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MULTIAGENT SYSTEMS(yoav shoham)
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MULTIAGENT SYSTEMS
Algorithmic, Game-Theoretic,
and Logical Foundations
Yoav Shoham
Stanford University
Kevin Leyton-Brown
University of British Columbia
Revision 1.1
M
ultiagent Systems is copyright © Shoham and Leyton-Brown, 2009, 2010. This version is
formatted differently than the book—and in particular has different page numbering—and has
not been fully copy edited. Please treat the printed book as the definitive version.
You are invited to use this electronic copy without restriction for on-screen viewing, but are
requested to print it only under one of the following circumstances:
• You live in a place that does not offer you access to the physical book;
• The cost of the book is prohibitive for you;
• You need only one or two chapters.
Finally, we ask you not to link directly to the PDF or to distribute it electronically. Instead, we
invite you to link to http://www.masfoundations.org. This will allow us to gauge the level of
interest in the book and to update the PDF to keep it consistent with reprintings of the book.
i
To my wife Noa and my daughters Maia, Talia and Ella —YS
To Jude —KLB
Contents
Credits and Acknowledgments xi
Introduction xiii
1 Distributed Constraint Satisfaction 1
1.1 Defining distributed constraint satisfaction problems 2
1.2 Domain-pruning algorithms 4
1.3 Heuristic search algorithms 8
1.3.1 The asynchronous backtracking algorithm 10
1.3.2 A simple example 12
1.3.3 An extended example: the four queens problem 13
1.3.4 Beyond the ABT algorithm 17
1.4 History and references 18
2 Distributed Optimization 19
2.1 Distributed dynamic programming for path planning 19
2.1.1 Asynchronous dynamic programming 19
2.1.2 Learning real-time A
∗
20
2.2 Action selection in multiagent MDPs 22
2.3 Negotiation, auctions and optimization 28
2.3.1 From contract nets to auction-like optimization 28
2.3.2 The assignment problem and linear programming 30
2.3.3 The scheduling problem and integer programming 36
2.4 Social laws and conventions 44
2.5 History and references 46
3 Introduction to Noncooperative Game Theory: Games in Normal Form 47
3.1 Self-interested agents 47
3.1.1 Example: friends and enemies 48
3.1.2 Preferences and utility 49
3.2 Games in normal form 54
3.2.1 Example: the TCP user’s game 54
iv Contents
3.2.2 Definition of games in normal form 55
3.2.3 More examples of normal-form games 56
3.2.4 Strategies in normal-form games 59
3.3 Analyzing games: from optimality to equilibrium 60
3.3.1 Pareto optimality 61
3.3.2 Defining best response and Nash equilibrium 62
3.3.3 Finding Nash equilibria 63
3.3.4 Nash’s theorem: proving the existence of Nash equilibria 65
3.4 Further solution concepts for normal-form games 73
3.4.1 Maxmin and minmax strategies 73
3.4.2 Minimax regret 76
3.4.3 Removal of dominated strategies 78
3.4.4 Rationalizability 81
3.4.5 Correlated equilibrium 83
3.4.6 Trembling-hand perfect equilibrium 85
3.4.7 ǫ-Nash equilibrium 85
3
.5 History and references 87
4 Computing Solution Concepts of Normal-Form Games 89
4.1 Computing Nash equilibria of two-player, zero-sum games 89
4.2 Computing Nash equilibria of two-player, general-sum games 91
4.2.1 Complexity of computing a sample Nash equilibrium 91
4.2.2 An LCP formulation and the Lemke–Howson algorithm 93
4.2.3 Searching the space of supports 101
4.2.4 Beyond sample equilibrium computation 104
4.3 Computing Nash equilibria of n-player, general-sum games 105
4
.4 Computing maxmin and minmax strategies for two-player, general-sum games 108
4.5 Identifying dominated strategies 108
4.5.1 Domination by a pure strategy 109
4.5.2 Domination by a mixed strategy 110
4.5.3 Iterated dominance 112
4.6 Computing correlated equilibria 113
4.7 History and references 115
5 Games with Sequential Actions: Reasoning and Computing with the Extensive Form
117
5.1 Perfect-information extensive-form games 117
5.1.1 Definition 118
5.1.2 Strategies and equilibria 119
5.1.3 Subgame-perfect equilibrium 121
5.1.4 Computing equilibria: backward induction 124
5.2 Imperfect-information extensive-form games 130
5.2.1 Definition 130
Uncorrected manuscript of Multiagent Systems, published by Cambridge University Press
Revision 1.1 © Shoham & Leyton-Brown, 2009, 2010.
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