EFFECT OF KNOWLEDGE OF REWARD AND IMPosm

Dmszsw OF REWARD upon SOCIAL CONTACTS AND;   : '5 . , M

BARGAH‘fiNG PM A THREE-PERSON COALITION GAME

Thesis for the Degree of 'Ph. D.
MECHiGAN STATE UNEVERSIW
GERRET EARL DE YOUNG

1971 '

LIBRARY

Michigan State
University '

 

This is to certify that the

thesis entitled
Effect of Knowledge of Reward and Imposed
Divisibiiity of Reward Upon Social Contracts
and Bargaining in a 3-Person Coalition Game

presented by

Gerrit DeYoung

has been accepted towards fulfillment
of the requirements for

Ph-D. degreeinfisthQJmL

MEXJWLIM

Major professor

Date—WH—z-é-H-BP-

0-7639

 

 

 

 

ABSTRACT

EFFECT OF KNOWLEDGE OF REWARD AND IMPOSED DIVISIBILITY
OF REWARD UPON SOCIAL CONTACTS AND BARGAINING
IN A THREE-PERSON COALITION GAME

BY

Gerrit Earl De Young

In deterministic coalition formation games, initial
contacts have been previously found to be directed most
frequently to the less powerful of two potential coalition
partners; successive contacts were directed more equally to
each of the partners. The dominant hypothesis explaining

this effect has been the elimination of confusion expla-

 

nation, which suggested that subjects quickly concluded
that, since a coalition of any two potential partners
resulted in victory in a deterministic game, each partner
could be considered to have equal resources. Alternative

explanations included the cumulative score hypothesis,

 

that players who were behind on the reward dimension
discriminated against the leader on that dimension, and

the utility of response variability hypothesis, that

 

subjects in repeated contact attempts vary their responses
through boredom or fatigue, or from considerations of

equity. No previous attempt to obtain responses of subjects

Gerrit Earl De Young

in a series of probabilistic coalition formation games had
been made, but the above hypotheses were adaptable to the
probabilistic game also.

The present Experiment I permitted the elimination
of confusion during a series of ten games, allowing
extensive experience with the game while preventing any
effect of accumulated reward. There was clearly no
tendency for contacts or reward divisions to be divided
more equally between the stronger and weaker candidates
in contrast with the elimination of confusion explanation.
The results of Experiment I appeared to be consistent with
only the accumulative score explanation.

Experiment II was designed to test the accumulative
score explanation by allowing one group of subjects to
choose coalition partners with knowledge of their own and
other potential partner's reward, while other subjects
chose partners with knowledge of only their own reward.
Subjects in both conditions apparently attended to only
their own reward, since the effect predicted for the group
which knew only their own reward occurred also in the
group which had knowledge of each partner's reward. This
may have been due to a memory factor introduced by a verbal
announcement of the rewards, or may have been caused by a
subtlety in the instructions.

Previous studies had found that subjects in a

deterministic version of a coalition formation game tended

Gerrit Earl De Young

to choose partners based on a criterion of "parity," such
that they would maximize the amount they had contributed
to a coalition, while subjects in a probabilistic version
tended to make choices based on a criterion of "security,"
such that they would try to form the coalition with the
maximum total resources. These findings were confirmed in
the present study; the probability of choosing the weaker
potential partners was consistently greater in the
deterministic version. When a specific divisibility of
reward was imposed in previous studies, it had been found
that coalitions were formed such that the propositions of
the resources contributed by each coalition partner
approximated one or the other of the positions of the
reward as it was required to be divided. No such trend was
observed in this experiment; apparently even relatively
small differences in the amount of resources contributed
by each partner were sufficient to specify to the subjects
which partner should receive the greater, and which the
lesser, portion of the reward. Consistent with the above
finding was the fact that when no specific reward division
was imposed, the proportion of the reward obtained by the
weaker candidate was consistently less than 50% but greater
than the proportion of the resources which he contributed
to coalition.

Mathematical models were constructed involving

mathematical expressions for the parity, security, and

Gerrit Earl De Young

cumulative reward factors; the results were intuitively
reasonable. The model involving only the parity factor
consistently fit the deterministic version data better
than either the model involving only the security factor
or the model involving a combination of the parity and
security factors. Likewise, the model involving only the
security factor consistently provided the best fit of the
probabilistic version data. The cumulative reward factor
consistently improved the prediction of data in which

subjects were aware of their accumulated reward.

EFFECT OF KNOWLEDGE OF REWARD AND IMPOSED DIVISIBILITY
OF REWARD UPON SOCIAL CONTACTS AND BARGAINING

IN A THREE-PERSON COALITION GAME

BY

Gerrit Earl De Young

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Psychology

1971

DEDICATION

To Peg, without whom this dissertation might

have been possible but not worthwhile.

ii

ACKNOWLEDGMENTS

It is difficult to make explicit the full extent
of my gratitude to the members of my committee, since they
have been helpful for several years in many different ways.
From Dr. Jim Phillips, I have received constant encourage-
ment and suggestions for a fascinating research area in
which to work. He has taken much of his time to acquaint
me, as he has all of his graduate students, with the full
range of responsibilities of a professional psychologist
and the importance of careful and well analyzed research.

Dr. Charles Wrigley has consistently influenced
the course of my early career, beginning even before the
start of my graduate program. His insistence upon the
importance of progressive techniques combined with his
desire to see them applied to whatever areas in which his
students might be interested have provided an ideal work-
ing situation. His personal interest and careful advice
will be long remembered.

Dr. Tom Connor's knowledge and interest in the
area of this dissertation have been helpful in many ways.

He has given his time and knowledge freely to offer a

iii

fresh viewpoint and valuable suggestions for improvements.
Dr. Raymond Frankmann's knowledge of mathematical models
and techniques have made the several discussions with him
interesting and valuable.

I have enjoyed knowing the members of my com—
mittee, and I look forward to continuing relationships

with them as colleagues and, hopefully, as friends.

iv

LIST OF
LIST OF
Chapter

I.

II.

III.

IV.

TABLE OF CONTENTS

TABLES . . . . . I . . . . . .
FIGURES. O O O O O O O O O I 0
INTRODUCTION 0 O O O O O O O O O
The Range of Triadic Coalition
Situations . . . . . . . . . .
Alliances in Mixed Motive Games . . .
The Coalition Formation Process . . .
Mathematical Models of Coalition
Formation . . . . . . . . . .
EXPERIMENT I . . . . . . . . . .

Statement of the Problem . . . . .
Method and Procedure. .
Results . . . . .
Discussion . . . . . . . . . .

EXPERIMENT II . . . . . . . . . .

Statement of the Problem . . . . .
Method and Procedure. . . . . . .
Results . . . . . . . . . . .
Discussion . . . . . . . . . .

MATHEMATICAL MODELS . . . . . . . .

Games Not Involving Accumulated Reward
or Imposed Divisibility of Reward . .
Games Involving Accumulated Reward . .
Discussion . . . . . . . . . .

SUMMARY AND CONCLUSIONS. . . . . . .

Successive Games . . . . . . . .
Imposed Divisibility of Reward . . .

iv
vi

Page

23
35

35
37
41
64

68
68
70
77
96
102
103
104
110
112

112
114

Chapter

Division of Reward . . . . . . . . . 115
Miscellaneous. . . . . . . . . . . 116

BIBLIOGRAPHY . . . . . . . . . . . . . . 118
APPENDICES
Appendix
I. Distribution Presentation Orders . . . . . 122
II. Sample Set of Response Sheets . . . . . . 123
III. Sample Bargaining Sheet . . . . . . . . 133

Iv. Instructions 0 O O O O O O O O O I O 134

vi

10.

LIST OF TABLES

Analysis of Variance for Probability of
Choosing Weaker Candidate, Experiment I .

Frequency of Coalitions in Two Versions of
Convention. . . . . . . . . . .

Analysis of Variance for Reward to Weaker
Candidate, Experiment I . . . . . .

Combinations of Reward and Resource Points,
i.e., Payoff Distributions, Used in
Experiment II. . . . . . . . . .

Order of Games Involving Accumulated Reward,
Imposed Divisibility of Reward, and Each
Resource Distribution Number, Experi-
ment II. . . . . . . . . . . .

Analysis of Variance for Choice of Weaker
Candidate as a Function of Condition,
Version, Resource Point, and Payoff
Distribution, Experiment II . . . . .

Probability of Contacting Weaker Candidate
in Each Condition and Version of Experi-
ment II, Summed Over All Payoff Distri-
butions. . . . . . . . . . . .

Analysis of Variance for Choice of Weaker
Candidate as a Function of Reward Divisi—
bility, Version, Distribution, and
Resource Point, Experiment II . . . .

Analysis of Variance for Reward to Weaker
Candidate, Experiment II . . . . . .

Number of 50-50 Splits, Mean Proportion of
Reward to Weaker, and Standard Deviation
of Reward to Weaker Candidate in Games Not
Involving Imposed Divisibility of Reward,
Experiment II. . . . . . . . . .

vii

Page

42

49

65

72

73

79

81

82

86

88

Table Page

11. Minimum Chi-squares of Models Based on Parity
and Security Factors in Version CV and UV
Games Not Involving Accumulated Reward or
Imposed Divisibility of Reward, Experi-
ments I and II Combined . . . . . . . 104

12. Minimum Chi-squares of Models Based on Parity
and Security Factors in Version CV and UV
Games Involving Accumulated Reward,
Experiment II . . . . . . . . . . 107

viii

Figure

1.

2.

10.

11.

LIST OF FIGURES

Flow Chart of Steps in the Coalition
Formation Process . . . . . . . . .

Observed Probabilities of Choice of Weaker
Candidate and Model I Predicted
Probabilities . . . . . . . . . .

Observed Probabilities of Choice of Weaker
Candidate and Model II Predicted
Probabilities . . . . . . . . . .

Resource Point by Version Interaction Means,
Experiment I. . . . . . . . . . .

Probability of Choice of Weaker Candidate in
Two Versions of the Political Convention
Game as a Function of Game Number, Experi-
ment I. . . . . . . . . . . . .

Amount of Money to Weaker Candidate and 99.9%
Confidence Intervals as a Function of Game
Number, Experiment I . . . . . . . .

Histograms of Reward to Weaker Candidate in
Each of Ten Games, Experiment I . . . .

Summary Histogram of Reward to Weaker Candi-
date in all Ten Games, Experiment I . . .

Parity Division and Actual Reward to Weaker
Candidate in Each Coalition, With 99%
Confidence Intervals, Experiment I .« . .

Summary Histogram of Reward to Weaker Candi-
date in all Games Not Involving Imposed
Divisibility of Reward, Experiment II . .

Offers and Demands Made in Each Trial of the
Bargaining Phase in all Games Not Involving
Imposed Divisibility of Reward, Experi-
ment II . . . . . . . . . . . .

ix

Page

12

30

34

45

48

52

54

61

63

91

93

Figure Page

12. Offers and Demands Made in Each Quarter of
Bargaining Sequences in all Games Not
Involving Imposed Divisibility of Reward,
Experiment II . . . . . . . . . . . 95

13. Backward Curve of Offers and Demands Made
on Each Trial, Experiment II . . . . . . 98

14. Observed and Predicted Probabilities of
Choice of Weaker Candidate, in Games Not
Involving Imposed Divisibility of Reward
or Accumulated Reward, Experiments I and
II Combined, All Vote Distributions. . . . 106

15. Observed and Predicted Probabilities of
Choice of Weaker Candidate in Games
Involving Accumulated Reward, Distri-
bution 60, Experiment II . . . . . . . 109

CHAPTER I

INTRODUCTION

The Range of Triadic Coalition
Situations

 

 

A triad can be viewed as an interrelated social
system containing three elements, each element consisting
of a person or an organization acting as a unit. The
analysis of triadic situations has ranged from Simmel's
(1950) discussion of the role of triadic conflict in the
maintenance of social structure through more recent
laboratory examinations of triadic situations and coa-
lition formation (Vinacke and Arkoff, 1957; Caplow, 1959).

Caplow (1968) postulated that triads are the
underlying building blocks of all social organizations.
He identified triadic conflict and coalitions as being
basic to primate dominance hierarchies (p. 41), family
life (p. 62), industrial organizations (p. 128), civic
politics (p. 142), international war (p. 150), and
national government (p. 155).

Edwards (1927) compared the English, American,
French, and Russian revolutions and presented an early

theory of revolution based upon a triadic analysis. A

society ready for revolution is typified by unrest and a
sense of repression, and can be divided into groups of
exploiters, intellectuals, and the exploited. The
exploiters control the majority of the resources of such

a society. The intellectuals, including the lawyers,
teachers, and religious leaders, typically are paid by the
exploiters and are expected to justify and transmit the
system from one generation to another. The exploited are
the group of productive laborers who are also paid by the
exploiters. A major symptom of an approaching revolution
according to Edwards is the defection of a majority of the
intellectuals who begin to sympathize with the exploited
class. As a result of this coalition, three main factions
emerge: the conservatives, the moderate-reformers, and the
radicals. The conservatives are composed of the exploiters
who still control most nonsocial resources; some of the
intellectuals and some of the exploited divide between the
moderate-reform and radioal factions according to their
concept of the required institutional changes. When the
conservatives become convinced that they can no longer
govern alone, they attempt to preserve some of their former
power by forming a coalition with the moderate-reformers,
and by instituting a program of reform. In the case of an
abortive revolution, this program could succeed. In the
revolutions Edwards studied, the moderate-reformers

quickly repudiated this coalition and formed a new coa-

lition with the better organized radicals. The new

coalition typically attempts to weaken the conservatives
by harrassment including imprisonment, confiscation of
property, and deprivation of political rights; the con-
servatives typically respond by emigrating and attempting
to foment foreign intervention. The radicals then
institute a reign of terror to eliminate the moderate-
reformers and to further suppress the conservatives.
Finally, the radicals gain full control, the reign of
terror is terminated and an attempt is made to form a
stable government.

Caplow (1968) suggested that an international
balance of power could be defined as a stable power dis-
tribution in a triad without coalitions.

The situation includes three or more organised

collectivities contending for advantage in the same

area, not subject to a common sovereign, and capable

of making war. Over some appreciable interval of

time peace prevails and no coalition is formed

(p. 152).
Several factors familiar from triadic theory contribute to
the maintenance of the balance of power and to the pre-
vention of the formation of a coalition. (1) If a coa-
lition were formed, the two coalition partners following
a victory over the third power would still be independent
powers with incompatible interests. If no two of the
three original powers were originally equal in strength,
no coalition should be formed since eventual defeat would

be certain for the weaker of the two coalition partners

after the elimination of the third power. (2) Ideological

differences often prevent otherwise advantageous coalitions.
(3) Stronger powers refrain from direct attacks against a
weaker neighbor, since such an attack would compel the
third power to join the weaker power to prevent his own
eventual defeat. Caplow (1968) cited historical evidence
that such a balance of power is typically unstable, and sug-
gested several mechanisms to account for the establishment
of a coalition and the resulting upset of the balance of
power. (1) A weaker power may preemptively form a
dangerous coalition with a stronger power if it believes
a coalition between the other two powers is imminent. For
example, Stalin believed that a coalition between the
Allies and Germany was imminent in 1939; the Soviet Union
therefore preemptively entered into a nonagression pact
with Germany before that could occur. (2) A weaker power
may miscalculate, offer to form a coalition with a stronger
power, and therefore help to defeat the third power before
being defeated itself. (3) One power may increase its
relative strength until it alone can defeat the two other
powers, eliminating the need for a balance of power.
(4) Balance of power triads may be linked in such a manner
that one element of a triad is an element in several other
triads. Thus the relative strength of some elements may
be indeterminable, and events in one triad can upset a
balance of power in another triad.

These analyses are presented to suggest the range

of application of triadic and coalition formation theory.

Factors which were used such as the reluctance of a weaker
element to form a coalition with a stronger because of the
control the stronger element can assert form a part of
rigorous triadic theories. It will be necessary in the
succeeding sections to further refine the concepts used in
such analyses and to focus upon the issues of special

relevance to laboratory research.

Alliances in Mixed Motive Games

Luce and Raiffa (1957) identify the existence of
a "game" with the existence of a conflict of interest.
Schelling (1958) divided two-person games into (1) pure
coordination, (2) pure conflict, and (3) mixed-motive
games. Gamson (1964) extended the classification for
situations involving more than two participants.

Pure coordination games, exemplified by the
interaction between a set of partners in a game of bridge,
are characterized by the existence of a solution which
maximizes the return for all players. In the classical
sense, a pure coordination game should not be interpreted
as a game at all, since no conflict of interest exists.
Any relevance of this situation to game or bargaining
theory typically derives from the introduction of some
impediment to communication, creating problems in coordi—
nating and mobilizing the resources required for the

achievement of the goal (Schelling, 1958).

The pure conflict game is distinguished by the
fact that no participant can gain more by joining with
others than he can gain by himself. An example of a two
person conflict game is chess, while the n-person situ-
ation, the pure n-uel, a generalization of the duel to
the n-person case (Cole and Phillips, 1969), is an example
of a pure conflict game, as are some n-person zero-sum
games.

The mixed-motive game is characterized by the
existence of both elements of cooperation and elements of
conflict. The usual example of a mixed motive game is the
two-person Prisoner's Dilemma Game, in which the par-
ticipants can make either cooperative or uncooperative
responses. In the n-person case, coalition games are
examples since there is something to be gained by cooper-
ation with some, but not all, participants. A dis-~
tinguishing feature of the mixed-motive game is that a
defection from a cooperative solution may conceivably
increase the defector's payoff, while such a result would
be impossible in the pure coordination game. This feature
of mixed-motive games is responsible for the fact that so-
called cooperative solutions to such games are highly
unstable. Such solutions do, however, exist.

A general class of cooperative solutions for
mixed-motive games is given by Phillips (1967). In

general, a cooperative solution is referred to as an

alliance, i.e., an ordered pair <Ci, Aj> in which C1 is
some sub-set of the n players and Aj is some agreement.

The null agreement differs from other possible agreements
simply in that the AO interactive process, i.e., no
negotiation, between the members of Ci is required to
achieve it. It should be pointed out in this connnection,
however, that the null agreement refers to a non-interactive
cooperative solution in a genuine mixed-motive game and not
to the mere successful cooperation in a game of pure
coordination. In general, the usefulness of the concept

of an alliance is limited to mixed~motive games since they
are unnecessary in pure coordination games and impossible

in pure conflict games.

In addition to the null agreement, Phillips notes
other types of agreements which constitute the basis of
more potentially cohesive alliances. For example, some
subset of players may agree to abstain from certain
behaviors as in some sort of tariff agreement or non-
aggression pact, or to perform certain other behaviors as
in some trade agreement or mutual defense treaty. The
fact that these two types of agreements become indis-
tinguishable when applied to a 2x2 matrix game is unim-
portant since this is due to the restriction of the range
of behaviors to only two alternatives for each player.

DeYoung and Phillips (1970) suggested a possible

parameterization of Alliance Theory in terms of the work

of Browning (1969) which deals with that aspect of

collective decision making generally known as log-rolling.
In this example, consider some set S of decision makers
and some set I of issues on which the decision makers
will vote, that is, for any issue i e I there exists
a partition of S, say Si = {S+, 50’ S} such that if
n(S+) > n(S-), where n(-) is an enumerative measure of
the set, the issue is said to have passed, and such
that if n(S+) < n(S_) the issue is said to have been
defeated. Assume further that there exist two functions
fi+(S) and fi_(S) that assign utilities to each member
of S in either case. That is, f-+(S) maps the members
of S onto a set r of utilities if, with respect to
issue i, n(S+) > n(S_) and fi_(S) maps the members of
S into r if, with respect to i, n (S ) < n(S_). Thus,
the situation can be characterized by the set F of
ordered pairs of functions <fi+, fi-> such that there
is a one-to-one correspondence between the set F and
the set I. Clearly there exist a wide range of F sets
which make this a mixed-motive game.

Browning proposed two outcomes of political
bargaining which can be interpreted as "cooperative
solutions," (1) the minimization of variability of utility
between players, and (2) the maximization of social welfare,
i.e., the maximization of sums of utilities over players.
Browning's model is formulated in terms of dyadic bargain-
ing, so that it can be conceptualized in terms of agreement
by two members of S to vote against certain subsets of the
issues I, even if this is contrary to either of the
player's individualistic interests. The number of issues
involved in the agreement could be used as a measure of
the intensity of the agreement, and DeYoung and Phillips
(1970) suggested the hypothesis that the solidarity or
cohesion of an alliance is directly proportional to the

number of issues on which there is agreement.

Alternatively, a pair of players may want to agree to vote
in favor of certain issues or, although not explicitly
included in Browning's model, it is possible that a pair
may want to agree not to log-roll on a set of issues upon
which their interests coincide. In the terminology of
Phillips (1967), it is possible in any of these instances
to speak of a set of potential agreements, Ak, where Ak
refers to an agreement on k issues. The null agreement,
A0, takes on the natural meaning of an agreement on exactly
zero issues. DeYoung and Phillips (1970) concluded:

If, in a game of the Browning type, it were possible
for some subset Cj of players to agree on every i e I,
then this group would be recognized as a highly
cohesive, maximally polarized sub-group of S. If, in
addition to the explicit issues, this group were to
invent or recognize some further transcendent issue
such as some method of side payment which would
guarantee to each member of C- some fixed proportion
of the payoff to the entire group, we would recognize
the sub-group to be further strengthened. We might
desire some special designation for this type of
agreement and for the resulting type of alliance.
Phillips (1967) provides these designations. The
agreement, in virtue of the guarantee of fixed pro-
portionality is termed a common-fate agreement, and
the alliance is designated a coalition. While this
usage of the term coalition is perhaps inconsistent
with many of the ways that term has been used in the
past, it is consistent with its meaning in a great many
of the experimental studies of coalition formation
(Vinacke and Arkoff, 1957; Kelley and Arrowood, 1960;
Gamson, 1961 a, b; Chertkoff, 1966; Phillips and Nitz,
1968; Nitz and Phillips, 1969; Cole, in press), and
is at least roughly compatible with the usage of Luce
and Raiffa (1957).

A coalition is treated, therefore, as a very
intense form of alliance which is one of a number of
possible relatively cooperative solutions to some mixed-

motive game. We will next be concerned with the

10

interactive process that leads to such a coalition. This
process and its outcomes as described in the succeeding

sections will form the focus of this dissertation.

The Coalition Formation Process

The coalition formation process typically involves
an attempt by each potential coalition member to combine
with some other potential coalition member in order to
share in certain payoffs or profits of the successful
coalition. A review of previous coalition formation
literature suggests that an analysis of the type of coa-
lition formation process to be considered here can be made
in terms of the following.

1. Each potential coalition member may attempt
to contact another, yielding a set of initial
contact probabilities from each individual
to each of the other individuals.

2. If no reciprocal contact is made, another set
of contact attemptsmay be initiated.

3. When a reciprocal contact is achieved, a
period of bargaining between the contacters
may ensue, in which matters pertaining to the
functioning of the proposed coalition
(division of profits of coalition, voting on
certain issues, etc.) would be expected to be

discussed.

 

 

 

 

4. If the bargaining is successful, i.e., if both
members agree to the results of the bargaining,
the coalition would be formed and could be
expected to perform the function for which the
coalition was proposed.

5. If the bargaining was not successful, the
process could be expected to return to step
2.

A flow diagram of the proposed process is presented in
Figure l.

A wide variety of factors could be expected to
affect the coalition formation process, either during the
contact period or during the period of bargaining. The
factors which would be expected to be primarily associated
with the contact process itself would be such things as
differential resources or differential status among
potential coalition members (Caplow, 1959; Vinacke and
Arkoff, 1957), the divisibility of the payoff (Phillips
and Nitz, 1968; Nitz and Phillips, 1969), certainty of the
payoff (Chertkoff, 1966), degree of control of resources
(Cole, in press), ideological factors, and past experience
with coalition formation (Kelley and Arrowood, 1960;
Chertkoff, 1966). Those factors which would be expected
to be primarily associated with the bargaining process
could be subclassified along the lines suggested by

Figure 1 into internal and external factors. Among the

12

 

Situation is
recognized as
mixed-motive

1

Members attempt to

 

 

 

 

 

 

contact one another

Reciprocal contact

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

established
Bargaining
initiated
l v
Bargaining Bargaining Bargaining
Terminated Terminated Terminated
Unsuccessfully Successfully Unsuccessfully
due to (condition due to
internal factors established) external factors

 

 

i

 

 

Continue with the Business
of the Coalition

 

 

Figure l.--Flow Chart of steps in the Coalition Formation

Process.

 

13

external factors would be such things as threats and
counteroffers by players not involved in the potential
coalition (Kline, 1968), or external events which upset
the present distribution of resources (Azar, 1969).
Factors internal to the bargaining process include such
things as bargaining style (Bond and Vinacke, 1961),
personality characteristics of the bargainers (Vinacke,
1969; Nitz, 1969), or considerations of fairness or equity

(Messe, 1969).
A. Coalition Formation Paradigms

One paradigm used to study coalition formation is
the "parchesi game" (Vinacke and Arkoff, 1957; Kelley and
Arrowood, 1960). In this game, several players have
different weights such that when dice are rolled, the
number rolled is multiplied by the player's initial weight
to determine the number of moves each player can make
toward the goal. Coalition formation enters into this
situation since each player (having a weight of NA) can
attempt to join with either the weaker player (with a
weight of NW) or the stronger potential partner (with
a weight of N3) in order to ensure that the coalition
members will reach the goal before the third player.

The pure truel, a special case of the n-uel in
which each of three players is given varying resources

and then is given the opportunity to remove part of either

14

of the other two players' resources, has been described
earlier as a pure conflict game. Cole (1969) devised a
mixed-motive form of the truel by allowing the participants
to form a coalition before the start of the game. The
effect of the coalition was the formation of a non-
aggression pact i.e., each coalition partner would attack
only the player outside the coalition until that player

was removed from the game by the total loss of his
resources.

Recent research in the field of coalition for-
mation (Chertkoff, 1966; Phillips and Nitz, 1968; Nitz and
Phillips, 1969; Nitz, 1969) has involved the use of a
Political Convention Situation, in which a subject is told
that he represents one candidate for nomination. The

subject's candidate purportedly has N votes, which is

A
less than the majority of the votes represented at the

convention and necessary to receive the nomination. The
subject is also told that there are two other candidates
for the nomination, a weaker candidate with NW votes and
a stronger candidate with NS votes (Nw < NS), neither of
whom controls a majority of the votes at the convention.
Since no candidate can win without joining with one of

the other candidates, the subject is asked which of the

other two candidates he will approach first to try to form

a coalition.

15
B. Initial Contact Probabilities
Effect of "Parity" and "Security"

Two factors, "parity" and "security," have been
found to be especially relevant to the subject's initial
choice of the weaker or stronger candidate. The concept
of parity, as discussed by Gamson (1961), suggests that
"any participant will expect others to demand from a
coalition a share of the payoff proportional to the amount
of resources which they contribute to a coalition."
Clearly, a candidate would contribute a higher proportion
of votes, therefore expecting to receive a greater payoff,
if he formed a coalition with the weaker candidate. The
postulate that subjects will tend to choose partners
according to the parity norm, combined with the postulate
that subjects attempt to maximize their share of the
payoff, leads to the prediction that the coalition formed
will be that winning coalition which has the fewest
resources of the possible winning coalitions; this line of
reasoning has been formalized as Minimum Resource Theory
(Gamson, 1961). Such a tendency has been observed by
Vinacke and Arkoff (1957) with respect to formation of
coalitions, and by Chertkoff (1966) with respect to initial
contacts. 0n the other hand, the coalition must be
"secure" i.e., must have enough resources to afford as
great a chance of winning as possible. Security could be

expected to be a factor in the Political Convention

16

Situation since subjects would be aware that a certain
amount of attrition in delegate strength is a possibility
in real life political conventions. Clearly, although the
subject's expected payoff would be maximized by choosing
the weaker coalition partner, the security of the coalition
would be maximized by choosing the stronger coalition
partner (Cole, 1969).

Since these two factors are maximized by two
incompatible choices, the salience of each factor can be
manipulated. For example, in connection with his statement
of the security principle, Cole (1969) introduced a
probabilistic variation of the truel in which the removal
of an attacked player's resources was determined with a
probability less than one. Chertkoff (1966) made the
attainment of a reward in the political convention paradigm
dependent upon the results of a national election following
the convention; in two conditions the likelihood of victory
in the national election was stipulated to be greater if
the strongest candidate were nominated.

DeYoung and Phillips (1970) discussed an Un—
committed Vote variation of the political convention
paradigm. N uncommitted votes were introduced in addition

U

to the N Nw’ and NS votes. The requirement for a winning

A!
coalition was that the coalition includes a majority,

. (N + N + N + N + 1)
i.e., at least A w 28 U votes. In general,

 

the formation of a coalition in the Uncommitted Vote

l7

situation did not guarantee a majority of votes, but simply
improved the chances of obtaining a majority of votes after
the uncommitted votes were split between the candidates.

In this situation as in the probabilistic games described
above, the emphasis placed upon the parity factor should

be decreased since in order for there to be a payoff, the
votes in the coalition must actually be sufficient to win
the nomination.

In contrast with the deterministic situation in
which the tendency to form the cheapest winning coalition
typically results in the exclusion of the candidate with
the greatest proportion of the resources (the "strength is
weakness" effect), data from the probabilistic coalition
formation situations suggest that players tend to choose
the stronger partner (the "strength is strength" effect)
in order to maximize their chances of winning; this result
is clearly in accordance with the security principle.

Effect of Divisibility of Reward Upon
Initial Contacts

Nitz and Phillips (1969) suggested that intra-
coalition compatibility, which could depend upon the ease
with which a reward for coalition formation could be
divided between the members, could be a factor in initial
coalition formation contacts. If the reward were only
unequally divisible, a coalition between two candidates

who differed in their amount of resources should be more

18

likely since the difference in resources could suggest to
the participants a norm for determining which participant
should receive the greater share, and which should receive
the smaller share of the reward. However, if the reward
were to be shared equally, a coalition between equals
should be relatively more likely. In confirmation of this
hypothesis, Nitz and Phillips found that when the unequally
divisible reward was the simulated nomination of a governor
and lieutenant governor, §S contacted the weaker candidate
significantly less often when the weaker candidate was
equal to the S in resources, compared with a condition in
which the weaker candidate had fewer resources than the S.
Equally important, gs contacted the weaker candidate less
often when the two were equal in resources in the unequal
divisibility of reward condition compared with a condition
in which the reward could be divided in any desired
proportion.

An alternative effect is intuitively reasonable
under conditions of imposed divisibility of reward, however.
If the reward is of sufficient importance to the partici-
pants that it outweighs any potential difficulties in
negotiation, the participants in an unequally divisible
reward condition may choose that potential partner who
would be more likely to accept the smaller share of the
reward, i.e., the weaker potential partner, regardless of
considerations of relative compatibility. When equality of

reward is imposed however, no participant can increase his

19

share of the reward by choosing the weaker potential
partner; he can only increase his chances of acquiring the
reward by increasing the security of the coalition. He can
achieve this goal by choosing the stronger potential
partner even if the candidate has considerably greater

resources than his own.
C. Successive Contact Probabilities

Kelley and Arrowood (1960) and Chertkoff (1966),
both using deterministic versions of coalition formation
games, found that, while contacts in initial coalition
attempts did occur between the weaker potential coalition
partners with more than chance probability, contacts in
later coalition attempts tended to be directed more equally
to each of the two potential coalition partners.

Four hypotheses have been advanced to explain
this change in response pattern:

1. gs may have quickly learned that since a
coalition with either of the other candidates
in the Committed Vote version resulted in
victory, each candidate could be considered
to have equal resources (Kelley and Arrowood,
1960). Vinacke, Crowell, Dien, and Young
(1966) have partially discredited this
hypothesis by conducting an experiment in
which they informed §$ directly that a coa-

lition with either candidate resulted in

 

 

 

 

 

20

victory. They interpreted their data as
demonstrating that the strength is weakness
effect was not weakened;

Since bargaining resulted in an accumulation
of reward over trials or games, a new resource
dimension (i.e., the accumulated payoff)
resulted. (a) The stronger candidate, after
being left out of earlier coalitions,
therefore quickly became the weaker candidate
on the new, more salient resource dimension.
Hoffman, Festinger, and Lawrence (1954) and
Bond and Vinacke (1961) have found that
players who were behind on the reward di-
mension tended to form a coalition against the
player furthest ahead on that dimension. (b)
The accumulated reward could affect each
player even if he attended only to his own
reward. If at one point, an S were relatively
strong on the accumulated reward dimension for
example, it is possible that he would attempt
a "gambling" strategy, choosing the weaker
vote resource player more frequently in an
attempt to acquire an even more disproportion-
ate proportion of the reward. Such an effect
could occur independently of the high reward
candidate's current status on the vote

resource dimension. There has been no

21

previous examination of the effects of one's
own reward in the coalition formation
situation.

3. Aside from the previously discussed factors
affecting the utility of each choice, there
may also be a utility of response variability.
Ofshe and Ofshe (1968) have developed and
tested a mathematical model for "stable state"
social choices involving this mechanism.

Ofshe and Ofshe suggest that besides "boredom

or fatigue,‘ equity considerations enter into
this utility of response variability.

4. In the case where successive choices are made
because there was no mutual choice on the
previous trial, the S's subjective expected
probability of a reciprocated choice could be
expected to decrease.

Differential predictions follow from these
hypotheses. If hypothesis 1, that §s reevaluate power
relationships, is correct, the payoff should be divided as
suggested by the parity norm after initial contacts,
replicating Vinacke and Arkoff (1957). After successive
contacts however, the §s should tend to divide the reward
evenly since at this point they would perceive each par-
ticipant's relative power as equal. A second implication

of this hypothesis, together with the suggestion that

there is less emphasis on the payoff factor in the

22

Uncommitted Vote Variation, is that there should be less
response change in successive contacts in the Uncommitted
than in the Committed Vote Variation. That is, a re-
evaluation of the payoff due a candidate should have less
effect on choices in the Uncommitted Vote version since
the security of the coalition is of greater importance in
that version.

If the accumulated reward were responsible for
the disappearance of the strength is weakness effect as
suggested by hypothesis 2, that effect should occur only
when (a) the §s are aware of the amount of the reward
acquired by the players at each point of the current vote
resource distribution, and when (b) the amount of the
reward acquired is inversely related to the number of
votes, since Bond and VinaCke (1961) found a tendency for
coalitions to form between those behind on the reward
distribution. On the other hand, the response change over
successive contacts should be negated by concealing from
the SS the amount of the payoff which had been acquired as
the game progressed.

If there exists a utility of response variability
as suggested by hypothesis 3, the model of Ofshe and Ofshe
(1968) should be adaptable to describe the development of
response variability over trials. One clear implication
is that the utility of response variability should be

greater when successive trials are employed rather than

23

successive games. Also, in contrast with hypothesis l,_
hypothesis 3 suggests that the degree of any response
change in the Uncommitted Vote variation should be
approximately equal to the degree of response change in

the Committed Vote variation, since the effect of a utility
of response variability would be approximately equal in
both variations.

Mathematical Models of
Coalition Formation

 

 

A. Shelly and Phillips (1966)

Shelly and Phillips (1966) suggested that each
individual in a triad evaluated the other two members in

terms of their value as a coalition partner. According to

V (W)
. . _ A
thlS theory, as in Luce (1959), Pw — VA(w) + VA(Sywhere

Pw represents the probability that A chooses the weaker

 

potential coalition candidate, and VA(w) and VA(s)
respectively represents the "values" to player A of choosing
the weaker and stronger candidate.

The model was derived for a deterministic coa-
lition formation paradigm in which the person with greater
perceived power is less likely to be included in a
coalition, the "strength is weakness" effect. Therefore,
VA(w) was defined in terms of perceived power M(A) and

M(w) of player A and the weaker potential coalition

M(A)
M(A) + M(w)’

 

partner respectively. In particular, VA(w) =

24

and as a first approximation M(A) was taken to be equal to

N Therefore,

A.

 

The model was tested empirically by comparing its
predictions with the observed data from studies by
Chertkoff (1966), Vinacke, and Arkoff (1957), Vinacke
(1959), Stryker and PSathas (1960), Shelly (1967), and
Phillips and Nitz (1968). The predictions of the model
were supported by data from the Chertkoff (1966) study but
were not supported in the other studies. It should be
noted however that the model has no parameters, and the
assumption that M(A) = N is an extremely strong as-

A

sumption.
B. Chertkoff (1967)

Chertkoff (1967) mathematized a revision of a
theory by Caplow (1956) for the deterministic condition
and the power division NA>NB>NC. Caplow's theory was based
on the assumption that players will attempt to enter coa-
litions in which they have control, i.e., coalitions in
which they have more resources than their coalition
partners. Caplow predicted that there would occur an

equal number of AC and BC coalitions, while predicting no

AB coalitions. As implied by the phrase "strength is

25

weakness" however, BC coalitions occur more frequently than
AC coalitions. Chertkoff's revision was intended to
correct this discrepancy.

Chertkoff (1967) revised Caplow's theory by
emphasizing the choices confronting each of the players
A, B, and C. Since player A can dominate either player B
or player C, either player should be equally attractive to
player A, and therefore he should contact either player B
or player C 50% of the time. Likewise, player C can
control neither player A nor B, and therefore C should
contact either player A or player B 50% of the time. On
the other hand, player B can only dominate C, and therefore
C should be chosen by B 100% of the time. Assuming that
the players make their choices independently of each
other, these probabilities can be multiplied to obtain the
probability of a mutual choice. The probability of an AB
coalition is .5 x 0., therefore no AB coalitions are
predicted; the probability of an AC coalition is .5 x .5,
therefore the probability of an AC coalition is .25; and
the probability of a BC coalition is .5 x 1.0, or .50.
In the remaining 25% of the cases, no coalition would be
expected on the first contact attempt. Therefore,
Chertkoff's revision of Caplow's theory predicts twice as
many BC as AC coalitions, which is more compatible with
the frequently observed "strength is weakness" effect.

In contradiction with this revision, AB coalitions do

26

occur; since no AB coalitions are predicted, x2 tests
cannot be conducted to determine the significance of this

discrepancy.
C. Ofshe and Ofshe (1968)

Ofshe and Ofshe (1968) emphasized another factor
other than the power division in a triad. After a series
of social choices, a utility of response variability might
develop, reinforced by boredom with a fixed response
strategy and by a desire for an outcome equitable to all
participants. Their model, based upon maximizing the
subject's expected utility is applicable when some basis
exists for presuming a stable subjective probability of
response reciprocation and for presuming a knowledge of the
participants' expected reward upon a reciprocated contact.
The subject's expected utility of a given choice on each
trial was defined as a function of his expected monetary
reward and of his utility of response variability.

Differentiating their expression for utility of
social choice with respect to Pw' Ofshe and Ofshe observed
that that utility was maximized when Pw = (1/4)(aw1rw +
cans) + 1/2 where Pw represents the probability of choosing
the weaker candidate, NW and Us represent the probability
of a reciprocated choice by the weaker and stronger
candidate respectively, and aw and as are parameters to be

estimated from the data. This model was designed to be

27

applied to stable-state data, i.e., data obtained after the
subjects have had extensive experience with the experimental

situation.
D. DeYoung and Phillips (1970), Model One

The first mathematical model discussed by DeYoung
and Phillips (1970) was developed for the prediction of
initial contact probabilities and was of the form: Pw =
ovPARW + B - SEC where P is the estimated probability of

W W
the subject initially contacting the weaker of the two
coalition partners and a and B are parameters estimated
from obtained data. a represents a "weight" or emphasis
placed by subjects upon the parity factor, while B
represents a quantification of the emphasis placed upon
the security factor. PARW is an index of the advantage to
the subject from the standpoint of payoff in choosing the
weaker over the stronger coalition partner. SECW is an
index of the relative advantageousness in choosing the
weaker as compared with the stronger coalition partner
from the standpoint of the security of the coalition.

The mathematical form of PARW was derived from
the parity norm, according to which player A in a winning
coalition with the weaker player would expect the weaker
player to demand a proportion of the payoff equal to

NW NA

fiX—x—fifi, leav1ng a share of w for player A. If

NA-l-N
player A formed a coalition with the stronger of the

possible coalition partners, his expected share of the

28

NA . NA
payoff would be -———-——-which would be less than ———————u

 

 

 

NA + NS NA + NW
Therefore,
NA
NA + NW
PARw = N + N
A A
NA+NW NA+NS

was taken as an index of the advantage, with respect to
the payoff, to player A if he chose the weaker instead of
the stronger potential coalition partner.

Similar reasoning was used in deriving the form of
SECw. The strength of a coalition with the weaker partner

NA+NW

can be represented by NS , i.e., the amount of resources

in the coalition divided by the amount outside the coa-
lition. Similarly, with respect to a coalition with the

stronger partner, the strength can be represented by

NA + NS
Nw

forming a coalition with the weaker partner, with respect

. Therefore, an index of the advantageousness of

to security is

 

 

Initial contact data has been collected by
Dr. James Phillips and his students for both the Committed

Vote and Uncommitted Vote variations by mean of a Political

29

Decision Questionnaire. Ten resource distributions were

used; each resource distribution was constructed such that
s if NA > 100
In order to test the applicability of Model One

NA+NW+NS= 300 and NA =§
to the collected data, a separate a was estimated for each
variation while a single B was estimated for both vari-
ations, such that the squared deviation between the
estimated and observed probabilities was minimized. The
initial probabilities of choosing the weaker candidate
along with the Model One estimated probabilities are

presented in Figure 2.
E. DeYoung and Phillips (1970), Model Two

The second model developed by DeYoung and Phillips
(1970) for the Political Convention Situation was an

extension of the Shelly and Phillips (1966) model. Again

VA (W)

Pw = VA (w) + VA (5); where Pw represents the probability

 

that A chooses the weaker potential coalition candidate,

and VA(w) and VA

to player A of choosing the weaker and stronger candidate.

(5) respectively represent the "values"

This model hypothesized that V

A (W) is proportional

to
l. The amount of payoff - PAR; - to be expected
from choosing the weaker candidate;
2. The security of the coalition - SEC¢ — i.e.,

the coalition's expected probability of

3O

 

       

 
   

1.00“
Observed
._._._——— Predicted by Model 1
.801 _' 1°209__ __ ————— NT = 300
P(W)
.6oi

 

 

B = .295

1 l
77 85 §2 104 N(A) 1é2 111 149

Figure 2.--Observed Probabilities of Choice of Weaker
Candidate and Model 1 Predicted Probabilities

31

success in winning the nomination given the
formation of a coalition with the weaker
candidate;
3. The subjective likelihood of a reciprocal
choice - “w - by the weaker candidate; and
4. The expected ease of bargaining - Bw - with
the weaker candidate, assuming a reciprocal
choice.
The above hypothesis suggested an expression for
VA (w); VA (w) = KwPARé - SECé - “w - Bw’ where K is a
constant of proportionality. This expression was designed
to be consistent with the intuitively reasonable suppo—
sition that if any of the above four factors is zero, the
value to player A of choosing the weaker partner is zero.

The parity norm suggested an expression for the

NA
— ' = —— .
expected payoff factor PAR N + N . The expre551on
could not apply in the Uncommitted Vote Variation, since

the maximum expected payoff could be

_1A__
+ 1

2

II
Mid

no matter with which candidate player A forms a coalition.

Although it would be difficult to suggest an expression for
PARw and PARS in the Uncommitted Vote Variation, the above

reasoning can be used to justify the assumption that

I
£¥§§fi%:= l in the Uncommitted Vote Variation.

32

Although data are lacking with respect to the
subjective expected security of a coalition, the perceived
security of a coalition was taken to be proportional to the
quotient of the number of votes in the coalition and the

number of votes outside the coalition, i.e., for compu-
NA + NW

 

tational purposes SECw =
Since no investigation had been made of “w and

Bw, these factors were combined with K; to yield Kw.

Therefore,

P _ VA(W) _ Kw - PARW - SECQ
_ —' . 1 . i ' '
w VA(w) + VA(S) Kw PARw SECw + KS PARS SECS.

 

 

This equation was simplified by dividing both the numerator
and the denominator of the right hand side by Kw - PARQ -

SECé, yielding

 

P = l
w l + K PAR' ° SEC'
5 s s

K PAR' ' SEC'
W W W

 

Ea
Kw

parameter a to estimate using this model. For the Com-

Replacing by a demonstrates that there is only the one

mitted Vote Variation therefore,

 

 

Pw= N 1N +N=1+iN
A . A 5 .Ji
1 + a NA + NS NW NW
NA '(NA + Nw)

 

33

For the Uncommitted Vote variation,

 

 

 

 

 

Pw = l + a PAR' l (N + N )
s A S
NU + NW
PAR‘; NU N S

= l
l + a (NA + NS) (NU + NS)
(NA + NW) TNU NW)

For either variation, since an analytical solution is
unavailable, a was estimated using a numerical least
squares procedure.

The predictions of this model compared with the
same data presented in the previous section are shown in

Figure 3.

34

1.001-

Observed
-— -- Predicted by Model 2

.80 ‘-

 

P(W)

.60‘“

 

 

 

77 85 92 164 132 111 149

Figure 3.——Observed Probabilities of Choice of Weaker
Candidate and Model 2 Predicted Probabilities

CHAPTER II

EXPERIMENT I

Statement of the Problem

 

Three major hypotheses were discussed in the
previous chapter for the disappearance of the strength is
weakness effect in successive Committed Vote version coa—

lition formation games. The elimination of confusion

 

explanation, that S's reevaluate power relationships after
experience with the game, was advanced by Kelley and

Arrowood (1960). The cumulative score explanation, that

 

players who were behind on the reward dimension discrimi-
nated against the leader on that dimension, was supported
by data obtained by Hoffman, Festinger, and Lawrence (1954)

and Bond and Vinacke (1961). The utility of response

 

variability notion was suggested by Chertkoff (1966), and

 

a mathematical model based on this explanation was advanced
by Ofshe and Ofshe (1968).

If an experiment were conducted such that §s could
have no knowledge of the accumulated winnings of themselves
or of the other participants, the three explanations yield

different predictions for the social contact and bargaining

35

36

data obtained over successive games. (1) The elimination
of confusion explanation predicts the disappearance of the
strength is weakness effect in the Committed Vote version
over time, but predicts stability of the strength is
strength effect in the Uncommitted Vote version. With
respect to the bargaining phase, the elimination of
confusion notion predicts that, initially, coalitions will
tend to divide the payoff in proportion to the resources of
the coalition partners but that, over time, the payoff will
tend to be divided equally between the two partners. (2)
The cumulative score explanation predicts stability of the
social contact and bargaining data over games in both the
Committed Vote and Uncommitted Vote versions. (3) The
utility of response variability explanation predicts a
tendency to random responding, i.e., a probability of
choosing the weaker candidate approaching .5, in the latter
games of both the Committed Vote and Uncommitted Vote
versions, but no departure from stationarity of reward
division.

Experiment I was designed to test these three
explanations. The necessary experimental control was
obtained by (l) ensuring that choices were made without
the Ss' knowledge of which individual represented each of
the other candidates in each game, and by (2) preventing
the accumulation of any reward over games; rather, one

game was to be chosen at random for the division of the

37

reward, making each game reward-independent and equally
important to the SS. If, as suggested by the interpre-
tation of Vinacke et al. (1967), experience with the coa-
lition formation paradigm did not lead to a change in
response pattern over games, Experiment I was to provide
sufficient data for the application of the mathematical

models.

Method and Procedure

 

Subjects. Seventy-two males from introductory
psychology courses at Michigan State University participated
in this experiment, resulting in 24 groups of 3 §$- gs
were unsystematically assigned to groups on the basis of
their order of appearance at the laboratory.

Resource Distributions. Two criteria were used

 

in choosing the resource distributions:

1. There should be consistency within versions,
specifically all coalitions in the determi—
nistic Committed Vote (CV) version should be
winning coalitions and no coalitions in the
probabilistic Uncommitted Vote (UV) version
should be automatically winning coalitions;
and

2. The resource distributions should cover a
large range of resource points at approxi-

mately equal intervals.

38

In accordance with the two criteria, the resource
distributions chosen were (80, 90, 130), (60, 100, 140), and
(70, 110, 120); for brevity, these will be referred to as
distributions 80, 60, and 70 respectively. In the CV
version, the 300 votes represented by each distribution
were the only votes in the convention. In the UV version,
600 total votes were represented at the convention, 300 of
them uncommitted to any candidate.

Experimenters. Two male Es were used in this

 

experiment. Each §_was trained by participation as a
subject in the experiment at least twice, and by par-
ticipation as an E in at least ten practice sessions. In
order to minimize demand characteristics and experimenter
bias, one of the Es was kept uninformed as to the theory
and purpose underlying the experiment.

Design. Each group of three gs was run in ten
games. The CV version was alternated with the UV version
through the ten games; for half the groups of three gs,
the first game of the series was a CV version, while the
first game was an UV version for the other half of the
groups. Six different orders of distribution, i.e.,
distribution type 60, 70, or 80, were used. These orders
are detailed in Appendix I. The six distribution presen-
tation orders combined with the two possible CV - UV
alternation orders resulted in twelve different game
presentation orders. Of the 24 total groups, each E ran

one group using each of the twelve game orders.

39

Experimental Apparatus. The Political Convention

 

game was administered at a table divided into three
sections which has been used in previous research on
coalition formation (Kline, 1968). The experimenter was
seated facing the SS on the other side of the table.

Each S was given a set of response sheets, on
which he was to indicate which of the other potential
coalition members he wished to contact on a given trial.
Besides an area for a response to be made, each sheet had
the current trial number, the number of votes the S's
candidate controlled, the total number of votes in the
convention, the number of votes each of the other two
candidates controlled, and the number of total votes in a
coalition were the candidate to choose either of the
other two coalition partners. The order of the two
alternative coalition partners was randomized on the sheet,
so that on one half of the sheets the stronger partner was
listed first, and on the other half of the sheets the
weaker partner was listed first. An example of one
possible set of response sheets is contained in Appendix II.

Bargaining sheets were used so that coalition
partners could successively offer each other part of the
reward for forming a coalition. A bargaining sheet is
contained in Appendix III.

Procedure. Upon their arrival at the experimental

 

room, the S5 were seated at the divided table and the

40

nature of the game was explained. The instructions were
tape recorded to insure uniformity; a copy of the in-
structions is contained in Appendix IV. After a practice
game, involving a distribution of 2, 3, and 4 votes, the
Ss played ten games, assigned and conducted as described
in the Design section.

Before each game, each S drew a packet of response
sheets at random to indicate the number of votes which his
candidate would control throughout that game. Each S also
knew how many votes each of the other candidates controlled,
but he did not know which of the other Ss represented which
candidate. Each S_made a choice on the response sheet
independently of the other Ss, and returned the response
sheet to the S through a slot in the partitioned table.

If no mutual choice was made, the S marked the next
response sheet and the game continued until there was a
mutual choice. The bargaining took place on the bargaining
sheets after two of the candidates chose each other as
coalition partners. No communication other than the
written offer was allowed during the bargaining phase of
the experiment.

After ten games were concluded, the number of the
one game which involved the actual $3.00 payoff was
selected by a random drawing. Since no coalition in the
UV version had a majority of the 600 possible votes, the

determination of the winner(s) immediately followed the

41

drawing if the game drawn was an UV game. The determi-
nation was made by splitting the 300 uncommitted votes
among the three candidates. Thus, the Ss in the coalition
received the votes that they controlled plus the sum of
the previously uncommitted votes assigned to each of the
two candidates whose representatives formed the coalition.
The Ss were made aware of the above procedure before they
played the game, and it had been pointed out that since
there were 300 votes outstanding, it would be possible for
the S left out of the coalition to receive all of the

$3.00 in a version UV game.

Results

Table 1 contains the overall analysis of variance
of the initial contact data in terms of the experimenter,
version (CV versus UV), distribution, and resource point
independent variables. All four effects are clearly fixed
effects. The conservative "never pool" rule discussed by
Winer (1962) was adopted, but in any case the large number
of degrees of freedom in the error term would make
negligible the effect of pooling nonsignificant inter-
actions into the error term. While the design of the
experiment suggests that a repeated measures analysis
would be appropriate, the random drawing of response
sheets resulted in the observation of some S5 at only
2 of the 3 possible resource points, the missing cells

making a repeated measures analysis impossible.

42

TABLE l.--Analysis of Variance for Probability of Choosing

Weaker Candidate, Experiment I.

 

 

Source SS DF MS F
Total 179.26 719
Experimenter 4.58 l 4.58 20.96 <.0005
Version 17.71 1 17.71 81.11 <.005
Distribution .53 2 .26 1.21
Resource Pt 1.18 2 .59 2.70 .07
E X V .003 l .003 .01
E X D .40 2 .20 .91
E X R .78 2 .39 1.78
V X D .63 2 .31 1.44
V X R 1.21 2 .60 2.77 .07
D X R 1.32 4 .33 1.51
E x V X D .63 2 .32 1.45
E X V X R .11 2 .05 .24
E X D X R .41 4 .10 .47
V X D X R .05 4 .01 .05
E X V X D .51 4 .13 .59
Error 149.30 684 .22

 

43

Therefore, an unequal n's factorial analysis was conducted,
forfeiting the power of the repeated measures design but
using the multiple games per group simply as a device for
the collection of more extensive data.

The analysis of variance in Table 1 indicates
that the experimenter effect was highly significant.
Whereas the overall probability of choosing the weaker
candidate was .48, the mean for the informed S_was .57
contrasted with a mean probability of choosing the weaker
candidate of .38 for the S who was uninformed as to the
purpose of the experiment. There was a highly significant
different between the two versions of the Political
Convention Game. The probability of a choice of the
weaker candidate was .63 in version CV and .33 in version
UV. No other effects were significant at the .05 level.
The effect of comparative power of a candidate (resource
point) approached significance (F2,684 = 2.70, p < .07).
The weakest candidate chose the weaker potential coalition
partner with a probability of .42, compared with a
probability of .47 of the middle candidate choosing the
weaker, and a probability of .54 for the strongest candi-
date choosing his weaker potential coalition partner. A
test of trend showed that the linear trend with respect
to resource point also approached significance with no
quadratic trend. The version by resource point inter-
action also approached significance (F2,684 = 2.77,

p < .07). Figure 4 shows the version by resource point

44

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46

interaction means. A t-test showed that the weaker candi-
date was more often chosen when that candidate's votes

were listed first, i.e., on the left, (t = 2.5, p <

718df
.05); that probability was .52, compared with .44 when the
weaker candidate's votes were listed on the right.

Table 2 contains the frequency of each of the
three possible coalitions in each version of the Political
Convention game, summed over all ten games. Most coa-
1itions in version CV were moderate-weak coalitions, and
the fewest coalitions were strong—moderate coalitions;
this finding is indicative of the strength is weakness
effect. Comparing the observed coalition frequencies with

an all-equal frequency null hypothesis yields ngf = 10.4,

p < .01. The table demonstrates an even stronger strengh

2
def

In support of this finding, Figure 5 suggests

is strength effect in version UV, = 30.0, p < .001.
that the probability of contacting the weaker candidate

did not become more random in the later games of the series
in either version CV or UV. In order to test this
hypothesis statistically, the observed frequencies of
contacting the weaker candidate were compared with the

hypothesis of a constant frequency of contacting the

weaker; for version CV, ngf = 2.05, n.s., and for version
UV, ngf = 5.58, n.s. In addition, the correlations in

version CV and version UV between the probability of

choosing the weaker candidate and the game number were

47

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48

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49

TABLE 2.--Frequency of Coalitions in Two Versions of
Convention.

 

Type of Coalition

 

 

Version
Strong- Strong— Moderate— 2
Moderate Weak Weak x
Committed Vote 24 44 52 10.4*
Uncommitted Vote 64 41 15 30.0**

 

Note: The expected value of each cell is 40, each chi-
square has 2 df.

*p < .01

**p < .001
each equal to .08. Therefore, the probability of choosing
the weaker candidate did not vary systematically over
time.

The utility of response variability concept
suggested that, especially over successive trials within a
single game, the conditional probability of choosing the
weaker candidate given that the weaker candidate was
chosen on the previous trial, i.e., the probability of
perseveration, should decrease in both version UV and CV.
In particular, the existence of a utility of response
variability would be indicated after several trials of a
game if that conditional probability were to decrease
below the overall probability of a choice of the weaker
candidate; at the same time the overall probability of
choosing the weaker candidate should approach .5 in both

version UV and CV. The data were summed into blocks in

50

order to have enough observations at each data point;
Block 1 was composed of trials 2 and 3, and Block 2 included
trials 4 through 7. The overall probability of choosing
the weaker candidate in version CV decreased from .63 at
trial 1 to .47 in Block 1 and .47 in Block 2; the perse-
veration probability decreased from .75 in Block 1 to .56
in Block 2. In version UV, the overall probability was
.33 on trial 1, .33 in Block 1, and .46 in Block 2; the
perseveration probabilities were .55 in Block 1 and .56
in Block 2. The stability of the game to game overall
probabilities of choosing the weaker candidate has already
been discussed in connection with Figure 5. The associated
game to game perseveration probabilities also showed no
systematic increase or decrease.

Figure 6 contains the bargaining data over games
for all distributions and both versions. There clearly
was no increase in the amount of reward to the weaker
candidate in the later games of the series; since the
99.9% confidence interval does not include a 50-50 division
of the reward at any point, there is over 99.95% confidence
that the true mean reward lies below a 50-50 division. A
x2 analysis comparing the number of 50-50 splits over the
ten games with the hypothesis of a constant number of

50-50 splits complements this finding, = 2.44, n.s.

2
X9df
The histograms in Figure 7 present the game by

game bargaining data in more detail. The most striking

51

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53

Figure 7.-—Histograms of Reward to Weaker Candidate in Each
of Ten Games, Experiment I.

54
l6«

Game 1

12~~

Frequency
00

 

 

 

 

4,
0 l

N.A. 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
16 _ Dollars to Weaker Partner

Frequency

 

 

N.A. 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
Dollars to Weaker Partner

16-1-

12*

Frequency

84k

 

161

Frequency

 

 

1.0

1.0

55

Game 3
X = 1.42
S = .128

 

 

I

1.1 1.2 1.3 1.4 1.5 1.6

Dollars to Weaker Partner

1 l
1.7 1.8

Game 4

1.38
.184

m><|

 

 

1.1 1.2 1.3 1.4 1.5
Dollars to Weaker Partner

 

er

(I)

Frequency

 

12

Frequency

56

 

 

 

 

 

 

 

N.A. 1.0 1.1 1.2 1.3 1.4 1.5 1.6 117 173
Dollars to Weaker Partner
Game 6
2:137
s = .154
. l —]" I
N.A. 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Dollars to Weaker Partner

57

 

 

 

 

 

 

 

 

12-
m 8 Game 7
o
c
o
g
0‘ ‘i
a I
m = 1.39 ,
4 = .160 a
0 T’ i
N.A. 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
Dollars to Weaker Partner
12~
6‘8—
5 Game 8
s
q
o
H
[1...
4 = 1.38
—_|||||W = 1.24
o i I
.A . . l. 5 1. 6 1. 7 1.8
Dollars2 to lWeaker4 Partner
Fig. 7

/_ 1

on
T

Frequency

 

12(-

Frequency

00
I

 

 

 

58

Game 9

1.40
.104

m>q

 

 

 

I

1.1 1.2 1.3 1.4 1.5 1
Dollars to Weaker Partner

I I j

.6 1.7 1.

Game 10

1.39
.128

X ><|
II II

8

 

 

 

I
1.1 1.2 1.3 1.4 1.5
Dollars to Weaker Partner

1 I
1.6 1.7

.T
1.8

 

59

feature of the histograms is that the variances of the
agreements reached appear to decrease over games; however
the usual Bartlett's test is inappropriate for two reasons.
Bartlett's test is seriously affected by non-normality and
the data do not appear to be normally distributed; also,
the variance of the third game is small so that Bartlett's
test of homogeneity of variance would yield no assurance
that a significant B indicated a decrease in variance over
games. However, the average variance in the first four
games was .033 compared with an average variance of .025
in the middle three games and an average of .014 in the
last three games. The hypothesis that there is no differ-
ence between these mean variances can be examined by the
more robust test suggested by Sheffe (1959, p. 83). The
obtained F(9,7) = 1.22 does not permit rejectance of the
null hypothesis. Figure 8 presents a summary histogram of
the same bargaining data summed over all games.

While the above analyses suggest that bargaining
is relatively stable over games and that Ss do not at any
time fail to discriminate against the weaker coalition
partner, the question remains as to whether Ss behave
literally in accordance with the parity norm, i.e.,
whether Ss arrive at reward divisions proportionate to
their relative contributions to the strength of the coa-
lition. Figure 9 shows the parity division, the actual
division of the reward, and 99% confidence intervals about

the actual reward division in each coalition, averaged

60

Figure 8.-—Summary Histogram of Reward to Weaker Candidate
in All Ten Games, Experiment I.

61

120 —

110 ”

100 '

90 ‘

1.39
.161

70 ‘

Frequency

50 ”

40 -

20 r

10 '

 

 

N.A. 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Dollars to Weaker Partner

Fig. 8

62

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63

 

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64

over all games and versions. At only one point does the
99% confidence interval include a parity division of the
reward. (The 95% confidence intervals, not shown, also
include a parity division of the reward at only this
point.)

Table 3 contains the analysis of variance of the
bargaining data in terms of the experimenter, Version,
distribution, and coalition (moderate-weak versus strong-
weak versus strong-moderate) independent variables. No
main effects or interactions were found to be significant,
justifying the univariate breakdown of the data in the
preceeding bargaining data analyses. The lack of a sig-
nificant coalition effect or distribution by coalition
interaction complements the data in Figure 9. Ss apparently
not only achieved agreements significantly different from
those suggested by the parity norm, but the trend of the
agreements reached did not vary in accordance with the
parity norm, i.e., the weaker candidate apparently received
as much of the reward in a strong-weak coalition as he
would have received from a coalition with the middle candi-

date.

Discussion

 

In the present experiment, in which an attempt was
made to prevent any estimate of the previous winnings of
any player, the strength is weakness effect was found to

persist over a series of ten games in a deterministic

 

65

TABLE 3.--Ana1ysis of Variance for Reward to Weaker

Candidate, Experiment I.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Source SS DF MS F SIG
Total 5.9160 239~
Experimenter .0178 l .0178 .7616
Version .0007 1 .0007 .0289
Distribution .0129 2 .0064 .2739
Coalition .0683 2 .0342 1.4559
E X V .0001 1 .0001 .0029
E x D .0783 2 .0391 1.6679
E X C .0075 2 .0038 .1606
V X D .0332 2 .0166 .7073
V X C .0229 2 .0114 .4876
D X C .1672 4 .0418 1.7808
E X V D .0330 2 .0165 .7025
E X V C .0220 2 .0110 .4679
E X D C .1035 4 .0259 1.1023
V X D C .1155 4 .0289 1.230
E X V D .2154 4 .0538 2.2951 .10>p>.05
Error 4.7885 204 .0235

 

66

version of a coalition formation game, while the strength
is strength effect was replicated in a probabilistic
version (Table 2). The analysis of the data provided
strong evidence that neither initial contacts (Figure 5),
nor reward agreements (Figure 6) became more equal to each
of two prospective coalition partners during the series of
games. These results stand in direct contrast to the
results reported in the Kelley and Arrowood (1960) and
Chertkoff (1966) studies in which cumulative scorekeeping
was allowed. The present study suggests that the results
of Kelley and Arrowood (1960) and Chertkoff (1966) are
therefore best accounted for by the cumulative reward
explanation of the increasing randomness of initial contacts
reported in those studies.

It is clear from the bargaining data that the
weaker candidate received less than half of the shared
reward (Figure 6), but that the degree to which this reward
share was less than half did not depend upon the exact
proportion of votes which the weaker candidate contributed
to the coalition (Figure 9, Table 3) in contrast with the
implications of Gamson's (1961) parity norm.

No systematic change in the variances of the
bargaining agreements was found over games. Data con-
sistent with the utility of response variability concept
were reported for trial to trial (within games) contacts;

such data are also consistent with an explanation in terms

67

of a decreased subjective expected probability of contact
reciprocation, however. That is, since more than one trial
within a game was necessary when there had been no mutual
choice, Ss may have changed their choices when they no
longer expected their preferred coalition partner to

choose them in return.

CHAPTER III

EXPERIMENT II

Statement of the Problem

 

Experiment II was designed to investigate further
the factors responsible for the disappearance of the
strength is weakness effect. Since in Experiment I Ss
did not ignore the power divisions of the potential coa—
lition partners in the final games within a series of
games in contrast with the Kelley and Arrowood (1960)
hypothesis, the effects of accumulated reward remained to
be investigated. As previously stated, Bond and Vinacke
(1961) and Hoffman, Festinger, and Lawrence (1954) found
that players who were behind on the reward dimension tended
to ally against the leader in the reward division. The
effect of the Ss knowledge of his own reward has not
previously been separated from that effect however. The
effects of accumulated reward would be a valuable addition
to a quantitative theory of coalition formation.

An adequate quantitative formalization of the
effect of accumulated reward would require the observation
of a sufficient number of choices made under each of a

limited number of accumulated divisions of the reward.

68

69

In Experiment II the necessary limitation of the numbers
of types of reward division was ensured by announcing the
amount of reward acquired by each candidate after each
third game only, by employing imposed divisibility of
reward, and by using several techniques which made it
difficult for Ss to accurately assess their accumulated
reward before the announcement was made.

Two experimental conditions were used in Experi-
ment II. In the Complete Knowledge (CK) condition the
accumulated reward was announced after every three games
in terms of the amount previously earned by the player
representing each power position in the next game;
therefore, although each player would not know which indi-
vidual represented each potential coalition partner in
the succeeding game, each player would be able to choose
a partner and bargain with him under knowledge of both the
number of votes controlled by him and his previously
accumulated reward. In the Incomplete Knowledge (IK)
condition, the accumulated reward was announced in terms
of the players who represented each power position in the
previous game so that each player would only be aware of
the division of the reward, his own accumulated reward,
and the vote distributions in the succeeding game.
Therefore, Condition IK would test for any effects caused
by a player's own relative reward accumulation as opposed
to interactive effects of one's own and other's accumulated

reward.

70

Method and Procedure

Subjects. Two hundred and sixty-four male
subjects from introductory psychology courses at Michigan
State University volunteered for participation in this
experiment, resulting in 88 groups of 3 Ss. Thirty of the
groups were run in Condition IK, and 58 in Condition CK.

Resource Distributions. The same resource
distributions were used in this experiment as were used in
Experiment I.

Design. Each group of three Ss was run in twelve
games. As in Experiment I, the CV version was alternated
with the UV version of the Political Convention situation;
for half the groups of three Ss, the first game of the
series was a CV version, while the first game was an UV
version for the other half of the groups.

The accumulated reward distributions announced
after every third game was restricted to one of three
predetermined reward distributions. Each game involved a
reward of 25¢; the possible reward distributions, one of
which was announced after every third game, were (25, 25,
25), (20, 25, 30), and (15, 25, 35). The number of reward
distributions was able to be restricted without the
awareness of the Ss for reasons outlined in the Procedure
section.

Straightforward combinatorial techniques reveal
that the above set of reward distributions can be combined

with the three resource distributions to yield 39 unique

71

combinations. Therefore only one resource distribution was
used in the fourth, seventh, and tenth games where the
effect of accumulated reward was measured, and a subset of
only five of the possible thirteen combinations of the
(vote) resource and reward distributions was selected to

be the set of "payoff distributions" in order to obtain an
adequate number of observations under each combination or
"payoff distribution." Distribution 60 was the resource
distribution chosen to be used at the first, fourth,
seventh, and tenth games in order to test the effect of
accumulated reward; that distribution is distinguished by
the equal intervals of votes between the weakest, middle,
and strongest player. The payoff distributions used are
detailed in Table 4, which indicates that in payoff
distribution 1, for example, the person who controlled 60
votes in the succeeding game had won 35¢ for the last

three games, the person who controlled 100 votes had won
25¢, and the person who controlled 140.votes had won 15¢ in
the last three games.

Games 2, 3, 5, 6, 8, and 9 which were presented
between the games used to measure the effect of accumulated
reward involved an imposed divisibility of reward. For
each group, in half of the 6 games a 50%-50% division of
the reward was imposed; in the other 3 games a 70%-30%
division of the reward was imposed. Distributions 80 and

70 which were used in the 6 intervening games are resource

72

TABLE 4.--Combinations of Reward and Resource Point, i.e.,
Payoff Distributions, Used in Experiment II.

 

Payoff Distribution

 

 

Point
Resource
l 2 3 4 5
60 35¢ 15¢ 20¢ 30¢ .25¢
100 25¢ 35¢ 25¢ 20¢ 25¢
140 15¢ 25¢ 30¢ 25¢ 25¢

 

distributions in which intracoalition compatibility could

be demonstrated by coalitions between the relatively equal
players (80-90; 110—120) or the relatively disparate players
(80-130; 70-120). Table 5 summarizes the design of this
experiment with respect to accumulated reward, imposed
divisibility of reward, and the resource distributions

used in each of the 12 games.

Two conditions were employed to separate the
usually confounded effect of an Ss knowledge of his own
reward from the effect of the Ss knowledge of his own and
the other players' reward. In Condition CK, the amount of
money credited to each player for each set of three games
was announced after distribution of the ballots for the
succeeding game in terms of the number of votes controlled
by each player in the succeeding game. In Condition IK,
the accumulated reward was announced in terms of the

number of votes controlled by each player in the previous

73

 

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74

game. The announcement was made twice to ensure accurate
reception by the Ss. In both conditions therefore, the

Ss were aware of the reward distribution for the previous
three games, and were able to compute their own total
reward from the first game. In Condition CK only, the

Ss also knew the number of votes controlled in the next
game by the player with each of the announced portions of
the reward. Only one S_was used throughout the experiment.

Experimental Apparatus. The table and the

 

response sheets were identical to those used in Experi—
ment I. The bargaining sheets indicated that offers should
be made in terms of percentages, but were otherwise
identical to those used in Experiment I.

Procedure. Upon their arrival at the experimental

 

room, the Ss were seated at the divided table and the
nature of the game was explained. The instructions were
tape recorded to ensure uniformity; they were similar to
the instructions for Experiment I, except that they
explained that a 25¢ reward would be paid for each game,
that an imposed percentage division of the reward would be
announced before some games, and that the amount of reward
won by each player in the previous three games would be
announced after every third game "in order to keep Ss
informed of their progress during the session." After a
practice game, involving a distribution of 2, 3, and 4

votes, the Ss played twelve games, assigned and run as

75

described in the Design section. The games were played

similarly to those in Experiment I with the following

exceptions:

l.

A running total of each S's winnings was kept
and each S's winnings for the previous three
games was announced before games 4, 7, and

10. One of only five payoff distributions was
announced before each of those games as
detailed in Table 4 of the Design section.

The type of imposed divisibility of reward in
games 2, 3, 5, 6, 8, and 9 was announced to
the Ss before they made their initial choices
of coalition partners. In the bargaining
phase of the 70-30 division games, only offers
consistent with that game's imposed divisi-
bility of reward, either 70% or 30% were
allowed to be made. No bargaining was
necessary in 50-50 division games.

The reward bargained for in each game totaled
25¢ instead of $3.00.

Bargaining was conducted in terms of a percent
of the 25¢ instead of a monetary amount.
Offers and counteroffers were made in terms

of multiples of 5%.

After all games were played, the Ss total

recorded winnings were summed and paid, in

76

contrast with Experiment I in which the Ss'
total earned depended upon the results of
only one of the games.

It was stated in the Design section that the
number of announced accumulated reward distributions was
limited to three. The following degrees of freedom in the
experimental procedure permitted this manipulation: (l)
Agreements were made in terms of percentages, while
announcement of the reward was made in terms of money
earned, rounded to the nearest 5¢; (2) Ss were not
informed as to whether they had won the 50%-50% imposed
reward division games since no bargaining was necessary in
those games; and (3) It was possible for the player left
out of the coalition to be assigned all of the 25¢ reward
for version UV games since the 300 uncommitted votes in
that version could be divided such that any player could
obtain a majority of the votes without being in a coa-
lition.

In order to test the degree to which the Ss were
aware of the deceptive devices used in the experiment,
after all games were run, Ss were asked to challenge any
of the announced reward distributions so that "any errors
which the S_made can be corrected before payment of the
reward." Any challenge was met by the S with an expla-
nation of the rewards in the three games leading to the

announced reward distributions. If no satisfactory

77

explanation were available, the necessary "corrections"
were made and the Ss were given a questionnaire to fill
out while the total rewards were being computed. The
questionnaire asked each S the amount he expected to
receive at the end of the session in order to test the S's
attentiveness to his total accumulated reward; it also
asked how likely he thought it was that deception was
involved in the experiment; and if he thought there was
deception, exactly what he thought the nature of the
deception was. Each S was also asked to rate how hard
he tried to win the games, using a scale ranging from 1
to 10.

The Ss were then informed that since the reward
distributions were not determined completely by their own
choices and bargaining, they would be given either the
amount which they had earned or $1, whichever was greater,
and in exchange were asked to refrain from discussing the
experiment until they received a letter explaining the
nature and purpose of the deception used and an overview
of the results of the experiment. The payoff manipulation
resulted in a mathematically possible reward range of

$3 - $5 per group.

Results
The analysis of the initial contact data was
divided into two parts. The effect of accumulated reward

was tested using games in Distribution 60; therefore the

78

data from these games were used in the first analysis of
variance. The second analysis of variance was concerned
with the effect of imposed divisibility of reward, and
therefore was conducted on games in Distributions 70 and
80.

The analysis of variance for the probability of
contacting the weaker candidate as a function of condition
(CK versus IK), version, resource point, and payoff
distribution is contained in Table 6. The first game in
each series and all games in payoff distribution 5, the
distribution which assigned 25¢ to each player, were
played under conditions of reward equality to all Ss.
Since there was no significant difference between choices
in these games, their data were combined for comparison
with the 4 payoff distributions involving reward inequality.
The only significant main effect was for version; whereas
the overall probability of choosing the weaker candidate
was .50, that probability was .67 in version CV and .33
in version UV.

The cumulative reward prediction for payoff
distribution 1 would be a disappearance of the strength is
weakness effect in condition CK, while the "gambling
strategy" hypothesis prediction would be an intensification
of the strength is weakness effect in conditions IK. The
predictions for other payoff distributions differ between

conditions also. Therefore, the occurrence of both effects

79

TABLE 6.-—Ana1ysis of Variance for Choices of Weaker Candi-
date as a Function of Condition, Version, Resource Point,

and Payoff Distribution, Experiment II.

 

 

 

 

 

 

 

 

 

 

 

 

 

Source SS DF MS F SIG
Total 263.98 1055
Condition .005 1 .005 .02
Version 23.17 1 23.17 107.62 <.0005
Resource Point .75 2 .37 1.73
Payoff Distribution .80 4 .20 .93
C X V .62 1 .62 2.89 .10>p>.05
C X R .91 2 .45 2.11
C X P .38 4 .10 .44
V X R .13 2 .06 .29
V X P .67 4 .17 .77
R X P 4.25 8 .53 2.47 .01
C X V X R 1.51 2 .75 3.50 .03
C X V X P .31 4 .08 .36
V X R X P 1.92 8 .24 1.12
C X R X P 1.79 8 .22 1.04
C X V X R X P 1.13 8 .14 .66
Error 214.46 996 .21

 

80

in their respective conditions or only one effect in one
condition would be indicated by a significant condition by
payoff distribution by resource point interaction. That
interaction was not significant; however the payoff distri—
bution by resource point interaction was significant,
suggesting that one of the predicted effects occurred in
both conditions. Summing over payoff distributions, the
probabilities of choosing the weaker candidate for the
candidates with the most, intermediate, and least reward
were .55, .53, and .40 respectively, consistent with the
gambling strategy hypothesis. A Scheffe test indicates
that the smallest of these probabilities is significantly
different from the other two, which do not differ from
each other. Since the condition by version by resource
point interaction was significant, Table 7 contains the
same data broken down by condition and version; 3 of the
4 sets of probabilities are in the predicted order. Since
there are six possible arrangements of 3 items, and since
there are 4 ways in which to select 3 from 4 sets, the
probability that a result at least this discrepant from
randomness is due to chance alone is less than .02. The
one variation from the pattern predicted by the gambling
strategy hypothesis involves a reversal of the probabilities
for the highest and middle reward levels.

Table 8 contains the analysis of variance for the

initial contact data as a function of reward divisibility

81

TABLE 7.-—Probabi1ity of Contacting Weaker Candidate in
Each Condition and Version of Experiment II, Summed Over All
Payoff Distributions.

 

Relative Amount of Money Possessed
by Candidate Making Choice

 

Condition - Version

 

Most Middle Least
(35¢ or 30¢) (25¢) (15¢ or 20¢)
IK - CV .80 .58 .48
IK - UV .42 .39 .21
CK - CV .67 .76 .58
CK - UV .35 .29 .28

Average .55 .53 .40

 

82

TABLE 8.--Ana1ysis of Variance for Choice of Weaker Candi-
date as a Function of Reward Divisibility, Version,
Distribution, and Resource Point, Experiment II.

 

 

 

 

 

 

 

 

 

 

 

Source SS DF MS F SIG
Total 528.00 2111
Reward
Divisibility .93 2 .46 2.10
Version 55.23 1 55.23 250.58 <.0005
Distribution .69 l .69 3.15 .10>p>.03
Resource Point 1.42 2 .71 3.23 .04
RD X V .48 2 .24 1.09
RD X D 1.47 2 .73 3.33 .04
RD X RP 1.62 4 .41 1.84
V X D .14 l .14 .64
V X RP 1.36 2 .68 3.09 .05
D X R .08 2 .04 .18
RD X V X D .47 2 .23 1.07
RD X V X RP .18 4 .04 .20
RD X D X RP 1.03 4 .26 1.16
V X D X RP .49 2 .25 1.12
RD X V X D X RP 2.70 4 .68 3.07 .02
Error 457.61 2076 .22

 

83

(no imposed division versus 50-50 imposed division versus
70-30 imposed division of reward), version, distribution
(70 versus 80), and resource point. Two alternative
predictions were advanced for the effect of imposed reward
division. (1) The first prediction involved the concept
of intracoalition compatibility. If the reward were to be
shared equally, contacts between candidates more nearly
equal in the number of votes controlled, i.e., mutual
choices by candidates 80 and 90 in Distribution 80 and by
110 and 120 in Distribution 70, should be relatively more
likely. However, if the reward divisibility were highly
unequal, contacts between candidates who were more unequal
in their amount of resources, i.e., between 70 and 110 or
between 70 and 120 in distribution 70 and between 80 and
130 or between 90 and 130 in distribution 80, should be
more likely since the difference in resources could suggest
a norm for determining which participant should receive the
greater share of the reward. This prediction would be
supported by a significant reward divisibility by distri-
bution by resource point interaction. (2) If the reward
were of sufficient importance to the participants that it
outweighed any potential difficulties in negotiation
caused by intracoalition incompatibility, the participants
in an unequally divisible reward condition should choose
the weaker potential partner since he would be more likely

to accept the smaller share of the reward. When equality

84

of reward is imposed however, no participant can increase
his share of the reward by choosing the weaker potential
partner, he can only increase his chances of acquiring a
share of the reward, particularly in version UV, by
increasing the security of the coalition. .This prediction
would be supported by a significant reward divisibility
effect or reward divisibility by version interaction.
Table 8 shows that neither of these predicted
effects was significant. Also, the number of offers
required before an agreement was reached in the 70-30
reward division condition did not support the intracoa-
lition compatibility notion. If coalitions between
relatively equal partners, i.e., coalitions (110, 120),
and (80,90), were less compatible than coalitions between
more disparate partners, longer bargaining sequences should
be required in the relatively equal coalitions before an
agreement was reached. The mean number of offers for the
coalition (110, 120) to reach an agreement was 5.5 compared
with 4.3 in the other two coalitions; the associated
t82 df equaled 1.05, n.s. In Distribution 80, the mean
for coalition (80, 90) was 7.6 compared with 4.9 for the
other two coalitions, yielding t50 df = 1.08, n.s. No
similar comparison was possible for the 50-50 reward
division condition since no bargaining was necessary in

that condition to reach an agreement.

85

The data for successive trials within a single
game were summed into blocks for comparison with the data
from Experiment I; Block 1 was composed of trials 2 and 3,
and Block 2 included trials 4 through 7. The probability
of choosing the weaker candidate in version CV decreased
from .67 on trial 1 to .56 in Block 1 and .57 in Block 2;
the conditional probability of choosing the weaker candi-
date given that the weaker candidate had been chosen on
the previous trial was .76 in Block 1 and .74 in Block 2.
In version UV, the overall probability of choosing the
weaker candidate increased from .33 on trial 1 to .48 in
Block 1 and .46 in Block 2; the conditional probabilities
increased from .72 in Block 1 to .77 in Block 2.

The analysis of variance for the reward to the
weaker candidate for games not involving imposed divisi-
bility of reward, in terms of version, distribution, and
coalition formed is presented in Table 9. The overall
average reward to the weaker candidate was 43%. The effect
of distribution was significant at the .01 level; the
weaker candidate in Distribution 60 received 42% of the
reward, while the weaker candidate in each of the other
two distributions received 44%. The effect of coalition
formed was also significant at the .01 level. The weak
candidate in a strong-weak coalition received 41% of the
reward, while the weak candidate in a moderate-weak coa-

lition received 42% and the moderate candidate in a

86

TABLE 9.--Analysis of Variance for Reward to Weaker
Candidate, Experiment II.

 

 

 

 

Source SS df MS F Sig
Total 3.4234 52.7
Version .0106 1 .0106 1.756
Distribution .0571 2 .0286 4.716 <.01
Coalition .0623 2 .0311 5.131 <.01
V X D .0056 2 .0028 .462
V X C .6282 2 .0141 2.323 .10>p>.05
D X C .0339 4 .0085 1.399 .10>p>.05
V X D X C .0167 4 .0004 .689

Error 3.0899 510 .0061

 

87

strong-moderate coalition received 45% of the reward. A
Scheffe test on these means demonstrated that the reward
in the strong—moderate coalition was significantly differ-
ent from that in the other two coalitions, which did not
differ significantly from each other.

There was a high positive relationship between the
first percentage demand of the weaker candidate and the
number of offers necessary to reach an agreement, r526 df =
.45, p < .0005, and negative relationship between the
first percentage offer to the weaker candidate by the
stronger candidate and the number of offers necessary to
reach agreement, r526 df = -.26, p < .0005. However, there
was no linear relationship between the number of offers

made and the final agreement reached, -.03.

r526 df =
Table 10 contains a summary of the game by game
bargaining data for those games not involving imposed
divisibility of rewards. There was no significant increase
in the frequency of 50-50 divisions of the reward,
x: df = 5.51, n.s. The mean reward agreements varied only
between 41% and 44% to the weaker candidate over games.
Grouping the variances into blocks of two, i.e., games 1
and 4 in Block 1, games 7 and 10 in Block 2, and games 11
and 12 in Block 3, and applying the test developed by

Scheffe (1959, p. 83) showed that the decrease in variance

over games was not significant, F(5,3) = 2.80. A summary

88

 

A.m.c .om.m n Am.mvmv

 

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COHDMH>mQ pumccmum
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mo cofluuomonm can:
Aomoc sHmom " MGMXV
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NH Ha OH 5 a H

 

.HH ucmEHHmmxm

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enmecmum eam .nmxmme on enmsmm no connnoaonm ammz .mnflfimm omuom mo Hmnssz--.OH mamas

89

histogram of bargaining agreements reached is presented in
Figure 10 for comparison with the same data from Experi-
ment I.

The sequence of offers made by coalition partners
in all games not involving imposed divisibility of reward
is presented in Figure 11. The occasional decrease in the
offers and increases in the demands in later trials could
be caused by the termination of bargaining in groups which
reached agreement in earlier trials leaving the data of
groups whose offers and demands were more discrepant.
Assuming that the same processes are involved in the
bargaining sequence regardless of length, and that the
process is simply proceeding at a faster rate in bargaining
sessions with fewer trials, a more accurate view of the
bargaining process can be obtained by dividing the
bargaining sequence for each game into intervals.

Figure 12 contains the bargaining data for the first
offer, the offers made one quarter of the way through the
sequence, at the midpoint of bargaining, at the three
quarter point, at the trial preceeding the last, and at
the trial where agreement was reached. The largest change
in offers and demands occurred between the next to the
last and last trial; the data would appear to be well
described by a parabolic function. Similarly, the graph
of the backward trial by trial bargaining data, where

trial 0 represents the trial on which agreement was

90

Figure lO.--Summary Histogram of Reward to Weaker Candidate
in All Games Not Involving Imposed Divisibility of Reward,
Experiment II.

91

160

I I

Frequency

 

 

N.A..20 .25 .30 .35.40 .45.50 .55.60 .65 .70 .75
Offer to Weaker Candidate

Fig. 10

92

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ommcm mcflcflmmumm one mo HMHHB comm CH mpmz mpcmEmQ cam mummmo||.HH ousmfim

93

as .mnm

 

 

 

HMAHB
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p _ _ 1 p l _ _ _ L w L p _ o
lvON
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1
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x 3 w p a ..om e
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105
1 om

 

94

.HH ucmEHmexm .pnmzmm mo muHHHQHmH>HQ pmmomEH mcw>ao>cH uoz mmEmw Ham CH
moocmsqmm mcflcflmmnmm mo Hmuumso 30mm CH opmz mpcmamo can mummmoul.ma musmflm

95

 

NH
HI
ummq ummq e\m «\H e\H H
_ _ r l _ t (J N.
memmz on Hmmmo L m.
pcmammumd
mane
H 1 v.
1m.
memmz mo pcmE-.

 

.mHm

premea go uoraiodoxd

96

reached and trial -1 represents the trial immediately
preceding the trial on which agreement was reached, appears
to be parabolic in form (Figure 13).

The total amount of reward expected by the Ss was
approximately normally distributed about a mean of $.98
and a mode and median of $1.00 with a standard deviation of
.26, and was correlated .35 (262 df) with the total amount
actually earned. The mean likelihood of deception score
on a scale from 1 to 10 was 4.37, with a standard devi-
ation of 2.50 and a median and mode of 5. With respect to
the degree to which Ss tried to win, where a score of 1
meant that the S did not try at all to win as much as he
could and 10 meant that he tried as much as possible, the
mean report was 8.16 with a standard deviation of 1.93, a
median of 9 and a mode of 10. There was little correlation
between the degree to which Ss tried to win and the amount
of money expected, amount earned, or their rating of the
likelihood of deception, r262 df = .07, .07, and -.02
respectively. There was little correlation of the S's
rating of the likelihood of deception with the amount of
money expected, r262 df = .04, and a small but significant
correlation of the deception-rating with the amount

actually earned, r262 df = .12, p < .05.

Discussion

 

There was no evidence in this experiment that Ss

in condition CK responded differently to their accumulated

97

 

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.Hmaue 30mm co one: mpcmEmo can mummmo mo m>n50 Unm3xommnl.ma musmflm

98

pcmEmmumd Hmcflm Eonm pumzxomm mcflucsou .quEdz Hmflue

MH .mHm

 

H- m. m- a- m- G. e- an m- OH- HH- NH- MHI
_ _ _ H _ _ IF p L P _ F _

1 OH.
- om.

m

1 om. o

d

O

1

Al‘-

I.

O

.u

1 ow o

I.

H

a

M

p.

1 om. m
1 ow.
1 on.

 

99

reward than did Ss in condition IK. Rather, Ss in both
conditions adopted a gambling strategy, tending to choose
the stronger vote resource candidate significantly more
often when they had received the least reward than when
they had received the most or intermediate amount in the
previous three games. Ss in both conditions apparently
attended to only their own reward, perhaps as a result of
the verbal rather than written mode of announcement. If
this restriction of attention occurred in condition CK, it
is clear that it was not forced by the experimental
procedure since the announcement was given twice to ensure
accurate and complete reception; it was perhaps inad-
vertently encouraged by the instructions, however, since
the Ss were told at the beginning of the experiment that
the accumulated reward was announced so that they could
keep track of their progress.

The degree of intracoalition compatibility did not
affect the number of offers required to reach an agreement,
nor, in contrast with the results of Nitz and Phillips
(1969), did it affect the initial contacts. That experi-
ment required a choice between one candidate exactly equal
in power or another candidate somewhat disparate. Ap-
parently, in this experiment even a small difference in
power was sufficient to specify to the Ss the proportion of
the reward which they should receive.

The game by game bargaining data (Table 9) revealed

no systematic changes in mean reward to the weaker

100

coalition partner, the number of even divisions of the
reward, or the variance of agreements reached. The weaker
candidate in strong-weak coalitions received significantly
less reward than in a moderate-weak coalition which was
consistent with the parity norm, but did not receive less
than the moderate power candidate in a strong-moderate
coalition, nor was the distribution by coalition inter-
action significant as required by the parity norm.

The correlational data involving the number of
offers made are intuitively reasonable, demonstrating that
more offers were required to reach an agreement if either
the first offer to the weaker candidate was relatively low
or the first demand of the weaker candidate was relatively
high. However, the final offer, i.e., the agreement
reached, did not depend upon the number of offers made.

The overall probability of contacting the weaker
candidate over successive trials within games approached
.5, which was Consistent with a utility of response
variability hypothesis. The associated conditional
probabilities did not decrease as required by that
hypothesis however.

As stated previously, no communication other than
the written offer was allowed during the bargaining phase,
and Ss only knew which coalition was formed as well as the
information already gained from the contact phase.

Therefore, any influence of previous agreements and

101

explicit threats or promises with respect to future
agreements between the same participants was prevented.

It was also clearly impossible for external factors, such
as threats or promises from the participant not in the
coalition, to influence the sequence of offers made.
Therefore, the bargaining sequence data (Figures 12 and 13)
present an estimate of the shape of the pure bargaining

process.

CHAPTER VI

MATHEMATICAL MODELS

It has been suggested that three factors have
affected initial contact choices in the present experiment.
The concept of parity (Gamson, 1961) has been advanced in
connection with the "strengh is weakness" effect in
deterministic coalition formation games. The security
principle (Cole, 1969) has been advanced to explain the
"strength is strength" effect in probabilistic coalition
formation games. The above two factors have been combined
in mathematical models developed by DeYoung and Phillips
(1970). Finally, the "gambling strategy" has been related
to initial contacts under knowledge of accumulated reward
in Experiment II. The present chapter investigates the
influence of the formulations of each of these factors
upon the fit of the model to the data. Model II of DeYoung
and Phillips (1970) has been selected as a vehicle for
these comparisons since, unlike Model I, it is clear that
differences in adequacy of various factors and combinations

of factors in accounting for the experimental data can be

102

103

ascribed to the formulations of the factors and not to
differences in the number of free parameters estimated.
Games Not Involving Accumulated

Reward or Imppsed Divisibility
of Reward

 

 

 

The effect of the parity and security factors are
examined here in all games in which no accumulated reward
or imposed divisibility of reward was announced, these
games being most similar to those whose data have been
previously presented in Figure 3. In order to obtain as
much data as possible at each point, the data from games 1
through 10 of Experiment I was added to that from games 1,
11, and 12 of Experiment II.

The model involving both the parity and security
factor is given in Chapter I. The model involving only the

parity factor reduces to

 

 

 

 

l . l
P = while P = . _ _
w l + 01(Nw + NA) w l + a(NS + NA) (NT NA NW)
(N5 + NA) (Nw + NA)-(NT-NA-NS)

expresses the model involving only the security factor.
The parameter a was estimated using a numerical minimum
chi-square procedure. The obtained chi-squares for the
models involving each factor are presented in Table 11.
For comparison, the model of Shelly and Phillips (1966),
identical to the above model involving only the parity
factor with a = 1, yielded chi-squares of 21.231, p < .01

in version CV, and 75.323, p < .001 in version UV. The

104

TABLE ll.--Minimum Chi-squares of Models Based on Parity

and Security Factors in Version CV and UV Games Not Involv-

ing Accumulated Reward on Imposed Divisibility of Reward,
Experiments I and II Combined.

 

 

 

Version
Factors
CV UV
Parity 4.056* 11.409***.
Security l4.516**** 8.013**
Parity — Security 10.365*** 10.983***

 

Each.x2 had 8 df.
*.85>p>.75
**.45>p>.35
***.25>p>.15
****.10>p>.05
best fitting model for version CV involved only the parity
factor, while the best fitting model for version UV
involved only the security factor. The predictions of
these two models and the observed probabilities are shown
in Figure 14. The correlations between predicted and

observed probabilities were .39 in version CV and .28 in

version UV.

Games Involving Accumulated Reward

 

The games in which accumulated reward was announced
were 4, 7, and 10 in Experiment II. The data from payoff
distribution 5 and game 1 were combined since both were
distribution 60 games involving equality of accumulated

reward. The chi-squares for each possible model are

105

Figure 14.--Observed and Predicted Probabilities of Choice

of Weaker Candidate, in Games Not Involving Imposed

Divisibility of Reward or Accumulated Reward, Experiments I
and II Combined, All Vote Distributions.

.80

.70

.60

.50

.40

.30

.20

Probability of Choosing Weaker Candidate

.10

I

I

I

 

 

106

Version
CV

Version
UV

Observed

— - — Predicted

 

Fig.

14

I
60

l r 1 I I 1 I I
70 80 90 100 110 120 130 140

Votes

107

presented in Table 12. The accumulated reward factor was
formalized by multiplying VA(W) by the amount of reward
which A had accrued divided by .25, the average reward.
TABLE 12.--Minimum Chi-squares of Models Based on Parity

and Security Factors in Version CV and UV Games Involving
Accumulated Reward, Experiment II.

 

 

 

 

 

Version
Factors CV UV
Without With Without With

Accum. Rw. Accum. Rw. Accum. Rw. Accum. Rw.
Parity 7.556* 5.372* 10.398*** 8.959**
Security ll.l92*** 8.147** 9.079** 6.749*
Parity-
Security 9.665*** 6.788* 9.079** 10.96l***

 

Each x2 has 14 d.f.
*.99>p>.90
**.90>p>.80
***.80>p>.65
For each model, the addition of the accumulated
reward factor improved predictions in each version. The
best fitting model involved the parity but not the security
factor in version CV, and the security but not the parity
factor in version UV.- These predictions and the observed
data are presented in Figure 15. The correlations between
the predicted and observed probabilities were .55 in

version CV and .67 in version UV.

108

Figure 15.--Observed and Predicted Probabilities of Choice
of Weaker Candidate in Games Involving Accumulated Reward,
Distribution 60, Experiment II.

 

109

 

Fig.

 

~99I-
.8Cr
/\
/\ /
/ / \\ I ‘. "\
07d” / \ / \ \ / \ .
/ — —VerSion
\ \ / CV
\ I \l
.60— \ ’
\
.50—
/\
.40— \ ‘
/ \ / \
\ / \ I a
/ \ \ / \ /
.30— / - / \ / \ / Version
\
\ / \/ UV
v
.20~
.lO— Observed
" - _ Predicted
' j’ I I I I I I I I I I I I I I
60 100 140 60 100 140 60 100 140 60 100 140 60 100_140 Votes
I.25.25.29 LBS-25.15) I15.35.25/L20.25.3y Q0.2o.2_5j Money
r r
Game 1 and Payoff Payoff Payoff Payoff
Payoff Dist. 1 Dist. 2 Dist. 3 Dist. 4
Dist. 5

15

110

Discussion

 

The present investigation of the factors influencing
social contacts support the contention of DeYoung and
Phillips (1970) that the security factor was sufficient to
account for the data from version UV; in both Table 11 and
Table 12, the fit of the model based upon only the security
factor, the relevant factor according to DeYoung and
Phillips (1970), was superior to models based upon the
parity factor or upon a combination of the parity and
security factors. The present analysis also suggested
that the parity factor was sufficient to account for the
data from version CV, and the model based on parity was in
fact superior to models based on the security factor or on
a combination of the security and parity factors.
Therefore, a simplified model involving only the parity
factor in version CV and only the security factor in
version UV would appear to be preferable to the model
advanced by DeYoung and Phillips (1970).

Although methods exist (Atkinson, Bower, and
Crothers, 1965) to test whether the simplified model is
significantly superior to the alternative models, it is
clear that the improvement was often small and the test
was not applied. Rather, the simplified model is recom-
mended by its more parsimonious explanation of the data, by
the fact that each factor is used in the version for which

it was originally advanced, and by the consistency of its

lll

superiority both in games involving accumulated reward and
in games involving neither accumulated reward nor imposed
divisibility of reward. Similarly, when models incorpo-
rating the accumulated reward factor were applied, small

but consistent improvements in predictions were observed.

CHAPTER V

SUMMARY AND CONCLUSIONS

Successive Games

 

The major purpose of the dissertation was the
examination of Kelley and Arrowood's elimination of con-
fusion explanation for the disappearance of the strength
is weakness effect in successive Political convention
games. Vinacke, Crowell, Dien, and Young (1966) eliminated
confusion by presenting gs with information about possible
strategies, including the strategy which viewed all
participants as being equal in power since any two could
win. They interpreted their data as demonstrating that
the elimination of confusion explanation was not adequate.

The present Experiment I permitted the elimination
of confusion during a series of ten games, allowing
extensive experience with the game while eliminating any
effect of accumulated reward. There was clearly no
tendency for initial contacts or reward divisions to be
divided more equally between the stronger and weaker
candidates in contrast with the elimination of confusion

explanation.

112

113

Although Experiment II was designed to allow §S
to choose coalition partners based upon both their own and
other candidate's reward, gs apparently attended to their
own reward only. This conclusion is supported by the
analysis of variance of the initial contact data (Table 6),
which showed that the level of accumulated reward had a
significant effect upon initial contact choices. However,
this effect did not differ between a condition in which
gs knew only their own reward and another condition in
which gs knew every player's accumulated reward.

It is probable that two factors are largely
responsible for the gs' restriction of attention to their
own reward. The announcement of the accumulated reward
was verbal, introducing a memory factor into the experiment
even though the announcement was repeated. Secondly, the
instructions stated: "The amount that you have won will be
announced every few games so that you can keep track of
your progress." Many gs may have interpreted this statement
as referring to the second person singular, even in the
condition in which they were given information about more
than one candidate's winnings.

These data therefore do not allow inferences to be
made about the effect of knowledge of the other candidates'
previous reward upon initial contacts. The "gambling"
effect that was observed as a function of an g's own reward

was conceptually similar to the effect of resource point in

114

Experiment I. In each case, the weakest candidate, whether
the weakness was in terms of previous reward (Experiment II)
or in terms of votes (Experiment I), chose the weaker vote
resource candidate less often than did the strongest candi-
date. Therefore, it appears that the money resource
dimension can have effects similar to the vote resource

dimension as suggested in the introduction.

Imposed Divisibility of Reward

 

When the possible reward division was limited to
either an extremely unequal (70%-30%) division or an
exactly equal division (50%-50%) previous research (Nitz
and Phillips, 1969) would suggest that initial contacts
would be made such that intracoalition compatibility would
be maximized. Specifically, there should be a tendency
toward the formation of the coalition whose parity division
of the reward most closely approximated the imposed reward
division. The intracoalition compatibility effect was not
observed in this experiment, nor did relative incompati-
bility of the coalition lead to difficulties in the
bargaining process in terms of the number of offers
necessary to reach an agreement. These results suggest
that intracoalition compatibility will affect initial
contacts only when the number of votes of one of the
potential coalition partners is exactly equal and that of
the other is not equal to the number of votes of the

candidate making the choice. Thus, relative intracoalition

115

compatibility, as in this experiment, does not appear to

affect initial contact choices.

Division of Reward

 

Gamson's (1961) parity norm suggested that the
reward for coalition formation should tend to be divided
in proportion to the resources contributed by each partner.
Kelley and Arrowood's (1960) elimination of confusion
explanation implied that the reward should tend to be
divided equally between the coalition partners after they
have had an opportunity to understand the game. The
present experiments support neither of these predictions.
The reward to the weaker candidate was everywhere less
than 50% and was significantly different from a parity
division for 8 of the 9 possible coalitions in Experiment I.
Rather, the observed reward to the weaker candidate was
consistently intermediate to the parity and even reward
division. In neither experiment did the reward to the
weaker candidate even vary from coalition to coalition as
specified by the parity norm (Tables 3 and 9).

The sequence of offers and demands made in this
experiment was relatively free of contaminating variables.
No communication other than the written offer was allowed
during the bargaining phase, and it was impossible for
external factors to influence the bargaining process. The

bargaining sequence under these conditions would appear to

116

be well described by a parabolic function (Figures 12

and 13).

Miscellaneous

 

The utility of response variability concept for
successive trials within games received little support
from these experiments. The probability of contacting the
weaker candidate did approach .5 in later trials within a
game in both versions of both experiments as required by
this hypothesis; however, this result would also be
observed if there was a decrease in the S's expectation of
response reciprocation by his preferred coalition partner.
However, response variability would be maximized if the
probability of perseveration decreased in later trials
within a game. Such a decrease was observed in only
version CV of Experiment I.

The present data did not support a utility of
response variability hypothesis for changes in game to
game initial contacts. However, the SS in the Kelley and
Arrowood (1960) and Chertkoff (1966) studies maintained
the same power positions in the same game from one game to
the next, making their experiment more similar to a series
of trials in the present experiment with bargaining
following each trial. The SS in the present experiments
played at different power positions in successive games

which had different resource distributions.

117

The results of comparing the mathematical models
were intuitively reasonable. The model involving only the
parity factor consistently fit the version CV data better
than either the model involving a combination of the
parity and security factors or the model involving only
the security factor. Likewise, the model involving only
the security factor consistently provided the best fit
of the version UV data. For each such model, the fit was
very good by a minimum chi-square criterion. The proba-
bility of failing to reject the null hypothesis of no
difference between the predicted and observed data ranged

from .45>p>.35 to .99>p>.90 for the best fitting model.

BIBLIOGRAPHY

BIBLIOGRAPHY

Atkinson, R. C.; Bower, G. H.; and Crothers, E. J. An
Introduction to Mathematical LearningfiTheory.
New York: Wiley, 1965.

Azar, E. E. The quantification of events for the analysis
of conflict reduction. Paper presented at the
Seventh North American Peace Research Society
Conference, Ann Arbor, Michigan, 1969.

Bond, J. R., and Vinacke, W. E. Coalitions in mixed-sex
triads. Sociometry, 1961, g1, 61-75.

 

Browning, R. Computer simulation of political bargaining.
Paper presented at the Seventh North American
Peace Research Society Conference, Ann Arbor,
Michigan, 1969.

Calfee, R. C., and Atkinson, R. C. Paired associate models
and the effect of list length. Journal of Mathe-
matical Psychology, 1965, a, 245-265.

 

Caplow, T. Further development of a theory of coalition
in the triad. American Journal of Sociology,

Caplow, T. Two against One: Coalition in Triads.
Englewood Cliffs, New Jersey: Prentice Hall, Inc.
1968.

 

Chertkoff, J. M. A revision of Caplow's coalition theory.
Journal of Experimental Social Psychology} 1967,
3' 172-177.

Chertkoff, J. M. The effects of probability of future
success on coalition formation. Journal of Experi-
' mental Social Psychology, 1966, a, 265-277.

 

Cole, S. G. Coalition preference as a function of vote
commitment in some dictatorial political convention
situations. Journal of Conflict Resolution, 1970
(in press).

118

119

Cole, 8. G., and Phillips, J. L. An analysis of uelative
conflict. Paper presented at Seventh North American
Peace Research Society Conference, Ann Arbor,
Michigan, 1969.

DeYoung, G., and Phillips, J. L. Mathematical models for
coalition formation. In Phillips, J. L. and
Conner, T. (Eds.) Studies in conflict, conflict
reduction, and alliance formation. Technical
Report 70-1 of the Cooperation/Conflict Research
Greoup, in preparation.

Edwards, L. The Natural History of Revolution. Chicago:
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Gamson, W. A. An experimental test of a theory of coa-
lition formation. American Sociological Review,
1961b, 26, 565-573.

Gamson, W. A. A theory of coalition formation. American
Sociological Review, 1961a, 26, 373-382.

 

Gamson, W. A. Experimental studies of coalition formation.
In L. Berkowitz (Ed.), Advances in Experimental
Social Psychology, Volume 1. New York: Academic
Press, 1964.

 

 

Hoffman, F. J.; Festinger, L.; and Lawrence, D. R.
Tendencies toward group comparability in compe—
tition bargaining. Human Relations, 1954, 1,
141-159.

 

Holland, P. W. A variation on the minimum chi-square test.
Journal of Mathematical PsychologYJ 1967, 1,

Howard, N. The theory of meta-games. General Systems:
Yearbook of the Society for General Systems
Research, XI, 1966, 167—186.

 

Kelley, H. H., and Arrowood, A. J. Coalitions in the

triad: Critique and experiment. Sociometry,
1960, 2;, 231-244.

 

Kline, D. K. The effect of bargaining sequence and type
of payoff upon coalition structure and stability
in the triad. Report 68-2 of the Human Learning
Research Institute, Michigan State University,
1968.

Luce, R. D. Individual Choice Behavior: a Theoretical
Analysis. New York: Wiley, 1959.

 

Luce , R.

Messe,

120

D., and Raiffa, H. Games and Decisions. New
York: Wiley, 1957.

 

L. Concept of equity in bargaining. Paper

presented at the Seventh North American Peace
Research Society Conference, Ann Arbor, Michigan,
1969.

Nitz, L. H. Strategies under non-transferable utility.

Unpublished Doctoral Dissertation. Michigan State
University, 1969.

Nitz, L. H., and Phillips, J. L. The effects of divisi-

Ofshe,

bility of payoff on confederative behavior.
Journal of Conflict Resolution, 1969 (in press).

 

L., and Ofshe, R. "Utility and choice in social

interaction. I--Theory," Center for Research in
Management Science, Research Report No. 2, 1968.

Phillips, J. L. Alliance structures in the triad: Notes

on the psychology of interpersonal alliances.
Paper presented at a colloquim, Miami University,
1967.

Phillips, J. L., and Nitz, L. Social contacts in a three

Scheffe,

person "political convention" situation. Journal
of Conflict Resolution, 1968, 12, 206-214.

 

H. The Analysis of Variance. New York: Wiley,
1959.

 

Shelling, T. C. Strategies of Conflict. Journal of

Shelly,

Shelly,

Siegel,

Simmel,

 

Conflict Resolution, 1958, 2, 203—264.

 

R. "A test of a stochastic model of coalition
formation under two conditions of newal structure."
Human Learning Institute, Technical Report No. 14,
1966.

R., and Phillips, J. L. "A social contact model
for coalition formation." Human Learning Research
Institute, Technical Report No. 6, 1966.

S. Choice, strategy, and utility, New York:
McGraw-Hill, 1964.

 

G. The Sociology of George Simmel. Kurt H. Wolff,
trans, ed. New York: Glencoe Press, 1950.

 

121

Straker, S., and Psathas, G. Research on coalitions in the
triad: Findings, problems, and strategy.
Sociometry, 1960, 22, 217-230.

 

Vinacke, W. E. Sex roles in a three-person game.
Sociometry, 1959, 22, 343-360.

 

Vinacke, W. E. Variables in experimental games: Toward
a field theory. Psycholggical Bulletin, 1969, 12,
293-318.

Vinacke, W. E., and Arkoff, A. Experimental studies of
coalitions in the triad. American Sociological
Review, 1957, 22, 406-415.

Vinacke, W. E.; Crowell, D. C.; Dien, D.; and Young, V.
The effect of information about strategy on a
three-person game. Behavioral Science, 1966, 22,
180—189.

 

Winer, B. J. Statistical Principles in Experimental Design.
New York: McGraw-Hill, 1962.

APPENDICES

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APPENDIX I

DISTRIBUTION PRESENTATION ORDERS

APPENDIX II

SAMPLE SET OF RESPONSE SHEETS

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APPENDIX III

SAMPLE BARGAINING SHEET

I}: k

 

I want

APPENDIX III

SAMPLE BARGAINING SHEET

 

votes votes
Leaving you I want Leaving you
3 $ $

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

133

APPENDIX IV

INSTRUCTIONS

APPENDIX IV

INSTRUCTIONS

Assume you are the manager for a candidate in a
political party convention. No candidate has a majority
of the votes in the convention, but no candidate can be
elected unless he can control a majority of votes. It will
therefore be necessary for you to approach other candidates
to attempt to join together in order to have a better
chance of winning the nomination. It is assumed that
these contact attempts take place before each convention
ballot. When on any ballot two candidates decide to
approach each other, their managers are expected to discuss
such matters as the proportion of the payoff which each
shall receive in case the partnership is successful in
winning the nomination. We are interested in how people
choose partners in this situation when they are trying to
earn as much for themselves as possible.
(BEGIN PASSING OUT BALLOTS FOR PRACTICE GAME)

The details of this particular political convention
are given on the ballots which are being passed out to you

now. Look closely at the ballots you have been given.
The first line tells you that there are 9 votes in the

convention, with no votes uncommitted. Some conventions

134

135

which you will participate in will have uncommitted votes
in them. The next line tells you your candidate has a
certain number of votes. One of you represents a candidate
with 2 votes, another of you represents a candidate with 4
votes, and the third candidate has 3 votes. The next line
shows that 5 votes are required for a majority in this
convention. A majority, or at least this number of votes,
is required to win.

The next line, the line that says, "The other
candidates have" contains the number of votes held by each
of the other two candidates besides yourself. When the
game is played, you are to circle one of these two numbers
to indicate the candidate with whom you would like to try
to form a partnership on each ballot.

The next line simply repeats your number of votes,
and the bottom line tells you how many total votes the
partnership would have if you formed a partnership with
the candidate whose votes are above it in that column. On
the first ballot choose one of the two candidates listed.
If there is no mutual choice on the first ballot, we will
continue through successive ballots until there is a mutual
choice. Before the game starts, do you have any questions
about the way the game is played?

(PAUSE)
This will be a practice game. You will have ten

seconds to make your choice. vPlease do not try to

136

communicate with each other, but simply make your choices
on the ballot and pass it through the slot in front of you.
(AFTER COMPLETION OF PRACTICE GAME)

There was a mutual choice on this ballot; the
person representing the candidate with 2 votes chose the
candidate with 2 votes, and the candidate with X_votes
chose the candidate with 2 votes. If no two of you had
chosen each other, we would have gone on through ballots
__, __ and so on until there was a mutual choice. The
next game will be played similarly to the first game,
except different numbers of votes will be used and some
games will have uncommitted votes in them. Make your
choices carefully since one of these games will be chosen
at random for the division of $3.00. The winner of the
game will have an opportunity to bargain over the division
of the money. After all games have been played, one of the
games will be chosen at random; and if a partnership has
won that game, the bargain made after that game will be
binding.

(LET §S CHOOSE BALLOTS AT RANDOM. ADMINISTER EACH GAME AS
BEFORE. GIVE SS AT LEAST 10 SECONDS TO EXAMINE THE GAME
BEFORE THE FIRST BALLOT.)

(ANNOUNCE BEFORE EACH GAME INVOLVING UNCOMMITTED VOTES)

Notice that this game has 300 uncommitted votes in
it. If this game is chosen for the division of the $3.00,

the 300 uncommitted votes will be split between the

137

partnership and the third candidate to determine who will
receive the money. If the person left out of the partner-
ship gets enough of the uncommitted votes to have a
majority, he would win the $3.00 all by himself. Otherwise,
the money will be divided between the partners according

to the agreement they have made.