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AN ECONOMIC MODEL OF PLEA BARGAINING IN :;:.:-:::é:.;.;f-_ I; :3
THE CR'M'NRL CCURT SYSTEM

  

 

 

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thesis entitled . *'
AN ECONOMIC MODEL OF
PLEA BARGAINING
IN THE CRIMINAL COURT SYSTEM

presented by

Judith Ann La chman

has been accepted towards fulfillment
of the requirements for

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,9) \ ABSTRACT
Il)\ AN ECONOMIC MODEL OF
3\ PLEA BARGAINING
.. R- ‘ IN THE CRIMINAL COURT SYSTEM
By

Judith Ann La chman

Plea bargaining, the informal process by which prosecution and
defense settle a criminal case without trial, is the dominant method of
disposing of criminal cases in U. S. metropolitan courts. Both the
process and outcome of plea bargaining lend themselves to analysis
within an economic framework: each party makes rational choices based
upon its estimate of the opportunity 'cost of a guilty plea, that is, the
expected outcome and associated costs of trial.

This dissertation presents a theory of plea bargaining—~the deri-
vation of the reservation prices of the prosecutor and defendant, the
offer-response pattern, and the outcome of bargaininq--and tests that
theory using micro cross-sectional data.

In the theory, each of the parties to negotiations is believed to
calculate the outcome and utility anticipated from trial, and then the
minimal concession which would be required from his opponent in order

to make the prOSpect of a plea of guilty equally attractive. For each

Judith Ann La chman

party, the trial option is represented by one constraint and the guilty
plea option by another. The "switch function" which results from re-
peating the above calculation for alternative values of the independent
variable is in effect the demand function for one good—-a plea of
guilty--stated in terms of the price of a substitute good, trial. Each
opportunity set is a production possibility frontier derived from functions
describing case outcome in terms of variable inputs to production of a
strong case (additional sleuthing and research efforts) and the relative
strength of the two parties. Concession is interpreted to be the
deliberate underutilization of the strength of one's own case.

The integration of the above items within the model leads to two
observable theoretical results. First, the derivation of the switch
functions--each of which divides the strength-outcome space into a set
Of acceptable bargain outcomes and a set of unacceptable ones--lends a
specific funCtional definition to the concept Of a Pareto Optimal set of
outcomes of negotiation; the set of negotiated outcomes should fall with-
in the region bounded by the switch functions. Due tO the lack of sta-
tistical tools, this can only be "tested" by visual observation of data
plots.

Second, a split-the-difference theory of bargained outcomes is
proposed, and the outcome function that suggests is a linear combin‘ation
of the two switch functions. Estimation of this outcome function serves

not only to test the general validity of the model, but also to determine

the proportions in which the bargainable difference is split.

Judith Ann La chman

In order to estimate outcome as a function of relative strength, it
is necessary first to create an index of relative strength based upon the
factOrs believed to be critical to the determination of strength at trial.
This is done using probability-of-conviction estimates made by prose-
cutors together with information on characteristics of the defendant and
of the offense, and the initial endowment of evidence. A logistic form
is employed, and predicted values for the probability estimates are used
in subsequent empirical work.

Regression .results obtained for the relative strength indicator
suggest that delay is beneficial to defendants, as is having a less
harsh judge, or knowing the complainant well. In contrast, having other
charges pending or being age 18 or older work to the defendant's disad-
vantage. The most significant results here relate to the race and jail
status of the defendant, , and interactions between these variables.
Taken singly, the effects of each of race and jail on strength at trial
are as expected: blacks fare worse than whites, and pretrial incarcera-
tion works against the defendant. However, the effect of race reverses
for defendants in jail awaing trial. Whether this occurs because of ra-
cial discrimination in favor of blaCks, or of class discrimination against
higher income blacks, or of racial discrimination against blacks--by
incarceration of "good risk" blacks while their white counterparts await
trial at liberty--must await further research.

The empirical results on the outcome function tend to support the

premises of the theory--for example, that outcome level rises with the

Judith Ann La chman

relative strength of the prosecutor--as well as to affirm the outcome
function itself. The search for a point of discontinuity in the outcome
function, anticipated because of the defendant's supposed specializa-
tion either in trial improvement or in consumption—-yields positive re—
sults for all variants of the outcome function estimated. In spite of the
roughness of the data, and of the somewhat precarious manipulations
made in deriving the level of disposable resources for individuals, the
results obtained tend to link higher income figures with lower plea sen-
tence maximums. This may be reflective of a tie between defendant
resources and plea outcomes or it may be ghosting a relationship which
is more complex. It is possible that this variable serves as a proxy for
the social and economic amenities of the various neighborhoods in which
the defendants live and from which the aggregate Census data was drawn.
In addition, this poses a number of questions about the interrelation-
ships between income distribution and criminal activity.

Finally, in estimates of the resulting outcome function, the split
of the Pareto optimal set is found consistently to be one favoring the

prosecutor, in the ratio of seven to three. Although defendants receive

sizable reductions in charge for pleading guilty, when we consider the
amount that is actually "up for grabs, " that is, the Pareto optimal set,
the prosecutor faces relatively better in plea negotations. Comparing
this finding with the fact that on average, potential sentences are re-
duced to about 38 percent of their original levels, would seriously raise
the question of prosecutor "over-charging" in the prebargaining stages

of the criminal justice process.

AN ECONOMIC MODEL OF
PLEA BARGAINING IN THE
CRIMINAL COURT SYSTEM

By

Judith Ann Lachman

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Economics

1975

Copyright by
JUDITH ANN LACHMAN
1975

ACKNOWLEDGMENTS

I am deeply indebted to Professor William Haley, the chairman
of my dissertation committee. To take on the job required courage; to
pursue it required not only professional expertise, but also much
patience and perseverance; and to execute it so well required action
above and beyond the call of duty, the sharing of creativity, curiosity,
and most insightful criticism. A debt of this sort cannot be adequately
acknowledged, much less repaid; it can only be passed along to others--
surely an awesome task--together with the encouragement and support so
generously given to me.

The contributions of Professors Daniel Saks and Jan Kmenta
to the development of both the dissertation and its author were
vital ones too numerous to cite. These follow a long procession of
gifts to me of their intellects, time, and enthusiasm spanning my
graduate school career.

Professor Zoltan Ferency's resourcefulness and advice concerning
the criminal justice institutions and law were invaluable both in the
formulation of the theory and in the gathering and use of the data. He
was always most willing to answer the often naive questions of a person
uninitiated in the ways of criminal law.

My conversations and classes with Professor Warren Samuels
constituted mental journies into many realms of economics; their legacy

was a new set of intellectual tools, a renewed addiction to economics,

iii

and a belief in the area of law and economics as a new territory to be
explored.

My colleagues Stewart Hodges, Rich Cohn, and Franco Modigliani
at M.I.T. shared with me their reactions and criticisms, and offered
much encouragement along the way. I also benefitted greatly from com-
ments offered at sessions of the Labor Seminar and the Economics and
Law Seminar during my year at the Sloan School.

Data concerning Recorder's Court were made available to me
through the soaperation of Presiding Judge John Murphy, Prosecutors
William Cahalan, Dominick Carnovali, John McConnell, and Peter
Bartholomew of Recorder's Court and the Wayne County Budget Office.
Donald McIntyre and other personnel of the American Bar Foundation
were most helpful in answering my questions and allowing me the use
of their materials from the 1958 Study of Criminal Justice Systems
and other items in the Foundation library. The assistance given by
these persons was indiSpensable.

The art of persuasion, as applied to computers, was exercised
and taught by my comrades-in-study Earl Knudson and Jan Palmer, who
also lent their support and interest to the fruits of those labors.

The path to graduate study in economics has been a somewhat
devious one. Much responsibility for my switch into economics
belongs with the late Charles Ferguson, whose theory classes sparked my
enthusiasm, and whose interest, persistence, and moral support made
the change possible. I received much on-the-job training while
working for the Bureau of Research at the State of Michigan; to my

colleagues and "teachers" there-~Drs. Vernon Smith, Gary Appel,

iv

Bruce Stuart, Leland Hall and to Ronald Fine--I owe much in return. At
that same time, classes and talks with Professor Milton Taylor provided
a critical link with my academic endeavors. During my third year, the
Center for Urban Affairs at MSU provided financial support as well as
opportunities for teaching, research, and the general interchange of
ideas.

In truth, the path to graduate study in economics-~or at least, to
somewhere--began much longer ago, and the persons who initiated it have
been with me all along the way. To my parents--who have tolerated one
crazy project after another, who have even been accomplices in some,
who have shared in my joys and embraced me even in my failures-~I bear
a debt of much kindness and love.

Responsibility for remaining errors is all my own.

TABLE OF CONTENTS

List of Tables
List of Figures

Glossary of Symbols

1. Introduction

2. A Mbdel of Plea Bargaining: A Theory of the Prebargaining
Posture of Prosecution and Defense, and the Outcome of
Negotiations
2.1 The Problem of Indeterminacy
2.2 The Prosecution
2.3 The Defense
2.4 The Outcome of Bargaining

3. A Mbdel of Plea Bargaining: Empirical Testing and Results
3.1 The Transition to an Empirically Testable Model
3.2 The Data and Its Discontents

3.3 The Disposition

4. Conclusions

Appendices
A.l Data Summary Tables
A.2 Selected Scatter Plots

A.3 Additional Regression Results

References

vii

ix

xi

10

28

58

84

92

99

122

134

141

149

153

157

10.
11.
12.
13.
14.
15.
16.

17.

LIST OF TABLES

Distribution of defendants by type of charge

Distributign of defendants by length of maximum
sentence (v)

Estimation of the outcome function (with no constant
term)

Estimation of outcome function using the difference
between the maximum sentence for the guilty plea of—
fense and the maximum for attempt on the original
charge as the dependent variable.

Estimation of outcome function with constant included
and guilty plea maximum as dependent variable.

PrOportion of cases in left-hand regime for different
cuts of the domain

Recorders Court Judges: Statistics derived from 1972‘
Annual Report

Original offenses charged: years maximum sentence

Guilty plea offenses: years maximum sentence

Jail status of defendants

Race of defendatns

Race and jail: interaction variable

Time until pretrial conference and total time until trial
Age of defendants

Defendants having other cases pending

Relationship of defendant and complainant

Previous arrests

vii

106

107

125

126

129

131

141

142

143

144

144

. 145

145
146
147
148

148

18.

19.

20.

21.

Estimation of relationship of relative strength
(probability of conviction) to characteristics of the
defendant and of the offense

Estimation of relationship of relative strength
(probability of conviction) to characteristics of

the defendant and of the offense. "Two-plus" defendants
are included.

Estimation of guilty plea maximum sentence with resources
of both parties held constant

Estimates of differences between guilty plea and attempt

charge maximum sentences, with resources of both parties
held constant

viii

153

154

155

156

9.

10.

11.

12.
13.
14.
15.

16.

17.
18.

19.

LIST OF FIGURES

Title
An exchange relationship: utility maps and outcome
possibilities
Reservation prices as functions of relative strength

The effect of alternative values of r , the relative
strength of D to P

Prosecutor equilibrium, taking Case V to trial
Effect of change in disposable resources R
Time delay effects

Positions of F and G for low values of v
Positions of F and G: the v-intercepts

Prosecutor equilibrium on trial and guilty plea
opportunity sets

Prosecutor indifferent between the modes of case resolution

Defendant's trial constraint under alternative values
of r"

Defendant specialization in goods v* or c

Defendant non-specialization

Defendant specialization in case improvement; in consumption
Defendant's trial and plea opportunity sets

Possible relationships of trial and guilty plea opportunity
sets

The adjusted guilty plea constraint
Case 3 and movement from Case 3 by increasing r0"

Increases in to" lead from Case 3 to Case 2 to Case 1

ix

13

14

33

34

35

36

40

41

42

44

64
64
65
65

67

69
71
72

73

20.

21.
22.
23.
24°

25.

26.

27.

28.

29.

30.

31.

32.

Title

The defendant's switch function in terms of V* for r" §_r"

m,

The defendant's switch function, with V*m a function of r"
The defendant's switch function, with Vb as a function of r"
Specialization in V* and possible reversal with time passage
The Pareto optimal set and other regions

Switch functions and outcomes resulting from differences
among defendants

Leftward shift in outcome function due to greater prosecution
resources

Alternative paths for a case, and distribution of cases by
mode of resolution

Example of a discontinuous function estimating well a
continuous one

White defendants: relative strength and guilty plea maximum

Nonwhite defendants: relative strength and guilty plea
maximum

Defendants with arrest records: relative strength and guilty
plea maximum

Defendants whose charges were reduced to a misdemeanor:
relative strength and guilty plea maximum

75
75
82
84.

87
89
102
132

149

150
151

152

SYMBOL

D

P

CC

81, 82, 8,; Sq
K19 K2: K3; Kg
1, II

r(S)

“I
(a1(ra),-aII(ra))

Yo
U(X,7)

U*(X)

Case V

Case X

v,x

UIU(v,x)

GLOSSARY OF SYMBOLS

REPRESENTS

 

SECTION

FIRST APPEARING

 

The defense

The prosecution

Contract curve

Initial endowments of goods
Subsets of contract curve
Parties to exchange

Relative strength of parties as
function of initial endowment

Apples owned by party j

Bargaining (Pareto optimal)
interval corresponding to Sc

Object of distributive bargaining
(Constant) level of all other goods

Utility as a function of the
amounts of x and y

Incremental utility function

The case P and D are both
concerned with

A composite of all the cases other
than V in the prosecutor's
portfolio

Outcome of cases v,x, e.g., sentence
measured in years

Prosecutor's utility function

xi

2.1
2.1
2.1

2.1

2.1
2.1

2.1

2.1

2.1

2.1
2.1

2.2

2.2

2.2

2.2

F(v,x,R)-0

rl

e

di
r'(e,d1,d2,...,dn)
Fl

(vo(r').xo(r')) and U0

R'IR'(R)

G!

REPRESENTS
Costs per day of maintaining cases
Vand X respectively

Number of days cases await
disposition

Costs of executing a trial for case
V; for Case X

Total budget of prosecutor
Fixed costs of prosecutor's office

Disposable resources of P
(assuming case V goes to trial)

Trial opportunity set, stated as an
implicit function

Relative strength of D to P, as
percieved by the prosecutor;
larger values of r' reflect a
stronger case for the defense

Initial endowment of evidence

Factors relating to the defendant,
other than e, i=1,...n

Relative strength of D to P stated
as a function of e and the di

The trial opportunity set, with r'
assuming a higher value than before

Solutions for utility maximization
of P on trial opportunity set

Time until prosecution of case V is
dropped by P

Time until case is resolved by a
plea of guilty

Opportunity set when case V is
resolved by a plea of guilty

Disposable resources when case V is
resolved by plea of guilty

The guilty plea constraint with

alternative values of one or more
of its parameters

xii

SECTION

2.2

2.2

2.2
2.2

2.2

2.2

2.2

2.2
2.2

2.2

2.2

2.2

2.2

2.2

2.2

2.2

2.2

2.2

2.2

SYMBOL
u!
0
VPDXP
r!
O
r.
m
V
In
xln
ICVghzo
(t' a V)
vm ‘ Sph")
C1. ..’3
k
Mva “x; M
w
.12
M'
c

REPRESENTS m

Utility‘ of P in equilibrium on 2.2

the guilty plea

Values of v, 3x obtained when P 2.2

maximizes his utility subject
to G ‘

The initial level of r' on the 2.2

trial constraint

Thealevel of r' exercised by D 2.2
at which P would be indifferent
between a plea of guilty and
going to trial

Value8.of V, X obtained via plea 2.2

of guilty, for which P is indiffer-
ent between a plea of guilty and

trial resolution

The indifference curve (at utility 2.2
level U ) stated in implicit form
0
Switch point (where W; depends upon 2.2
'
ro
The prosecutor's switch function 2.2
For example given, parameters of 2.2
P's utility function
Counterpart for case}! of r' 2.2
Amounts of variable input M to 2.2
production by P or a strong case
2.2

Price per unit of M

Amount of good M equal to R/w (trial) 2.2

Amount of good M equal to R/w (guilty 2.2
plea)

All other goods to which the defen— 2.3

dant can allocate his resources,
other than toward improvement of

his case

xiii

SYMBOL
vs
vs
I?

REPRESENTS

The good V redefined to be a
desirable good

Improvement in v relative to
some benchmark

A benchmark value of v, for
example, the maximum value
for D

The, defendant's total resources,
if going to trial

Wealth (or initial level of re-
sources)

D's income not dependent upon
jail status

Income; dependent upon jail status

Jail, status: J-l if D in jail, 0
otherwise

Defendant's fixed costs of trial
Defendant's time-dependent costs

Defendant's disposable resources
for trial

The. defendant's trial opportunity
set, stated in implicit form

Relative- strength of P to D as
viewed by defendant

The; function f with alternative
parameter values

c-intercept of f
v*-intercept of f

v*—intercept of f'

xiv

sew

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3
2.3

2.3

2.3

2.3

2.3

2.3
2.3

2.3

SYMBOL

r'.
P romise

I."

r.‘
‘leesc

REPRESENTS

 

Utility of defendant, .(0,c )

specializing in consumptign
(on trial constraint)

Utility of defendant, .(V3,0)
Specializing in case-improvement

(on trial constraint)

D on f

58!:

Opportunity set

c-intercept of g

0

r érgisc)

N
r rdisc)

variable

XV

Maximum of {61, u'7-

0

Maximum. of {ul, “2}
0 o

The: value of r" such that u dull-I12;
either type of specializaEioa
will yield same maximum utility to

Defendant'ss plea bargain Opportunity

D's. disposable resources on guilty

} =3 “'11
0

Utility from c-specialization on 3
Utility from v*-specialization on f

Level. of r" exercised by P, which
is promised to be less than r3

Level. of r" at which D is indifferent
between modes of case resolution

Value of r" at which u -u(')

0

Portion of D's switch function (for

Portion of D's switch function (for

The entire switch function for D

The restatement of r' and r" as one

SECTION

2.3

2.3

2.3

2.3

2.3

2.3

2.4

SYMBOL REPRESENTS SECTION

 

S (r) New notation for the prosecutor's 2.4
p switch function
U*(v ) PTQSGCUtOI'S incremental utility 2.4
o m function
u*(vm) Defendant's, incremental utility 2.4
o function
?\;L-h WeiShts_ accorded the P's and D's 2.4
switch functions, respectively
fb,fw Outcome functions of a defendant, 2.4
compared by race possibilities
(in figure 25)
y Y =- 1 for r"_€ rdisc and X= O 3.1
otherwise
hp(r), hd(r) Functions which permit both switch 3.1

functions to be stated in terms
of a single variable r for relative
strength of the parties

iv

Sum of sentences on all pending 3.2
j-l OJ charges (n is number of charges)
21 Characteristics of the defendant 3.2

and of the offense (previously
labeled "e" and "d1")

q(zl, ..., 2d) A linear function of the 2 used 3.2
in estimating the relati%e
strength variable

r Probability_ of conviction on original 3.2
p offense, as estimated by prosecutors
xl Term from P's switch function without 3.3

h coefficient

x2 First; term of D's switch function, 3.3
without (l-7I) coefficient

x3 Second term (which includes dummy 3.3
variable for regime) of D's

xxr‘l

SYMBOL

81.32.83

ij, 811, X33

REPRESENTS

Coefficients. of xl,x2,x3,
respectively

The- terms v , a , x3 multiplied
by the regression regime dummy
variable xj

H- H
o rdisc

Cuts. in regression regime,
J'l. see, 10

Dummy 'variables for parts of
regression regime, j-l, ..., 10

xvii

SECTION

3.3

3.3

3.3

3.3

3.3

1. Introduction

 

In contrast to popular belief and to the Constitutional right to
trial by jury, the large majority of criminal cases in the United States
are settled by prosecution and defense without trial.1 Although clothed
in controversy during occasional law school discussions and media ti-
rades, the practice of plea bargaining has elicited little attention
outside those circles and has continued to be the common mode of case
resolution.

Legislation and judicial rulemaking have rarely addressed the
issues of the controversy until quite recently,2 and then primarily to
give official approval to ongoing practice, rather than to actually
create the institution in law. Plea bargaining has been--and it ap‘
pears, will remain--a highly informal proceeding, allowing the parties
great latitude both in structbring the negotiation process3 and in de-
termining an acceptable outcome.

In both its pre-bargaining posture and in its behavior during
the course of the negotiations, each party pursues a special manifes-
tation of its own self—interest. The defendant.(D) seeks to minimize
his expected loss resulting from the disposition of his case (time in
prison and other forms of loss) and so he considers entering a plea of
guilty in return for the dropping or reduction of some of the charges
against him.“ The prosecutor (P) attempts to maximize the total convic-
tions among his cases,5 together with his goals of disposing of all

cases in the docket, and taking into consideration both the seriousness
1

of the charges and any other factors relevant to his priority judge—
ment about them.6 Although sacrificing the possibility of obtaining

a more favorable outcome at trial, each party via pretrial settlement
eliminates the risk of losing at trial, and frees his own time and ad-
ditional resources for other purposes.

Both the prebargaining posture of the parties and the outcome
of negotiation can be described as the results of economic processes in
which individuals make rational choices among alternative courses of
action on the bases of anticipated utility gains or losses. In the
following pages a model is constructed which derives the prebargaining
stance of each of the parties, and states the set of possible guilty
plea settlements as a function of the relative strength of the prose-
cution and defense. The model is then tested using data on individual
defendants in felony cases, with the resulting estimates serving both
to affirm the model in a general sense, and to indicate the degree to
which the parties are able to extract from one another the reservation
price of a guilty plea.

' This chapter serves to introduce some of the issues toward which
the theory is directed and provides a sketch of the court system as
depicted in the economics literature.

The second chapter is devoted to the development of a theory of
plea bargaining: First the problem of indeterminate outcomes--the
legacy of bilateral monopoly problems--is discussed, and a rationale
for adoption of a split-the-difference theory is proposed for the bar-
gaining problem in the context of the court system. In the following

two sections, the choices of prosecutor and defendant are placed in the

traditional framework of utility theory, with each party allocating its
resources between improvement of case outcome on the one hand, and all
remaining goods on the other. For the defendant, the remaining goods
are cOnsumption goods, savings, and other uses of resources, and for
the prosecutor the remaining goods are the outcomes on the rest of his
caseload.

Each party chooses which of two alternative constraints to ob-
serve, where trial defines one of the constraints and a plea of guilty
defines the other. However, if one party initially prefers. the trial
option, the second party can entice him to bargain by adjusting the
"price" of a guilty plea-~by agreeing not to exercise the full strength
of his case in court--so as to make the first party indifferent between
the two alternatives. Between the concessions necessary to make ac-
ceptable a plea Of guilty are the set of pareto Optimal outcomes.

The determination of this set and the consequences of adoption of a
split-the-difference theory of negotiation are discussed in the last
section of the second chapter.

In the third chapter, the model is adapted for use of individual
case data obtained from the Recorder's Court of Detroit. The first
section outlines the additional assumptions necessary to ready the model
for estimation. The following section describes the data and gives
some information about the two parties to the case as well as intro-
ducing the construction of a variable for relative strength of the par-
ties. The third section presents and discusses the results from esti-
mation of the outcome function determined earlier.

Finally, the fourth chapter summarizes and interprets the

theory and empirical findings in terms of policy considerations and

areas for further research.

Until quite recently, economists have devoted scant attention to
issues of criminal justice, and within the new and growing literature
on the economics of criminal justice, almost no research has been done
on plea bargaining and related questions of court management.7 One ma—
jor exception to the latter generalization is William Landes's seminal
article, "An Economic Analysis of the Courts," (1971) which is discussed
below.

In beginning his paper, Landes promises first to identify the
variables which relate to the choice made by P and D between a pretrial
settlement (plea agreement) and trial. He then sets out to show that
"the decision to settle or go to trial depends on the probability of
conviction by trial, the severity of the crime, the availability and
productivity of the prosecutor's and defendant's resources, trial ver-
sus settlement costs, and attitudes toward risk." (Landes, 1971:61)

As a first attempt at expressing decisions of actors in the court sys-
tem in terms Of economic control variables, the article is a pathbreak—
er and provides a wealth of signposts for future research, some of
which Landes has followed himself (1973). However, as a model of the
trial-versus-guilty plea decision, his work is less convincing than if
it relied upon a more realistic view of the legal institutions and the
traditions surrounding them.

One major weakness in Landes's conception of these insti-

tutions is his perception of the relationship of the negotiated

settlement to the full trial, where the relationship involves not only
the order of events in time, but also the role of the trial option in the
the determination of the bargained outcome. In his model, Landes views
the bargained outcome as entirely separate from the trial-versus—guilty
plea decision, and he creates a model in which the latter decision is
made first, independent of the result of the former process; that is,
Landes's prosecutors and defendants decide first on whether to go to
trial, and then later decide the conditions of settlement. This is
analogous to a model of labor negotiations in which the parties first
make an irrevocable decision on whether to have any strike associated
with the new contract and then, if they happen to decide against striking,
sit down to work out the terms of the contract.8

This conception of the trial decision has several crucial
defects. First, as the comparison with labor negotiations suggested,
Landes's model fails to recognize the intimate relationship of the con-
flict situation, trial, to the attainment of the negotiated agreement,
the plea of guilty: the former indicates failure to achieve agreement.
This last idea embraces two critical components, namely, that conflict
signals a form of failure, and that conflict follows attempts at non-
conflict settlement, rather than the other way around. In addition,
Landes's conception fails to recognize the importance of the ability to
choose the trial course of resolution--at least the ability to use the
threat of going to trial—-as a source of power in bargaining; for with-
out the ability to reject a bargain proposal of the other party (in the
world of an Edgeworth box, the ability simply not to exchange goods)

neither party could induce the other to accept any proposal it might

make, and both could remain intransigent at the levels of their initial
demands.9 However, added to problems involving the logic of this assump-
tion of the Landes model is the fact that the actors and the institutions
which constitute their stage simply do not relate to one another in the way
Landes describes. With the exception of a very few highly publicized
cases, every case is viewed by a prosecutor as a potential plea bargain;
and with the exception of a very few defendants, the decision about
whether to go to trial results not from a simple decision to plead guilty
because it is cheap, but from a comparison between the outcome
anticipated from trial and the best negotiated outcome he could

wrangle from negotiations. This sparring for position frequently
continues until the very mOment that a trial would Otherwise begin;

only after agreement has been consummated or finally rejected has

the trial decision actually been made.

Finally, Landes makes the assumption that whenever a Pareto op-
timal bargain exists, i.e. whenever there exists at least one such out-
come, then the parties will settle. This is to assume away the inter-
esting part of the question,10 namely, how do defendants decide to
accept or reject a bargain and consequently avoid or participate in a
trial? Because of the assumptions made and the underlying conception
of the institutions involved, the Landes model is of limited assistance
here in constructing a model of plea bargaining. '

The view of the institutions and of the process of plea bargain-
ing which are assumed in this dissertation are, briefly, as follows.
Having gone through the preceding stages of the criminal justice

machinery--arrest, investigation, and charging——a case enters the

prosecutorial stage, where the trial versus plea bargain choice is made
by the parties. Although institutional practice variesfrom one court
to another,11 at some time while the case rests in the prosecutor's
Office the opportunity for pretrial bargaining arises. In the past
this has frequently been carried out in hurried whispers before the
commencement of trial. Now, in some courts such as in Detroit, the
practice has been to institutionalize the bargaining session by arrang-
ing a pretrial conference as part of the normal processing of a case. In
any event, it is fair to say that most any defendant in a large urban
court has the opportunity to enter into negotiations if he wishes to

consider pleading guilty.

Although some bargaining sessions are characterized by "hard-
line" bargaining and the exchange of specific offers from one side or
another, many are marked by a great deal of information-exchanging,
which serves to bring about a consensus or the determination of a small
interval within which most of the offers will reside. Although the
trial Option always looms in the background, discussion is centered on
the special factors which mitigate the desirability of a harsh sentence
(such as would be possible at trial on the original charge) and on the
consequences of pleas to alternative charges.

Thus, once much of the preliminary information has been exchanged,
the problem is essentially a search for the set of pareto optimal out-
comes and then,via negotiation, a resolution within that interval. The
theory of plea bargaining presented here rests upon this conception of
the process by which a pretrial settlement is made,or rejected in favor

of going to trial.

8

NOTES TO CHAPTER 1

1Estimates on the prOportion of guilty pleas taken in criminal
courts vary from 95 percent to 69 percent, according to McIntyre.
(1967:132). Newman (1966) reviews some of the studies producing
those figures, and Morse and Beattie (1932) provide one example.

2For examples of these official stamps of approval, see
People v. West (1970), Santobello v. New York (1971), and McCoy v.
U.S. (1966).

3This is especially true for the prosecutor. As part of the
executive branch in the criminal justice system, he can exercise
discretion in the administration of guilty plea negotiations, subject
to legal and political constraints. The concept and justification of
prosecutorial discretion are discussed briefly in People v. West
(1970) and in Remington (1969, 1972).

 

4The Newman article (1956) and book (1966) list and compare
various modes of bargaining for pleas and the alternative mediums of
exchange.

5These conditions placed on the utility function are justified
by two considerations: Upon assuming his position, a prosecutor
pledges "...not to convict, but to see that justice is done." Never-
theless, his conviction rate is frequently a primary topic of dis-
cussion during election campaigns, with a low one Often cited by the
Opposition as evidence of a "soft" or "ineffective" job having been
done by an incumbent.

A second reason for believing the prosecutor's utility to be
as indicated is the reliance of the U.S. justice process upon the
adversary system for the presentation of evidence and its interpre-
tation. Such a system requires that a prosecutor scale a role
barrier in order to introduce information which would tend to favor
the defense at trial. The adversary model of the criminal justice
system is discussed in more detail by Klonoski and Mendelsohn
(1970).

6For example, the deterioration of evidence over time might
impose time-varying preferences for the sequencing of cases through
time.

7What has been done so far appears primarily in a new
publication, the Journal of Legal Studies and in issues of the
J.P.E. during the last four years.

 

8A8 another example, Landes's conception is similar to a belief
that nations at war first end the conflict and then work out the terms
of the treaty.

9This would be similar to the assumptions made (though not of
specific institutions, nor of the relation of conflict to negotiated
agreement) by John Cross (1969) in his model of the bargaining process.

10There should always existzaPareto optimal outcome so long as one

assumes the linear utility functions necessary for use of the von
Neumann-Morganstern index (which Landes utilizes) because both parties
save the money which would otherwise go toward the preparation for
trial--and after-trial resources are the only consideration to Landes
defendants and prosecutors. The outcome anticipated from trial, plus

or minus quite a bit, would enclose an interval of Pareto optimal
outcomes. ‘

11Institutional variation is discussed in section 3.2.

2. A Model of Plea Bargaining: The Pre-bargaining Posture of

Prosecution and Defense and the Outcome of Negotiations

 

2.1.The Problem of Indeterminacy

 

Within the body of microeconomic theory there arise a number of
situations--most notably those involving bilateral monopoly or Oligo-
poly-in which one is unable to prescribe a determinate solution to
the outcome of an exchange process. Until the appearance of the early
bargaining theories in the 1930's, the furthest an economist would take
the analysis of such a relationship was to set boundaries on the out-
come, based upon the outcome each party would have accepted in the com-
petitive case, and close with a statement that although economic
theory can suggest what the demands of the parties will be, the final
outcome depends upon the strength of the bargaining parties, the latter
being an issue outside the scOpe of economics.1

Such a statement is lacking in two important respects. First,
as evidenced by the newly-burgeoning literature on bargaining contri-
buted by economists, it is possible and indeed desirable that the dis-
cipline of economics countenance such issues. Second, the statement
is misleading in its implication that, should other social sciences be
able to furnish an economist with information on or quantification 0f
the strength Of the two parties, the economist would then be able to
determine the outcome, i.e. that an evaluation of relative strength

would be sufficient in the way of additional information to transform

lO

11

an indeterminate situation into a determinate one.

These two difficulties became intertwined as most of the bar-
gaining theorists set off in search of some measure of relative strength
which could be. defined wholly in economic terms .2 This new
variable could then be imposed upon the less-indeterminate-than-before
"solution"--usua11y conceived of as an interval within which the out-
come would lay --and yield an answer to the problem. The bargaining
theorists essentially agreed with the earlier microeconomists, that the
outcome of bargaining situations is determinate once a variable for
relative strength is included; they merely disagreed as to which dis-
cipline should undertake the awesome task of measuring that variable.
It is crucial to note that (l) with few exceptions, the bargaining
theorists considered the model complete and determinate once a variable
for relative strength had been included, i.e. whereas the microeconow
mists had considered the determination of the final outcome (within the
previously determined bounds) to be entirely outside the scope of eco-
nomics, the bargaining theorists considered it to be entirely within;
and (2) the bargaining theorists did not perceive any problem of simul-
taneity in the determination of the bounds on the outcome and the rela-
tive strength of the parties. Both of these perceptions are evidenced
in the development of the prototype theory: the existence of an inter-
val Of indeterminacy is presumed, with bounds already set by recourse
to the competitive model; and then a measure of relative strength is
introduced as rather a d33§_g§_machina, resolving the dispute within
the given interval?

The natural result of such a conception of the bargaining

12

problem was a belief that if indeterminacy persisted even after the
introduction of the strength variable, then one or another of the as-
sumptions of the augmented model must not have been satisfied. Thus we
see Hicks (1963), Foldes (1964), Cross (1969), and others inves-
tigating the effects of insufficient information on bargaining. Other
economists explored the possibility that the negotiators are not op-
timizing their supposed objective functions, but are, perhaps, torn by
differences of goals within their ranks--as Ashenfelter and Johnson
(1969) assume for labor unions--or, as posited by deMenil (1971) that
the two parties are not engaged in distributive bargaining pure and
simple but rather experience a positive sort of interdependence such
as the vested interest of a labor union in the profits of the company.
In other words, the response to the persistence of indeterminacy of
outcome in.the augmented model . was not to question whether the model
was in fact determinate, but to search for reasons that the model
might be inapplicable.

The pursuit of inapplicability was quite successful, and as a

result, as noted by Hamermesh (1973), "Bargaining theory contains very

few interesting propositions that can be tested empirically." This is

due both to the restrictive assumptions eventually required (in an area
meant to complement the restrictive assumptions of the competitive
model!) for imposition of the bargaining theories and to the kinds of
predictions the theories typically yielded, i.e. they frequently did
not specify outcome in terms of the units of the good in question.
Since some of the empirical work using bargaining theory will be

discussed in a later chapter, we return now to the question of

-13-

indeterminacy within the framework of the model.

Although it is conceivable that one would encounter a bargaining
situation in which the strength of the parties is independent of the
other variables within the modelf' ’ we would expect this to be the ex—
ceptional occurrence. For the belief in the universality of that situ-
ation is equivalent to acceptance of the proposition that regardless of
initial endowments of the bargain good or of input factors, regardless
of the forms of the objective functions, and regardless of the levels
of other variables in the theory, the relative strength of the parties
remains unaffected at its initially-determined level; and conversely,
that a given level of strength is in no particular way associated with

the existence or size (however measured) 6f the bargaining zone.

In order to discuss this issue in a bit more specific a setting,

consider the set of possible exchange relationships depicted by the

Edgeworth box below:

 

 

 

 

 

<———11's apples II
S __. C m
S +J
3 ‘ . / 3
I 2 -‘l‘_ K32! c
,\\ .
m ‘:;V In
‘5 53.731. \» - r.
c / \ H
/
n
L c _ it
I I'fls apples -——-)-

Figure 1. An exchange relationship: utility maps and outcome possi-
bilities.

where K1, R2, and K3 are the subsets of the contract curve CC correspond-

ing to initial endowments 81’ $2, and 33, respectively, and the utility

-14

functions of I and II are represented in the usual fashion. If we were
to ask an unsuspecting subject to compare the relative strength of the

parties at 82 versus at 33, we should not be surprised to receive the

reply that I is relatively stronger at 83 than he is at 82. To give an

exaggerated example of this, we might assume that at S both parties

2

have enough food to subsist, while at S I considers additional food

3.
of either variety to be a luxury, and II is on the verge of starvation.
If we define r(S) to be the relative strength of the parties, with r
increasing with I's strength, we could restate our subject's conclu-
sion as r(S3)) r(52).

If we take any line having a nonnegative slope (with respect to
either I's or 11's origins) we can order the endowment—strength variable
r by the ordering of the corresponding values of one of the goods. For

emmple, if we draw the line determined by S and S3, we have ar/baI

2

(where a represents apples) greater than zero, and in addition, for each
g‘ point on the line, we can determine a corresponding exchange interval

K‘g and thus additionally create a correspondence of r-values with

Pareto-optimal intervals:

‘aII(r3)

 

 

 

r2=r(82) r3=r(S3)
relative strength-—o

Figure 2. Reservation prices as functions of relative strength.

15

where ( ai(nx)”a11(‘¥9 ) represents the bargaining interval correspond—
ing to 3‘; Note that when the set of goods to be divided consists of

only one good, the line-ordering of the Sthas unambiguous meaning, and

S“. is measured in units of the good under discussion.5

Such a sketch serves to underline some basic results of the
Edgeworth- Box construct. First, it reiterates the fact that for a
given level of relative strength--at least to the extent that it relates
to endowments--there corresponds not one, but many possible bargain I
outcomes. Within the interval corresponding to a particular level of
r, we would expect the final outcome to be a function of a number of'
variables, at least some of them relating to factors outSide the scope
of economics, such as the interpersonal skills and experience of the
bargainers. We would consequently anticipate that any deterministic
theory proposed by an economist would either take such considerations
into account directly by including relevant variables or some general
index to represent them; account for them indirectly by suggesting some
approximation to outcomes which is sensitive to changes in the under-
lying behavioral variables; or content himself, with the aid of vari-
ables for which he does have information, to perform the valuable task
of narrowing the interval of indeterminacy?

Second, we note that a given bargain outcome can correspond to
many levels of relative strength (although we generally expect a posi-
tive relationship,) and thus allow for the possibility of attaining an
acceptable bargain even though the parties have incorrect or insuffi-
cient information, an occurrence which surely arises in the real world.

For example, if I overestimates his strength at r3 and also drives a poor

16

bargain compared to what he feels he could; and if II correctly
estimates the relative strength to be r2, they can both settle

happily at outcome a in Spite of their differing estimates.

I
It is also curious—-but consistent with the results of the Siegel
and Fouraker (1960) bargaining experiments--that under some
circumstances such as those of the illustration just given,
increasing the information available to one or both of the parties may
actually widen the bargaining interval and result in greater uncertainty
of outcome, as well as possibly yielding a less favorable outcome to
the party who was previously the more ignorant of the two!

As mentioned earlier, the deterministic bargaining theories ad-
dress themselves to the distribution of an amount of a good within an
already-specified Pareto-optimal interval. For the examples given a-
bove, the alternative to this distribution by means of negotiation was
simply the failure to redistribute. In other bargaining situations,
the "conflict" or nonnegotiated outcome may have associated with it
various costs, outcomes, and levels of uncertainty relating to both
costs and outcomes. Although the questions of relative strength and
determination of bounds on outcomes remain the same, the problem of
measuring these factors may become considerably more complicated. A
number of bargaining theorists allow for this in shorthand fasion by the
use of what we shall call an incremental utility function.

Very simply stated, the incremental utility function is a func-
tion of one variable--the object of the distributive bargaining, x--

formed by stating the utility function as a function of x when x is

l7

attained via bargaining (i.e. not by conflict) and subtracting the util-
ity value associated with conflict, that 15:

U*(X) = U(X.yo) - U(xo.yo)

where x is the outcome of negotiations, yo represents the (constant)
level of all other arguments of the utility function, U(xo,yo) is the

*
utility associated with conflict outcome x0, and U (x) denotes the

incremental utility function. The latter (as well as the former) term
may be difficult to summarize in a single dimension, but that is a
necessary step. The incremental utility function is then a one-dimen—
sional measure of the utility of x minus the constant utility value
associated with trial.

Different theorists perform a variety of permutations on these
incremental utility functions and the values they take on for particular
bargain alternatives. For example, Nash (1950) predicts that, subject
to a set of restrictive assumptions concerning the incremental utility
functions, the outcome will be the distribution of x which maximizes
the product of the incremental utility functions of the parties. Zeu-
then (1930) hangs the fate of the parties of a pair of "risk willing-
ness" expressions (later adapted by Pen, 1952) which Harsanyi has
shown to yield the Nash outcome (1956). Cross (1969) has shown that
the Cartter (1959) expressions for "bargaining attitude" are simply the
inverses of Zeuthen's risk-willingness expressions, and hence yield the
Nash outcome.

These theories share two common deficiencies: first, they fail to
suggest why or how the parties could be expected ever to arrive at the

predicted outcome point,7 i.e. they give no hint as to the process or

18

dynamic involved in reaching consensus at that particular point; and
second, they give the potential user of their theory no help in the dif-
ficult job of restructuring a specific bargaining problem in a manner
amenable to application of the theory, particularly the task of develop-
ing the incremental utility functions.8 Their solution to the bargain-

ing problem, if not useless, is at best unsatisfying.

Several other theorists have suggested rather mechanistic resolu-
tions to the bargaining problem. For example, Cross (1965, 1969) posits
learning behavior on the part of the bargainers; but due to his assump-
tion that both parties are completely intransigent in their required
outcomes, his theory eventually rests on the contradictory9 pair of
axioms: (1) Although each party expects the other to accede completely
to its demands in due course, it (2) then revises its expectations in
response to moves by its opponent (but then repeats the' naive assump—
tion that its new demand will be completely fulfilled). Bental and
Comay (1973) suggest an adaptive expectations model which is a bit
more helpful than Cross's, butrequires that the parties have a great
deal of information at hand.

To summarize, each theory of bargain outcome discussed in the
preceding pages bears one or more of the following deficiencies:

(l) The theory neither relies upon nor is necessarily consonant with
any concept of the dynamics of negotiation}0 a criticism most readily
leveled at the early (static) deterministic theorists; (2) The theory
describes the process of negotiation quite well, but fails to predict
outcome--just the opposite of the first deficiency--a problem especial-

ly common to the behavioral theorists and prOponents of commitment stra-

l9

tegies; (3) The theory is able to predict outcome and recognizes the

stepwise nature of the bargaining process, but does not adequately ac-

count for the interaction between the parties, escaping that problem

by effectively assuming away the two-party character of the bargaining.

Since most of the empirical work to date has been tests of this third
set of theories, the deficiency is particularly disappointing.

It appears that most of these theories have limited usefulness
as descriptive models of the bargaining process, and consequently of
plea bargaining. We are left with what might be called a basic indeter-
minacy: we would expect that in most bargaining situations, there is
the possibility that two parties, both reasonably well-informed and
rational, might disagree ex_§gtg_on the outcome of their negotiations.
As pointed out by both Bishop (1964: 412) and Cross (1969: 33) (al—
though in different contexts,) were that not true, we would find our-
selves observing few instances where bargaining took place at all--
for both parties would immediately accept the inevitable outcome and
skip the bargaining.

An alternative approach to the bargaining problem which has been
largely ignored11 --perhaps because of its simplicity--is the straight-
forward maxim of splitting the difference between the reservation prices
of the two parties, where the split is broadly construed to include any
proportional split, not necessarily that of equal shares. A split-the-
difference theory of bargained outcomes is consistent with both the be—
havioral theories of negotiation and with observed practice. Such a
theory is also consonant with a concept of a negotiated outcome as the

result of an interactive process of concession, based upon three

20

principal characteristics of bargaining: (l) Bargaining is executed
by a sequence of alternating moves between the two parties; (2) the

sequence of each party's positions is a strictly monotonic one, with
the possible exception of the rejection of final offers; and (3) the
amount of one concession depends upon the size of the preceding con-

cession by the other party.

The first characteristic is simply a recognition that the even-
tual outcome is attained by a sequence of offers and counteroffers,~
each reflecting a new concession. This response pattern implies that
the outcome is unlikely to be on the boundary of the pareto optimal
zone, for that would require zero concession repeatedly by one party

or the other.12

It also implies that a commitment or "stay pat" stra-
tegy (Schelling, 1960; Saraydar, 1971b) employed by either party would
make a conflict situation almost inevitable, so that instances we observe
of negotiated settlements would tend to exclude situations in which a
commitment strategy was used.

In addition to the bargaining process involving a number of moves
by each party, "good faith" bargaining is generally considered as de-
manding some concession with each move. "Certainly, Party will consider
who made the last move and whether that move was regarded by the other
as an"adequate' move...0ne side may not be happy with the amount of
concession made by the other, but he is under considerable pressure to
respond with some concession of his own." (Walton and McKersie 1965: 87)

Finally, the concession-response pattern requires that significant

concession by one party be reciprocated by the other. That is, a certain

21

concession by one party implies a lower bound on the absolute value

of the concession offered in the next move by the other party, meaning
that there is likely to be some overall relationship between the size of
1's concession and the size of II's; for example, it may be the pattern
in negotiations between I and II for a two-unit concession from I to be
matched by a three-unit concession from II. Summed over all moves, this
response pattern would result in an outcome which is in roughly those

same prOportions.13

Although the characteristics cited above lend support to a split-
the-difference theory as applied to a specified bargain interval, it is
yet to be shown that a fixed proportional split should apply to any ar-
bitrary bargain interval, given that it.is valid for one. In fact, it
is in doing so that the elusive nature of the relative strength concept
is highlighted.

We have postulated a bargaining situation in which for a given
pareto optimal interval the proportion of the difference accorded each
party depends upon a set of behavioral and institutional factors. The
interval to which the analysis was applied is associated with some
value of r, the relative strength variable discussed earlier. The broad-
er concept of relative strength has thus been introduced as a two-
faceted concept, represented by strength as dependent upon initial posi-
tion (the endowment strength variable) and strength as exhibited by
the bargaining conventions and skill factor. These two manifestations
of strength are not independent of one another, for clearly the value

of the first determines the set of acceptable distributions of the good.

22

The bargain conventions and skill variable becomes more central to
the determination of outcome once the endowment strength aspect has set
the environment in which the skill factor enters.

Since the Split of a given Pareto optimal interval depends upon
certain traditions and upon the skills of the parties, the question is
whether those factors influence the split of the bargainable interval
differently for alternative choices of theiintervals. Since neither
bargaining conventions nor the interpersonal skills and experience of
thebargainers is dependent upon the level'of endowment strength,14 the
answer here supplied is no.

Thus the idea that the bargainable difference, whatever it is, is
divided in some proportion which is uniform for alternative values of r
would appear to be a reasonable basis for a theory of outcome. It is
this theory that is adopted in the model of plea bargaining to follow.

We now face the task of defining the region of indeterminacy in
the setting of the plea bargain-versus-trial choice, where the former
is the result of negotiation, and the latter of the failure to reach a
mutually acceptable bargain. 15 This is accomplished by developing the
analogs of the incremental utility functions for zero increment, and
allowing the relative strength of the parties to vary. For each, a
"switch function" results, defining the minimally acceptable outcome
necessary in order to induce the party to give or accept (as the case
may be) a plea of guilty. The region bounded by the switch functions--
for some ranges of relative strength, the empty set--defines the set of

possible outcomes of successful negotiations. And within this region,

23

if one is willing to posit a rule or theory of outcome, the resulting

plea bargain or trial Option choice can be predicted,

24
NOTES T0 2.1

IlFor example, Charles Ferguson writes:
Our general conclusion is that price and quantity is inde-
terminate in cases of bilateral monopoly...[T]he information
the economist has is not sufficient to determine the precise
market solution.
The additional information necessary to determine the outcome he con-
sidered to be "anterior to the realm of economic analysis." (1966:248)
(underscoring in the original.)

Pen expresses the need for an economic theory which is deter-
minate, and does so in rather stark terms(l952:25):
The term 'indeterminate' is used in this [mathematical] sense
by most economists. It then means nothing more than that
their system of equations, that is, their theory, is inadequate
to offer an explanation of the magnitudes to be examined.

A typical example of this is the 'theory of limits' of
bilateral monopoly...It is superfluous to explain once more
that the price is indeterminate within these limits; this
merely amounts to saying that the investigation has been bro-
ken off before it came to a successful conclusion.

\
3Dunlop (1944; 119), for example, classifies the factors deter-
mining bargaining power into three categories:

(1) Tastes of workers and employees with respect to wages and
manhours bought and sold, and institutional factors--
such as property rights and wage-hour legislation--which
influence demand and supply conditions;

(2) Market conditions, expecially the degree of competition
in the labor market, the product market, and the market
for complementary and competitive factors of production;

(3) Pure bargaining power: the ability to get favorable bar-
gains apart from market conditions.

4For example, suppose that a bargaining interval is determined
entirely by technical factors, that disinterested negotiators are hired
and paid on identical incentive schemes for each unit of the good above
or below the appropriate bound which is attained at bargaining, or for
obtaining a conflict outcome. Further, we must either require that each
level of relative strength be associated with bargaining intervals of
fixed length, or specify attainment in terms which adjust to the length
of the bargaining interval.

Note that otherwise it is necessary to also order S1 relative to

6Cross scoffs at this last role, since in his view, the issue
is simply one of lack of information (1969: 25, 28)

This assumption of perfect knowledge is not really a sim-
plification of the bargaining process; indeed, we take it to
be arecitation of one of its most fundamental properties. It
is precisely the lack of knowledge that permits two individ-
uals to form differing’ opinions as to how a situation which
requires agreement will turn out...An assumption of perfect
knowledge will reduce the whole process to a set of mechanical
rules from which we cannot possibly gain any insight into the
dynamic mechanism of concession, or into the process whereby
the parties formulate and change their expectations. (under-
scoring in original)

 

7For example, the Nash solution is not a solution to the bargainr .
ing problem, but merely narrows the set of possible outcomes, and does
that without reference to any dynamic or process which would cause the
parties ever to arrive on that function--or even to have a motivation to
do so. As Comay3_t__a_l_ have pointed out (1974), the Nash solution has
come to be regarded more as a normative solution than as a positive one
(thus rescuing it from the responsibility of being empirically verifi-
able). However, even this kindlier interpretation of his theory leaves
Open the question of why that solution would be viewed as preferable by
the "exchange community" of the two parties, by some ephemeral outside
world (i.e. socially desirable to a larger community, if not optimal in
the incomplete model of two-party negotiation), or to either of the par-
ties. Surely each party is maximizing his own utility, and within the
Pareto optimal region, would strive to gain an outcome on the other par-
ty's switch function (in the terminology of labor negotiations or oligo-
poly, at the reservation price of the other party) and only in the excep-
tional situation would this coincide with the Nash function. As a com-
munity of two, P and D--or any other two bargainers, for that matter--
would have little reason to maximize the product of their utility gains
from bargaining if both could be happier with an alternative distribution.
Finally, we have no hint as to why some sort of outside world might find
that solution to be socially desirable.

8The unsuspecting user of the theory may well find himself faced
with one of the following problems: (1) The incremental utility func-
tions only lend themselves to statement as implicit functions, thus
rendering the theory's required manipulations extremely cumbersome, if
not impossible; or (2) Solutions (distributions of the good which is
the objective of the bargaining) that satisfy the conditions of the
Nash outcome may not exist, or if they do exist, may not be real num-
bers. Further, the fact that for certain pairs of utility functions the
outcome must always be on the boundary of the pareto optimal interval
revives one's reservations about the role of the variable for relative
strength in the determination of that interval, and of the outcome with-
in the interval.

26

19There is some disagreement concerning whether the Cross behavior-
al assumptions are contradictory or merely paradoxical. The controversy
is exemplified in Coddington's review (1970) of the Cross book.
For whatever relief it affords, it is here noted that the assump-
tions cannot be contradictory in the first period, at least.

10This is analogous to offering a point as a market equilibrium

without clear definitions of the demand and supply functions, or with-
out a cobweb theorem offering a reasonable explanation for the equili-
brium point being an attainable one.

11The one notable exception to this is a study done by Daniel

Hamermesh (1973) in which the author, while attempting to test the

Nash theory, instead tests for a fifty-fifty split of the difference be-
tween labor's and management's first offers. Since he is testing for
the applicability of the Nash theory (which requires that i£_the two
parties have identical utility functions, then they must divide the
pareto optimal interval in half) and finds it to be probably inappli-
cable, he does not test the Nash theory itself. In Hamermesh's opinion,
the Nash theory is a split-the-difference theory--in the admittedly
unlikely circumstance that it should be applicable to a bargaining sit-
uation--and so he believes that he has tested the Nash theory as well as
a split-the-difference theory. I briefly discussed that interpretation
with Robert Bishop, who disagrees with Hamermesh, stating that the Nash
theory is not a split-the-difference one. To support Bishop's view:

the Nash theory rests upon four stated axioms and (depending upon one's
approach to assumptions made en route to a theorem) two or more assump-
tions less prominently set forth. The failure of any particular bargain
situation to satisfy an axiom implies that the Nash theorem on outcomes
cannot be applied; this is because the antecedent conditions have not
been satisfied, and follows from simple logic. Hence it is here con-
tended that the result of the Hamermesh test--even were other metho-
dological problems absent--is not to test the Nash theory, but rather to
show why that theory cannot be tested (n1 his data with his assumed
utility functions.

'12Walton and McKersie (1965: 92) point out that "The most difficult
aspect of making a concession at any point during negotiations is to be
sure that it is coupled to a concession from Opponent."

’131n this example from Ann Douglas's study (1962: 278) a concession

by the union has effectively telescoped several moves, thus requiring
similar action from management. A mediator cautions management to re-
sist following their initial plan of small-step concessions, in order to
respond to the union in kind:
Now look. That union has got right into the realistic area.
If you come up with a cent or 2d [sic] you are going to
damage these negotiations beyond repair. Now you know you
have to settle this and you know that you have a--a limit.

27

The question if-—I know what you're trying to do. You're
trying to get to that limit without jeopardizing your bar-
gaining position, and I'm here to help you do that, but don't
make it harder on me. The union has made such a substantial
move, and now you know they're in the realistic area. If you
come out with a penny, you're gonna insult them. You're
goin' to make them feel that you're not bargaining in good
faith. If I were you I would throw out at least §9Z_of what
you're going to give them--and I didn't know what you were--
what the company was going to give them.

 

‘laflowever, for alternative values of the endowment aspect of the .
strength variable--i.e. violating the underlying ceteris paribus assump-
tion--there may be differences. For example, the particular negotiators
on both sides of General Electric contract bargaining may be considerably
different than those who would be hired if G.E.'s strategy were to be
radically different from the bulourism ‘ which has characterized it in the
past.

 

15This is in contrast to Landes, who conceives of a bargain as

the failure to go to trial. Fortunately, he is alone in his perception
of bargaining as the failure to have a conflict. Unfortunately, with a
few exceptions-—Hicks, Cross, and the nondeterministic theorists such
as Walton and PICKer818.—-few perceived of conflict as signifying fail-
ure in negotiations.

28

2.2 The Prosecution

 

The prosecutor, in addition to his role as one of the parties
in the trial arena, fulfills an administrative function in the judi-
cial process. Having limited resources, he is confronted with more
cases than he can investigate thoroughly and pursue fully through
the complete criminal justice process. Consequently, he must make
a choice as to the allocation of his resources among the various
cases}' so as to obtain the best outcome overall, or effectively make
such a decision by default.2

His utility is then a function of the outcomes of the set of
cases before him:

U = U (v,x)

where v denotes the outcome of Case V, which we single out for detailed.
consideration, and x represents a composite of the outcomes of all the
other cases under the assumption that the prosecutor (P) allocates his
resources among those remaining cases in an optimal fashion.3 (Here,
outcome may be measured in a number of ways. For example, it might be
the actual sentence given, measured in years; or it might be an index
of various aspects of the outcome, such as a weighted average of a fine
and a prison sentence.) The prosecutor (P) then seeks to maximize his
utility subject to the constraint imposed by his budget and his ability
to use the resources at his disposal in a productive fashion.4

The prosecutor's utility is then a function of two variables,

v and x, and we make the usual assumptions about the shape of his util-

ity function with respect to these variables, i.e.

29.

AU/av > 0, «sZU/bv2 < O,

aU/arx ) 0, and «sZU/ax2 < 0.

Given his utility function, the prosecutor confronts a constraint
on his total resources: his budget for the relevant time period (and,
although alternative values for the budget allotment are considered be-
low, it is assumed that whatever its value, B, it is fixed for the time
period under consideration.) Within his budget, P finds himself subject
to certain fixest' costs, for example-those~assoeiatedewith_Owningwand
maintaining the physical facilities, certain "housekeeping" personnel
(i.e. those not directly involved in the handling of cases) and other
commitments which are his legacy from the preceding period.

The remainder of his budget is devoted to costs which are vari-
able in the sense that they can be attributed to the processing and dis-
position of particular cases, for example, the time of attorneys and
other supporting staff. However, use of the term "variable" is a bit
misleading here fOIVthe following reasons. The total amount of these
resources is fixed (within the time period we are considering, for ex-
ample, the budget year,) since they are defined to be the difference
between total resources--which are fixed--and costs which are within
the traditional definition of fixed Operating costs. Hence, "variabil-
ity" might be more accurately labeled "attributability".

Within the total of "attributable" resources are three sets of
costs.‘5 Considering the costs involved in the processing and dis-
position of a single case (at least to the extent that P contributes to

determination of the disposition) we first note the costs associated

with the processing of a case, independent of outcome. Specifically,

30

if and B8 are the costs per day of maintaining cases V and X

Bsf,v f,x

respectively, and tv and tx are the number of days to which the former

costs apply, the attributable resources are reduced by the amount

thsf,v + tstf,x

where the case-maintenance costs are what will be called semi-fixed
costs--costs which do not vary with output, but do vary with time.

The second subset of attributable resources is the set of costs
associated with the execution of a trial;7' these are labeled Tv and
Tx’ and their sum further reduces the available resources.

The remaining resources are ones that are truly disposable re-
sources: those which P can choose to allocate to the improvement of
one case or another. Denoting the total budget by B, the fixed costs of
the prosecutor's office by Bf, and disposable resources by R, we have
that

R I B e Bf - tvBsf,v - tstf,x - Tv — Tx'

Assuming that all cases come to trial:3 it is only the dispose
able resources R, which the prosecutor is free to spend in pursuit of
the maximization of his utility, the other portions of his budget having
been already committed to the various costs mentioned above.

Before discussing the trial Opportunity set which relates to R,
let us consider some ways in which the level of R is affected by alter-
native values of its component variables. First, and most obvious, R
is increased by increases in the total budget, B, or decreases in the

fixed costs of running the institution, B Second, cuts in pretrial

f.

delay for v or x, or reduction of the maintenance costs for the trial

31

docket would result in a higher R. Finally, reduction in the costs of
trial(s) would make available a larger amount of disposable resources.
-Although scant reference will be made in future pages to the determina-
tion of R, it should be noted that alternative values for the underlying
variables can be expected to effect shifts in R and hence yield alter—
native trial Opportunity sets?

Within the limitation Of his disposable resource, R, the prosecu—
tor can choose to allocate his resources between X and V, and we would
expect there to be some tradeoff between the two. That is, if F(v,x,R)=0
is the trial Opportunity set, stated in implicit form, we anticipate

that

bF/dv
- OF'ax <1 0

(applying the implicit function theorem.) Should F be a linear func-
tion, as in the example at the end of this section, the left-hand ex-
pression in the inequality would be simply the slope of the constraint.

However, we cannot expect F to be linear in the general case,
for the following reasons. The units of v and x obtained by commitment
of R resources are not goods purchased by P at a constant unit price,
but rather, each case outcome is the result of a production process in-
volving inputs such as the initial endowment of evidence, additional
sleuthing by or on behalf of the prosecution, contributions to the
strength of the case made by expert witnesses and technical resource
persons, the skilled efforts of the prosecuting attorney, etcetera.

If we assign to this production-of-a-strong-case function the
same attributes we typically assign to other production functions--pos-

itive and diminishing marginal productivity with respect to each face

32

tor--we obtain cost functions which bear the usual properties. If we
then further require that the total amount of resources devoted to v

and x be equal to R, we can express the set of attainable pairs (v,x)
with x as a function of v. It is this function—-actually, a production-
possibility frontier (ppf)—r.which was called F(v,x,R) above.

For our purposes, F has two parameters which are of special im-
portance. The first of these, R, has already been discussed. The
second, which we label r', is a measure of the relative strength of the
defense (D)to the prosecution, with larger values of r' reflecting a
stronger defense case than do lower values. (In the special case of a
linear Opportunity set, r' is proportional to the absolute value of the
slope of the line.)

Relative strength, as measured by r', is actually many factors in
disguise, since prosecutors make their estimates of relative strength

based upon a number of considerations: r represents the subjective

evaluation by P of the strength of the defendant's case relative to that

10.

of his own. Included in this estimate is, at least initially, an

educated guess as to what the other party'isTlikely-to$do intthe way 6f
case development, as well as a hopefully more accurate guess about what
he himself is likely to do in his preparation and presentation of the
case at trial .11

Subjective as such an evaluation may appear, it is possible and,
according to attorneys from both sides, the usual practicegjzv to make
exactly such appraisals based upon certain objective criteria as: the
previous criminal record of the defendant; various demographic charac-

\

teristics of the defendant, such as race, age, sex, relationship to the

33

labor force, jail status; and other factors. In addition, they consider
the endowment of evidence-~and in particular, the initial endowment of
evidence received by the prosecutor from the police, such as positive
identification of the defendant by witnesses, legally obtained confes—
sions, etc. These are items of information which could be brought out
at trial, especially if the defendant should take the stand and testify

in his own behalf, and are factors which tend generally to favor one

side or another}3 Letting e represent the initial endowment of evi-
dencegl' and the d1 the other factors relating to the defendant,

v = v
r r G:,d1,d2,...,dh).

Clearly, alternative states of these variables can be expected to af-

fect the level of r'.15

The effect of higher values of r' is to associate with each val-
16'
ue of v :1 lower value of x. This is illustrated graphically be-

low by F':

(0.xo)

F
J

 

 

Figure 3. The effect of alternative values of r', the relative strength
of D to P.

34

It is now possible to describe the prosecutor's allocation de-
cision in terms Of the familiar utility maximization, subject to a con-
straint which is possibly nonlinear.17 Let F(v,x,r')=o be the opportun-
ity set, stated implicitly, on which we consider alternative (but pre-
determined) values of r'. If U(v,x) is the prosecutor's utility
function, we seek to maximize U subject to F. Since we have assumed
that the opportunity set is (nonstrictly) concave with respect to the
origin, and that the utility function is Of traditional form, the out-
come Of P's constrained maximization problem yields a unique solution,

(vo(r'), xo(r')) for case outcomes, and consequent level of utility, U0.

 

 

Figure 4. Prosecutor equilibrium, taking case V to trial.

The effects of changes in r' upon the prosecutor's allocation of
resources and upon expected case outcomes can be traced through their
effects on F, to determine alternative values of v, x, and U. Decreases
in r', however incited by one or more of its contributing variables,

(as well as increases in R) should result in a higher value of U, and of
v, x, or both.:“3 For example, the existence of previous arrests of D,

other things being equal, should thus result in a lower value of r' and

a consequently higher value of v and U than would the absence of such a

35

record. Similarly, information that the defendant is nonwhite, or of
other factors favorable to the prosecution would cause the value of r'

19

to be lower than otherwise. Increases in e are unfavorable to the

defense, and hence lower r' and raise vazo

In analogous fashion, reduction in pretrial delay, or of the
costs of trial (for example, recent experiments on the use of juries
of six members) would result in a lower value of R, shift F outward,

and would lead P to adjust his allocation and attain a higher expected

value of v and U,21 as illustrated by F' below:

 

 

 

Figure 5. Effect of change in disposable resources R.

Thus it is clear that the cost effects of pretrial delay are unfavor-
able tO the prosecutor.

In addition to the cost effects, however, the passing of time
may have evidence effects as well. Having been endowed with a cache
of evidence or information by the police, P faces the possibility of
partial loss of that evidence as a function of the time interval between
the offense or gathering of evidence, and the time of its utilization

at trial. It is generally agreed that evidence in favor of either

36:

party diminishes with the passing of time, although it is conceivable
that it could increase, independent of any evidence-gathering effort
by a party.22

The relevant question, then, is not whether usable evidence de-
clines over time, but rather what the impact of this decline is upon
the parties relative to one another; and the net effect is generally
construed to favor the defense. 23 The evidence effect of delay is
represented in the construct thus far developed as a decrease in e and
a consequent increase in r', since r' is a function of e.

The sum of the two time (delay) effects is determined by the

reduction in R due to the increased case-maintenance costs, and the in-
crease in r' due to the decreased availability of earlier existing evi-

24 The result of these two effects is to assign to each value

dence.
of x a lower value of v than would have been assigned with a shorter

delay period, represented graphically by a downward shift and/or clock-

wise rotation of the opportunity set F:

 

 

 

Figure 6. Time delay effects.

37

Consequently, the prosecutor obtains lower values of v and U, (with
the possible exception--rather unlikely-~of V being an inferior good in
the relevant range of R.)

So far in this section, a construct has been proposed to explain
allocative decision making by a prosecutor, who with limited resources
attempts to obtain the best possible outcomes v and x at trial. Dif-
fering factors relating to each case--the initial endowment of evi-
dence, expected time until trial, and characteristics of the defendant
and of the offense, as well as P's ability to develop a strong case
based on these factors--determine the opportunity set which is open to
him for a given level of disposable resource. However, this description
of a prosecutor's choices is unrealistic for two interrelated reasons:
First, if a prosecutor were to bring every case to trial, he would be
unlikely to have any disposable resources left for making the deci-

sions we have described; in fact, he would probably run out of re-

sources before disposing of even a quarter of the caseload.25

The natural result of such a shortage of funds is to drop some cases
in order to do a better job with the others. The power to make such
selection is within the scope of the prosecutor's discretion and this
discretion is exercised at a number of levels of the criminal justice
process prior to the judicial stage.26 If P were to drOp case V, for
example, at time td zfigtv, the scheduled time for trial, the effect
would be to increase P's resources by

Bsf,v (t:V - rd) + TV

This new level of resources could be allocated among the remaining

38

cases, while V would be assigned the value v = 0 (unless the indexing
of v were so constructed as to include the psychic costs of waiting for
an outcome, in which case v might be greater than zero even for cases
ending in a prosecutor's drOp, dismissal, or acquittal.)

A second Option for disposing of cases other than the full-trial
alternative originally introduced, is for some of P's cases to be re-
solved by a defendant's plea of guilty, thus avoiding the cost of trial
and the more extensive preparation which trial would require ofiP.27
In addition, in some courts, separate machinery is set up to negotiate
and process pleas of guilty, thus considerably reducing the time from

the offense to disposition of the case. In the event of a plea of

guilty at timetp R would increase by

- _ + 'r
Bsf,v’° (tv tp) V

where we assume that the trial time necessary for the defendant to

make his plea is sufficiently small to save P virtually all of T". In

the event that D pleads guilty, v - v If we further allow v

to range as low as zero, the first case, that of a zero non-trial out?
come, can be subsumed under the second, and viewed as a limiting case.
Were it not for these two avenues of flexibility, the prosecutor would
have scant ability to pursue any of his cases, even high priority ones,
beyond the outcome attainable at zero level of disposable resources.28
What are some of the characteristics of a guilty plea, from the
point of view of the prosecutor, and how ought these to be interpreted

within a model such as the one unfolding here? The typical resolution

of a case by plea of guilty involves the defendant's entering of the

39

plea in return for either a promise of leniency in sentencing (where
the units of v are measured as, say, the maximum penalty possible,) or
a reduction or dismissal of some of the charges against him. Which of
these alternatives occurs depends upon the particular court in which
the case would be tried--i.e. restrictions imposed by applicable state,
local, or federal laws, internal rules, the institutionalization of
certain traditions (such as the belief that a reasonable bargain is
the reduction from charge "Z" to charge "Attempted Z")--and with-
in those restrictions, upon the discretion of the prosecutor.29

The view of many prosecutors that accepting pleas of guilty con-
stitutes "compromising with the defendant" indicates a belief that the
bargained outcome is in general less than the expected outcome of trial,
although perhaps not uniformly so. Nevertheless, in cases resolved by
guilty pleas, the utility to the prosecutor of that outcome must exceed
that of the trial alternative, or he would have insisted upon trial.

Considering the outcome opportunity set to be a function different

from F--labeled G, and referred to below as the plea opportunity set--
we can make some inferences about this new function and its relation to
the trial Opportunity set. First, because of the saving in trial costs
and possibly from reduction in pretrial delay, the level of plea re-
sources R' - R'(R) exceeds R, and hence the x-intercept of G exceeds
its counterpart on F. (The relation between the F and C values for the
v-intercept will depend upon r' and other factors mentioned below, and
hence is not determined at this point.) Second, G must dominate F (i.e.

lie to the northeast of F) for at least some values of v-—or else G

40

would never be preferable to F, and no rational prosecutor would find
occasion to accept a guilty plea. Coupled with our first inference,
that the x-intercept of G is greater than that for F, the second would
suggest that G dominates F for low values of v, if not for higher

ones:

o,F
\

 

 

V

Figure 7. Positions of F and G for low values of v.

Finally, the fact that prosecutors believe most plea bargains to result
in a lower value of v than would trial, implies that for some values of
v, F must dominate c;30 for. (resorting to proof by contradiction) were
that not true, i.e. if G always dominated F, a plea bargain would re-
sult in the typical reduction of v only in the event that P considered

V to be an inferior good..'3‘1 Thus for some values of v--to be compa-

tible with the first two inferences, for some large values of v-~F dom-

inates G:

41

«ng

JP

V

 

 

Figure 8. Positions Of F and G: the v-intercepts.

This last statement, and particularly its graphical interpretation of
the v-intercepts, is not surprising: it says that the best outcome
obtainable from trial is at least as great as the best obtainable from
a plea of guilty.

The set of factors which determine G is the same as that which
determine F, except that the difference between tv and t may en—
ter into the determination of F for some defendants in some courts (in
particular, if the court maintains separate pre—bargaining and pre-
trial queues.) Given the values of the relevant parameters, it is pos-
sible to state F and G, and one may even say that G is the plea con-
straint which corresponds to F, since a change in any of the values of
the contributing parameters would yield a new pair of functions, F'
and G'.

As he did earlier for F, P determines the highest utility level
he can reach while observing G as azconstraint, i.e. he determines val-
ues of x and v such that

aU/AV = aG/ev

37173;:— 367bx

42

In addition to obtaining the equilibrium values for v and x, the prose-
cutor can associate with the plea option a utility value Us, which may
or may not compare favorably with U0, the utility value he associates
with trial. Clearly, if U6 2,Uo, P will choose the plea of guilty as

his preferred method for disposing of the case. An example is sketched

below.

 

 

 

Figure 9. Prosecutor equilibrium on trial and guilty plea constraints.

If Ué,‘ U6, the trial option would be the more attractive one
for P, if the only alternative to F is believed to be G. However, G
may not constitute the only alternative to the trial constraint, once
we allow the possibility of the defendant's not bringing to bear the
full strength of his case, as he would otherwise be expected to do at
trial. Instead, in pleading guilty the defendant typically presents
very little evidence of his innocence. Recall that both F and G were
derived under the assumption that the relative strength of the two
parties is r'. Now suppose that, although the value is r', the defen-
dant were to agree as part of a plea bargain, to not exercise the full

strength of his case; that is, he promises to use only r' which is less

43

than r3. Applying the earlier discussion of the effects on the Oppor-
tunity set resulting frOm alternative values of the parameters, such
a pledge by D would result in a counterclockwise rotation of G, and

possibly result in making a plea of guilty preferable to trial.
The above interpretation of the kind of compromise involved in a

plea bargain is quite straightforward. Typically, when the two parties
go to trial, they "throw everything they've got" at the other party

and consequently, r'

can be properly construed to be the appropriate
level of strength used in the derivation of F. However, the evidence
presented at the time of a guilty plea is not at all the volume from
either party which would be mustered for trial upon a plea of not
guilty--else there would be staged a full trial, in effect. In fact,
each party will present very little, usually just enough to suggest
that the charge to which the defendant is pleading is an appropriate

one for the circumstances of the case.‘32

By not presenting his strong-
est evidence at the time of pleading, the defendant who pleads guilty
has agreed to exercise a low value of r' (as viewed by P) in spite of
the fact that the true value (the maximum, i.e. r3) may be higher.33
The extent to which D would have to reduce r' in order to in-
duce P to consider a plea of guilty--that is, the magic level of r',
ré, necessary in order to make P indifferent between observing F based
upon r; and observing G based upon r$--can be determined within the
present construct. Since F remains fixed regardless of the value of
r; (in fact, r; depends upon rs) the utility of trial outcome, U0, re—
mains constant, and provides a benchmark for evaluating the relative

attractiveness of plea outcomes resulting from the values of r; which

are under consideration in the search for ré. Uo thus determines a

44

function Ir' (v,x) = 0, the equation of the indifference curve at that
utility levgl. To determine the appropriate value of r; we restate G
to include r' explicitly (although r; is not a variable, but only one
possible value of the parameter r'.) With the plea opportunity set now
written as G(v,x,r') = 0, we seek vm, xm, and r; satisfying the follow-

ing conditions:

(i) Ir;(vm.xm) = 0 .

(ii) G (vm,xm,r$) = 0 , and

(iii) ¢_I(vm.X) 3g<vm.xm.rt;l)

 

6v - 6v
0
%(Vm.xm) _:_:§{_(vm.xm.rm)

That is, we seek the value of r; and coordinates (vm,xm) for which the

adjusted constraint G is tangent to the indifference curve Ir" The

0
solution is graphed below:

 

 

V

Figure 10. Prosecutor indifferent between the modes of case
resolution.

45

To briefly recapitulate the geneology of r; and its descendants,
we note that r; is a given value of the variable representing relative
strength in case V as viewed by the prosecutor. This value, along with
a corresponding one for X, the composite of P's other cases, are para-
meters in functions which describe the production-of-a-strong-case-for-
trial. Given a fixed amount of disposable resources, R, and a suitable
translation from units of the various inputs for strong-case production,

the output X can be stated in terms of the output V; and this function,
a production-possibility frontier, is the function Fr,(v,x) a 0. A

similar function, Gr,(v,x) - 0 describes the opportunity set available

if the case is resolged by a plea of guilty, rather than by a full trial;
although R enters G as a parameter, it does so in the guise of R'(R),

the disposable resources level made possible by savings from bypassing
trial and perhaps a shorter pre-disposition queue.

When P performs his utility maximization routine subject. alter—
natively,.to F and to G, he chooses to observe the opportunity set which
affords him the higher level of utility. However, there exists a level
of r', namely ré which, when substituted for r' in G, causes adjustment
of G such that P is indifferent between disposing of the case via F or
via G. At this point, one can consider a number of juxtapositions, one
with another, of the many special values derived by the process just «
outlined. For the initial value r; yielded v0, x0, and U0 when
subjected to F; and U0 in alliance with G led us to values r&, vm, and xm.

Although we could choose to represent this set of related values by an

ordered n-tuple (r5, v , x

U , r , v , x ), it is the relation of r'
o o m m m o

’ o

with vm in which we have especial interest. We refer to the ordered

pair (rs, vm) as a "switch point" for it expresses the minimum level of

46

v necessary (when obtained by a plea of guilty) to pursuade P to ac-
cept a plea of guilty rather thanvtb pursue the route to trial.

4 Two other pairings Of entries from the seven-tuple are of pass-
ing interest here. One is the difference,(ré - I91) which describes the
compromise necessary from D in order to induce P to consider acceptance
of a guilty plea as a viable alternative. Of course, D may eventually

concede a great deal more, or none at all, as the bargaining progresses;

(r3 - ré) only specifies the initial compromise necessary to convince P
that D is quite serious about a plea.34 Second, we take note of the
difference between v0 and vm, which measures the difference in outcome
units which P experiences by switching from F to Gr" Whereas the
difference (r; - ré) represents sacrifices by D, suth as the promise to
not exercise his rights to trial (including judgement by panel of peers,
the right to confront witnesses, etc.,) a weakened position if he should
decide to appeal the verdict or sentence, and others, here (v0 — Ym)
represents the associated outcome loss to P.

Having shown that for a given level of relative strength (as per-
ceived by P) we can derive the corresponding plea bargain constraint
and level of vm which would make the prosecutor indifferent between
trial and settlement, it remains to be shown how these items--and in
particular, the level of vm--are derived for differing levels of the
strength variable r' (allowing alternative values for r3). We seek to
derive a new function, which we label a "switch function" for P, for

which each point is a switch point, as was (rs, vm). Although we could

define the switch function as a function of any subset of {;',..., x4},

47

we again choose to focus on the relationship between the strength of
the case (at trial, rs) and the minimally acceptable plea necessary to
induce P to switch from obeying the trial constraint to obeying the
bargaining one. The resulting function can be interpreted to be a der
mand function for one good--a plea of guilty at level vm--as a func-
tion of the "price" of a substitute good, outcome obtained via trial?5
Deriving the switch function point-by-point, we would reason in
the following manner: a lower value of r; would imply a higher level
of U0; that in turn would require a lower value of r; and result in a
higher value of vm. We would thus expect that as trial strength of the
prosecutor vis-a-vis the defense falls, the outcome in terms of v would
increase also. Then to the new function vm - Sp(r') we attribute a
negative first derivative, but cannot determine the sign of the second
derivative without knowledge of the forms of the utility function and

trial-and-plea-opportunity sets.36

Following is an example in which a prosecutor's switch func-
tion is derived from a Cobb-Douglas utility function, and simple pro-
duction function for evidence and resource constraint. All were
chosen for use here because they possess the properties required by
the theory, yet are simple enough to survive the necessary mathemati-
cal manipulations.

Suppose a prosecutor has, or pledges himself to

(i) a utility function U(v,x) - vqxe, where Oéq, (5 $1;

(ii) production functions for v and x:

V - %3 V and x - %-. M

48

where M." and Mx denote the amounts of variable input M
which is a mush of all variable inputs to the production
by P of a strong case;

(iii) a total disposable resource limitation of R if V is
resolved via trial, or R' if by plea of guilty. At a
price of w per unit of M, this allows the total variable
input to x and v to be M -= R/w for trial, or _M_' - R'lw
for pre-trial settlement.

First it is necessary to derive the production possibility fron-
tiers which constitute the trial and plea opportunity sets. From (iii)
we have that Mg +Mx - M_for trial and M§ + M.x - M' for a plea.
Since derivation of G is precisely parallel to the derivation of F,

only the latter will be performed fully here. Rewriting the trial re—

source limitation as MQ - M -Mx’ and substituting into the production
function, we obtain Mk a M_- r'v. Substituting that expression into
the production function for x yields

r'v + kx =QM
for the function F. Similarly, G can be written

r'.v + kx = M'.
p —

Next, solving the utility maximization problem for U(v,x) sub-

ject to F yields

«w (w.
“(4413)? ' “Tangy—1.,

 

49

and consequent utility level

,0- - (fiawfi)? (g)?

The problem then reduces to one of solving the following three

 

equations:

(a) r$v + kx - M:
1/ -‘/(3
(b) x - Uo Pv (which is another way of writing U(v,x))

U
m _ dx_ , where x(v) is the indifference curve of equation
dv

(b). This last equation may be rewritten

r' = k . 5.111/(5 v-QIP-l
m ‘3 0
Substituting from (b) and (c) into (a), we obtain

91' ”P12“

v - U

k (2%.» 1)

1/4
O

 

and further substituting for U0 the value obtained earlier, as well as

dollar resources for units of M,

v (R/w) (OL+(3)/a.. _(_X_.«__ . (r,)-1 .
m (R'lw) 6/0“ a”?
As the reader can verify, the utility function meets the condi-
tions set forth in the theory, i.e.
aU/av > o , 32113.3( 0
QU/ax > O , and 320/1x2< 0.

50

Further, the linear opportunity sets are nonconvex with respect to the
origin, and, since 21'} M, the plea constraint dominates the trial con-
straint for small values of v.

The switch function which results is also as suggested by the
theory: the level of the guilty plea is inversely related to the defen-
dant's strength (de/ar' (0); for R' fixed, vIn increases with R. The
latter reflects not only the general desirability of having disposable
resources, but also the effect of pre-trial delay, which serves to
reduce R. Finally, if the trial date (and hence R) is held constant,
and the bargaining, or plea date is allowed to vary, large values of
R' may be undesirable to P; large values of R' in relation to R reflect
the need for reallocation of time between the queues so that bargaining
occurs closer to the date of the potential conflict. This last state-
ment is consistent with the catalytic effect of impending trial dead-

lines.

51
NOTES T0 2.2

1Here, the prosecution is assumed to be one unit and is referred
to as if it were one person in spite of the fact that many people are
involved in the processing of the case and the various decisions made
along the way. As with industrial relations and bargaining, a natural
question is whether the various members of each unit have coincident or
divergent vested interests, and the answer is probably that there exists
some conflict. In particular, the interest of the prosecutor in ob-
taining convictions--his rate on those will become a campaign issue--
may conflict with other public goals, such as minimizing personal injury
incidence in a certain part of the city. The various potential conflicts
are outlined in Lester Thurow's discussion of "Equity Versus Efficiency
in Law Enforcement" (1970).

Nevertheless, at least within the prosecutor's office, it is
reasonable to conceive of the prosecutor as a single entity, for the
court deals with it as such. First, Justice Burger for the Court, then
Justice Douglas concurring:

"...[A]t this stage the prosecution is not in a good position
to argue that its inadvertent breach of agreement is immater-
ial. The staff lawyers in a prosecutor's office have the
burden of 'letting the left hand know what the right hand is
doing' or has done. That the breach of agreement was inad-
vertant does not lessen its impact."

"The staff of the prosecution is a unit and each member must
be presumed to know the commitments made by any other member."
Santobello v. New York 404 U.S. 257 (1971).

 

2'An example of the decision made by default is provided by the
execution of an ill-fated "sixty-day law" in Florida. Passed several
years ago in the wake of the death of a juvenile incarcerated while
awaiting trial, the new law required that defendants not tried within
sixty days of their arrest be released. After enaction of the law, the
courts apparently continued along on a first-come-first-served basis.
When the first group of defendants was released under the law, the pub-
lic outcry was great, as many accused felons prepared to join them on
the streets; the law was quickly made "inoperative."

3 Theoretically, there is also a problem of time-sequencing in the
disposition of the cases relative to one another: while one case is
being resolved under the prosecution's attention, others deteriorate as
evidence becomes harder to preserve and to find as time passes. Be-
cause the deterioration may be uneven from one case to another, each
ordering of the cases in time may constitute a choice among goods which
are in fact different goods. It has been suggested to me that this is

52

analogous to the problem of the physicist who finds that the substance
with which he is dealing (particularly in nuclear physics) is itself
changing through time. However, the time-sequencing problem is ignored
here for the present time.

4'The units of measure for case outcome may be given several repre-
sentations; for example, the number of years of prison sentence, or a
general index of loss of prestige, fines, probation conditions and prison
sentence.

5 In textbooks one is able to find definitions of fixed costs
which could yield contradictory classifications of certain kinds of
costs. Here, if we interpret output to mean efforts at improving case
outcome, some of the costs we have labeled as semi-fixed ones are
those which do not vary with output, but do vary with time.

6
In U.S.v. Wiley (1960), Chief Judge Campbell outlined some of

the costs of a case up until the time of sentencing:
For example, in the case before me, the official total cost
to the government...was $11,250. The Assistant United States
Attorney, upon representation spent three days in preparation
for Wiley's trial and one day on trial. Several investigators
likewise spent three days in preparation for trial and one
day at the trial. I myself spent one half day in preparation
for trial and then one day on trial and accordingly the Court
was totally occupied for a day and a half. Though the expense
and time that would have been saved the government if Wiley
had pled guilty is fairly obvious, I suggest that if this had
been a criminal trial lasting weeks or months as many of our
trials actually do, then the saving of time and expense to
the government would have been far more striking.

 

7'For the moment it is assumed that all cases do come to trial.

This assumption is discussed further below.

8’This assumption is made here in order to simplify the analysis,
and is relaxed later in the section, when the alternatives of dropping
cases (or recommending dismissal) and pleas of guilty are introduced as
possibilities.

'Each of these possibilities has its counterpart in familiar
discussions of the "crisis in the courts" as some like to call it. Poli-
ticians and others have called for increased court funding, cutting
back the "fat" in the bureaucracy, reducing delay ("Justice delayed is
justice denied," as the adage goes.) The last two cost-cutters repre-
sent areas of reform which are "hot" and have been exposed to recent
experimentation. Although the cost of pretrial detention is frequently

! All!
HurratnrsptCIunlll‘

53.

part of the budget of an institution other than the prosecution, the
resources thus spent represent a net public expenditure, and subject to
accounting limitations, resources saved by reduction of pretrial deten-
tion (either a reduction in the number of defendants in jail, or in the
period of incarceration) should make possible an increase in the dis—
posable resources of the prosecutor. Some cities and states have changed
their record-keeping to systems which not only allow lower maintenance
costs, but also permit greater coordination among the several law enforce—
ment agencies. Finally, many governmental units are making proposals

for change which would reduce the number of "false alarm" trials, at
which witnesses, attorneys, juries, and judges must appear and wait

(and be paid) only to find that trial is postponed for one reason or
another; the effect of such proposals is to reduce Tv and Tx'

0In general, when P faces what he considers to be a tough case,
he would expect a worse outcome, ceteris paribus, than he would antici-
pate from an easy one, i.e. av/‘cr'< 0. However, it should be noted
that although incantations of ceteris paribus evoke the above result on
the opportunity set F, they will not necessarily yield that same result
when applied to the solution of the utility-maximization problem, as
should become clearer once P's choice mechanism is presented below.

 

 

11
'Note that any revisions by P of this estimate will change the
benchmark by which he judges the strength of his Opponent.

12 See'McIntyre (1965) and Newman (1966) for discussion. The pro-
secutor* in Detroit confirmed that they do this; in fact, a blank labeled
"Probability of Winning" appears on the case file forms.

'Certain pieces Of information--the race, approximate age, sex,
custody, social status--would be known by a jury in any event if D went
to trial. Others--previous arrests and employment status, for example--
could only be brought out by questioning the defendant himself. It is
Often helpful to D for him to take the stand in his own behalf, but in
doing so he raises the possibility of other damaging facts becoming
known.

1("According to Phillip Schaenman of the Urban Institute, a study
being conducted there indicated that little sleuthing by police (for
themselves or for the prosecution) took place after arrest, although
occasionally there might be some additional work done on the eve of A
trial. For the most part, the evidence gathered around the time of ar-
rest was the total evidence available to P for trial.

The theory of plea bargaining presented here has little to say
about the form of this function, and thus can be no more specific than
to include a description of the directional effects of each of the con-

tributing variables. For example, we could require the di to be stated

54‘

(i.e. coded) so that3r/3d1<0 for all i.

16 For example, if F is the function, v2 + (x-r')2 - (x -r')2,
x, v, r' z 0 and r' restricted to be less than x , the opporgunity set
is an arc of a circle with radius (x -r') and center at (0,r'). As r'
increases, F will rotate clockwise to F', as illustrated in the text.

17 This characterization of the Opportunity set as the result of a
production process is not new in its application to micro-analysis.
Gary Becker (1965) developed such an opportunity set to describe the set
of choices of a consumer in allocating his resources between market-in-
tensive and (consumer-) time-intensive goods. It is also curious but
disappointing to note that although Harsanyi conceived of a production
relation underlying a bargaining relationship, he failed to carry the
notion any further in his theory:

Power in a schedule sense can be regarded as a 'production
function' describinghow a given individual can 'transform'
different amounts of his resources (of his working time,
his money, his political rights, his popularity, etc.) into
social power of various dimensions (of various strengths,
scopes, amounts and extensions). (1962)

Unless v or x is an inferior good, the effects of a decrease in
r or increase in R are as noted in the text. It is highly unlikely that
x (the rest of the caseload, taken as a whole) should be regarded as an
inferior good. It might, however, be true for individual cases, such
as v. If, say, v is deemed not very serious (e.g. a littering offense)
and pursuit of more serious cases is out of financial reach perhaps be-
cause Of lack of access to lab facilities, etc., the effect of an in-
crease in R could be to shift prosecution efforts from the "small stuff"
to more serious offenses. As a non-court example of this, the Boston
police claim that with only $80 per narcotics officer per month for
purchases, it is impossible for them to do a thorough job in policing
narcotics traffic (a minimal purchase would be $65) and therefore they
concentrate their efforts on other types of offenses. This is in con-
trast to the situation in New York, where the figure is several thousand
dollars Per agent er month. Agctelevision documentary on the Boston
drug trade, 'The eptember Factor," April 1974.

'Verification of the effects of these factors exist: among the
literature of sociology and criminology, and is borne out by the data
used from Detroit. See section 3.2 and the data appendix.

2OThis effect of the endowment variable highlights the fact that
the ability to attain certain case outcomes at trial depends intimately
upon the earlier work of the police.

55

21'Under certain circumstances the changes in r' and/or R could
result in a major shift of resources from x to v (or from v), although
such a shift would require special conditions on the utility function.
This might happen in a highly publicized case, as the details became
known and pressure was exerted on the prosecution to raise the priority
of the case.

22'A hypothetical situation would be one in which two witnesses to
a crime later meet, piece together their partial understandings of the
incident, and appear on the prosecutor's doorstep.

23 In the strategy of the defense in nearly every criminal case
delay is given a prominent place. It often means absence of
witnesses, and in the case of repeated continuances, disgusted
and angry witnesses for the state, the memory of the witnesses
becomes dimmed, the prosecutor may be changed, he may lose in-
terest as the case grows stale, and in many ways a decided ad-
vantage for the defendant in the average case may be had by
the simple expedient of putting off the trial.

warren and Beatty (1932) referring to the Missouri Crime Survey.

24

'Witnesses die, forget, or become unavailable; the heroin in
the French connection case being stolen are examples of this.

25'The costs of trial relative to those of a plea of guilty form
a ratio of eight or ten to one. One result of the high proportion of
guilty pleas--and of institutions which have adapted themselves to the
use of that resource-saving method of disposition-- is a sense of des-
peration at the suggestion that all cases be given full trials. In a
magazine interview (Mills, 1971) New York defense attorney Martin Erd-
mann described a time when he felt a client was being treated unfairly
and threatened to insist on trial for all the clients of the Legal Aid
Society where he worked, the latter accounting for the defense of 60
percent of the city's criminal caseload. The prosecution compromised.

26 The discretionary role derives from the separation of powers
doctrine in the U.S. Constitution. See Remington casebook (1969) and
supplement (1972) for sequence of cases affirming and explicating
prosecutional discretion.

An example of the usefulness of such discretion is given by Katz-
man (19 : 436) for the police:

The existence of many people vulnerable to arrest in a neigh-
borhood can be a valuable resource to the policeman who can
stock up on 'potential arrest.'...A patrolman may deliberately
not detect a petty criminal...who in exchange might provide
information about more serious crimes in the future.

56

27 The use of negotiated pleas of guilty is a controversial
issue. The ABA has taken a position favoring such pleas so long as
they are obtained through certain procedures which the Association be-
lieves safeguard the rights of the defendant and the interests of
the community. (See Standards for Pleas of Guilty, 1967).

Arguments in favor of guilty pleas are the efficiency of the sys-
tem (assuming the defendant is actually guilty), and the belief that
admission of the crime by a guilty defendant indicates a willingness
. to COOperate and an interest in rehabilitating himself. Opponents
argue that the purpose of the trial process is not efficiency, but
justice, and encouraging defendants to waive the right to trial does
less than insure the latter. They further argue that the agreement
to relinquish the right to trial often has the effect of losing other
rights as well, such as the Fifth Amendment right not to incriminate
oneself. A theory which considered the output of a court to be not
only dispositions of cases fed into it, but also the explication and
preservation of individual rights, including a right to safety, might
be able to accommodate arguments from both sides of this issue.

Alternately, if zero disposable resources were the way of life
for prosecutors, the role of and processes employed by prosecutors
would in effect determine a different institution from what currently
exists in most cities.

29 Remington and others point out (1969: 603) that
There are several motivations on the part of the prosecutor
and the courts in compromising with the defendant. A major
one...is avoiding the cost in time and effort of trying
cases unnecessarily. Another reason is the uncertainty of
conviction.

30‘There is an unlikely but possible problem here: if the second
derivative of G were sufficiently large (sufficiently greater than that
Of F) and if we were to allow the v-intercept of G to be greater than or
equal to that of F, we could encounter two intersections of the oppor-
tunity sets, with the result that plea bargains would yield better out-
comes for P at the extremes, and the mediocre or medium-lowbpenalty
cases would be better taken to trial. Given P's assumed risk aversion,
this would be a surprising conclusion.

31 Since P cannot regard all of his cases to be inferior goods,

the presence of some such cases merely strengthens the above argument
as regards the others.

57

32’One interesting example is of a plea bargain for reduction of
charge from breaking and entering at night to breaking and entering
during the day--in which case the time of the offense would not be
stressed!

33
This is consonant with Harsanyi's definition of power in bar-

gaining: "The common-sense notion of social power makes it the ability
to achieve certain things--an ability that the person concerned is free
to use or to leave unused." (underscoring his) (1962: 87)

34 As one prosecutor phrased it, this compromise is necessary to
indicate that "we're really talking about a plea."

3SIn the sense that either a plea of guilty or a trial satisfies
the legal conditions for disposition of a case, and in the sense that
neither P nor D desires more than one of these to occur for a given
offense, these "goods" are perfect substitutes for one another. How-
ever, in terms of characteristics and some of the consequences of pleas
of guilty, such as limitations on the ability to appeal, the relation-
ship is not so simple.

36From the vantage point of this stage in the development of
the thesis, one can see some of the difficulties encountered in
attempting to apply theories involving the use of the incremental
utility functions. Since the determination of the Pareto optimal
interval--so far, we have determined the endpoint contributed by the
prosecutor--is dependent upon r', the incremental utility function
is in fact a function of two variables, (v—v ) being the obvious- one
addressed by the theory, and r' being the one "hidden" in vm.

58

2.3 The Defense

 

Unlike the prosecutor, the defendant cannot choose to drop the
particular case in which he is involved, but does have a choice not
only in whether he pleads guilty or not,1 but also--whichever way
he pleads to the charges against him--to decide how vigorously he will
fight the case. He makes this latter decision (which in turn affects
the former one, also) based upon his attitudes toward different trial
outcomes and the opportunity set which he confronts for trial.

The defendant's utility is a function of the outcome of the
case in question and of the amounts of other goods (and services, and
savings, etc.) We represent the former again by v, and the latter by
c. To this utility function u(v,c) we ascribe the usual properties

concerning changes in u as a function of changes in c, i.e.

2
%>0 and :g—‘clz<o

Clearly, v is undesirable to the defendant, and so we expect bu/8v(0;
the sign of the second derivative is somewhat less obvious, however.
For desirable goods, such as c, our assumption of diminishing
marginal utility is equivalent to an assumption of increasing immunity
or indifference toward incremental quantities of the good as a function
of the amount one already has: an additional unit of c means less to
a person who already has a lot than it would were he to have little to
begin with. The counterpart assumption for an undesirable good such
as v is that the more of v a defendant already has, ths less differ-
ence an additional unit makes in his utility level. Thus for a per-

son confronted with a one-year sentence, adding another year

59

would cause a greater utility decline than would an added year if his
anticipated sentence were already twenty years. If we accept this
premise, we have effectively assumed that BZu/avz 2 0--an assumption
which, although appropriate in this context, is unusual within tradi-
tional economic theory,”2 and holds some rather different implications
for outcome than would its Opposite.3 In particular, it is likely
that when he subjects this utility function to some opportunity sets/
constraints, P will find that in order to maximize his utility, he

must choose between two corner solutions; this possibility is only
mentioned here to foreshadow results to come much later in this section.

Since it is infrequent that one discusses utility in terms of
both desirable goods and undesirable ones.4 ' a substitution for v is
made by defining a new good V* whose units, v*, measure improvement
in v relative to some benchmark, such as the maximum possible level of
v: v* = G'- v, where 3’13 the benchmark amount of V. Thus defined,

V* is a desirable good, that is,?5ufi3v*)7 0, but retains the positive
second derivative from before?

We also require that the indifference curves in v*-c space have
negative first derivatives and have second derivatives which are ‘ex—
clusively negative or positive throughout the domain of v and c?’
Further, since the case in which the second derivative is positive is
precisely the analog of the situation encountered by the prosecutor,
the discussion here concerns itself with the case in which the second
derivative is negative?

The trial opportunity set confronted by the defendant is similar

in its development to its counterpart for the prosecutor, with the

60

major exception that the level of resources available for allocation
between c and v* is no longer a predetermined amount received from an
outside source such as a city council or state government, but rather
is dependent upon a number of factors relating to the defendant and to
his case. Paramount among these factors are D's source and amount of
income, pre—trial jail status, and expected times for trial and for
pre-trial bargaining (or other method of handling a plea of guilty.)
For the period up until trial, D's total resources are:

RT=WO+Wltv+(l—J).Y.tv.

where RT represents total resources; Wo his wealth or initial level of
resources; W1 is income that is not dependent upon jail status, such as
income from certain transfer programs or from forms of non-human in—
vestment ,per unit time ; Y is D's income which is dependent upon his
jail status such as wages, unemployment compensation, or stipend
from job training programs' per unit time; J is his jail status, with
J - 0 if D is out of jail pending trial, and J = 1 if he is in jail;
and tv, as in the preceding section, represents the expected time until
trial, where the time of arrest is considered to be zero.
As did the prosecutor, the defendant experiences both costs
which are fixed (assuming he goes to trial), which are labeled Bo;
and ones which are time-dependent, associated with maintenance of his
case, such as attorney fees above the retainer , labeled B1.
Taking the defendant's disposable resources, 3 to be the dif-
ference between his total resources RT and the costs associated with

pursuing his case,

61

3=WO+th+(1-J)th—Bo—Bt

1 l v
which can be restated as

R-Wo-BO+(Wl-Bl)tv+(l-J).Y.tv

where R_represents the defendant's disposable resources; Bo and B are

l
the fixed and time-dependent costs of his defense; W6 is his initial
resource level; W1 and Y are his time-dependent sources of income,
which are respectively independent of and dependent upon jail status;
J is unity if D is in jail and zero otherwise; and tv is the time from
arrest until trial. Although it is possible for some of these values to
be zero or less (for example, Y equal to zero in the case of a person
living entirely from wealth, or R negative for a person in debt) we
redefine §.to be zero in the event that a negative 3 would otherwise
result from the above computations in the beliefs that (l) the borrow-
ing capacity of persons who are defendants in criminal trials is usual-
ly quite limited; and (2) due to judicial decisions of recent years,
indigent defendants have been given the right to assigned counsel, in
which case Bo and B1 should be close to zero if B would otherwise be
less than or equal to zero?
This representation of the level of D's disposable resources per-
mits easy analysis of the effects of alternative values of each of the
variables which comprise 3, For the moment ignoring the evidence ef-
fects of delay, clearly a defendant who is employed and out of jail
pending trial will have a very different attitude toward delay than

would that same defendant if employed and in jail or if receiving no

jail status-dependent income (and in jail.) Thus, it is impossible

62

to state the resource effects of delay upon D without additional know-
ledge about the origin of his income and about his jail status.

The effects of alternative values for the other contributors to
.R are Obvious: Other things being equal, a defendant has greater re-

sources if he has a higher income (of whatever origin, whether W or Y),

1
greater wealth, lower case maintenance costs, and lower fixed trial
expenses? ’ One can also see the effects of various subsidies upon

the level of D's disposable resources. For example, if the defendant's
family is on welfare—-although the amount of the payment might decline
somewhat if D were in jail and his allotment subtracted from the total--
his level of disposable resources would remain largely unaffected by

his jail status, assuming he has a negligible amount of other jail
status-dependent income. 133 another example, subsidies to defendants in
the form of legal services would cause R to increase, as noted above;
however, such programs may limit the ability of a defendant, who to
qualify had to claim indigence, from contributing any of his resources
to the cause of bettering his trial outcome, i.e. he might be forced

by institutional.rules to allocate all of his resources to c.

Having disposable resources R, as determined above, the defen-
dant chooses to allocate 3 between c and v*, and does as subject to his
trial Opportunity set f (again, this function need not be linear, but
is assumed to be nonstrictly concave with respect to the origin) which,
as with F, is derived as a production possibility frontier. Underlying
f are production functions for v* and c, but the production relation
for c is merely an identity function. Stated in implicit form, the

trial opportunity set is f(v*,c) = 0, with

63

15f40v*
’—_af/ac (o.

The defendant's trial opportunity set depends, in addition to
v*, c, and 3, upon the relative strength of the parties, this time as
viewed by the defendant, which is here labeled r". As was done earlier

with the prosecutor, r"

is defined to be large when the opposition is
stronger than for the lower values of r". Note, however, that a strong—
er case for D is associated now with low values of r" and high values
of r'. Like the prosecutor, the defendant makes an assessment of his
own initial strength and his enthusiasm for exploiting its possibili-
ties; consequently, alternative values of r" are reflective of alterna-
tive estimates of P's strength by D.

The location of f changes in response to differing possible val—
ues of r" in the same manner as did F due to alternative values for r'
in the earlier analysis of P's Opportunity sets. These changes in r"
are precipitated by differing states of the same factors which affected
r', namely the initial state of the evidence, and characteristics of
the defendant and of the offense (although the resulting change in r"
is in the opposite direction from its corresponding effect on r'.) The
weaker is P's perceived strength, the lower is r"; if we compare f at
one value of r" with the same function f' based upon a lower value, the

contrast would be as sketched below:

64

 

 

v*

 

v* v? F

0
Figure 11. Defendant's trial constraint under alternative values of
r" .

The defendant seeks to maximize his utility subject to the Op-
portunity set f; however, depending upon the specific trial constraint

and utility function, D may be able to do this only by specializing in

one good or the other:

fF-z_1’

(V*.0)
Figure 12. Defendant specializatiod’in goods v* or c.

 

 

Were the second derivative of the opportunity set (dzc/dvz)f to be
greater than that of the indifference curves--graphically, the constraint
would have a "tighter" curvature than would the indifference curves-—
such specialization would not be necessary, and the maximization prob—
lem would revert to one sketched below, which is conceptually the same

as the situation described in the preceding section for the prosecutor

65

(and hence that case is not pursued here.)

 

 

 

(v5.0) ((7330)
Figure 13. Defendant non-specialization.

However, if the second derivative Of f is less than that of the
indifference curves, a search for extrema other than at the boundaries
would yield a point of minimum, rather than maximum, utility for D. If
this latter situation obtains--for example, in the event that f is lin-
ear--D will choose the specialization that yields the higher level of
utility. Let u: - u(0,co) and u: - u(v3,0). Then uo - max {mini} ,
and uo is the utility D associates with trial. Examples of situations
which would give rise to specialization in v* or c, respectively, are

drawn below.

 

 

 

 

2 2 2
no 111 1.12
“ ‘\
l
(09cc) 1 x‘ ‘\u\ “2
t. \I‘ 1
‘11
f \
U1
0 V*
“2;":

 

Figure 14. Defendant specialization in case improvement; in consumption.

66

Clearly, the value of r" may be critical in determining the de-
fendant's approach to dealing with his case; for the first example a-
bove, by choosing successively larger values of r", could be transformed
into the situation depicted by the second example, thus causing D to
reverse his specialization, even though all other factors have remained
unchanged. Thus, differing values of components of r", in particular
values of e, the stock of evidence, could result in vastly differing
outcomes.v*, even though the difference in the r"—values might be very

small.11

Allowing r” to vary between values such as it took on in the

examples above, we can determine the special values r; of r" such that

1.1 -u

o for all r"(r'8' (specialization in v*)

OHON

and u = u for all r")r; (specialization in c.)

0
Then for r" - r3, uo - u: - u:, and either mode of specialization will
yield the utility-maximizing result. 12
Having considered his various outcomes to be obtained under the
assumption that he observes the trial opportunity set f, the defendant
also makes similar judgements concerning an alternative opportunity set,
which represents the set of possible outcomes if he should plead guil—
ty. 13 Pleading guilty offers a number of possible advantages to D:
in addition to reducing the maximum sentence he could receive, and to
avoiding some of the undesirable publicity and psychic costs of trial,
the defendant is able to conserve some of the resources which would

otherwise have been spent on the costs of case maintenance (for the

post-plea bargaining but pre-trial period) as well as the costs

67

associated with the trial itself. Thus the level of disposable resources

on the guilty plea Opportunity set g is

'I - .-
3 Wo+(W1Bl)tp+(lJ)Ytp,

where tp is the time from arrest to the entering of a plea of guilty.

The difference between R_and‘R' is then

3-3' -BO+ (WI-Bl) (tv-tp)+ (l-J) Y (tv-tp) ,

which we expect will be greater than or equal to zero, implying that g

will have a larger c-intercept than does f.

The defendant performs the same exercise of utility maximization
subject to g that he executed when f was being considered as the con-
straint. Since, (for the same reasons we included the restriction on
the prosecutor) we expect the v*-intercept of g to be less than or equal

to the intercept for f, we concern ourselves here only with the utility

value associated with (0,c;), in the expectation that u; - max (h;1,u;2)‘-
ugl, 14 where uél - u(0,c$) and u$2 - u(v3',0). The graph of the two

Opportunity sets which D confronts is the same picture as that of P's
Opportunity sets, save for the labels. Below is the graph for D, with
the utility levels indicated for each intercept.

(0.65)‘

(0.60)

 

 

 

T'

(V;',O) (V300)

Figure 15. Defendant's trial and plea opportunity sets.

68

After determining the utility levels obtainable at trial and by
plea of guilty, uo and u; respectively, the defendant chooses the Op-
tion which affords him the greater utility level. However, as before,
if u;<:uo, appropriate initial concessions by P again effected by
failure to exercise the full strength of his case, could make a plea of
guilty sufficiently attractive to induce the defendant to so plead. In
taking a closer look at D's comparison of trial outcome with that of a
plea, and at the determination of the concession necessary should the
latter Opportunity set not be an instant success, three cases are con-
sidered: (1) r3 >r'a', the special level of r" on f at which both trial
and consumption-specialization would result in the same utility to D;
(2) rg(r; and u31> u:; and (3) r3<r'8' and uéICug. These are shown
together in the figure below; then each is analyzed in the pages which
follow. As can be discerned from the sketches and from the discussion,
varying the values of r" and c; relative to co is sufficient to trans-

form one case into another.

69

   

 

 

 

 

(coca)
(0.co)
6,me (vim) (v3.0) (ma)
Case 1: to" ) r"8 Case 2: r3 4 :3 °
u'1>u2
O O

L

(0.C')
O

(0,:o

    

 

 

(v5.0) (vac)
O

, u n l 2
Case 3. r0 < r', and u; ('uo

Figure 16. Possible relationships of trial and guilty plea Opportunity
sets.

In the first of these situations, the defendant chooses to
specialize in c if obeying the trial constraint (since r3) r'B'.) and so
the availability of the guilty plea option merely offers him the 0p-
portunity of being able to Obtain more c-dwith the sameVboutcome--
than if he went to trial. 15

In the second situation also, when rg<jrg and when obtaining the

utility gain possible from pleading'guilty would cause a reversal in

70

D's specialization from v* to c, the defendant is clearly better off

by observing opportunity set g. In both this and the previously de-
scribed situations, no concession by the prosecutor is necessary in or-
der to make g attractive to the defendant: in the first case, D had not
intended to exert much (resource) effort toward improving his case in
any event, and so any resource saving made possible by avoiding trial
represents pure gain to him; and in the second, the resource saving
more than compensates D (in terms of utility units) for the less desir-
able outcome in. terms of v*.

It is in the third situation that a plea of guilty is not immedi-
ately preferable to going to trial, and in which some--in fact, perhaps
quite a large-~concession from P is necessary in order sobring accep-
tance of a guilty plea into the realm of the possible. Since in this
case rg‘(rg, the defendant is specializing in trial improvement (V*) on
f, and expects outcome v3 to be the result. Since uc',1( u:, the mere
fact of the resource saving possible on g relative to f is not suffi-
cient to pose g as a viable alternative to the latter, and concession
by P would be necessary to change D's current preference for f. Since

specialization will take place in either event on both constraints,

P must concede to D an outcome set which is at least as good as the out-

come D expects to Obtain at trial. This compromise is accomplished by

P's promise not to exercise his full strength r3, but rather to exercise
I!

some lesser amount rpromise' In effect, the compromise necessary-~with

the level of r" utilized by P being rg--that is the one which causes

the v*-intercept of g to be equal to that of f, namely v3. 16 It is

important to note that v3 does not necessarily denote an outcome of

71

zero, when stated in terms of v, the undesirable good. As originally
defined, v: was simply the best trial outcome that could be attained
via f, and hence an offer of outcome v3 by P may in fact be an offer
which does not let D off "scot free". 17
Searching for the magic value of r" at which D is indifferent
between Observing f or observing g leads to r;, the value of r" for
which g passes through the point (v3,0); that is, the value of r" which

results in the same outcome (not just the same utility level) as the

trial specialization on f, illustrated below.

 

 

 

(v3.0)

Figure 17. The adjusted guilty plea constraint.

is less
promise

Once that value of r; has been promised,ui.e., once r

than or equal to r;, D will be willing to consider pleading guilty at

v: - v3. Since this was also the outcome expected if D went to trial,
we can say that for cases in this third class, the outcome will be

*0 H H 0
v0, for rpromisef- r1u . this outcome can be attained by a plea of guilty,

and otherwise by trial. 18 Thus the switch function for D is identical

to the normal demand function for situations in the third class.

72

It is yet to be shown the manner in which the three cases dissect-
ed above descend into one general situation from which .two key parameters
are permitted to assume alternative values. To do so, we return to the
third case and recall that originally, i.e. prior to any concession by
P, uo was greater than us, where both depended upon the original value
of r", r3. That is, for the given value of r3, D preferred to observe
the trial constraint and to specialize in trial outcome improvement
while doing so. New let r3 increase until uo falls to us, represented
graphically by rotating f in a clockwise manner until the v-intercept
of f meets the same indifference curve as the c-intercept of g (the lat-
ter has remained constant.)

(o’cé) (Oscé).

(osco) (09cc)

    

 

 

 

 

'. A
( so: ( *,0) . * 5 V* 0
'3 v9 . ((1:84)) (1:56))
Case 3 Moving from Case 3

Figure 18. Case 3 and movement from Case 3 by increasing r3.

This can be accomplished algebraically by solving the equation of the

indifference curve at utility level u; for its v*-intercept, then solv-
ing the equation of f for the value of r" resulting in that same v*-

intercept. Labeling this value of r" by rdisc we can see that as r"

assumes values larger than rdisc , Case 3 no longer obtains, but rather,
19

Case 2.’

73
Allowing r" to take on still higher values, we arrive at Case 1.

The sequence of these moves is sketched below.

1 3, Case 3 (not drawn for Cases 1 and 2)
(0.co)

 

 

 

(0.Co) , Case 1
f, Case 2
f, Case 3
(v3, 0) (v3.0) (v3.0)
Case 1. Case 2 Case 3

Figure 19. Increases in r3 lead from Case 3 to Case 2 to Case 1
(Indicated by clockwise-pointing arrows.)

As can be seen from the preceding figure, D would engage in trial im-
provement specialization in Cases 3 and 2. However, in Case 2, that
preference on the f constraint is eclipsed by his switch to opportunity
set g, and hence consumption specialization on the latter constraint.

As r" is further increased and Case 1 is the relevant situation, the
preference now on f for consumption specialization is again overshadowed
by the dominance of g: this is the case in which the availability of

the guilty plea option does not reverse D's specialization, but merely

increases his disposable resources.
To summarize the conclusions from the case-by-case analysis: In
case 3, D is initially specializing in trial outcome improvement on f

and plans to have the charges against him disposed of at trial. If the

74

prosecutor is to interest D in pleading guilty, P will not only have to
make some concession, but will have to offer D a charge at least as low
(as what D otherwise expects to obtain from trial. Then for each level
of r3, v; - v3, and the switch function (in.Case l) traces out the or-

dinary "demand" function for v; in terms of r3, i.e.

Sdl(r")

 

'l
r. a r" H
5 disc

Figure 20. The defendant's switch function in terms of v*, for
r" é rdisc'

where the r; and v3 for each point of the switch function correspond to
the r" parameter and v*-intercept of a trial opportunity set.

Further, since the other two cases resulted in consumption spen
cialization on g, v* - 0 for both. Since Eases l and 2 both entailed
values of r3 less than r; - r"

disc.

defendant's switch function Sd2 uniformly zero (i.e. v*‘- 0) for

, we have the right-hand part of the

r"
1‘)!-disc

The switch function for the defendant is then v* - 0 for
r3 )‘min {r 3', r's') and v* - v3 otherwise, i. e. for r"( min (1” r1, r'}.
For the lower half-domain, the switch function is identical to the

demand for v; as a function of r3, and so we know that dv$ldr"‘i(h

75

for the upper half-domain, the switch function is a straight line.

«F7
Sdl‘v;(r")

"eh——

Figure 21. The defendent's switch function, with v5 a function of r".

 

 

r

Re-transforming v* back into v (to facilitate combining the
theory develOped here with that concerning the prosecutor), we write
D's switch function as

vm - Sd(r") - v - v;(r"), sketched below.

v I Sdz'vm(r")

l
l
I

/

r"

 

<1

 

Figure 22. The defendent's switch function in vm as a function of r".

In the next few pages, an example Of the defendent's switch
function is derived for a particular utility function and linear
constraint. It was not only necessary for the utility function to
exhibit the prOperties required by the theory -- such as appropriate
signs of partial derivatives -- but also that it be manipulable
throughout the steps required for the derivation, and that the final

function attained not require statement in implicit form. The result

76
of considerable searching, the following function was the simplest one

found which met the above criteria. Let D have the utility function,
u(v,c) = (v+1)“1 + ln(c+l)

satisfying the conditions that
Q2
ac>°

2
3c

2
and _c_i£>0’ dc<0.

And suppose also that he confronts
pcc '+ r" (G'- v) = 3'

- u —_ _ v
and pcc + rpromise (v v) .3

as trial and guilty plea opportunity sets, respectively, where these
Opportunity sets can be derived from production functions of the form
used in the example of section 2.2.

For a first step, it is necessary to state ui, ui, and u'1 for
the given utility function and constraints. Since the model provides
that D always specialize in v-reduction or in c, it is necessary only
to solve for the corner solutions and for the point of discontinuity
which severe Case 3 from Cases 1 and 2.

Recall that for values of r" falling into Case 3, the switch
function was the set of ordered pairs (r3, v0). Substituting from
the trial constraint,

-_-I " -
vo v R/ro (assuming Pc 1)

and so, allowing alternative values of V0 and r3,
. n =—_ n
Sd1 v(r ) v .R/r

"L n
for r ‘trdisc°

77

On the trial constraint, the utility derived from specialization

in c is
ul = u(V c ) a (Var1 + ln(c +1)
0 ’ o o

and from specialization in v—improvement is

2 -l
uo - u(v0,0) = (vo+l)

where V0 is the intercept of f. Substituting from f into u: and u:
the expressions
c = R and v = G'- R/r"

o —- o -—

(assuming Pc = 1) we have

u =- (?+1)'1 + 1n(R+l)

and u

0'00?"l

- (3 - yr" + 1)"1

Similarly, on the guilty plea constraint, specialization in c
yields utility level

:1. _ I a "" “1 I
uo u(v, co) (v+1) + ln(co+l).

Substituting from g into u'l,
o

uél - (7741)"1 + ln(_R_'+l),

to use the notation from the theory.
For"r")"r£isc,ixthe defendant.is specializing in consumption and

v - 3, since trial outcome is surrendered completely. Thus Sd2 = v(r")

disc'

Combining these results, the defendants switch function is

- 3 for r")*r

- H _ - _ u H n
Sd1 v(r ) v R/r for r firdiac

‘ _ n a '— - n n
_Sd2 v(r ) v for r :>rdisc

It remains for us to determine the value of rdisc' As defined

78

earlier, raisc - r3 was the special value of r" at which D was indif-

ferent (on the trial constraint) between specializing in trial outcome

improvement or in consumption. To determine rdisc we set us equal to

uO and solve for r", which is the value we seek. The requirement that

u; - u: is equivalently stated

1

v-R/r" m1 +1

 

+ ln(c+l) I + ln(Rf+l)

-1_
3+1
where the value of V0 corresponding to r; is (3'- R/rg) on the g con-

straint. Combining terms and solving for r2,

 

 

 

R
V-.’ +1=_ _11
I. '
rdisc (v+l) + ln(R +1)
so
__ 1 - _} = ln(Rf+l),
v-R/rdisc +1 v+l
and

(3+1) — (Lg/raises)

 

_, __ ln(R'+1) .
(v+1) (veg/r318c+l)

Combining terms in the numerator of the lefthand side of the equation,

3h."

 

 

 

d1“ - ln(R'+1) .
(v+l) (v+1-R/rdisc)
. R/r"
Then disc v+l _ R/rgigc
(v+l) ln(R'+1)
so that R "
rdisc (v+1-/rdisc)

(3+1) 1n(1_1_'+1)

n '_ H _
rdiscv + rdisc -3

79

and 1 __
_ + 1 R = r31 (v+l)
(v+1) 1n (1331) 3°

 

Combining terms on the left and dividing by (3+1),

r" _ l+(v+l) 1ng'+1)
disc (3+1)2 1n@'+1)

 

g .

In terms of the graph notation of the theory

1 + (3+1) ln(cc')+l)
. C

 

r" -
“8° (3+1)2 ln(cé-i-l) °

N - N
This gives us the point of discontinuity rS rdisc'

The switch function of the defendant has been found to be

;'— R/r” for r"f-

 

H
S a —- rdisc
d —
H H
v for r > rdisc
'— v
where n 1 + (v+l) ln(co+l)
r = . c .
disc 0

(3+1)2 1n (c3+1)

. Again, as with the example given for the prosecutor, the characteristics

of the model can be verified here. Clearly, when r" is large, i.e.

when the prosecutor's strength is great, the outcome will be large also.

disc

surrender outcomes comprise more of the set of possible outcomes, thus

When the defendant's resources are great, r is larger, and the non-
bettering the plea bargain outcomes for at least some values of the in-

dependent variable representing relative strength.

Joining the theory of the defendant with that of the prosecutor,
the outcomes of plea bargains can be circumscribed and, subject to im-
position of a bargaining theory, predicted, as is done in the section to

follow.

NOTES to 2.3

1His right to trial is guaranteed by the Sixth Amendment. Al-
though it is most always possible for a defendant to plead guilty, this
act must meet the conditions of being intelligent and voluntary, as
determined by the judge. And although it is typical that the plea be
made upon either the promise or upon the expectation of some sort of
leniency in some subsequent stage of the criminal justice process (such
as sentencing), such leniency is not guaranteed. See Remington (1969).
Finally, the rights of the defendant to have his plea of guilty accepted
and honored--given that the plea is intelligent and voluntary-—have
been affirmed by the courts. McCoy (1966), Santobello (1971).

 

2The case in which azu/av2<0 is explored in an appendix. The
results are analogous to those for the prosecutor, which we derived
earlier.

3At a seminar in which I spoke on this topic, I was surprised to
have an example of this aspect of a utility function--that is, increas-
ing immunity--supplied to me by a colleague. Having had his car stolen
(and returned) ten times in the past year, he said he found that the
disutility of each additional incident declined as the number continued
to increase!

40ne notable exception--in addition to that of mean-variance
portfolio analysis--is the labor-leisure choice of an individual. Howe
ever, a perusal of that literature in microeconomic theory revealed no
discussion of the sign of the second derivative, nor even a convincing
argument as to why the indifference curves should bear the shape which
is typically assigned them in textbook drawings, i.e. positive first
and second derivatives.

5This can be verifigd by substitution of v* - 3'- v into the
earlier assumption, where v represents the maximum level of v possible.

6Functions satisfying the conditions for first and second deriv-
atives of utility with respect to c and v* may exhibit a number of pos-
sibly awkward characteristics in their indifference curves. For example,
the sign of their second derivative may be positive for low levels of
U, than zero, then negative; it is even possible for the function to
have points of inflection along a single indifference curve. For this
reason, the utility function considered here is restricted to be one
having strictly convex or concave indifference maps. For other cases
the analysis can be extended and suitably modified to obtain the desired
results, though the task of doing so may be considerably more laborious.

7The defendants are thus the "plungers" which Tobin (1958) dis—
cussed and then declared inoperative. For those earlier plungers,

~81

however, the choice was between return and risk; consequently, a plun—
ger's attitude toward the bad good revealed quite explicitly his atti-
tude toward risk, namely, that though he disliked it, he became more
immune to additional doses as his "stock" of it increased. In contrast,
the defendant to which this utility function belongs, is more immune to
additional v as his amount in hand increases; and because we have as-
sumed a positive second derivative of utility with respect to v, the
defendant is ng£_risk—averse, but rather favors risk with respect to v.

8One interesting paragraph in the American Bar Foundation study

of several court systems, including the Recorder's Court of Detroit,
noted that, "The most common criterion used in selecting attorneys for
the indigent is need.,.'Young and Old attorneys needing money will do

a very thorough job of investigating, preparing, and arguing the case.'"
(McIntyre 1967: 128).

9Conceivably, most any of the factors could be related to others
pertaining to the case, for example, the seriousness of the offense to
the costs of trial; or income with the type and seriousness of the of-
fense, since certain income groups have higher likelihood of being in-
volved in particular classes of offenses (e.g. white-collar workers
with embezzlement), some are more likely to be out of jail pending
trial, etc. Because introduction of these interrelationships would
vastly complicate a model already suffering from that malady, they are
omitted here. They could be introduced later, and while they would
render the derivation of the results of the model more laborious, their
introduction should not substantially alter the results.

10Similarly, if an employed defendant goes to jail and his family
subsequently goes onto assistance, his disposable resources might ac-
tually increase. With the exception of this last possibility, the insti-
tuting of release-on-recognizance programs, in which defendants do not
have to post bond (but do have to promise to return for trial!) could
be expected to increase disposable resources of defendants.

11For example, suppose D has no time-dependent income and no legal

fees (and that there is no prohibition of his expenditure on case-improve-
ment.) The resource and evidence effects of delay may cause the special-
ization to reverse under different values of t , e.g. f could rotate
outward sufficiently to reach (vf,0) as its infercept, and thus result

in specialization in v*:

82

(o,co)7

  

 

 

 

(V333) If ' (vi 0)

Figure 23. Specialization in v* and possible reversal with time passage.

12The derivation of r" is an follows: Since we have assumed the
signs and magnitudes of the second derivatives to be such that only cor-
ner solutions are possible for utility maximization, and since we have
fixed the c-intercept of f (by assuming a given level of R) we can
determine the level of utility obtainable by specialization in c, which
subsequently determines the corresponding indifference curve for that
utility level. On that indifference curve, we determine the v*-inter-
cept; and with that information, one simply "solves" the function f for
the value of r" for which f passes through the desired v*-intercept.

13Normally, the offense to which he would enter a guilty plea would
be one having severity less than or equal to that of the original charge.
However, under some circumstances a defendant might even plead to a
higher charge, although that would be the exceptional situation, if he
wished to avoid a label such as that of a sex offender and is willing
to take a chance on receiving a harsher (or equally harsh) sentence
in order to do so.

14Were that not the case, i.e. if u; - “$2, then the plea constraint
would have to lie whollynbelow the'trtal cOnstraint, contrary to our
understanding of the relation of co to c;.

15This may be interpreted as representing the choice of the de-

fendant who pleads guilty purely to save money, since the outcome is not
necessarily better (or worse) than that which he expects from trial.

16Such an action is possible and rational for a prosecutor to

undertake, for even after conceding to D the outcome D would otherwise
expect from trial, P would have conserved hi§_resources and-~as is la~~
ter shown-«may still be obtaining a better outcome than what he (P)
would have expected from trial.

83

17This is consistent with the idea of there being certain common

points of settlement. For example, one of the prosecutors interviewed
in Detroit said that in pretrial negotiation, they try not to go below
the corresponding attempt charge (e.g. for larceny in a building, not
below attempted larceny in a building) unless there are special circum—
stances which warrant additional concessions. Restated in terms of
gains to the defendant, D can expect a reduction of approximately the
difference between the original charge and the corresponding attempt
charge, with the amount of the concessions being greater or less, de-
pending upon special circumstances and the bargaining skill of the at-
torneys (or whomever else might be involved in the bargaining.) If the
outcome D expected from trial was equal to that obtained from a plea

to attempt, he will surely be interested in discussing further the pos-
sibility of a guilty plea.

18
For circumstances which align with those of this third class,

the procedure for determining r; is as follows: knowing the form and
c-intercept of f, one solves for r" the equation(s) representing g,
with the values v* - v* and c - 0 (and possibly also v* - 0 and c a c',

dfipigding upon the fora of g) substituted in. The resulting valug of o
r r O
- m

19
It should be noted that the set of r" for any of these cases

could conceivably be the null set. In particular, if c is sufficient-
ly close to c', there will be no r" satisfying the conditions of case 2.

For example, f c <10; due to a longer pretrial queue, case 2 would
revert to Case 1.

84

2.4 The Pareto Optimal Region and the Outcome of Negotiations

 

In the preceding two sections, functions were derived which ex-
press the minimally acceptable set of bargains for each of the parties
considering the option of a plea of guilty. Each function can be seen
as dividing the relative strength-outcome plane into two regions, one
of which constitutes the set of acceptable plea bargains (for given val-
ues of relative strength) and the other of which represents plea outcomes
sufficiently undesirable to make trial the preferred method of dispos-
ing of the case.

With apprOpriate adjustment of the variables r' and r" used above
to represent relative strength so that each can be stated in terms of
one variable r,1 we can portray these two functions on the same graph
and easily view the four possible permutations of acceptability of out-
come: (1) Both parties prefer a plea bargain to trial; (2) both parties
prefer a trial to a plea of guilty; (3) the prosecutor prefers a plea of
guilty to trial, but the defendant does not; (4) the defendant prefers
a plea of guilty to trial, but the prosecutor does not. The four re-

gions are indicated in the graph which follows:

Sp(r)
__ (3) (22_

V’j' ————— T

S (r)
v o
m I,“ d

(4)

4

 

__ w.

r-o
Figure 24. The pareto Optimal outcome set and other regions.

85

Clearly, the pareto optimal region is region (1) and we would ex—
pect in general that outcomes would fall within this region. The func-
tions bounding the region are the switch function Sd(r) and Sp(r) de-
rived in 2.2 and 2.3, with both stated in terms of one relative strength
variable r.

In addition, in the jargon of section 2.1, the switch functions
represent the incremental utility functions at levels of zero increment
for the party whose switch function is under consideration. That is,
since the switch functions represent the ordered pairs (r,vm) for which
the appropriate party is indifferent between resolution by plea or by
trial, the additional utility gained by a plea at that level is zero.
Forays into the pareto Optimal region from the boundary will yield in-
creases in utility to one party and decreases to the other, but both

incremental utility functions will have associated with them positive

 

(or on the boundary, zero) utilities at any point in (1).
For the prosecutor, the incremental utility function is

U*(vm) = U(vm,xm) — U(vo,xo)

where the notation is as before, i.e. v0 and x0 are the trial outcomes
obtainable (and are given dependent upon a certain fixed value of r) and
xIn is the outcome on the remainder of P's caseload. For the defendant,
the incremental utility function is

u*(vm) - u(vm,c$) - u(v$,0)

and c; is the amount of good C obtainable under conditions of a plea of
guilty and v; is the level of v obtainable via trial. In the event that

1 .—
uo)u:, the latter term on the right would be u(co,v).

86

The pareto optimal region can then be stated in two alternative
ways: It is the set of pairs (r,vm) such that Sp(r)§vm§._Sd(r) if
Sp(r)f§Sd(r); or equivalently, it is the set of (r,vm) for which , for
each value of r, U*(vm),>. 0 and u*(vm)aO.

Adopting a split-the-difference theory of bargain outcomes for
the reasons cited in 2.1, we anticipate the set of possible bargain out-
comes for differing levels of r to be

vm - )Sp(r) + (1-)\)Sd(r)

where B:l-?!is the proportional split of the pareto optimal set. That
is, we take the outcome function to be a linear combination of the two
switch functions, with weights according to the split of the amount of
v to be distributed. Further, since the defendant's function cannot

be assumed continuous in the general case, but rather is piecewise con-
tinuous in the two sectors of its domain, the outcome function needs

to be stated in a manner similar to that of D's switch function:

h Sp(r) + (l-A) Sd1(r) for r"é r;

m ASP“) + (1-71) Sd2(r) for r") r;

where Sd1 and Sd2 are the pieces of the defendant's switch functions
in the left- and right-hand sectors of the domain of r and r; is the
special level of r" at which D is indifferent between specializing in
trial outcome improvement or in consumption.

Of interest at this point in the development of the model is the
question of how the Pareto optimal region and the outcome function shift
in response to the suggestion of alternative values of the key para-

meters 0

87-

Factors which contribute to the determination of relative strength
for a given defendant will determine the actual value of r which is rel-
evant (and consequently determine the specific bounds on the plea possi-
bilities): alternative values of component variables of r cause "move-
ment" along the switch functions and the outcome function. For example,
if a defendant adds another arrest to his record during the period pend-
ing resolution of the case which we are considering, we can expect this
fact to be reflected in a higher value of r, dictating a new reference
position on the outcome function. The same would be true for other
determinants of r, such as the amount of evidence initially at hand.

For these parameter changes, neither the location of the Pareto optimal
region no: of the outcome function has changed--only the relevant loca-
tion in the region or on the function has changed.

However, when two defendants, identical but for the parameter of
one of the contributors to r, are compared to one another, the Pareto
optimal region and corresponding outcome function would appear to shift.
For example, if we believe it more advantageous to be a white defene
dant than to be a black one, we would expect the region and outcome func-
tion for the latter to lie to the left (northwest) of that for the
former, as illustrated below.

'v‘

 

 

 

Figure 25. Switch functions and outcomes resulting from differences among
defendants. Dotted curves denote D's switch function; dashed curves, P's.

“38.

Thus for each level of r, the black defendant will fare worse than the
white one, although the outcome will clearly be sensitive to the amount
of evidence and the other factors which determine the level of r. To
the extent that discrimination exists, it does not supplant considera-
tion ofevidentiary and other factors (such as previous record) relevant
to court decisionmaking; rather, it shifts the scope of the alternatives
to which those considerations are addressed. A similar analysis can be
applied to the impact of other characteristics which vary among defen-
dants, and the earlier analysis to alternative situations for one defen-
dant.

Changes in variables introduced through the opportunity set can
be traced through the determination of the switch functions to assess
their effect on outcome. Other things being equal, supplying the pro-
secutor with greater resources would translate both the trial and plea
Opportunity sets upward; not only is the outcome from trial expected to
be higher, but the equally-desirable plea bargain increases as well.
This is true for each value of r and so the increase in resources has
shifted the entire switch function for the prosecutor upward (to the
left). Since no such corresponding change has occurred for the defen—
dant, his switch function remains in its former position, and the pareto
optimal region contracts. If the proportion of the split remains the
same as assumed before, the net effect of the higher level of resources
is to shift the outcome function to the left--although not by as much as

the shift in P's switch function:

.39

 

 

Figure 26. Leftward shift in outcome function due to greater pro-
secution resources. ‘ '

Similarly, increases in defendant disposable resources, through negative
tax or subsidy programs or whatever, would shift Sd to the right, re-
sulting in an expanded bargain region and an outcome function to the
right of its original position.

As noted in the section on the defendant, the effect of time can-
not be specifically determined without quite a bit of information on
the defendant, and then could vary among defendants. The effect on the
prosecution is to shift his switch function to the left: neither the
cost effects nor the evidence effects of delay is helpful to hims The
defendant, on the other hand, may find one or both of those effects to
be beneficial; we especially expect the evidence effect to favor him.
Suppose that the effect on the defense is damaging: Sd shifts to the
left. Since Sp has shifted to the right, the combined effect from both
parties is difficult to determine and will depend upon the comparative
damage inflicted by the delay. On the other hand, if delay is helpful

to D, his switch function shifts to the right along with P's and the

.90

outcome function does so also. This latter situation seems to be the
belief among law enforcement officials as well as among defense attor-
neys.

Continuing the same manner as above, it is possible to analyze
the effect of differing circumstances confronted by the prosecution and
the defense. Those which favor the defense shift his switch function
(and ceteris paribus, the outcome function) to the right; ones which

favor P shift his switch function (and ceteris paribus, the outcome func-

 

tion) to the left. Depending upon the joint effect on the two switch
functions, the Pareto Optimal region itself shifts, expands or contracts,
or does both. With the implicit assumptions involved in adoption of a
split-the-difference theory of outcome--especially those concerning the
probability distribution of outcomes within a bargain interval for a
fixed value of r——one can posit the outcome function to be descriptive

of the set of possible settlement points for pleas of guilty.

~91

NOTES to 2.4

1Further discussion of possible transformations follows later in
this section and the next chapter. For the moment, let one example
suffice: if we define r = -r' = -l/r", we can speak of the outcome as
viewed by each party in terms of the variable r alone.

3. A MOdel of Plea Bargaing: Empirical Testing and Results

3.1 The Transition to an Empirically Testable Model

 

In the preceding chapter a model was posited to determine the
bounds on the outcome of mutually agreed upon pleas of guilty. In ad-
dition, under the assumption that the amount of v within the pareto
optimal interval was to be split between the parties in a fixed pro—
portion, an outcome function for v in terms of r, the relative strength
variable, was determined. In this chapter both of these descendants
of the switch functions are subjected to empirical testing using the
specific relations introduced in earlier sections as examples for the
prosecution, the defense, and the outcome of two—party negotiations.

The first task, that of testing the validity of the switch func-
tions as bounds on the set of plea bargain outcomes, presents signifi-
cant problems due to the lack of appropriate statistical tools. Much
of the thrust of applied statistics and econometrics has been devoted
to the search for ways of expressing relationships in terms of mea-
sures of central tendency and corresponding measures of dispersion
from them. In contrast, little attention has been paid to the com-
plementary issue of placing outside bounds on sets, and attempting to
characterize relationships by functions which (together) contain the
set of relevant points, rather than by one function which approximates
them.

There are two notable exceptions to the indictment of the para—
graph above. The techniques of quality control and its subsidiary con-
cerns attempt to define bounaries on regions of acceptable outcomes,

but require foreknowledge of the function whose approximation the

92

93

bounding functions enclose and the maximum acceptable tolerance. Hence
these are really problems of finding boundaries subject to constraints
which are tied to preViously known functions describing (desired) cen-
tral tendency: the bounding functions are determined from infor-
mation (or requirements) on central tendency, rather than from the data
directly. The second possible resource for determining bounding func—
tions is a technique suggested by Michael Farrell, who determined an
isoproduct curve by connecting the set of outside points (i.e. ones
lying farthest to the southwest on a graph), using crossectional av-
eraged data on manufacturing. 1 However, his method clearly could be
disastrous for use with rougher micro cross-sectional data, in which
one or more outliers could unfavorably affect the results--as would

be the case if applied to data on the trial versus guilty plea choices
made by individuals. Thus, the statistical literature does not pro-
vide much guidance for determining the validity of the theory of the
functions which bound bargained outcomes. Nevertheless, some graphical
evidence is presented in which cases are segregated by different cri-
teria and the relative strength-plea outcome relationship can be Ob-
served.

The second task, that of testing the formulation of the outcome
function derived from a split-the-difference theory of negotiation, is
a bit easier and is performed using multiple regression, following the
imposition of some additional simplifying assumptions. Since the func-
tion to be estimated is in general expected to be discontinuous--at
least nondifferentiable at one point--a modified search technique is

employed which is similar to the approach of Quandt (1958, 1960) in

91.-

his work on shifting regression regimes for use in determining the
point of discontinuity of the outcome function.

As developed in the second chapter, the outcome function for a
split of the Pareto Optimal intervals in proportions A: (l-)§) was

determined to be
v1n = fiSp(r) + (l-A) Sd(r).

Since Sd is defined differently when D is specializing in trial out—
come improvement than when he is specializing in consumption, the out-
come function must be stated differently for these two circumstances

also:

ASP“) + (l-)) Sdl(r) for rérdisc.

hSp(r) + (l-A) Sd2(r) for r) rdisc.

sc. 1’

is the defendant's switch function to the left (i.e. for low values

where rdi is the level of r corresponding to min r" r's'} and Sdl

of r) of the discontinuity, and S 2 is his switch function to the right

d
of that point.
For the particular utility functions and opportunity sets posited

in earlier sections, the outcome function is
vm - (R/w) “4"), & hp(r) + (l-A) 3
(R'/W)P/‘ “W?
+ (141)“: hd(r)

_ ” u
where '6' l for rérdisc.

and Y- 0 otherwise; wh'ere«'hp(r) and hd(r)
are defined so that hp increases and hd decreases with increases in r;

and where the definitions of all other terms are as before. As derived

in the example of section 2.3 the point of the discontinuity of 8d and

95

hence of the outcome function is

[(541) ln(c;+l) + l]

 

r" =
disc (3+1)2 1n(c;+1)

Since the function as stated above contains too many parameters
to be estimated at once, it is necessary to make assumptions concerning
the values of some of them, and about functional relationships among
others. Once those assumptions have been made, the outcome vm is a
function of U, R, R', 3, R, and r, all of which have known values for a
given defendant, subject to the adoption of some additional assumptions
such as those involved in sorting out variable from fixed costs in the
prosecutor's budget.

With the exception of X, each of the variables appearing in the
outcome function is subject to scaling problems. For the two representa-
tions of the prosecutor's resources, the problems (to the extent that
they exist) may be traced to assumptions made in determining the levels
at fixed, attributable, and variable costs, as well as to the defini-
tion of w, the price per unit of the good introduced as variable
input to evidence production. Although it would be desirable to esti-
mate the values ofdandF, rather than to assume them, it is not pos-
sible to do that with the information available, so the assumptions en-
tailed there are another source of possible error.

Similarly, judgements about the disposable resources of the de-
fendant under conditions of a trial or a plea bargain pose scaling
problems due to the initial incompatibility of most income data with
the definitions of R_and.§f employed here. The adaption for use in

applying this model requires assumptions about its utility function as

96

well as about the time period relevant to his decisions. The scaling
problems surrounding the use of 3 are less obvious: recall that 3 was
defined to be the worst outcome (as viewed by D) that could occur at
trial, defined by his trial constraint. In using data on maximum or
average sentences applicable to all defendants, we may not be using the
appropriate value of 3 for a particular defendant, or for the particular
charge against him. 2

Finally, the introduction of the h-functions poses problems not
only of scaling but of multicollinearity among the terms. Since no
theory of the determinants of relative strength has been prOposed here--
its contributing variables have been listed but not integrated within an
economic theory--it is impossible to say what a correct measure of that
variable should be. The familiar question of ordinal versus cardinal
measures is also relevant here. Nevertheless, an index of relative
strength is utilized in the estimation; the construction of that varia-
ble is discussed in section 3.2.

The spectre of multicollinearity appears because the same source
of information about relative strength is used in both the first and
third terms. This is in contrast to the theory of 2.2 and 2.3, in which
the first term uses the defendant's subjective estimate, and the third
term uses the prosecutor's. However, the use of one measure is neces-
sary not only for obtaining the required data, but also within the
theory (as mentioned in 2.4) for permitting the superimposition of both
switch functions upon the same space. The multicollinearity exacts its
toll in the variance of the.estimates of coefficients, which may or may

not be a hardship.

97

However, the appearance of r in both terms also adds some com-
plexity to the determination ofavm/ar, the incremental effect upon
outcome of a stronger prosecution case. This is discussed further in
section 3.3. Let it suffice to say here that with the exception of
‘awmflav, none of the partial derivatives is aconstant--and thus able to
be read from the estimated equation as the coefficient of a lone term in
the regression-—but rather, each is itself a function. Consequently, de-
termination Of the signs of partials in order to verify from empirical
results the contentions of the theory must involve additional computa-
tion. Nevertheless, if those computations are performed, some infer-
ences can be made about the effects of a number of variables upon plea
bargain outcomes.

Also, momentarily ignoring the possible scaling problem with 3,
it is possible to read directly from its coefficient the proportion of
the bargainable difference accruing to each of the parties. Since the
coefficient 3 is the proportion of D's reservation price which P is able
to exact, values between .5 and unity would suggest that the prosecutor
is faring better than is the defendant.

The results of the above tests lend themselves to interpretation
both in terms of the validity of the model as a whole, and in terms of
the value of the parameter A which describes the proportion by which
the two parties split the difference between their reservation prices.

The data available, problems encountered in the empirical work,
presentation of the results, and their interpretation appear in the

following sections.

98

NOTES T0 3.1

l.

Farrell chose the points to connect with one another by
checking to see whether inclusion of the i-th point would cause the new
set of connected points to jut in and out again, i.e. he required that
for the i-th point to be on the boundary, the function thus obtained
must retain its convexity with respect to the origin. Since he used
relatively nice averaged data, his results presented a semblance of
reasonableness, thus recommending the technique for use in similar set-
tings.

2For example, if it is customary for any offense carrying ten
years or less for a plea charge to carry half or less of the penalty on
the original, then for those offenses, the appropriate value of v
would be one half of the maximum on the original charge. This sort of
problem cannot be avoided where indeterminate sentences are used, with-
out going to the (possibly inaccurate) egtreme of interviewing defen-
dants to determine their opinions about v.

3The results on the latter question provide an interesting com-
parison with findings of Daniel Hamermesh (1973) in his test of a split-
the-difference model applied to labor-management contract negotiations.
Among other problems associated with the methodology used, Hamermesh
tested only for a split of exactly fifty-fifty. Nevertheless, he was
able to reject that hypothesis, finding that final settlements were
much closer to management's initial offer than to labor's. It would be
helpful to test his theory in a manner which would search for the
proper proportion to describe the split, or repeat the procedure he
used under alternative proportional splits. Since this is the only
empirical work of which I am aware that uses a split-the-difference
theory, there are no others to which the results determined in the
present work could be compared.

99

3.2 The Data and Its Discontents

Because much of urban crime--crimes directly affecting the
physical state of persons and prOperty (as opposed to more exotic of-
fenses such as skyjacking, or crossing state lines with intent to in-
cite a riot, which are federal offenses)--is the purview of state and
local law, usually administered by county or district courts, respec-
tively, there exists the potential for great variation from one insti-
tutional context to another. Consequently, the transition from the
theoretical model to one to be tested in a Specific institutional set-
ting must take into account the peculiarities of applicable laws, rules,
and traditions. 1 This section includes a brief sketch of the Re-
corder's Court of Detroit, from which the data used in the following
section was obtained; it also contains a description of how the data
were collected and utilized, a summary of some aspects of the individ-
uals whose cases were drawn in the sample, and finally, a list of
other steps taken to prepare the data for use in testing the theory
of plea bargaining.

Recorder's Court, born in nineteenth century America, is beset
in the twentieth century by a spectrum of challenges and problems not
unlike those found in other major cities. In fact, one might even say
that the Detroit Recorder's Court is close to the prototype of the
urban court. 3 Its geographic jurisdiction encompasses the portion
of Wayne County which is also within the city of Detroit (hence ex-
cluding most of the suburban and rural areas nearby); as one of two

county courts (the other in the nonDetroit portion of the county), its

100~

case jurisdiction is over offenses against the laws of the State of
Michigan. Along with many other American courts, Recorder's Court
found itself in the mid-Sixties mired with an excessive backlog of
cases, facing complaints from all parties to the operation of the crim-
inal justice system, including the public, police, defendants, prose-
cutors, lawyers, and others. The response to this dissatisfaction was
almost as diverse as the problems which spawned it, as visiting judges
were brought in to hack away at the backlog--outnumbering the regular
judges by three to one--and much general streamlining of procedures was
begun which is at least acceptable, if not commonplace, today. How-
ever, this reorganization has not eliminated the collection of prob-
lems which initially arrive on the court's doorstep, nor changed the
basic processes which are carried out by the prosecutor and the court.
One major life process of a court is the practice of plea bar-
gaining, the method by which the great majority of a court's cases are
disposed. For most of its history--which in the United States dates
back to the Salem witch trialsé—plea bargaining has been a process con-
firmed in law only by its mere existence, rather than by any justifi-
cation in statute law or case precedent. Although the last three years
have seen a change in this perspective on plea bargaining as an extra-
legal process, 5that metamorphosis was only beginning at the time Re-
corder's Court initiated its reforms of the plea negotiation process.
Although each case receives initial processing by several of a
large number of persons involved in entry-level research and evaluation
of cases, the actual discussion and possible settlement of cases before

trial is handled by a small unit of three persons who together with

lOl

supportive staff, specialize in exactly that work. Every felony case

is scheduled for a pretrial conference, as the session is called, and
although a defendant may request that no reduced plea be entered (if

he does not wish to consider a plea of guilty) most defendants or their
counsel attend the meeting, and most subsequently plead guilty. The
prosecutors who work in the Pre-Trial Division, as well as other par-
ticipants in the criminal justice system, believe that the "new" (seven-
year-old) system breeds greater uniformity in the handling of pretrial
settlements, as well as greater opportunity for flexibility when war-
ranted by the circumstances of individual cases. This flexibility
ranges from the very deliberate consideration of factors relevant to

the rehabilitative potential of the defendant, such as his age, previous
criminal record, relationship to the labor force, etcetera,to the
special difficulties experienced by out-of-towners in a city far from
home.

However, the three pretrial prosecutors also operate under a set
of rules concerning the general handling of cases in the absence of
special circumstances. Those rules, together with the coordination
possible in such a small group in which one member also works as an
administrator of the other two,provide for greater accountability than
was possible under the earlier setup in which a hundred prosecutors
carried out those tasks.

In addition to introducing centralization of plea negotiations,
the advent of the pretrial conference as an integral part of the trial
process was accompanied by other changes as well. For those defendants

who decide to plead guilty, the case can be sent to (abbreviated) trial

102

in a much Shorter period than would otherwise have been necessary. In
fact, for the period relating to the data used here, it was the prac-
tice of two judges to handle such cases more or less full-time, 6thus
creating a somewhat separate set of machinery for the remainder of the
court's processing of the guilty plea-defendant's case. The alterna-
tive routes for a case are sketched in the possibility tree diagram be-
low, along with the percentages of cases along each branch. 7 Figures
in parentheses denote percentages of those considered at the point of

branching, that is, they refer to the total caseload.

 

Plead '
guilty Trial 65.1
5-”
Case lJS/’/////
enters Dismiss #1 15.2
docket ////////
‘33\2\\\Not plead Dismiss #2 ' 12.7
guilty ’////////
\\\\\\\\Trial ‘0 Acquit 2.7
(19 7)\\33\i\\\ b.
Verdict ba‘
(7.4) “’3
gGuilty 4.7

Figure 27. Alternative paths for a case and distribution of cases

by mode of resolution.
Dismiss #1 denotes methods of dismissal before trial, which are dominated
by dismissals upon motion of the prosecutor. Trial in the case of the
defendant pleading guilty is a short and usually perfunctory affair, and
should not be equated with trial in the case of the defendant

who does not plead guilty. Dismiss #2 relates to dismissals after

103

cases have entered the trial process-~having left the sphere of prosecu-
tor discretion--and are derived principally from cases in which dis—
missal was made by the judge (not upon recommendation of P). Verdict
represents the situation in which a guilty or not guilty verdict is
rendered, and the last two branches are self-explanatory. It should

be noted that, although the Annual Report does not separate the sub-

 

groups, defendants may be found guilty of the original charge or of any
lesser included ones, 8 so that the clumping of these together im-
plies some loss of information concerning the alternative which a de-
fendant confronts when considering trial as an option. It is also
worth mentioning that if there is any likelihood that guilty defendants
tend to plead guilty more often than do not guilty ones, the lower-
paths in the preceding sketch will represent a more rosy picture than
an arbitrary defendant would face.

In Recorder's Court, the object of bargaining, exchanged for the
promise of a guilty plea, is usually a reduction in the charge to which
the defendant would plead; the average reduction for defendants in the
data set used was to slightly less than four-tenths of the original
charge, where the ratio taken is that of the maximum penalties of the
plea charge to original charge. However, concessions may-take other
forms as well, forms which represent "in kind" transactions and which
did not appear in the case files drawn in the sample. For example, it
is not uncommon for sizable concessions to be made by a prosecutor upon

assurance that a young defendant is willing to enter military service,

104

in order that a prison sentence not prevent or deter that activity.
Because it is possible for the entirety of concessions by either side
not to be fully apparent from perusal of case files, the results de-
rived from such data should be accepted with a small bit of reserva-
tion; however, they do tell most of the story.

The information used in this study was drawn from the docket

sheets of the Pre-Trial Office, and corresponding files of the Prose-
cutor (the main files). The first forty property cases were drawn from
the pretrial docket sheets for each of February, April, June, and
August of 1972 to comprise the original sample. Although details re-
garding the composition of this group are presented in the appendix, it
is noted here that the choice of dates was designed to largely avoid
seasonality problems (such as more of certain crimes at Christmastime,
etc.); to be after the initiation of a Standard evaluation form pro-
viding most of the informational items of use here; to be recent enough
to be of some relevance for policy consideration; and sufficiently old
for most of the cases to be closed, thus providing information on dis-
position.

It was decided that the sample Should be restricted to felony
property cases for several reasons. Paramount among those reasons
was the heavy reliance necessarily placed upon the complainant in per-
sonal injury cases: if the complainant refuses to cooperate at any
time, the case is essentially over. Further, for a number of offenses,
such as with rape, the complainant must be exonerated before the defen-
dant's guilt can be considered, thus adding a "stage" to the process

not included in the original model. Finally, since there is some

105

speculation as to the rationality of Offenders in many classes of per-
sonal injury offenses, 10 it was not clear that the model's most basic
assumptions could be met for randomly selected personal injury cases. 11
Although the recordkeeping of the prosecutor's office in Detroit
stands out among American cities as quite thorough, certain items of
critical interest here were occasionally left blank on the forms; be-
cause of these missing pieces of information and because of some other
minor problems detailed below,12 it was necessary to eliminate some
observations from use here. For the one hundred fifty-five cases
from which the description was drawn, a miniature new year was posited,
with the prosecutor's budget scaled accordingly (as best could be esti-
matedl3) and the prosecutor's utility function was assigned parameters
d- 1/155 and (3- 154/155 to reflect equal (multiplicative) case
weightings. It was further assumed that the costs of trial were composed
of the costs of executing the trial itself plus the costs per day to the
prosecutor of maintaining the case until the dates of the pretrial con-
ference, and the subsequent costs until trial.14 Together with informa-
tion on income derived from the 1970 Census, attributed to defendants
based upon residence address zip codes,15 the information available was

sufficient for engaging in empirical testing.

Some characteristics of the defendants in the sample who pleaded
guilty--with dismissal considered a guilty plea at outcome zero-—were
as follows: the average value of Va, taken to be the maximum penalty
on the original charge, was 15.1 years; the average of the maximum
penalties on the reduced plea offenses was 6.3 years, with the average
ratio of the latter to the former being .36 as alluded to earlier. The
distributions of original and reduced charges and the maximum sentences

for each are shown in tables one and two.

Table 1. Distribution of defendants by type of

Felonies
Auto-related
Robbery
Burglary

Larceny

Misdemeanors

Dismissals

106

Original Chaggg

8.4%
27.7
31.6

32.3

charge (in percentages).

Plea Chargg

 

44.5
31.0

11.0

107

Table 2. Distribution of defendants by length of maximum sentence (v)
(Distribution is in percentages.)

Years Maximum

 

 

Sentensea Original Charge Plea Charge
0 (dismissal) 10.3%
.28 10.3
1 1.3
2 6% 34 8
2 5 2.6 12 2
4 4.5 6 5
5 36.2 11 O
10 20.6 1.3
15 12.9 1.9
25b 1.3 .0
life 21.3 10.3

 

aNinety days was coded as .2 years

For two defendants, the sum of two separate offenses on same case
was 25.

108

Over half of the defendants, 53.5 percent, awaited their trials
in jail, in spite of the release-on—recognizance program to which the
court subscribes. 0f the defendants in custody, the proportion of
blacks 17 was within a half—percent of their proportion in the total
sample, which was 77.4 percent. For slightly over a quarter of the
defendants a weapon was involved in the associated incident, thus
providing--in addition to the distribution of maximum sentences-~an
indication of the seriousness of the offense.

As a group, the defendants were young, with most falling into
the eighteen-to-twenty-four age group: 17 percent were juveniles
(under age 18), 76 percent were age 24 or younger, and 90 percent were
age 40 or younger. In the great majority of instances--over 88 percent--
the complainant and defendant were not related by family ties or by
friendship. 18

Although 24 percent had at least one other case pending in the
court, only a quarter of those defendants had two or more. It was the
policy of the court if a defendant had other cases pending, that all

his cases be handled at once when the first reached the head of the

queue. In practice, negotiations then dealt with the entire iii voj

where n is the number of cases (including the one appearing in the sam-
ple) for a given defendant, rather than simply being concerned with v01
which we observed. Because it was typical practice for the defendant
involved in several cases to plead guilty to one offense and have the
others recommended for dismissal, the resulting v to which D pleaded

could be expected to exhibit greater variability for these Cases:

either D would plead guilty to another offense not in the sample and

109

the observed outcome would be zero in spite of the presence of other

factors which would tend to make us expect a positive outcome; or the
value would be particularly high relative to the original charge, and
the observed value of v0 would result in greatly understated value(s)
for the dimension(s) of the Pareto Optimal set. 19 Hence the effect

of other pending cases of an individual defendant is to increase the

variability of outcome on the case included in the analysis.

For the group of defendants in the sample, the average time from
the date of the offense (which is usually within a day or two of the
arrest date) to the pretrial conference was 70 days; however, the medi-
an was slightly over 27 days, indicating that most cases reached that
point in far less time than the mean would lead one to believe. From
the time of the pretria; conference to the trial date available with
the (later-) assigned judge was 72 days on average. Because this
interval was not necessarily (or even usually) known to defendants be-
fore the pretrial conference, 20 the average value was used in the em-
pirical work which follows.

For use in the testing of the model, the sample had to be pared
somewhat. The sample included only five women, which was an insuffi-
cient number upon which to determine the impact of the sex of the defen-
dant upon the outcome of plea negotiations. Further, since women are
reputedly treated with some deference by law enforcement agencies,
failure to take account of that factor could be detrimental to the va-
lidity of any test of the theory, as well as proving deceptive should
one desire to use the model for prediction.

Second, observations were eliminated for which values of critical

110

variables were missing from the data. The most prominent among these
"hatchet" variables was the prosecutor's estimate of the probability
of conviction, which is crucial for the determination of the relative
strength variable. In addition, for what one might call "two-plus"
defendants (where a case had more than one defendant), the value of
that piece of information was of questionable reliability, due to the
manner in which items were recorded for defendants other than the first
one listed for a given case, 21 so "two-plus" defendants were also ex-
cluded. Had those cases been maintained in the sample, they would
have imposed a curious weighting on the types of cases considered.
(For example, eight persons were accused of breaking and entering a
railroad car while it was still in the station, an otherwise unusual
occurrence.) Finally, observations for which the disposition of the
case was not given or for which it was sent to trial (and hence could
not have been resolved by pretrial settlement) were deleted. Dis-
missals were kept in the sample, and assigned the value zero for out-
come. The final set of observations used consisted of eighty-six
"first" defendants.

It was extremely fortunate for the empirical testing of the
theory that the Detroit data included an estimated probability of con-
viction, for had there been no such information available concerning
the priority or strength of the case as perceived by one party or anoth-
er, it would have been necessary-~and perhaps, quite precarious--to
construct such a variable, and to do so independent of any concept of
how either party makes such evaluations. Since information on the pro-

secutor's assessment of the probability of conviction was available, it

111

was possible to derive from this a function which could be used as an
index of the relative strength of the two parties. The derived index
of relative strength 9‘ (derived below) was believed preferable to the
prosecutors' probability estimates for use in later regressions for
several reasons. It seemed to be the pattern of the prosecutors that
estimates were always made in multiples of five, i.e. 60, 65, ...,100 22
(with the exception of one modest prosecutor who wrote 100's as 99_)
and that pattern made it difficult for such a proxy for r to be rightly
considered a continuous variable, a requirement inferred by the theory.
Since greater specificity on the part of the prosecutors would likely
have provided only spurious accuracy anyway, the scaling system they
used probably served its administrative purpose well. However, for our
use of r as a continuous variable and in order to decrease the harsh
lines otherwise drawn between cases in which the "true" r-values (as
perceived by a given party) are close to one another, it was decided

to use the predicted values of the probability of conviction, stated

as a function of the variables mentioned earlier as being relevant to
the determination of relative strength: jail status, race, race-times-
jail, total time from offense to trial, an index of the harshness of
the judge to which the case was assigned for trial, 23 whether the de-
fendant had other cases pending, whether the defendant and complainant
had family or friendship ties, and whether the defendant was a juvenile.
The resulting predicted values for the probability estimates were used
as an index of relative strength, and therefore should not be inter-

preted as 31mp1y-representing probabilities. "

Nevertheless, in order to retain the option of interpreting r in

112

such a way as to have the values bounded--which the probabilistic inter-
pretation conveniently required--it was necessary to use a bit more 613-
berate. function than one which was linear in each of the variables
listed in the preceding paragraph. A logistic function was used, hav-

ing the form

r = 1
(z ... z )
p 1 + eq 1! ’ d!

where the 21 are the characteristics of the defendant and of the ofe

fense, rp is the probability variable from the prosecutor's files, and
the function q(zl, ..., zd) is assumed to be linear in the parameters. 24

The numerator- is the upper bound on the function, which we have taken
here to be one.

By suitable algebraic manipulation, the logistic function can
be stated so that the exponent of e forms the right-hand side of the
equation,and the left-hand side is ln(lizp) which is the natural loga-
rithm of the odds in favor of the defendZnt. This transformation of
the logistic function permits estimation of a linear function rather
than of a nonlinear one. And since the logarithm is an increasing func-
tion, the results of the regression in which the dependent variable is
the log of the odds in favor of D lend themselves to easy interpreta-
tion in terms of the relative strength variable, for the ordering is
exactly opposite to that favoring the prosecutor. Once the coefficients
for the 21 have been determined, it is a simple task to transform them
back to the form of the logistic function and use the estimate of r

thus obtained in any subsequent work.

The results of the regressions on ln(lizp) are presented below,
P

113

with alternative specifications of the q-function to be found in the
appendix (the alternatives arising from different representations of

the age variable, as well as inclusion of some alternative variables.)

ln(lizp) - -l.l3 -1.24 Jail -.55 Race +1.05 Race Jail + .002 Total time

p (-3.55) (-1.98) (2.63) (1.07)
+ .004 J Plea - .36 Pend Chg + .30 Know Wit - .22 Age 17 + e1
(.56) (-l.94) (1.16) (-.98)
R2 = .23225

where the variables were given values as follows:

Jail equals 1 if D is in custody and 0 otherwise:

Race equals 1 if D is nonwhite and 0 otherwise;

Race Jail equals 1 if D is nonwhite and in jail, and 0 otherwise;

Total time is the number of days between the offense and pretrial
conference, plus 72 (the average from pretrial date to trial);

J plea indexes the harshness of the judge by the percentage of
defendants assigned to her or him who plead guilty (and so
J Plea decreases, the harsher is the judge);

Pend Chg is 1 if D has other cases pending and 0 otherwise;

Know Wit is 1 if D and the complainant are friends or relatives
and 0 otherwise;

Age 17 is 1 if the defendant is age 17 or younger and 0 otherwise;

and t values are given in parentheses.

Although not all the coefficients are significant at low signifi-
cance levels, most signs are as eXpected. Jail status and race have the
anticipated signs, with the coefficient of their interaction term indi-
cating that (l) the use of only the two variables (i.e. excluding the

interaction term) would overstate the effect on relative strength; and

114
(2) including the joint effect of race and jail results in the following
order with respect to the defendant‘s strength vis-a-vis the prosecutor,
in order from strongest to weakest: whites not in jail, blacks not in
jail, blacks in jail, whites in jail. The order of the latter pair of
sets has several possible eXplanations. One is that anti-black dis-
crimination in pretrial incarceration results in having blacks with
strong cases residing in jail while their white counterparts are
released. Another possible explanation is that of discrimination in
favor of blacks by the black-dominated juries in Detroit; this would be
consistent with some previous speculation about pro-black bias of
juries (Iimg, April 2, 1974) but would seem to conflict with the ordering
of persons not in jail, unless the discrimination is based upon other
factors as well, such as income, social class, or age. Finally, the
functional forms chosen and data used may be a source of this anomaly;
however, the significance of the coefficient possibly makes this caveat
less necessary.

Delay is beneficial to the defendant, as is having a less harsh judge;
in contrast, having other pending charges or being age 18 or older would
work to D's disadvantage. The sign of Know Wit would seemingly be
negative, since positive identification by a relative or friend would
appear to be an obstacle to D's defense; however, since the estimate is
being made by the prosecutor (recall that all of this is based upon his
estimate of the probability of conviction) it is possible that he
would consider the identification by a disinterested party -- who would

be less likely to recant at the time of trial, having little sympathy for

D -- to be more dependable.

115
The importance of the jail status of the defendant is deserv-

ing of special note, for it recurs elsewhere as well as here. Its
impact on the prosecutor's estimate of conviction,. and consequently
upon the estimated relative strength variable, occurs for'i several
reasons. The first to come to mind is likely to be the resource effect
for a defendant in jail is less able to earn, borrow, or otherwise ob-
tain resources for use on his defense; and if determined to be indi-
gent (which may result in part by his being in custody) and assigned
counsel, he may be considered worse off than were he able to choose his
own attorney. 26 However, the resource effect is accompanied by other
factors having perhaps even greater impact: (1) The defendant who is
out of jail is able to work as a surrogate employee of his attorney,
doing some of the legwork of preparation for trial which might other-
wise have been ommitted, or accomplished by paid employees at a higher
cost to D. (2) The defendant is able to perform a number of tasks which

would be difficult if not impossible for others to carry out, such as
recognizing witnesses by face (a particularly valuable ability if their
names are not known) and locating them in places they frequent, for D
is more likely to know their friends and habits than would his attor-
ney. (3) Finally, the effect upon juries at trials in which a defen-
dant is led in by a bailiff, rather than arriving of his own volition,
is to confirm the attitude that "he wouldn't be here if he weren't

guilty,’ a belief that is surely unfavorable to the defense. Thus the
strong and significant negative impact of the defendant's being in jail
is not a surprising finding.

With the determination of the index of relative strength, we

have sufficient information with which to estimate the outcome function

for plea bargains.

-116

NOTES to 3.2

One notable example of the contrast among various institutions
is provided by comparison of pretrial negotiations in Michigan with
those in Kansas, as pointed out in the American Bar Foundation study
of several state and local law enforcement systems. In Michigan--and
in Detroit, the source of the data used in this study--the offense
upon which arrest was made is carried forward from one stage of the
criminal justice process to the next (possibly with additional includ-
ed charges tacked on) until a pretrial settlement is reached, at which
time the prosecutor would take the action necessary in carrying out
his part of the agreement. In Kansas, however, the plea bargain is
made earlier in the process, and a researcher unaware of that would be
surprised to find that all defendants who plead guilty do so at exactly
the level of the "original" charge recorded in the prosecutor's files.
Clearly, in the latter situation, a data collector would have to delve
further back into the process to determine the original level of v
to which the level of the guilty plea should be compared. (MacIntyre
1967, Newman 1966).

2For more detail on the evolution of these modifications, the
reader may wish to consult the appendix containing the more grubby
aspects of that process, such as the step-by-step breakdown of the vari-
ous costs associated wi:h trial; for the preservation of reader inter-
est and sanity, the more laborious calculations are bypassed here in
favor of a summary presentation.

3In fact, as such, Recorder's Court was chosen for one of two
special case study sites for the Task Force on the Courts report to the
President's Commission on Law Enforcement and the Administration of
Justice. See The Challenge of Crime in a Free Society, by the Commis-
sion and the Task Force Reports: The Courts.

 

4One man was crushed to death under stones, while being offered be-
fore each additional'stdne was dumped upon the existing pile, the op-
portunity to confess his witch status and thereby be released.

5Some of the case decisions on this have been cited in the pre-
ceding chapter. One of the more important was Santobello v. New York
in which the Supreme Court ruled that a bargain made by a prosecutor must
be executed by him or his successor. In their discussion of the case,
the justices cite the necessity of plea bargaining to the Operation
of the criminal justice system, particularly because a requirement that
every case be tried would result in hopeless backlogs.

 

This and other similar generalizations in the text come from in-
terviews with two of the pretrial prosecutors in Detroit, from the.Annual_

117

Report of the court for 1972, and from reading and study in crim-
inal justice. Where possible, more detailed citations are given.

The reader can verify this statement on the allocation of
judges' time by consulting the statistics on judges in the data summary
appendix and noting the small percentage of time for a few judges spent
in trials (jury cases plus "waivers".)

The system of "plea judges" was changed in September 1972, when
a blind draw docket system was instituted. Under the new system, each
entering case was assigned to a judge (randomly determined) and placed
in her or his trial queue. The change in procedure was in part a
response to the problem (for the court) of judge-shopping, in which
some defendants would attempt to delay their cases, renege on plea set-
tlements, and utilize other steps in the hope of being assigned to
another judge. (Annual Report, 1972)

 

7See the Annual Report for 1972 for the raw figures, which are
also listed by finer categories of offenses. The percentages used in
the diagram were derived from those figures.

 

8For example, a defendant accused of first degree murder may be
found guilty of second degree murder.

9Another example of in-kind settlements is provided by the set-
tlement of some less serious offenses charged to out-of-towners: if
they wish to plead guilty early on (at arraignment, typically) they
may receive suspended sentences on the condition that they promise not
to return to the city. (A bit like in the Westerns, as one seminar
observer pointed out!)

LOThis caveat is in keeping with the now-established tradition in

the economics literature of excluding individuals who are not sure to
be rational, as that term is defined by economists. See, for example,
"Crime and Punishment: An Economic Approach" by Gary Becker (1968).

11
Drug cases were also excluded for the same reason; however,

since most drug offenses are violations of federal law, only two
observations were affected.

12A number of the data problems were within repair, as for exam-

ple, when the question about whether a weapon was involved was left
blank but the defendant was charged with armed robbery. However, a
number of items which would have been interesting to try as indicators
of the amount of evidence for the prosecutor were not always supplied,
for example, whether the defendant had made a confession.

3The set of cases drawn for this study constituted 1.33 percent
of the felony cases in 1972, which corresponded to a budget of
$58,551 for non-fixed costs for the miniature year. Totals to which

118

the part-year percentage was applied were given me by telephone from
the Budget Office, County of Wayne.

14Based on the assumptions listed in the appendix, the average

cost of trial was estimated at $1817 and a per day case maintenance
cost of one dollar was assumed, since various alternative intermediate
assumptions showed that to be a ballpark estimate and the true figure
was impossible to determine from the limited information at hand.

15Information on the defendant's income, welfare status, and em—
ployment are given on the Pre-sentence Report, to which I did not have
access. To substitute for this, the zip codes of defendant's resi-
dences were noted and levels of income were assigned to defendants
from the 1970 Census based upon zip code. However, the Census does not
give a median or even a meaningful mean income by zip code (the mean
income figure includes quite heterogeneous units) and so the numbers
used were the prOportions of the individuals in the zip codes who had
incomes above the poverty lines determined by the Social Security Ad-
ministration. These numbers were then adjusted to represent reasonable
dollar figures, additional assumptions were made concerning the amount
of savings at hand, and the defendants' disposable resources for trial
and guilty plea alternatives were computed using the expressions de-
rived in section 2.3.

 

16This is another example of differences in institutional prac-

tice which must be taken into account in the type of empirical analysis
performed here. From statements of persons in the prosecutor's office
and from the records I used, it was evident that the relevant consider-
ation in Detroit for negotiation of guilty pleas is the maximum penalty
associated with an offense, (with the exception of the "labeling" prob-
lems associated with sex crimes, child abuse, and a few other offenses.)
In contrast, according to one prosecutor, it is the practice in New
York and Chicago for plea negotiations to be concerned with the actual
sentence to be given, since prosecutors there frequently make recom-
mendations on sentence (or agree not to make a recommendation, as was
the arrangement with Mr. Santobello.) Although information on maxi-
mum penalties might be available for the latter two cities, it would
not be the relevant figure for use as the plea outcome, and probably
not for the original charge either.

17As is shown later in this chapter, the jail status of a defen-

dant overshadows the racial factor in importance, the latter being of
special concern for defendants who are in jail. Apparently the pres-
tige factor for defendants out of jail (attending trial of their own
accord) dominates the race difference which would otherwise favor the
whites.

119

18This is in contrast to "crimes of passion" such as murder,

which are more friendly affairs. In Detroit in three-quarters of the
murder cases, assailant and victim are relatives or friends.

19This becomes a special case of the phenomenon one might call

cross-case bargaining, in which the outcome for case A is sacrificed
somewhat by the prosecutor in order to obtain a more favorable out-
come from case B, with D's contribution to the cause-- assuming he is
case A --being his availability as a source of information which P
deems helpful to bettering his outcome from case B. Although far less
frequent than the bread-and-butter mode of bargaining described by this
model (single case bargaining which is pareto optimal within the two-
party community of P and D,) cross-case bargaining is perhaps the more
familiar of the two due to the wide publicity surrounding the Water-
gate-related cases.

20This was because the blind draw system for assigning judges

(and knowing one's place in the queue of the assigned judge) did not
begin until after the period of this data.

21A number of the case files involved two or more defendants at

the time of arrest. The status of the case was continually recorded
(usually) for one or possibly more of the defendants who survived var-
ious sources of attrition, such as transferral to other courts or other
counties or states, dropping of charges, guilty plea at arraignment,
etcetera. Forthe first survivor who continued through the regular plea
process, the records were generally quite thorough. However, it ap-
peared that the records of the others--labe1ed here as "two-plus" de-
fendants-~sometimes were filled in with the same entries as those of
the first defendant, particularly the entry for the probability of
conviction, which was typically evaluated for the first defendant and
then attributed to the others also. Since the evidence of guilt could
and often did vary widely among defendants on a single case, as did
other factors affecting outcome, it was surprising to find the proba-
bility of conviction to be identical for all defendants on the case.
In view of the pattern for repetition, it was believed that inclusion
of the "two-plus" defendants would add to the errors-in-variables
problems already present (that due to the many prosecutors making the
estimates and the limited set of discrete values possible for assign-
ment) and so only first defendants were used for the estimation of
the relative strength variable and regressions on the outcome function.

22The limited range for the probability estimates has two possi-

ble justifications. One is that the prosecutors who make the estimates
are insupportably Optimistic, and thus feel that even the worst case
has better than an even chance of winning.

The alternative explanation lies with a perception of the
prosecutorial stage of the criminal justice process as not only a

~120

processor.prior to trial, but as a subagency which receives cases
sifted out in the preceding stages of the process. Thus P does not see
cases in which there was no probable cause for arrest (although these
would receive low probability estimates) nor does he see arrests not
made because a police officer and potential defendant traded informa-
tion instead.

Thus if one believes that the previous stages in the process
are good sifting devices (or alternatively, that they are sufficiently
desperate for funds that priorities are set), then it is a logical con-
sequence that only the fittest of the potential cases survives to the
point of entering the prosecutor's files. This is equivalent to a sort
of bureaucratic Darwinism.

23As it turned out (because of the later institution of the blind

draw system) this piece of information was not usually known by defen-
sants until the pretrial conference, thus allowing scant time or no

time at all for the implications of the judge assignment to be taken in—
to account in the estimates of r. As revealed below, this variable

wins the contest for being the least significant of the ones entered

in the regression equation.

4Of course, q is not linear in the variables, since Race and
Jail are entered in interaction with one another, as well as being en-
tered separately.

25When run with all cases for which the necessary data existed

(including two-plus defendants) the results were close to the same as
those reported in the text:

In l-rp - -l.37 -l.15 Jail -.62 Race + .88 Race.Jail
rp (-3.24) (-2.11) (2.17)

+ .002 Total time + .01 J Plea - .298.Pend Chg

(1.09) (1.02) (*1.55)
+ .32 Know Wit - .23 Age 17 + e1

(1.17) (-1.02)
R2 - .1712 n - 109

Additional regressions are displayed in the data appendix.

26It would have been desirable to have information on whether the

counsel for defendants had been privately retained or had been assigned
by the court. However, several factors mitigate the need for such data.
First, there are four sets of attorneys who represent criminal defen-
dants: private attorneys who do not specialize in criminaldlhw aIOne;
attorneys who do specialize and are retained by defendants, private at-
torneys assigned to defendants and attorneys who practice through the

121

Defender's Association (which is funded in part by the Detroit Bar
Association and the fees paid by the court.) Since judges assign cases
to private attorneys who practice in the court anyway, the second and
third groups are largely the same. The first group is small if it ex-
ists at all. The contrast which would be of interest is between the
privately practicing attorneys and those from the Defender's office--
and the distinction there is not one of the income of the defendant.
Second, the number of privately practicing attorneys is small;
they are specialists and are physically present at the court much of
the day (thus limiting time for preparation.) From the notes of the
American Bar Foundation study, Michigan Field Reports (1958)
"There are only seven or eight bona fide "criminal lawyers'
in the city of Detroit...
"The primary reason, according to [a lawyer], for the small
number of criminal lawyers in Detroit is the fact that a
general practitioner simply does not have the time to devote
to criminal matters. Those attorneys not appearing in Record-
er's Court every day are required to spend at least one half
a day when the defendant is arraigned on the warrant. A full
day is required at a preliminary examination, another half
a day is required when a defendant is arraigned on the infor-
mation. On the other hand, the criminal lawyers, being in
Recorder's Court anyway, only consider a few minutes of their
time being taken up on arraignments. Preliminary examinations
can be conducted in a matter of 20 minutes or so. The civil
lawyers, therefore, find it necessary to charge much higher
fees than are required by the criminal lawyers. Since a vast
majority of the defendants in Recorder's Court are not able
to pay the fees required by the 'general practitioner' most
of the criminal cases are handled by the few criminal law-
yers."
Thus the absence of information on whether the defendant was as-

signed counsel is of relatively little harm as relates to the income of
the defendant.

122

3.3 The Disposition

 

In the preceding two sections, the model was adapted and the
data made compatible with the model for use in testing the outcome
function determined by a proportional split of the Pareto optimal re-
gion between prosecutor and defendant. Substituting the assumed para-
meter values, and representing the h-functions as hp = -l/? and hd = —?

the outcome function to be estimatedsis

155

R 154 ."E'1 + (14») . ?+ (1-70 ..x. 3 . ?+e

RI

 

vm =- -7\(.0065) .

which can be restated as
vml-alx1 +azx2+a3x3 +6
where al - -), a2 - l—h, a3 8 l-),, and the x1 are the first, second,

and third terms respectively, stripped of their a coefficients and E. is

i
randomly distributed. We expect that a2 will be a number between zero
and unity, and that al and a3 will be numbers such that‘avmflm. We

would also expect the absolute values of a1 and a3 to be less than one,

with their signs as noted, except that the scaling problem may distort
the magnitudes, as discussed further below.

If one recalls (l) the set of assumed parameter values, i.e., regard-
ing the prosecutor's switch function, that w - l, 0C" 1/155- (5" 154/155
(reflecting the weighting of P's cases); and regarding the defendant's
switch function, that pc - 1; and (2) the assumptions made in the
process of readying the data for use here, such as requiring g and R'
to be equal to one if they would otherwise be less, adapting the Census
poverty statistics for use as an income variable, and making other

assumptions entailed in the determination of the costs of trial

w123-

and case maintenance; then it is clear that some of these assump-
tions bear the potential for affecting the results of any attempt to
estimate the outcome function. The manner in which the h-functions are
stated will also precipitate this danger. These regrettable but nec-
essary assumptions-—which affect the magnitudes of x1, x2, and x3, and
are therefore referred to as "the scaling problem"-—wage warfare with
the empirical results on two interrelated fronts: first, they can es—
calate or de-escalate the scale of the x1 terms (especially x1 and x3)

and thus threaten the sign ofiavmfi39, since the sign of the latter re-

sults from the addition of two terms, possibly of opposite sign; and

H

disc. and thus through! sever one

second, they determine the value of r
regression regime from another.

The first liability--the general problem of variable scaling--
holds little hope for rescue, other than further attempts to adjust the

parameter values so that the resulting scales of the x1 seem "reason-

able". The second one, that of compensating for scaling problems in
determining the regression regime to which an individual should be as-
signed, holds out a bit more promise of a reprieve.’ The "cut variable"

Ywas defined by comparison of ’1'; with r'disc , and only the second of

these is affected by problems of scaling. 1 Instead of simply defin-

A
ing 2' to be 1 when r3 érd'isc and 0 otherwise, we define a new 'vari-

A
_ n _ n . _
able V; to r d isc. and cut variables 83 1 if y s kj and 0 other

- aljx1 + azsz + a3jx3j can be estimat-

- t 11?), and the
j—

wise. The outcome function vm

3

ed for several alternative values of k (where

:1 "31

results of these several regressions may be compared with one another

to determine the value of kJ which yields the best results, as indicated

124

by the values of the coefficient of determination and the signifi-
cance of the regression coefficients. 2 Once the best value of k3
(among those chosen, as the cardinality of the set of k1 must necessar-
ily be quite small) has been found, the difference between the best kj
and zero may give some indication of the scaling problem resulting from
the assumptions regarding 3; R], and pc, since the theory implies that
when the correct values of those terms are used, the best cut point is

the one for zero difference between‘?3 and r"

disc.’

The result of the regressions for ten alternative cuts of the r"
domain are given in Table 3, in.which the dependent variable is the
maximum sentence associated with the offense to which D pleaded guilty.
To account for the possibility that the appropriate measure of outcome
was the difference between the plea offense maximum and the maximum
for the attempt charge corresponding to the original charge '3, the
same regressions were also run with that difference as the dependent
variable, and those results are given in Table 4.

For both versions of the regression equation (i.e. for both of
the alternative dependent variables) the best results occur at F: .6,
with the plea maximum yielding a slightly better value of R2 than did
the plea/attempt difference, and the preference reversed for coefficient
significance. However, in both instances the coefficient of determin-
ation is quite close to one-half, and the t-values of all coefficients
are greater than two.

The fact that some cuts of the domain--actually any cut--yield
better results than the "no cut" case,would tend to confirm the theory

that the defendant specializes in trial improvement or in consumption.

125
TABLE 3 Estimation of the outcome function (with no constant term)

Dependent variable is maximum sentence associated with guilty plea
Offense. ‘

 

C t a a R2

u 1 a2 3

kl=-.25 -.510 . 632 3. 854 .5002
(-1.74) (9.33) (.29)

k2=0 -.510 . 632 3. 854 .5002
(-1.'71) (9.33) (.29)

k3=.25 -.546 . 637 2.748 .5014
(-1.77) (9.28) (.53)

k4=.50 -.792 . 670 3. 396 .5067
(-1.'98) (8.76) (1.09)

k5=.55 -1. 191 .723 5.923 .5213
(-2.59) (8. 87) (1. 94)

k6=.60 -1.314 . 738 6.282 .5233
(-2.66) (8.72) (2.03)

k7=. 65 -1.268 .727 5.288 .5141
(-2.22) (8.01) (1.57)

k8-.70 -1. 16 .707 3.932 .5055
(-1.58) (6. 84) (. 99)

k9=. 80 I -1.38 .700 4.671 .5059
(-1.51) (7.27) (1.02)

km: , -.322 .634 -1.077 .4998

no cut (-.21) (8.08) (-. 11)

 

Note: The first and last of the kj represent cuts extremely near and at
the bounds of r, respectively.
155

Above, x1 = . 0065R /R'154 . r

-1

ll
|SO <I
"">

2

X
X3

126

Table 4. Estimation of outcome function using the difference between
the maximum sentence for the guilty plea offense and the maximum
for attempt on the original charge as the dependent variable.

 

 

Cut a1 a2 a3 R F

None -.o4199 .571 -2.812 .4323 31.60
(-.347) (7.055) (-.298)

k1=-.25 -1.1316 .562 6.234 .4331 31.70
(-3.687) (8.051) (.449)

k2=.00 -1.1316 .562 6.234 .4331 31.70
(-3.687) (8.051) (.449)

k3-.25 -1.1849 .570 4.144 .4357 32.049
(-3.730) (8.057) (.774)

k4-.50 -1.6078 .627 5.724 .4529 34.348
(-3.946) (8.035) (1.792)

k5-.55 -2.1419 .697 8.798 .4826 38.701
(-4.627) (8.492) (2.858)

k6=.60 -2.3203 .719 9.306 .4861 39.39
(-4.667) (8.454) (2.990)

k7-.65 -2.2733 .706 7.977 .4667 36.32
(-3.928) (7.660) (2.336)

k8=.70 -2.0573 .670 5.629 .4444 33.19
(-2.737) (6.310) (1.348)

k9=.80 —1.7679 .612 3.499 .4354 32.00
(-1.873) (6.130) (.738)

Above, x1 - .0065R155/R'154 . 2-1
*2”;
x - BI?

3

127

Worded in a more rigorous manner, had the "no cut" case yielded the best
results, we would have had to conclude either that such specialization
does not take place, or that at least for the sample individuals, the
point of specialization reversal occurred outside the boundaries of

the set of observed v. 4 In any event, although this result tends to
be supportive of the theory, it cannot be viewed as the outcome of a
particularly strong test of that theory.

Turning to the regressions for f- .6 we find that the sign and
magnitude of a2 are as predicted; if there are no scaling problems with
3 (or if one can be assured that they are minor) the conclusion is that
the outcome of bargaining is comparatively closer to D's reservation
price than it is to P's. In fact, the bargainable difference is split
close to the ratio of seven to three in favor of the prosecutor. This
poses an interesting contrast with the split of the total amount of v,
labeled G, when we broaden our focus beyond the Pareto optimal set.
Recall that the average reduction in maximum sentence was to less than
four-tenths of the original maximum. If the whole interval from zero
to 6 were considered Pareto optimal, this major reduction in charge
would seem to imply that the outcome generally falls closer to P's
least desired outcome, leading us to believe that the defendant is the
stronger party in bargaining. However, once the region of bargainable
outcomes has been subjected to the condition of Pareto optimality, the
assessment of bargaining skill reverses: within the region bounded by
the reservation "prices" of the parties, the outcome falls closer to

the D's reservation price, thus making P the stronger bargainer.

128

Second, in considering 3vm/3r, which we predicted to be pos-
itive, it is necessary to engage in some arithmetic calculation in or-
der to determine the sign which the regressions would yield. Recalling

that

 

min/3,- .. A(.0065) . R,154 r"2 + (1-)() Y3

we can write

avm/ar ' -a1x1r-1 + a3YR .

Since the partial is a function of r as well as of other variables, we
would evaluate it at the means of the remaining variables; there is
potentially a conceptual problem involved in basing an evaluation on
the mean of‘Xfi which being dichotomous, is not well represented by its

mean. The problem is moot, however, since -a and 83 are positive (as

1
predicted) for both versions of the regression, and so avm/Dr must be
positive also, even when X'is assigned the alternative values of zero

and unity. Note that so long as a and a3 bear the signs anticipated

1
for them, scaling problems cannot affect the sign of Evin/Br. It is
when the signs of al or a3 fail to meet this precondition, as in the
additional regressions in the appendix,that the scaling problem can aC‘
tually become a problem. 6 Thus,with the h-functions used here, our
predictions on the sign of ‘bvm/ar are fulfilled also.

In deriving the outcome function as a linear combination of the
switch functions of the two parties (neither of which had a constant
term) we found ourselves with a function having no additive constant
term. To test whether that omission was in error, the same sets of

regressions were run without suppression of the constant, with the re-

sults reported in Table 5,8 As the reader can verify, the constant

Table 50
guilty plea maximum sentence as dependent variable.

129

Estimation of Outcome function with constant included, and

 

 

3

2

Cut Constant a1 a2 a3 R

None 39.355 -4.507 .636 .26.041 .4999 27.32
(.15) (-.16) (7.98) (-.16)

k1--.25 -.402 -.426 .633 3.754 .5002 27.35
(-.03) (-.l3) (8.05) (.27)

k2-.00 -.402 -.426 .633 3.754 .5002 27.35
(-.O3) (-.13) (8.05) (.27)

k3=.25 -2.512 -.024 ..644 2.848 .5015 27.50
(-.19) (-.01) (7.98) (.54)

k4=.50 .490 -.895 .669 3.409 .5067 28.07
(.03) (-.28) (7.908) (1.07)

k5=.55 -2.949 -.580 .731 5.953 .5216 29.80
(.20) (.19) (7.96) (1.93)

k6-.60 -6.412 -.413 .758 6.495 .5244 30.13
(-.43) (-.00) (7.77) (2.06)

k7-.65 -1.412 -.973 .731 5.286 .5142 28.93
(-.09) (-.31) (7.36) (1.56)

k8-.7O -l.257 -.895 .711 3.925 .5055 27.94
(-.08) (-.28) (6.41) (.98)

k9=.80 -5.007 -.393 .718 4.995 .5065 28.05
(-.33) (-.12) (6.43) (1.06)

where x1 - .0065R155/R'154.?'-1
x - 9
2 A
x -.§.t

130

was not significant at any cut, and additionally reduced the re-
liability of the coefficients of the other terms, from which we can
conclude that the version from the theory is the preferred form.

In summary, the form of the outcome function which provided the
best results was the one which observed most strictly the form of the
function constructed in the theory; the maximum sentence of the plea
offense and the difference of that variable from the maximum for attempt
on the original charge yielded similar results, both with values of R2
close to .5 and significant t-values for the regression coefficients.

The question of whether the function estimated is in fact dis-
continuous at some point was tested by comparison of the discontinuous
function at its best cut with the cut points at either end of the domain
of 23 for the function descended from the theory, the best cut (in fact,
any cut) was better than the alternative formulations, thus lending
support to the belief that defendants specialize in trial outcome im-
provement or in consumption.

The sign of the partial derivative of outcome with respect to
relative strength was positive, as predicted, and the coefficiena--
of the term 3 indicated that the prosecutor generally drove a much
harder bargain than did the defendant. It would appear that the esti-
mation of the outcome function not only verifies the formulation of the
mode1--subject to the assumptions made concerning parameter values and
budget costs--but is also helpful in evaluating the results of nego-
tiations in terms of the distribution of the bargainable amount of the

good V between the two parties to the case.

131

Table 6. PrOportion of cases in left-hand regime for different cuts
of the domain.

No cut .000
k1 = -.25 .012
k2 = 0 .012
k3 = .25 .058
k4 = .50 .291
k5 = .55 .395
R6 ' .60 .442
k7 - .65 .512
k8 - .70 .616
k - .80 .814

132

Notes to 3.3

1Recall that rdisc , the level of r" at which D would reverse

his specialization was defined t9 be
[(v+1) ln(c;+l)+1]

u =
rdisc

 

__ 2 . co
(vtl) ln(c3+l)

so that the scaling problem is introduced in two ways: (1) through the
values of co and c3 which are subject to assumptions about origins and
amounts of D's resources; and (2) the estimates of time until trial.

2The modified search technique employed here is reminiscent of
the work on time-varying parameters pioneered by Richard Quandt and
developed further by David Belsley (1973)r Alexander Sarris (1973), and others.
Quandt's approach was to cut the domain at each value of t, a
discrete-time variable and to separately estimate a linear function on

each regime thus defined. He then determined the best cut (value of t)
to be the one which minimized

7‘ _ S-ltaT-t
x d‘T
where 0" and6' denote the standard errors of estimate of the left- and
right-hand regimes, respectively, and «for the whole domain of t. See

Quandt(l958, 1960) and Belsley (1973) and Sarris (1973) papers.

 

Walton and McKersie (1965) contend that once common points of settle-
ment have been recognized ("once" referring to the historical pattern

of negotiations, not an event within a single bargaining session), as
the attempt charge may be regarded in Detroit, bargaining will often
lead to that common point. If they are correct, one might want to con-
sider the deviations fiom that common point, rather than simply the
settlement point alone, as the relevant outcome measure.

4On the other hand, failure to reject the hypothesis that the

discontinuity exists, as is the case here, can be taken to imply that
the discontinuity does exist, or else that the introduction of the dis-
continuity allows a better fit of another function (even though the
other function may be continuous.) For example, if the true function
were the nonlinear function drawn below, and we used two linear ones

to estimate, we should not conclude that the true function is discon—
tinuous, just because the discontinuous pair describes the true func-
tion well:
y 11

true function

 

 

 

X

Figure 28. Example of a discontinuous function estimating Well a con-
tinuous one.

133

'SComparison of the Split of 3 with that of the pareto optimal
set suggests another point as well. As discussed and sketched in
sections 2.1 and 2.4, the pareto—optimal interval changes in both loca—
tion and size, depending upon the level of_v; this implies that any
assessment of bargain outcomes based upon v alone may not only be in
error conceptually, but may be confirmed empirically if the observa-
tions tend to fall at one extreme of the domain of v. The construct
tested here does not fall victim to that problem in part because the
use of switch functions immediately restricts it to the pareto Opti-
mal set, and also because the proportion of the split is assumed to be
fixed throughout the domain of v.

6In fact, even for the additional sums, which are displayed in
the data appendix, the signs of av [61' are correct. The potential
problem lies in the marriage of thg scaling problem to that of the non-
independence of the first and third variables: the latter may reverse
one or more signs of coefficients; if that has occurred, then the
scaling of the terms to be "added" (in fact, one subtracted from the
other) becomes a crucial determinant of the sign of bvm/ar.

7The fact that a and a do not lie in the intervals [-l,01 and
{0,11, respectively, only serves to point out again the scaling prob-
lems inherent in the model as applied to this data.

The regressions for the other possible cut points appear in the
appendix.

4. Conclusions

 

In the beginning of this dissertation, various aspects of the
judicial process were interpreted within an economic construct. Each
of the two parties to negotiations, the prosecution and the defense,
was believed to calculate the outcome and utility he anticipated from
trial, and then the concession which would be required by his Opponent
in order to make the prospect of a plea of guilty equally attractive.
For each party, the trial option was represented by one constraint and
the guilty plea option by another. Thus the "switch function" which
resulted from repeating the above calculation for alternative values
of the independent variable was in effect the demand function for one
good--a plea of guilty--stated in terms of the price of a substitute
good, trial. Each opportunity set was a production-possibility fron-
tier describing the case outcome in terms of variable -
inputs to trial (additional sleuthing and research effort) and the
relative strength of the two parties. Concession was interpreted to be
deliberate underutilization of the strength of one's own case.

The integration of the above items within the model led to two
observable theoretical results. First, the derivation of the switch
functions--each of which divided the strength-outcome space into a set
Of acceptable bargain outcomes and a set of unacceptable ones-~lent a
specific functional definition to the concept of a pareto optimal set

of outcomes of negotiation. Hence, if one accepts the precept that

134

135

under any bargaining theory pareto optimality is prerequisite to a
trial settlement, the set of negotiated outcomes should fall within
the region bounded by the switch functions 1. Due to the lack of ap-
propriate statistical tools, this could only be "tested" by visual ob-
servation of data plots.

Second, a split-the-difference theory of bargained outcomes was
posited, and the outcome function that suggested was a linear combina-
tion of the two switch functions derived earlier. Estimation Of the
outcome function served not only to test the general validity of the
model, but also to determine the proportions in which the difference
was split. In addition, since the defendant's switch function was
expected in general to be discontinuous at some value of relative
strength, due to his supposed specialization either in trial improve-
ment or in consumption, the outcome function was expected to be dis-
continuous also (or at least nondifferentiable) at some point. Al-
though not providing a particularly strong test of these contentions,
since continuous functions are sometimes well estimated by discontinu-
ous ones, a search for the point of discontinuity could serve to reject
that hypothesis were the data to be best fitted by the continuous ver-
sion of the outcome function, i.e. by a version in which the discon-
tinuity fell outside the relevant portion of the domain.

The empirical results tended to support the premises of the
theory--for example, that outcome level rose with the relative strength
of the prosecutor--as well as to affirm the outcome function itself.
The search for a point of discontinuity yielded positive results for

all variants of the outcome function which were estimated, and the

136

split of the Pareto optimal set was found consistently to be one which
favored the prosecutor.

In order to estimate outcome as a function of relative strength
it was necessary to first create an index of relative strength based
upon the factors which were believed to be critical to the determina-
tion of strength at trial. For that purpose estimated values of the
prosecutor's estimate of the probability of conviction were used, where
the estimates were the predicted values of that variable. Those values
were Obtained by regressing the probability estimates against a list of
characteristics of the defendant and of the offense, and the initial
endowment of evidence, where a logistic form was used for estimation.
The signs of the coefficients were consistent with much of the inter-
disciplinary literature about the effects of those various factors upon
the treatment of individuals within the criminal justice system.

Subject to the usual caveats concerning the assumptions made in
developing the model and imperfections of the data, the results noted
above have a number of implications both for further research and for
policy formulation. The crucial factor of jail status permeates the
model and is confirmed in the empirical work; even the strong effects
of race in the determination of relative strength pale next to custody
status. Further, the custody status of the defendant typically is cru-
cial in terms of his ability to earn or receive income, suggesting that
both policy and academic research consider not only the separate effects
of such factors, but also their joint effects upon the outcome of al—
ternative processes for resolution of the case. Thus, for example,

current release-on-recognizance programs should be evaluated in terms

137

of their income (i.e. family welfare) and case-outcome effects as well

as in terms of capiases issued (documents issued for failure to appear
at trial.) Similarly, negative tax programs, unemployment compensation
and other sources of income to defendants warrant consideration of

other trial-related factors as well.

The assignment of income data based upon residential location
poses an interesting problem in interpretation of the empirical results.
In spite of the roughness of the data, and of the somewhat precarious
manipulations made in deriving the level of disposable resources for
individuals, the crosstabulations especially tended to link higher in-
come figures with lower plea sentence maximums. This fact may be re-
flective of a tie between defendant resources and plea outcomes or it
may be ghosting a relationship which is more complex. Since the Cen-
sus entry used was the probability of being above the poverty line--
and although we do not have data on individuals, criminal defendants
are reputed to be rather homogeneous in income leve1--it is possible
that this variable is in fact serving as a proxy for the social and
economic amenities of the various neighborhoods in which defendants live
and from which the Census data was drawn. In addition, this poses a
number of questions about interrelations between income distribution
and criminal activity. Clearly these questions offer a rich area for
further research.

The time-honored maxim, justice delayed is justice denied, was
borne out in the testing of the model, at least to the extent that the
form of the outcome function (and thus the underlying Opportunity sets)

is affirmed. Since the time factor enters both through the resource

138

constraints and via the evidence effect of delay, analysis of different
methods of rationing court time should take both into account. And
again, because each of these effects is interwoven with others in the
determination of the Opportunity sets, there are a number Of important
relationships deserving of further investigation.

Although defendants obtain sizable reductions in charge for
pleading guilty, when we consider the amount that is actually "up for

grabs,"

that is, the Pareto optimal set, the prosecutor fares relative-
1y better. The reasons for this-~whether the administrative centraliza-
tion peculiar to Detroit, the quality of defense attorneys, the resources
of the parties relative to one another, or whatever--are deserving of
more attention than their passing mention here.

The development and testing of a model such as this one bears
implications for other areas of research (in particular, for applica-
tion to bargaining theory) as well. The switch functions derived in
the theory provide a two-dimensional generalization of the concept of
the points of zero incremental utility for two bargaining parties. Se-
cond, the switch functions pose a concrete interpretation of the con-
cept of a negotiated outcome which is exactly as attractive as its con-
flict substitute: the ability to derive such a point or function had
always been assumed, but had not been demonstrated. The linking of
strength in bargaining with a production process may shorten the dis-
tance between the deliberative, earthbound nature of traditional micro-
economics and the ethereal aura of the bargaining theories, which as-
sume that all issues of micro theory have been settled before making

their appearance.

139

Although not a dynamic model of the bargaining process, the con-
struct introduced here is consistent with the ideas of the negotiated
outcome being the result of a series of concessions, one in response to
another, and of there being a dominant pattern of bargained outcomes--
as represented by the split of the Pareto Optimal set--between the two
parties to negotiation. Although the assumption that the outcome func-
tion is a linear combination of the switch functions is Open to criti-
cism, this same assumption assures that the two-party nature of the
bargain process is not sacrificed, as has often been necessary in or-
der to obtain a testable equation (deMenil 1971; Bental and Comay 1973;
Comay, Melnick, and Subotnik 1974; Ashenfelter and Johnson 1969;4and
others.)

Finally, use of a split-the-difference theory suggests that
Occum's Razor may be a useful tool in selecting a bargaining theory,
and also adds to the recent store of empirical work in an area of eco-
nomics long believed to hold little potential for such application.

Where to go from here? In addition to those cited above, some
topics for additional research include incorporation in the model of
uncertainty and the role of information; greater detail in the develop-
ment of the production function for case outcomes; an economic theory
about the determination of relative strength; the creation of models
of other stages in the criminal justice system and their linkages with
those of the courts; and the extension of the theory to an aggregate
model of the market for guilty pleas. Coupled with data on these same

issues, the existence of such theories would permit a panOply of addi-

140

tional questions to be broached concerning the process and incidence of

plea bargaining and other legal processes.

APPENDICES

APPENDIX A.1

Data Summary Tables

Table 7.

1972 Annual Reports

 

141

Recorder's Court Judges:

Statistics Derived from

 

 

 

 

 

 

 

Percent Percent -
Name Pleas Dismissals J/W J+W
1 Colombo 76 23 .86 -.08
2 Crockett 45 33 .23 -.63
3 Davenport 73 15 .73 -.16
4 Evans 64 29 .19 -.68
5 Ford 41 41 9.99* .92
6 011113 53 47 3.00 .50
7 Heading 58 33 3.30 .53
8 Leonard 88 10 .78 -.13
9 Maher 79 11 1.81 .29
10 Murphy 72 27 .71 -.17
11 Olsen 78 16 .29 -.56
12 Poindexter 78 15 1.62 .24
13 Schemanske 58 16 6.97 .75
Ave. Regular Judges 69 26 1.00 -.00(-.004)
Ave. Visiting Judges 60 33 1.28 -.12
Ave. Total Cases 64 30‘ 1.15 +.17

 

* The true figure is 24.43
J denotes jury trial; W waiver of jury.

142

 

 

 

 

Table 8. Original Offenses Charged: Years Maximum Sentence
Years Absolute Relative Cumulative
Frequency Frequency Adjusted Freq.
(Percent) (Percent)

2.00 1 0.6 0.6

2.50 4 2.6 3.2

4.00 7 4.5 7.7

5.00 56 36.1 43.9
10.00 32 20.6 64.5
15.00 20 12.9 77.4
40.00 35 22.6 100.0
Total 155 100.0 100.0
Mean 15.097
Median 8.984

Standard Error 1.119

Standard Deviation

13.937

143

Table 9. Guilty Plea Offenses: Years Maximum Sentence

 

Years Absolute Relative Cumulative

 

 

 

Frequency Frequency Adjusted Freq.
(Percent) (Percent)
0.0 16 10.3 10.3
0.20 16 10.3 20.6
1.00 2 1.3 21.9
2.00 54 34.8 56.8
2.50 19 12.3 69.0
4.00 10 6.5 75.5
5.00 17 11.0 86.5
10.00 2 1.3 87.7
15.00 3 1.9 89.7
40.00 16 10.3 100.0
Total 155 100.0 100.0
Mean 6.392
Median 2.104
Standard Error .939

Standard Deviation 11.696

Table 10.

Not in Jail

In Jail

Table 11.

White

Nonwhite

144

Jail Status of Defendants

 

 

 

Value Absolute Relative
Frequency Frequency
(Percent)
0.0 72 46.5
1 00 83 53.5
Total 155 100.0
Race of Defendants
Value Absolute Relative
Frequency Frequency
(Percent)
0.0 35 22.6
1.00 120 77.4
Total 155 100.0

 

145

Table 12. Race and Jail: Interaction Variable

Value Absolute Relative
Frequency Frequency
(Percent)

 

Nonwhite and in Jail 64 41.3

Others 91 58.7

 

Table 13. Time until Pretrial Conference and Total Time
until Trial (in days)

 

Until Pretrial Total Time

Conference Until Trial
Mean 70.11 142.15
Median 27.25 102.25
Standard Error 8.71 10.72

Standard Deviation 108.43 '133.50

146

Table 14. Age of Defendants

 

 

Absolute Relative Cumulative
Age Frequency Frequency Adjusted Freq.
(Percent) (Percent)

16 3 1.9 1.9

17 23 14.8 16.8

18 28 18.1 34.8

19 14 9.0 43.9

20 14 9.0 52.9

21 11 7.1 60.0

22 10 6.5 66.5

23 6 3.9 70.3

24 9 5.8 76.1

25 5 3.2 79.4

26 5 3.2 82.6

27 2 1.3 83.9

28 5 3.2 87.1

29 2 1.3 88.4

30 3 1.9 90.3
31-35 5 3.1 93.5
36-40 5 3.1 96.8

Over 40 5 3.1 100.0

147

Table 15. Defendants Having Other Cases Pending

 

 

 

Relative Cumulative
Number of Absolute Frequency Adjusted Freq.
Other Cases Frequency (Percent) (Percent)
0.0 118 76.1 76.1
1.00 30 19.4 95.5
2.00 4 2.6 98.1
3.00 l 0.6 98.7
4.00 1 0.6 99.4
5.00 1 0.6 100.0
Total 155 100.0 100.0
Mean .323
Median 0.0

Standard Deviation .720

Table 16.

Know Each Other
(Family Friendship
or Regular Contact)

Do Not Know Each Other

Table 17. Previous Arrests

None

One or More

148

Relationship of Defendant and Complainant

 

 

 

Absolute Relative
Frequency Frequency
(Percent)

18 11.6

137 88.4
Absolute Relative
Frequency Frequency
(Percent)

77 49.7

78 50.3

 

APPENDIX A.2

Selected Scatter Plots

The object of the first part of the theory develOped here was
to determine the functions which bound the set of Pareto Optimal plea
bargain outcomes. Since'there do not exist statistical tools which
test for achievement of that prOperty in the switch functions derived,
the following plots are Offered for observation. Plea bargain outcomes,
taken to be the maximum sentence on the pleaded offense, are graphed
on the vertical axis against the estimated relative strength on the
horizontal axis, where the latter increases with greater prosecution
strength. This same plot appears for different subsets of the sample
used in the regression analysis of section 3.3.

149

relative strength and guilty plea maximum

White defendants:

FELgflIrE: 2E).

34.0

39.:

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uo.c~

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swoon

00.0"

1vuza¢> umtaauuw o.

a . 1 ».;.4> cz.mm.z s 1 ”made, owe-ode
awann.5 1 .s. .ecam .vgq..on1 1 .a. .awuau.¢. .n.uc.o. 1 any so any can
«new..n 1 .u:¢¢.e_ze_n o.aa~.o 1 uuug.un a .nuoo.o 1.x. to..caweccu

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o o . — .o o
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APPENDIX A.3

Additional Regression Results

The first two sets of regression results display the alternative
specifications of the linear function used in estimating relative
strength from the data on the prosecutor's opinion about the probability
of conviction at trial.

The latter two tables present the results of regressions run in
the same manner as the ones shown in section 3.3, except that here the
resources of both parties are assigned their mean values. Resources
are held constant at those levels in order to reduce the amount of
cross-variation within the first and third terms, which are each the
product of several variables.

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TABLE 20 Estimation of Guilty Plea Maximum Sentence with Resources of

Both Petites Held Constant

155

 

2

Cut al 82 83 R F

None -.290 .635 -1.270 .4997 41.45
(-.19) (8.11) (-.13)

k1-125 -.509 .632 3.867 .5001 41.52
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k2-0.0 -.509 .632 3.867 .5001 41.52
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(-1.77) (9.27) (.53)

k4-150 -.790 .670 3.389 .5066 42.60
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k6-.60 -1.312 .738 6.273 .5231 45.53
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k7=.65 -1.265 .727 5.275 .5140 43.88
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k8-.70 -1.151 .707 3.895 .5053 42.39
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k9-180 -1.369 .700 4.637 .5057 42.46
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155/R,154 -1

where x - .0065R .r

1
x2 .
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and where R, R', and B are held constant at
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TABLE 21 Estimates of Difference Between Guilt
.Charge. Maximum Sentences,

156

y Plea and Attempt

with Resources of Both Parties held

 

Constant

Cut 81 a2 83 R2 F (all 4.0005)

None -0.602 .572 -3.133 .4321 31.58
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k1--.25 -1.131 .562 6.272 .4328 31.67
(~3168) (8.05) (.45)

k2-0.0 -1.131 .562 6.272 .4328 31.67
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k3-.25 -1.185 .570 4.171 .4356 32.03
(3.73) (8.05) (.78)

k4=.50 -1.606 .627 5.719 .4526 34.31
(-3.94) (8.03) (1.79)

k5-155 -2.141 .697 8.795 .4823 38.66
(-4.62) (8.48) (2.85)

k6-.60 -2.319 .719 9.299 .4865 39.33
(~4.66) (8.44) (2.98)

k7-165 -2.271 .706 7.965 .4663 36.26
(-3.92) (7.65) (2.33)

k8-.70 -2.048 .669 5.582 .4439 33.13
(-2.72) (6.30) (1.37)

k9-.80 -1.758 .611 3.453 .4350 31.96
(-1.86) (6.11) (.73)

where x1 - .OO65R155/R'154J”1
x-G
x3 -‘§.Y’

and where R, R', and g are held constant at

their respective means.

REFERENCES

157

REFERENCES

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American Bar Association, Advisory Committee on the Criminal Trial
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American Bar Foundation. Michigan Field Reports (Daily notes of inves-
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Ashenfelter, Orley A. and Johnson, George E. "Bargaining Theory, Trade
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Bartholemew, Peter Prosecutor, Pre-Trial Division, Recorder's Court,
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Becker, Gary "Crime and Punishment: An Economic Approach." J.P.E.
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Belsley, David "On the Determination of Systematic Parameter Variation
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Boulding, Kenneth Conflict and Defense: A General Theory. New York:
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Cartter, Allan M. Theory of Wages and Employment. Homewood, Illinois:
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Chamberlain, Neil Collective Bargaining;» New York: McGraw—Hill Book
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Coase, R.H. "The Problem of Social Cost." J. Law Econ. 3 (October 1960):
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Coddington, Alan "A Theory of the Bargaining Process: Comment." A.E.R.
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1 Review of The Economoics of Bargaining by John G.
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Comay, Yochanan; Melnick, Arie; and Subotnik, Abraham "Bargaining,
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Cross, John G. The Economics of Bargaining;_ New York: Basic Books, Inc.,
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Cummings, L.L. and Barnett, D.L. "Bargaining Behavior in a Symmetric
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Dahl, Robert "The Concept of Power." Behavioral Science 2 (1957):
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Danziger,Sheldon and Wheeler, David "Malevolent Interdependence, In-
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Dawson, Robert O. Sentencing: The Decision as to Type, Length, and
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de Menil, George Bargaining} Monopoly Power versus Union Power. Cam-
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Douglas, Ann Industrial Peacemaking. New York: Columbia University
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