Spatial Evolutionary Game Theory: Deterministic Approximations, Decompositions, and ...

155 Pages·2014·1.64 MB·English

University of Massachusetts Amherst
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Open Access Dissertations
9-2011
Spatial Evolutionary Game Theory: Deterministic
Approximations, Decompositions, and
Hierarchical Multi-scale Models
Sung-Ha Hwang
University of Massachusetts Amherst, [email protected]
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SPATIAL EVOLUTIONARY GAME THEORY: DETERMINISTIC
APPROXIMATIONS, DECOMPOSITIONS, AND
HIERARCHICAL MULTI-SCALE MODELS
A Dissertation Presented
by
SUNGHA HWANG
Submitted to the Graduate School of the
University of Massachusetts Amherst in partial fulfillment
of the requirements for the degree of
DOCTOR OF PHILOSOPHY
September 2011
Mathematics and Statistics
(cid:13)c Copyright by SUNGHA HWANG 2011
All Rights Reserved
SPATIAL EVOLUTIONARY GAME THEORY: DETERMINISTIC
APPROXIMATIONS, DECOMPOSITIONS, AND
HIERARCHICAL MULTI-SCALE MODELS
A Dissertation Presented
by
SUNGHA HWANG
Approved as to style and content by:
Luc Rey-Bellet, Chair
Markos Katsoulakis, Member
Andrea Nahmod, Member
Samuel Bowles, Member
George Avrunin, Department Chair
Mathematics and Statistics
To Seung-Yun
ACKNOWLEDGMENTS
I would like to thank my advisors, Luc Rey-Bellet and Markos Katsoulakis.
Without their helps this work would have been incomplete. Luc Rey-Bellet, who
alsoservedasamemberofmyeconomicsdissertationcommittee, taughtandsuper-
vised me in various ways. Since I started taking his graduate differential equations
course fall 2004, I had taken four courses with him. He introduced me into topics
like entropy production, large deviations, and so on. I would also appreciate his
willingness to meet me almost every week during three years of my dissertation
work. Markos Katsoulakis introduced the various methods of approximating spa-
tial stochastic processes. I am thankful to him for his generous support through
the National Science Foundation. Andrea Nahmod first recommended me to start
the master program in Mathematics Departments, when I was a Ph.D. student in
Economics Departments. I appreciate her insightful comments on my dissertation.
Samuel Bowles, like Luc Rey-Bellet, served twice for my dissertations committees.
I am thankful to his insightful and encourage remarks and comments on my disser-
tation work. Last but not least, I express my deep gratitude to my wife, Seung-Yun
Oh, and appreciate her support and sacrifice in spite of her own Ph.D. work.
v
ABSTRACT
SPATIAL EVOLUTIONARY GAME THEORY: DETERMINISTIC
APPROXIMATIONS, DECOMPOSITIONS, AND
HIERARCHICAL MULTI-SCALE MODELS
SEPTEMBER 2011
SUNGHA HWANG, B.A., SEOUL NATIONAL UNIVERSITY
M.A., SEOUL NATIONAL UNIVERSITY
Ph.D., UNIVERSITY OF MASSACHUSETTS AMHERST
Directed by: Professor Luc Rey-Bellet
Evolutionary game theory has recently emerged as a key paradigm in various
behavioralsciencedisciplines. Inparticularitprovidespowerfultoolsandaconcep-
tual framework for the analysis of the time evolution of strategic interdependence
among players and its consequences, especially when the players are spatially dis-
tributed and linked in a complex social network. We develop various evolutionary
game models, analyze these models using appropriate techniques, and study their
applications to complex phenomena.
In the second chapter, we derive integro-differential equations as deterministic
approximations of the microscopic updating stochastic processes. These generalize
the known mean-field ordinary differential equations and provide powerful tools to
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investigate the spatial effects on the time evolutions of the agents’ strategy choices.
The deterministic equations allow us to identify many interesting features of the
evolution of strategy profiles in a population, such as standing and traveling waves,
and pattern formation, especially in replicator-type evolutions.
Weintroduceseveralmethodsofdecompositionoftwoplayernormalformgames
in the third chapter. Viewing the set of all games as a vector space, we exhibit ex-
plicit orthonormal bases for the subspaces of potential games, zero-sum games, and
their orthogonal complements which we call anti-potential games and anti-zero-
sum games, respectively. Perhaps surprisingly, every anti-potential game comes
either from Rock-paper-scissors type games (in the case of symmetric games) or
from Matching Pennies type games (in the case of asymmetric games). Using
these decompositions, we prove old (and some new) cycle criteria for potential and
zero-sum games (as orthogonality relations between subspaces).
We illustrate the usefulness of our decompositions by (a) analyzing the gener-
alized Rock-Paper-Scissors game, (b) completely characterizing the set of all null-
stable games, (c) providing a large class of strict stable games, (d) relating the
game decomposition to the Hodge decomposition of vector fields for the replicator
equations, (e) constructing Lyapunov functions for some replicator dynamics, (f)
constructing Zeeman games - games with an interior asymptotically stable Nash
equilibrium and a pure strategy ESS.
The hierarchical modeling of evolutionary games provides flexibility in address-
ing the complex nature of social interactions as well as systematic frameworks in
which one can keep track of the interplay of within-group dynamics and between-
group competitions. For example, it can model husbands and wives’ interactions,
playing an asymmetric game with each other, while engaging coordination prob-
lems with the likes in other families. In the fourth chapter, we provide hierarchical
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stochastic models of evolutionary games and approximations of these processes,
and study their applications
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TABLE OF CONTENTS
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
CHAPTER
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Evolutionary game theory . . . . . . . . . . . . . . . . 1
1.1.2 Spatial stochastic processes . . . . . . . . . . . . . . . 3
1.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 Deterministic approximations of spatial stochastic
processes . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Decompositions of normal form games and statistical
mechanics . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.3 Hierarchicalmulti-scalemodels: Coarse-grainedMarkov
chains . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Future Research Agendas . . . . . . . . . . . . . . . . . . . . 13
2. DETERMINISTICEQUATIONSFORSPATIALEVOLUTIONARY
GAMES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1 Spatial Games and and Strategy-Revision Processes . . . . . 16
2.2 Meso-scopic Limits . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 Basic setup . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.3 Heuristic derivation of the differential equations . . . . 27
2.3 Existing Results . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 The proof of Theorem 2.2.4 . . . . . . . . . . . . . . . . . . . 33
2.5 Spatially uniform interactions: Mean-field Dynamics . . . . . 47
2.6 Equilibrium Selection and Pattern Formations . . . . . . . . . 52
2.6.1 Linear stability analysis . . . . . . . . . . . . . . . . . 53
2.6.2 Example: Two-strategy symmetric games . . . . . . . 56
2.6.3 Traveling front solutions and equilibrium selection:
Imitation versus Perturbed Best Responses . . . . . . 60
2.6.4 PDE Approximations . . . . . . . . . . . . . . . . . . 64
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