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Game Theory, Alive Game Theory, Alive Anna R. Karlin and Yuval Peres Draft January 20, 2013 Please send comments and corrections to [email protected] and [email protected] i We are grateful to Alan Hammond, Yun Long, G´abor Pete, and Peter Ralph for scribing early drafts of this book from lectures by Yuval Peres. These drafts were edited by Liat Kessler, Asaf Nachmias, Yelena Shvets, Sara Robinson and David Wilson; Yelena also drew many of the figures. We also thank Ranjit Samra for the lemon figure, Barry Sinervo for the Lizard picture, and Davis Sheperd for additional figures. Sourav Chatterjee, Elchanan Mossel, Asaf Nachmias, and Shobhana Stoy- anov taught from drafts of the book and provided valuable suggestions. Thanks also to Varsha Dani, Kieran Kishore, Itamar Landau, Eric Lei, Mal- lory Monasterio, Andrea McCool, Katia Nepom, Colleen Ross, Zhuohong Shen, Davis Sheperd, Stephanie Somersille, and Sithparran Vanniasegaram for comments and corrections. The support of the NSF VIGRE grant to the Department of Statistics at the University of California, Berkeley, and NSF grants DMS-0244479, DMS-0104073, and CCF-1016509 is acknowledged. Contents 1 Introduction page 1 2 Two-person zero-sum games 6 2.1 Examples 6 2.2 Definitions 8 2.3 Saddle points and Nash equilibria 10 2.4 Simplifying and solving zero-sum games 11 2.4.1 The technique of domination 12 2.4.2 Summary of Domination 13 2.4.3 The use of symmetry 14 2.4.4 Series and Parallel Game Combinations 17 2.5 Games on graphs 18 2.5.1 Maximum Matchings 18 2.5.2 Hide-and-seek games 20 2.5.3 Weighted hide-and-seek games 22 2.5.4 The bomber and battleship game 25 2.6 Von Neumann’s minimax theorem 26 2.7 Linear Programming and the Minimax Theorem 31 2.7.1 Linear Programming Basics 32 2.7.2 Linear Programming Duality 33 2.7.3 Duality, more formally 34 2.7.4 The proof of the duality theorem 35 2.7.5 An interpretation of a primal/dual pair 38 2.8 Zero-Sum Games With Infinite Action Spaces∗ 40 2.9 Solved Exercises 47 2.9.1 Another Betting Game 47 3 General-sum games 49 3.1 Some examples 49 ii Contents iii 3.2 Nash equilibria 52 3.3 General-sum games with more than two players 56 3.4 Games with Infinite Strategy Spaces 60 3.5 Evolutionary game theory 62 3.5.1 Hawks and Doves 62 3.5.2 Evolutionarily stable strategies 65 3.6 Potential games 68 3.7 Correlated equilibria 71 3.8 The proof of Nash’s theorem 75 3.9 Fixed-point theorems* 77 3.9.1 Easier fixed-point theorems 77 3.9.2 Sperner’s lemma 80 3.9.3 Brouwer’s fixed-point theorem 83 4 Signaling and asymmetric games 94 4.1 Signaling and asymmetric information 95 4.1.1 Examples of signaling (and not) 96 4.1.2 The collapsing used car market 98 4.2 Some further examples 99 5 Social choice 102 5.1 Voting and Ranking Mechanisms 102 5.1.1 Definitions 103 5.1.2 Instant runoff elections 105 5.1.3 Borda count 106 5.1.4 Dictatorship 107 5.2 Arrow’s impossibility theorem 107 5.3 Strategy-proof Voting 109 6 Auctions and Mechanism Design 114 6.1 Auctions 114 6.2 Single Item Auctions 114 6.2.1 Profit in single-item auctions 116 6.2.2 More profit? 117 6.2.3 Exercise: Vickrey with Reserve Price 118 6.3 The general mechanism design problem 118 6.4 Social Welfare Maximization 121 6.5 Win/Lose Mechanism Design 125 6.6 Profit Maximization in Win/Lose Settings 126 6.6.1 Profit maximization in digital goods auctions 127 6.6.2 Profit Extraction 128 6.6.3 A profit-making digital goods auction 129 iv Contents 6.6.4 With priors: the Bayesian setting 131 6.6.5 Revenue Equivalence 134 6.6.6 The revelation principle 136 6.6.7 Optimal Auctions 137 7 Stable matching 140 7.1 Introduction 140 7.2 Algorithms for finding stable matchings 141 7.3 Properties of stable matchings 142 7.4 A special preference order case 143 8 Coalitions and Shapley value 146 8.1 The Shapley value and the glove market 146 8.2 The Shapley value 150 8.3 Two more examples 152 9 Interactive Protocols 155 9.1 Keeping the meteorologist honest 155 9.2 Secret sharing 158 9.2.1 A simple secret sharing method 159 9.2.2 Polynomial method 160 9.3 Private computation 162 9.4 Cake cutting 163 9.5 Zero-knowledge proofs 164 9.6 Remote coin tossing 166 10 Combinatorial games 168 10.1 Impartial games 169 10.1.1 Nim and Bouton’s solution 175 10.1.2 Other impartial games 178 10.1.3 Impartial games and the Sprague-Grundy theorem 186 10.2 Partisan games 192 10.2.1 The game of Hex 195 10.2.2 Topology and Hex: a path of arrows* 196 10.2.3 Hex and Y 198 10.2.4 More general boards* 200 10.2.5 Other partisan games played on graphs 201 10.3 Brouwer’s fixed-point theorem via Hex 206 11 Random-turn and auctioned-turn games 212 11.1 Random-turn games defined 212 11.2 Random-turn selection games 213 11.2.1 Hex 213 Contents v 11.2.2 Bridg-It 214 11.2.3 Surround 215 11.2.4 Full-board Tic-Tac-Toe 215 11.2.5 Recursive majority 215 11.2.6 Team captains 216 11.3 Optimal strategy for random-turn selection games 217 11.4 Win-or-lose selection games 219 11.4.1 Length of play for random-turn Recursive Majority 220 11.5 Richman games 221 11.6 Additional notes on random-turn Hex 224 11.6.1 Odds of winning on large boards under biased play. 224 11.7 Random-turn Bridg-It 225 Bibliography 227 Index 231 1 Introduction In this course on game theory, we will be studying a range of mathematical models of conflict and cooperation between two or more agents. We begin with an outline of the content of this course. We begin with the classic two-person zero-sum games. In such games, both players move simultaneously, and depending on their actions, they each get a certain payoff. What makes these games “zero-sum” is that each player benefits only at the expense of the other. We will show how to find optimal strategies for each player in such games. These strategies will typically turn out to be a randomized choice of the available options. For example, in Penalty Kicks, a soccer/football-inspired zero-sum game, one player, the penalty-taker, chooses to kick the ball either to the left or to the right of the other player, the goal-keeper. At the same instant as the kick, the goal-keeper guesses whether to dive left or right. Fig. 1.1. The game of Penalty Kicks. The goal-keeper has a chance of saving the goal if he dives in the same direction as the kick. The penalty-taker, being left-footed, has a greater likelihood of success if he kicks left. The probabilities that the penalty kick scores are displayed in the table below: 1 2 Introduction goal-keeper L R L 0.8 1 R 1 0.5 penalty- taker For this set of scoring probabilities, the optimal strategy for the penalty- taker is to kick left with probability 5/7 and kick right with probability 2/7 — then regardless of what the goal-keeper does, the probability of scoring is 6/7. Similarly, the optimal strategy for the goal-keeper is to dive left with probability 5/7 and dive right with probability 2/7. In general-sum games, the topic of Chapter 3, we no longer have op- timal strategies. Nevertheless, there is still a notion of a “rational choice” for the players. A Nash equilibrium is a set of strategies, one for each player, with the property that no player can gain by unilaterally changing his strategy. It turns out that every general-sum game has at least one Nash equilibrium. The proof of this fact requires an important geometric tool, the Brouwer fixed-point theorem. One interesting class of general-sum games, important in computer sci- ence, is that of congestion games. In a congestion game, there are two drivers, I and II, who must navigate as quickly as possible through a con- gested network of roads. Driver I must travel from city B to city D, and driver II, from city A to city C. A D (1,2) (3,5) (2,4) (3,4) B C Fig. 1.2. A congestion game. Shown here are the commute times for the four roads connecting four cities. For each road, the first number is the commute time when only one driver uses the road, the second number is the commute time when two drivers use the road. The travel time for using a road is less when the road is less congested. In the ordered pair (t1,t2) attached to each road in the diagram below, t1 represents the travel time when only one driver uses the road, and t2 represents the travel time when the road is shared. For example, if drivers I and II both use road AB, with I traveling from A to B and II from B to A, then each must wait 5 units of time.
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