Strategic Interdependence, Hypothetical Bargaining, and Mutual Advantage in Non-Cooperative ...

223 Pages·2017·4.57 MB·English

The London School of Economics and Political Science
Strategic Interdependence,
Hypothetical Bargaining, and Mutual
Advantage in Non-Cooperative Games
Mantas Radzvilas
A thesis submitted to the Department of Philosophy, Logic and Scientific Method
of the London School of Economics and Political Science
for the degree of Doctor of Philosophy, September 2016.
Declaration
I certify that the thesis I have presented for examination for the PhD degree
of the London School of Economics and Political Science is solely my own
work other than where I have clearly indicated that it is the work of others
(in which case the extent of any work carried out jointly by me and any
person is clearly identified in it). The copyright of this thesis rests with the
author. Quotation from it is permitted, provided that full acknowledgement
is made. This thesis may not be reproduced without my prior written con-
sent. I warrant that this authorisation does not, to the best of my belief,
infringe the rights of any third party.
I declare that my thesis consists of 85,056 words.
Statement of conjoint work
I confirm that Chapter 2 is based on a paper co-authored with Jurgis Karpus
(King’s College London) and I contributed 50% of this work.
Mantas Radzvilas
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Abstract
One of the conceptual limitations of the orthodox game theory is its inability
to offer definitive theoretical predictions concerning the outcomes of non-
cooperative games with multiple rationalizable outcomes. This prompted
the emergence of goal-directed theories of reasoning – the team reasoning
theory and the theory of hypothetical bargaining. Both theories suggest that
people resolve non-cooperative games by using a reasoning algorithm which
allows them to identify mutually advantageous solutions of non-cooperative
games.
The primary aim of this thesis is to enrich the current debate on goal-
directed reasoning theories by studying the extent to which the principles
of the bargaining theory can be used to formally characterize the concept of
mutualadvantageinawaywhichiscompatiblewithsomeoftheconceptually
compellingprinciplesoforthodoxgametheory, suchasindividualrationality,
incentive compatibility, and non-comparability of decision-makers’ personal
payoffs.
I discuss two formal characterizations of the concept of mutual advan-
tage derived from the aforementioned goal-directed reasoning theories: A
measure of mutual advantage developed in collaboration with Jurgis Kar-
pus, which is broadly in line with the notion of mutual advantage suggested
by Sugden (2011, 2015), and the benefit-equilibrating bargaining solution
function, which is broadly in line with the principles underlying Conley and
Wilkie’s (2012) solution for Pareto optimal point selection problems with
finite choice sets. I discuss the formal properties of each solution, as well as
its theoretical predictions in a number of games. I also explore each solution
concept’s compatibility with orthodox game theory.
I also discuss the limitations of the aforementioned goal-directed reason-
ing theories. I argue that each theory offers a compelling explanation of how
a certain type of decision-maker identifies the mutually advantageous solu-
tionsof non-cooperativegames, butneither of themoffers a definitiveanswer
to the question of how people coordinate their actions in non-cooperative so-
cial interactions.
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Contents
1 Introduction 11
1.1 The Basic Elements of Non-Cooperative Game Theory . . . 14
1.1.1 A Formal Representation of a Normal Form Game . . 14
1.1.2 What do Payoffs Actually Represent? . . . . . . . . . 15
1.1.3 The Complete Information Assumption . . . . . . . . 19
1.1.4 Best Response . . . . . . . . . . . . . . . . . . . . . . 22
1.1.5 Best Response and The Nash equilibrium . . . . . . . 25
1.1.6 Rationality as Best-Response Reasoning . . . . . . . 29
1.2 Social Coordination and the Conceptual Limitations of the
Best-Response Reasoning Models . . . . . . . . . . . . . . . 32
1.3 Social Coordination Theories . . . . . . . . . . . . . . . . . . 37
1.3.1 Cognitive Hierarchy Theory . . . . . . . . . . . . . . 37
1.3.2 Stackelberg Reasoning . . . . . . . . . . . . . . . . . 42
1.3.3 Social Conventions Theory . . . . . . . . . . . . . . . 43
1.4 Strategic Reasoning and Mutual Advantage . . . . . . . . . 48
1.4.1 Coalitional Rationalizability . . . . . . . . . . . . . . 48
1.4.2 Goal-Directed Reasoning and Mutual Advantage . . . 52
1.4.3 Team Reasoning and Mutual Advantage . . . . . . . 53
1.4.4 Hypothetical Bargaining . . . . . . . . . . . . . . . . 55
1.5 The Structure of the Thesis . . . . . . . . . . . . . . . . . . 59
2 Team Reasoning and the Measure of Mutual Advantage 60
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.2 Team Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.2.1 What is Team Reasoning? . . . . . . . . . . . . . . . 63
2.2.2 Team Reasoning and Transformations of Payoffs . . . 65
2.3 Team Interests and the Notion of Mutual Advantage . . . . 68
2.3.1 Self-Sacrifice and Mutual Advantage . . . . . . . . . 68
2.3.2 Interpersonal Comparisons of Payoffs . . . . . . . . . 69
2.4 Team Interests as the Maximization of Mutual Advantage . 71
2.4.1 Formalization . . . . . . . . . . . . . . . . . . . . . . 73
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2.4.2 Two Properties of the Function . . . . . . . . . . . . 74
2.4.3 Reference points . . . . . . . . . . . . . . . . . . . . 75
2.4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . 77
2.4.5 No Irrelevant Player . . . . . . . . . . . . . . . . . . 79
2.4.6 Independence of Irrelevant Strategies . . . . . . . . . 80
2.4.7 Interpersonal Comparisons of Advantage . . . . . . . 81
2.5 Mutual Advantage and the Problem of Coordination . . . . 85
2.6 The Triggers of Team Reasoning . . . . . . . . . . . . . . . . 88
2.7 Conceptual Limitations . . . . . . . . . . . . . . . . . . . . 91
2.7.1 Implicit Motivation Transformations . . . . . . . . . 91
2.7.2 Stability Issues . . . . . . . . . . . . . . . . . . . . . 94
2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3 Hypothetical Bargaining and the Equilibrium Selection In
Non-Cooperative Games 98
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.2 Hypothetical Bargaining . . . . . . . . . . . . . . . . . . . . 104
3.2.1 Misyak and Chater’s Virtual Bargaining Model . . . 104
3.2.2 The Limitations of the Model . . . . . . . . . . . . . 106
3.3 The Ordinal Benefit-Equilibrating Solution . . . . . . . . . . 111
3.3.1 The Intuition Behind the Ordinal BE Solution . . . . 111
3.3.2 Formalization . . . . . . . . . . . . . . . . . . . . . . 116
3.3.3 Ordinal BE Solution Properties . . . . . . . . . . . . 119
3.3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . 122
3.4 The Cardinal Benefit-Equilibrating Solution . . . . . . . . . 126
3.4.1 The Intuition Behind the Cardinal BE Solution . . . 126
3.4.2 Formalization . . . . . . . . . . . . . . . . . . . . . . 131
3.4.3 The Properties of the Cardinal BE Solution . . . . . 133
3.4.4 Relation to other Bargaining Solutions . . . . . . . . 135
3.4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . 137
3.4.6 Hypothetical Bargaining in Pie Games . . . . . . . . 140
3.5 Possible Extensions of the Model . . . . . . . . . . . . . . . 144
3.5.1 Application to N-Player Games . . . . . . . . . . . . 144
3.5.2 Social Interactions and the Set of Feasible Agreements 150
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
4 Hypothetical Bargaining, Social Coordination and Evolu-
tion 155
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.2 Hypothetical Bargaining and Rational Choice . . . . . . . . 160
4.2.1 Hypothetical Bargaining as a Reasoning Algorithm . 160
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4.2.2 The Rationalization Problem . . . . . . . . . . . . . 163
4.3 Hypothetical Bargaining and Social Coordination . . . . . . 175
4.3.1 Hypothetical Bargaining and the Problem of Common
Beliefs . . . . . . . . . . . . . . . . . . . . . . . . . . 175
4.4 Hypothetical Bargaining and Evolution . . . . . . . . . . . . 180
4.4.1 Evolutionary Game Theory and Population Dynamics 180
4.4.2 Hypothetical Bargaining as an Evolutionary Adaptation187
4.5 The Non-Uniqueness Problem . . . . . . . . . . . . . . . . . 200
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
5 Conclusion 205
Bibliography 212
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List of Figures
1.1 Prisoner’s Dilemma . . . . . . . . . . . . . . . . . . . . . . . 17
1.2 Weak Dominance game . . . . . . . . . . . . . . . . . . . . . 31
1.3 Coordination game . . . . . . . . . . . . . . . . . . . . . . . 33
1.4 Hi-Lo game . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.5 Coordination game . . . . . . . . . . . . . . . . . . . . . . . 40
1.6 Battle of the Sexes game . . . . . . . . . . . . . . . . . . . . 45
1.7 ‘Weak’ Common Interest game . . . . . . . . . . . . . . . . . 50
2.1 Hi-Lo game . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.2 Prisoner’s Dilemma game . . . . . . . . . . . . . . . . . . . . 64
2.3 Monetary Prisoner’s Dilemma game (a) and its transforma-
tion (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.4 A variation of the Prisoner’s Dilemma game . . . . . . . . . 69
2.5 AnotherrepresentationofpreferencesinthePrisoner’sDilemma
game. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.6 Chicken (a) and High Maximin (b) games . . . . . . . . . . 77
2.7 Three versions of the extended Hi-Lo game . . . . . . . . . . 85
2.8 Coordination game with a conflict of players’ preferences . . 93
3.1 Common interest game . . . . . . . . . . . . . . . . . . . . . 99
3.2 Hi-Lo game . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.3 4x4 games with three weakly Pareto optimal Nash equilibria 108
3.4 Ordinal coordination game . . . . . . . . . . . . . . . . . . . 112
3.5 OrdinalcoordinationgamewithauniqueParetoefficientNash
equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.6 Ordinal Chicken game . . . . . . . . . . . . . . . . . . . . . 123
3.7 Divide-the-Cake game (even number of pieces) . . . . . . . . 124
3.8 Divide-the-Cake game (odd number of pieces) . . . . . . . . 125
3.9 Coordination game with three weakly Pareto optimal outcomes126
3.10 Coordination game with three weakly Pareto optimal outcomes138
3.11 Chicken game . . . . . . . . . . . . . . . . . . . . . . . . . . 139
3.12 Battle of the Sexes game . . . . . . . . . . . . . . . . . . . . 139
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3.13 An example of a pie game from Faillo et al. (2013, 2016)
experiments (Source: Faillo et al. 2013) . . . . . . . . . . . . 141
3.14 An example of a pie game depicted in Figure 3.13 as seen by
the two interacting players in one treatment. The positions of
the three slices were varied across treatments (Source: Faillo
et al. 2013) . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.15 3x3 pie game depicted in Figure 22 represented in normal form142
3.16 Three player coordination game played by three hypothetical
bargainers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
3.17 Three player coordination game, in which the matrix player
is not a hypothetical bargainer . . . . . . . . . . . . . . . . . 149
3.18 The Boobytrap Game . . . . . . . . . . . . . . . . . . . . . . 151
3.19 Two versions of the extended Hi-Lo game . . . . . . . . . . . 153
4.1 Common interest game . . . . . . . . . . . . . . . . . . . . . 156
4.2 Coordination game . . . . . . . . . . . . . . . . . . . . . . . 176
4.3 Coordination game . . . . . . . . . . . . . . . . . . . . . . . 192
4.4 Mixed motive game . . . . . . . . . . . . . . . . . . . . . . . 195
4.5 Mixed motive game under (a) replicator dynamics and (b)
best response dynamics . . . . . . . . . . . . . . . . . . . . . 196
4.6 Mixed motive game with efficient exploiters . . . . . . . . . 197
4.7 Mixed motive game with efficient exploiters under (a) best
response dynamics and (b) replicator dynamics. . . . . . . . 198
4.8 Extended Hi-Lo game . . . . . . . . . . . . . . . . . . . . . . 201
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List of Tables
2.1 Thelevelsofindividualandmutualadvantageassociatedwith
each outcome. . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.2 A summary of the results for the remaining games . . . . . . 78
3.1 Players’ foregone preferred alternatives . . . . . . . . . . . . 113
3.2 Comparison of players’ losses of attainable individual advantage123
3.3 Comparison of players’ losses of attainable individual advantage123
3.4 Comparison of players’ losses of attainable individual advantage125
3.5 Comparison of players’ losses of attainable individual advantage125
3.6 Comparison of players’ losses of maximum individual advantage138
3.7 Comparison of players’ losses of maximum individual advantage139
3.8 Comparison of players’ losses of maximum individual advantage140
3.9 Summary of Faillo et al 2013 results. Choices predicted by
the cardinal BE model are indicated by be. Choices predicted
by the team reasoning model are indicated by . . . . . . . 142
τ
3.10 Summary of Crawford et al 2010 results. The choices of
player 1 (P1) and player 2 (P2) are presented separately.
Choices predicted by the cardinal BE model are indicated
by be. Choices predicted by the team reasoning model are
indicated by . . . . . . . . . . . . . . . . . . . . . . . . . . 143
τ
3.11 Players’ levels of individual advantage, losses of maximum
individual advantage, and differences of individual advantage
losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
3.12 The ratios of each players individual advantage level to the
sum of players individual advantage levels, and the distance
between each players actual ratio and the strictly egalitarian
ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
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Acknowledgments
Several people have helped me to improve this thesis. I should first mention
myprimarysupervisor, JasonMcKenzieAlexander. Withouthisopenmind-
edness to my ideas about game theory, profound knowledge of game theory
and mathematics, as well as his exceptional attention to detail, my thesis
would have been considerably poorer. I am also indebted to my secondary
supervisor, Richard Bradley, whose critical yet encouraging comments and
unwavering support gave me the intellectual courage and determination to
develop my ideas about game theory. I would like to thank them both for
all the support they provided me with throughout these years.
Iwouldliketothankmyexaminers, RobertSugdenandFrancescoGuala,
with whom I had an intellectually challenging, yet ultimately very enjoyable
and educational exchange about the topics of this dissertation. I would also
like to thank Alex Voorhoeve, whose support and critical feedback on my
work was invaluable.
I am also grateful to my very good friends and colleagues, Jurgis Karpus
and Nicolas Wüthrich, whose feedback on numerous drafts of my work was
extremely helpful.
My fellow PhD students James Nguyen, Goreti Faria, Deren Olgun, Jo-
hannes Himmelreich, Catherine M. Greene, Aron Vallinder, Philippe van
Basshuysen and Tom Rowe have been great friends and companions in mo-
ments of joy and distress during my life as a PhD student. Special thanks
go to my good friend Susanne Burri, whose advices and encouragement were
of great help to me during my first year as a PhD student.
I thank my partner, Alina Mickevic, for her unwavering patience, under-
standing and loving support, especially during the last year of my PhD.
Most importantly, I am profoundly grateful to my parents, Aldona and
Vytautas, who encouraged me to pursue what I am truly passionate about.
Their unconditional love and support sustained me throughout my PhD and
proved vital for the completion of this thesis.
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