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A Course in Game Theory -- Solution Manual







Contents
Preface ix
2 Nash Equilibrium 1
Exercise 18.2 (First price auction) 1
Exercise 18.3 (Second price auction) 1
Exercise 18.5 (War of attrition) 2
Exercise 19.1 (Location game) 2
Exercise 20.2 (Necessity of conditions in Kakutani’s theorem) 3
Exercise 20.4 (Symmetric games) 3
Exercise 24.1 (Increasing payoffs in strictly competitive game) 3
Exercise 27.2 (BoS with imperfect information) 4
Exercise 28.1 (Exchange game) 4
Exercise 28.2 (More information may hurt) 4
3 Mixed, Correlated, and Evolutionary Equilibrium 7
Exercise 35.1 (Guess the average) 7
Exercise 35.2 (Investment race) 7
Exercise 36.1 (Guessing right) 8
Exercise 36.2 (Air strike) 8
Exercise 36.3 (Technical result on convex sets) 9
Exercise 42.1 (Examples of Harsanyi’s purification) 9
Exercise 48.1 (Example of correlated equilibrium) 10
Exercise 51.1 (Existence of ESS in 2 × 2 game) 10
4 Rationalizability and Iterated Elimination of Dominated Actions 11
Exercise 56.3 (Example of rationalizable actions) 11
Exercise 56.4 (Cournot duopoly) 11
Exercise 56.5 (Guess the average) 11
Exercise 57.1 (Modified rationalizability in location game) 11
Exercise 63.1 (Iterated elimination in location game) 12
Exercise 63.2 (Dominance solvability) 12
Exercise 64.1 (Announcing numbers) 12
Exercise 64.2 (Non-weakly dominated action as best response) 12
5 Knowledge and Equilibrium 13vi Contents
Exercise 69.1 (Example of information function) 13
Exercise 69.2 (Remembering numbers) 13
Exercise 71.1 (Information functions and knowledge functions) 13
Exercise 71.2 (Decisions and information) 13
Exercise 76.1 (Common knowledge and different beliefs) 13
Exercise 76.2 (Common knowledge and beliefs about lotteries) 14
Exercise 81.1 (Knowledge and correlated equilibrium) 14
6 Extensive Games with Perfect Information 15
Exercise 94.2 (Extensive games with 2 × 2 strategic forms) 15
Exercise 98.1 (SPE of Stackelberg game) 15
Exercise 99.1 (Necessity of finite horizon for one deviation property) 16
Exercise 100.1 (Necessity of finiteness for Kuhn’s theorem) 16
Exercise 100.2 (SPE of games satisfying no indifference condition) 16
Exercise 101.1 (SPE and unreached subgames) 17
Exercise 101.2 (SPE and unchosen actions) 17
Exercise 101.3 (Armies) 17
Exercise 102.1 (ODP and Kuhn’s theorem with chance moves) 17
Exercise 103.1 (Three players sharing pie) 17
Exercise 103.2 (Naming numbers) 18
Exercise 103.3 (ODP and Kuhn’s theorem with simultaneous moves) 18
Exercise 108.1 ( -equilibrium of centipede game) 19
Exercise 114.1 (Variant of the game Burning money) 19
Exercise 114.2 (Variant of the game Burning money) 19
7 A Model of Bargaining 21
Exercise 123.1 (One deviation property for bargaining game) 21
Exercise 125.2 (Constant cost of bargaining) 21
Exercise 127.1 (One-sided offers) 21
Exercise 128.1 (Finite grid of possible offers) 22
Exercise 129.1 (Outside options) 23
Exercise 130.2 (Risk of breakdown) 24
Exercise 131.1 (Three-player bargaining) 24
8 Repeated Games 25
Exercise 139.1 (Discount factors that differ) 25
Exercise 143.1 (Strategies and finite machines) 25
Exercise 144.2 (Machine that guarantees v i ) 25
Exercise 145.1 (Machine for Nash folk theorem) 25
Exercise 146.1 (Example with discounting) 26
Exercise 148.1 (Long- and short-lived players) 26
Exercise 152.1 (Game that is not full dimensional) 26
Exercise 153.2 (One deviation property for discounted repeated game) 26
Exercise 157.1 (Nash folk theorem for finitely repeated games) 27
9 Complexity Considerations in Repeated Games 29
Exercise 169.1 (Unequal numbers of states in machines) 29
Exercise 173.1 (Equilibria of the Prisoner’s Dilemma) 29
Exercise 173.2 (Equilibria with introductory phases) 29Contents vii
Exercise 174.1 (Case in which constituent game is extensive game) 30
10 Implementation Theory 31
Exercise 182.1 (DSE-implementation with strict preferences) 31
Exercise 183.1 (Example of non-DSE implementable rule) 31
Exercise 185.1 (Groves mechanisms) 31
Exercise 191.1 (Implementation with two individuals) 32
11 Extensive Games with Imperfect Information 33
Exercise 203.2 (Definition of X i (h)) 33
Exercise 208.1 (One-player games and principles of equivalence) 33
Exercise 216.1 (Example of mixed and behavioral strategies) 33
Exercise 217.1 (Mixed and behavioral strategies and imperfect recall) 33
Exercise 217.2 (Splitting information sets) 34
Exercise 217.3 (Parlor game) 34
12 Sequential Equilibrium 37
Exercise 226.1 (Example of sequential equilibria) 37
Exercise 227.1 (One deviation property for sequential equilibrium) 37
Exercise 229.1 (Non-ordered information sets) 39
Exercise 234.2 (Sequential equilibrium and PBE) 39
Exercise 237.1 (Bargaining under imperfect information) 39
Exercise 238.1 (PBE is SE in Spence’s model) 40
Exercise 243.1 (PBE of chain-store game) 40
Exercise 246.2 (Pre-trial negotiation) 41
Exercise 252.2 (Trembling hand perfection and coalescing of moves) 41
Exercise 253.1 (Example of trembling hand perfection) 42
13 The Core 45
Exercise 259.3 (Core of production economy) 45
Exercise 260.2 (Market for indivisible good) 45
Exercise 260.4 (Convex games) 45
Exercise 261.1 (Simple games) 45
Exercise 261.2 (Zerosum games) 46
Exercise 261.3 (Pollute the lake) 46
Exercise 263.2 (Game with empty core) 46
Exercise 265.2 (Syndication in a market) 46
Exercise 267.2 (Existence of competitive equilibrium in market) 47
Exercise 268.1 (Core convergence in production economy) 47
Exercise 274.1 (Core and equilibria of exchange economy) 48
14 Stable Sets, the Bargaining Set, and the Shapley Value 49
Exercise 280.1 (Stable sets of simple games) 49
Exercise 280.2 (Stable set of market for indivisible good) 49
Exercise 280.3 (Stable sets of three-player games) 49
Exercise 280.4 (Dummy’s payoff in stable sets) 50
Exercise 280.5 (Generalized stable sets) 50
Exercise 283.1 (Core and bargaining set of market) 50
Exercise 289.1 (Nucleolus of production economy) 51viii Contents
Exercise 289.2 (Nucleolus of weighted majority games) 52
Exercise 294.2 (Necessity of axioms for Shapley value) 52
Exercise 295.1 (Example of core and Shapley value) 52
Exercise 295.2 (Shapley value of production economy) 53
Exercise 295.4 (Shapley value of a model of a parliament) 53
Exercise 295.5 (Shapley value of convex game) 53
Exercise 296.1 (Coalitional bargaining) 53
15 The Nash Bargaining Solution 55
Exercise 309.1 (Standard Nash axiomatization) 55
Exercise 309.2 (Efficiency vs. individual rationality) 55
Exercise 310.1 (Asymmetric Nash solution) 55
Exercise 310.2 (Kalai–Smorodinsky solution) 56
Exercise 312.2 (Exact implementation of Nash solution) 56

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