37 27.67
Treatment 5 5 152 23.95 23.42
Treatment 6 2 61 15.73 21.64

 

   

 

 

 

Table 5: Analysis of Voter Heterogeneity

Treatment # Avg., Error Std. Dev., Avg., Std. Dev.,
Rates Error Rates ’69s 133's
.35 .28

 

 

 

 

 

 

Treatment 1

Treatment 2 .23

Treatment 3 .29 I]

Treatment 4 .18 n
] Treatment 5 .29

Treatment 6 .09

 

 

Table 6: Predicted and Observed Candidates' Effort Levels

 

Treatment # Average Std. Dev. Average Std. Dev.
Effort of Effort Predicted of Predicted
Effort Effort

 

 

 

 

 

 

 

 

Treatment 1 18.66

Treatment 2 19.44 5.78

Treatment 3 15.43 5.06

Treatment 4 14.75 3.31
]]:Treatment 5 12, 16 0

Treatment 6 11.16 0

 

 

 

 

 

 

194

Table 7: Regression-based Analysis of Effort Discrepancy

 

 

Variable (1) (2) (3)
OLS OLS-FE OLS-RE
Constant 20.10 - 19.34
(1.08) (1.64)
Highb 1.82 2.15 2.08
(0.91) (1.31) (1.21)
Highprod 0.85 0.78 0.79
(0.91) (0.93) (0.92)
Exper -0.25 -0.30 -0.28
(0.08) (0.07) (0.08)
Fixed Effects No Yes No
R-squared 0.02 0.02 0.02
n = 687

 

 

Notes: OLS-FE refers to the fixed-effects estimates of equation (1 . 12). OLS-RE
refers to the random-effects estimates of equation (1 .12). The numbers in
parentheses are standard errors.

Table 8: Relative Efficiency of Experimental Elections

 

 

 

 

 

 

 

 

 

 

 

Effort Upper Average Relative
Bound Effort Efficiency

Treatment 1 70.36 67.43 95.84%
Treatment 2 70.36 44.42 63.13%
Treatment 3 57.74 48.72 84.38%
Treatment 4 57.74 32.27 55.89%
Treatment 5 0 5.24 - 7]
Treatment 6 0 2.51 -

 

 

 

195

Table 9: Summary of Observations for One-Shot Sessions

_reatment # — of _verage _td. Dev.
_rials _ffort _f Effort

Treatment 1

    
  

 

 

Treatment 2

 

Treatment 3

 

 

Treatment 4

  

 

 

 

APPENDIX B

TABLES FOR CHAPTER 2

Table 10: Parameter Values (in francs) for the Experiment

_6:]EOE- 1 «a I In «:1» l Mio- l- Cw}

Treatment 1 0. 5
Treatment 2 6-00 2-0 0.. 5 1 10 1
Treatment 3 _300 20 1 10 1

Table 11: Equilibrium Set for One-Shot Experimental Elections

Treatment # Pooling Efficient Separating Efficient
Equilibria Pooling Equilibria Separating
(0, 0), (0, 1) (0, 0), (0, 1) (0, 3, 0» 1), (01 3, 0, 1)

(0,6, 0, 1)
(o, 0), (o, 1) (o, 6, o, 1), (0, 6, o, 1) "

 

 

 

 

      
   
     

Treatment 1

Treatment 2 (0, 0), (0, 1)>

(3, 1)
(0, 0), (0, 1),
(3, 1), (6, 1)

(O, 9, 0, 1)

Treatment 3 (0, 0), (0, 1) (01 9, 0: 1) (O: 9: 0: 1)

 

 

 

 

 

 

197

Table 12: Equilibrium Set for Repeated-Interactions Experimental Elections

_reatment # _ooling .fficient _eparafing .fficient
_quilibria _ooling _qullibria _eparating

 

 

 

 

 

Treatmentl (0, 0), (0, 1), (O, 0), (O, 1) (0, 3, 0, 1)
(3 1)
Treatment 2 (0, O), (0, 1), (O, O), (O, 1) (0, 3, O, 1)
(3, 1), (6, 1)
(O, O), (0, 1), Same as in
Treatment 3 (3, 1), (6, 1), (0, O), (O, 1) Treatment 2 (O, 3, 0, 1)

 

(9, 1)

Table 13: Summary of Observations for FI-OS and II-OS Sessions

Session Policy Outcome Policy Outcome Reelection
Characteristic (IL) (1.1) Probability

Treatment 1 (27, O, 0, 0) (1, 12, 0, 0) (~14, 10, _3 _)

Treatment 2
Treatment 3

(9,6,0, 0)
(13,1,2, 1)

(2, 6, 16, 1)
(1, 1, 1, 20)

(0.0, .33, 1.0, 1.0)
(0.0, 0.0, 0.0, 1.0)

 

 

ILL 25 ;
Treatment 1
Treatment 2
Treatment 3

(42, 1, 0,0)
(27, 12, 0,0)
(16, 8, 21, 0)

 

(2, 40, 5, 0)
(1, 8, 30,2)
(0, 7, 22, 6)

 

(.07, .93, 1.0, _)
(.18, .50, 1.0, 1.0)
(.07, .33, .84, 1.0)

 

198

Table 14: Rationality of Candidates and Voters in FI-OS and II-OS Sessions

Session Optimal Modal Reelection Conditional

Character- Policy Policy Probability Probability
istic Choice Choice of In

 

FI-OS:
Treatment 1 (.14, 1.0, _, _) (.04, 1.0, _, _)
Treatment 2 (0.0, .33, 1.0, 1.0) (.18, .50, 1.0, 1.0)
Treatment 3 (0.0, 0.0, 0.0, 1.0) (.07, .50, .33, .95)

 

II-OS:
Treatment 1 (.07, .93, 1.0, _) (.05, .98, 1.0, _)
Treatment 2 (.18, .50, 1.0, 1.0) (.04, .40, 1.0, 1.0)
Treatment 3 (.07, .33, .84, 1.0) (0.0, .47, .51, 1.0)

 

 

Table 15: Equilibrium Selection for One—Shot Elections (Consistent Responses)

Session Equilibrium Equilibrium Equilibrium
Characteristic Ranked 1 Ranked 2 Ranked 3

Treatment 1 ((0, 3, 0, 1), s, 90%) ((0. 6. 0, 1). s, 60%) ((0. 0), p. 10%)
Treatment 2 ((0, 6,0, 1), s, 63%) ((0, 9, 0. l). s. 25%) ((3. I). p. 10%)
Treatment 3 ((0, 9, 0, 1), s, 83%) ((0, 0), p, 33%) All Others

 

       
 
   

  

 
 
  

 
  

    

 

        
 

II-QS:

     
    

 
   
 

Treatment 1 ((0, 3, 0, 1), s, 86%) ((0, 6, 0, 1). s, 50%) ((3. I), p, 42%)
Treatment 2 ((0, 6, 0, 1), s, 65%) ((0, 9, 0, 1), S, 30%) ((3, l), p, 31%)
Treatment 3 ((6, 1), p, 45%) ((09 99 09 I): S: 26%) ((03 0): p9 19%)

 
       

 
 
     

Notes: "All Others" includes the three pooling equilibria: (0,1), (3,1) and (6,1).

 

 

 

 

 

199

Table 16: Equilibrium Selection for One-Shot Elections (Likelihood Methods)

Session Equilibrium Equilibrium Equilibrium (6,, €13)
Character- Ranked 1 Ranked 2 Ranked 3

istic

Best Equi-
librium

 

FI-OS:
Treatment 1 ((0, 6, 0, l), s) ((0, 1), p) (.04, .14)
Treatment 2 ((3, l), p) ((0, 1), p) (.15, .18)
Treatment 3 ((6, 1), p) ((3, 1), p) (.07, .05)

 

 

II- .
Treatment 1 ((0, 3, 0, 1), s) ((0, 6, 0, 1), s) ((0, 1), p) (.07, .11)
Treatment 2 ((0, 6, 0, 1), s) ((0, 1), p) ((3, 1), p) (.13, .19)
Treatment 3 ((6, l), p) ((3, 1), p) ((0, 1), p) (.17, .21)

 

 

 

Table 17: Summary of Observations for the FI-RI and II-RI Sessions

Session Policy Outcome Policy Outcome Reelection
Characteristic (1.) (In) Probability
Elzfl;

Treatment 1 (100, 4, 1, 0) (6, 98, 0, 0) (~14, 99, 0-0, _)
Treatment 2 (98, 30, 3, 1) (89 92: 379 0) ('07: 859 1'09 1'0)

     
 
  

 

  
     
   
 

Treatment 3 (81, 25, 1, 0) . (9, 73, 17, 15) (~13, -73, 95, 1-0)
11:31.;

Treatment 1 (113,3, 1, 0) (7, 66. 50. 1) (.17, .64, 1.0. 1.0)

Treatment 2 (89, 28, O, 0) (19 38: 533 16) ('08, 76: 109 10)

Treatment 3 (64, 38, 18, 0) (2, 52, 65, 10) (~06, .75, 92, 10)

 

 

 

 

200

Table 18: Equilibrium Selection for Repeated Elections (Consistent Responses)

       
   

 

 

Session # Trials # Trials # Trials
Characteristic (With (With Efficient
Separation) Separation)

FI-RI:
Treatment 1 9 9 9 0
Treatment 2 12 11 6 1
Treatment 3 10 8 6 2

II-RI:
Treatment 1 11 10 4
Treatment 2 11 10 2 1
Treatment 3 11 6 2

 

 

Table 19: Informational Efficiency of Experimental Elections

 

 

 

 

Treatment 1 Treatment 2 Treatment 3
Characteristic (%) (%) (%)
FI-OS 92.50 80.00 90.00
II-OS 92.22 85.00 63.75
FI-RI 87.56 84.76 81.98
II-RI 79.25 88.44 74.69

 

 

 

 

APPENDIX C

FIGURES FOR CHAPTER 3

 

 

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