Solutions Manual for An Introduction to Game Theory 2023 Edition by Martin J. Osborne

Solution manual for

AN INTRODUCTION TO GAME THEORY

M ARTIN J. O SBORNE University of Toronto

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Contents

Preface 1

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Introduction 1 Exercise 5.3 (Altruistic preferences) 1 Exercise 6.1 (Alternative representations of preferences)

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Nash Equilibrium 3 Exercise 16.1 (Working on a joint project) 3 Exercise 17.1 (Games equivalent to the Prisoner’s Dilemma) 3 Exercise 18.1 (Hermaphroditic fish) 3 Exercise 20.1 (Games without conflict) 4 Exercise 27.1 (Variant of Prisoner’s Dilemma with altruistic preferences) 4 Exercise 27.2 (Selfish and altruistic social behavior) 5 Exercise 30.1 (Variants of the Stag Hunt) 5 Exercise 31.1 (Extension of the Stag Hunt) 6 Exercise 31.2 (Hawk–Dove) 6 Exercise 33.1 (Contributing to a public good) 7 Exercise 34.1 (Guessing two-thirds of the average) 7 Exercise 34.2 (Voter participation) 8 Exercise 34.3 (Choosing a route) 9 Exercise 37.1 (Finding Nash equilibria using best response functions) 10 Exercise 38.1 (Constructing best response functions) 11 Exercise 38.2 (Dividing money) 11 Exercise 41.1 (Strict and nonstrict Nash equilibria) 12 Exercise 41.2 (Finding Nash equilibria using best response functions) 13 Exercise 42.1 (A joint project) 13 Exercise 44.1 (Contributing to a public good) 15 Exercise 47.1 (Strict equilibria and dominated actions) 16 Exercise 47.2 (Nash equilibrium and weakly dominated actions) 17 Exercise 48.1 (Voting) 17 Exercise 49.1 (Voting between three candidates) 17 Exercise 49.2 (Approval voting) 18 Exercise 50.1 (Other Nash equilibria of the game modeling collective decision-making) 19 Exercise 50.2 (Another mechanism for collective decision-making) 20 Exercise 51.2 (Symmetric strategic games) 20 Exercise 52.2 (Equilibrium for pairwise interactions in a single population) 20 v

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Contents

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Nash Equilibrium: Illustrations 21 Exercise 58.1 (Cournot’s duopoly game with linear inverse demand and different unit costs) 21 Exercise 59.1 (Cournot’s duopoly game with linear inverse demand and a quadratic cost function) 22 Exercise 59.2 (Cournot’s duopoly game with linear inverse demand and a fixed cost) 22 Exercise 60.1 (Variant of Cournot’s duopoly game with market-share maximizing firms) 25 Exercise 60.2 (Nash equilibrium of Cournot’s duopoly game and the collusive outcome) 25 Exercise 61.1 (Cournot’s game with many firms) 26 Exercise 62.1 (Nash equilibrium of Cournot’s game with small firms) 27 Exercise 63.1 (Interaction among resource-users) 27 Exercise 67.1 (Bertrand’s duopoly game with constant unit cost) 28 Exercise 67.2 (Bertrand’s duopoly game with discrete prices) 28 Exercise 68.1 (Bertrand’s oligopoly game) 29 Exercise 68.2 (Bertrand’s duopoly game with different unit costs) 29 Exercise 69.1 (Bertrand’s duopoly game with fixed costs) 30 Exercise 74.1 (Electoral competition with asymmetric voters’ preferences) 31 Exercise 74.2 (Electoral competition with three candidates) 31 Exercise 75.1 (U.S. presidential election) 32 Exercise 75.2 (Electoral competition between candidates who care only about the winning position) 32 Exercise 76.1 (Citizen-candidates) 33 Exercise 76.2 (Electoral competition for more general preferences) 34 Exercise 77.1 (Competition in product characteristics) 34 Exercise 80.1 (Direct argument for Nash equilibria of War of Attrition) 35 Exercise 80.2 (Variant of War of Attrition) 35 Exercise 81.1 (Timing product release) 36 Exercise 81.2 (A fight) 38 Exercise 85.1 (Nash equilibrium of second-price sealed-bid auction) 38 Exercise 86.1 (Second-price sealed-bid auction with two bidders) 39 Exercise 87.1 (Auctioning the right to choose) 40 Exercise 87.2 (Nash equilibrium of first-price sealed-bid auction) 41 Exercise 88.1 (First-price sealed-bid auction) 41 Exercise 89.1 (Third-price auction) 41 Exercise 90.1 (All-pay auctions) 42 Exercise 91.1 (Multiunit auctions) 43 Exercise 91.2 (Waiting in line) 43 Exercise 91.3 (Internet pricing) 44 Exercise 92.1 (Lobbying as an auction) 45 Exercise 97.2 (Alternative standards of care under negligence with contributory negligence) 45

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Exercise 97.3 (Equilibrium under strict liability) 47 4

Mixed Strategy Equilibrium 49 Exercise 101.1 (Variant of Matching Pennies) 49 Exercise 106.2 (Extensions of BoS with vNM preferences) 49 Exercise 110.1 (Expected payoffs) 50 Exercise 111.1 (Examples of best responses) 50 Exercise 114.1 (Mixed strategy equilibrium of Hawk–Dove) 51 Exercise 114.2 (Games with mixed strategy equilibria) 52 Exercise 114.3 (A coordination game) 52 Exercise 114.4 (Swimming with sharks) 52 Exercise 117.2 (Choosing numbers) 54 Exercise 118.1 (Silverman’s game) 55 Exercise 118.2 (Voter participation) 56 Exercise 118.3 (Defending territory) 57 Exercise 120.2 (Strictly dominating mixed strategies) 58 Exercise 120.3 (Strict domination for mixed strategies) 59 Exercise 121.2 (Eliminating dominated actions when finding equilibria) 59 Exercise 127.1 (Equilibrium in the expert diagnosis game) 60 Exercise 127.2 (Incompetent experts) 60 Exercise 128.1 (Choosing a seller) 61 Exercise 130.2 (Approaching cars) 63 Exercise 130.3 (Bargaining) 64 Exercise 132.2 (Reporting a crime when the witnesses are heterogeneous) 64 Exercise 132.3 (Contributing to a public good) 65 Exercise 136.1 (Best response dynamics in Cournot’s duopoly game) 66 Exercise 136.2 (Best response dynamics in Bertrand’s duopoly game) 66 Exercise 139.1 (Finding all mixed strategy equilibria of two-player games) 67 Exercise 141.1 (Finding all mixed strategy equilibria of a two-player game) 68 Exercise 141.2 (Rock, paper, scissors) 69 Exercise 141.3 (Election campaigns) 70 Exercise 142.1 (A three-player game) 72 Exercise 146.1 (All-pay auction with many bidders) 73 Exercise 146.2 (Bertrand’s duopoly game) 73 Exercise 147.2 (Preferences over lotteries) 74 Exercise 149.2 (Normalized vNM payoff functions) 74 Exercise 150.1 (Games equivalent to the Prisoner’s Dilemma) 74

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Extensive Games with Perfect Information: Theory 77 Exercise 156.2 (Examples of extensive games with perfect information) Exercise 161.1 (Strategies in extensive games) 78 Exercise 163.1 (Nash equilibria of extensive games) 78 Exercise 163.2 (Voting by alternating veto) 79 Exercise 164.2 (Subgames) 79

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Contents

Exercise 168.1 (Checking for subgame perfect equilibria) 80 Exercise 173.2 (Finding subgame perfect equilibria) 80 Exercise 173.3 (Voting by alternating veto) 80 Exercise 173.4 (Burning a bridge) 81 Exercise 174.1 (Sharing heterogeneous objects) 81 Exercise 174.2 (An entry game with a financially-constrained firm) 82 Exercise 176.1 (Dollar auction) 83 Exercise 177.1 (Firm–union bargaining) 84 Exercise 177.2 (The “rotten kid theorem”) 85 Exercise 177.3 (Comparing simultaneous and sequential games) 86 Exercise 179.1 (Subgame perfect equilibria of ticktacktoe) 87 Exercise 179.2 (Toetacktick) 88 Exercise 179.3 (Three Men’s Morris, or Mill) 88 6

Extensive Games with Perfect Information: Illustrations 89 Exercise 183.1 (Nash equilibria of the ultimatum game) 89 Exercise 183.2 (Subgame perfect equilibria of the ultimatum game with indivisible units) 89 Exercise 183.3 (Dictator game and impunity game) 89 Exercise 183.4 (Variants of ultimatum game and impunity game with equity-conscious players) 90 Exercise 185.1 (Bargaining over two indivisible objects) 93 Exercise 185.2 (Dividing a cake fairly) 94 Exercise 186.1 (Holdup game) 95 Exercise 187.1 (Agenda control) 95 Exercise 189.1 (Stackelberg’s duopoly game with quadratic costs) 96 Exercise 191.1 (Stackelberg’s duopoly game with fixed costs) 97 Exercise 192.1 (Sequential variant of Bertrand’s duopoly game) 97 Exercise 196.1 (Three interest groups buying votes) 98 Exercise 196.2 (Interest groups buying votes under supermajority rule) 99 Exercise 196.3 (Sequential positioning by two political candidates) 99 Exercise 196.4 (Sequential positioning by three political candidates) 100 Exercise 198.1 (The race G1 (2, 2)) 102 Exercise 201.1 (A race in which the players’ valuations of the prize differ) 102 Exercise 201.2 (Removing stones) 103 Exercise 202.1 (Hungry lions) 104 Exercise 203.1 (A race with a liquidity constraint) 105

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Extensive Games with Perfect Information: Extensions and Discussion Exercise 210.2 (Extensive game with simultaneous moves) 107 Exercise 210.3 (Two-period Prisoner’s Dilemma) 107 Exercise 211.1 (Timing claims on an investment) 108 Exercise 211.2 (A market game) 109 Exercise 212.1 (Price competition) 110

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Exercise 214.1 (Bertrand’s duopoly game with entry) 113 Exercise 216.1 (Electoral competition with strategic voters) 114 Exercise 217.1 (Electoral competition with strategic voters) 115 Exercise 220.1 (Top cycle set) 116 Exercise 221.1 (Designing agendas) 116 Exercise 221.2 (An agenda that yields an undesirable outcome) 116 Exercise 224.1 (Exit from a declining industry) 117 Exercise 225.1 (Effect of borrowing constraint in declining industry) 117 Exercise 227.1 (Variant of ultimatum game with equity-conscious players) 118 Exercise 227.2 (Firm–union bargaining) 119 Exercise 227.3 (Sequential duel) 120 Exercise 227.4 (Sequential truel) 121 Exercise 228.1 (Cohesion in legislatures) 123 Exercise 230.1 (Nash equilibria when players may make mistakes) 127 Exercise 233.1 (Nash equilibria of the chain-store game) 127 Exercise 233.2 (Subgame perfect equilibrium of the chain-store game) 127 Exercise 234.1 (Nash equilibria of the centipede game) 128 8

Coalitional Games and the Core 129 Exercise 241.2 (Stag Hunt) 129 Exercise 243.1 (Cohesive games) 129 Exercise 245.1 (Three-player majority game) 130 Exercise 245.2 (Variant of three-player majority game) 130 Exercise 245.3 (Stag Hunt) 130 Exercise 245.4 (Variant of Stag Hunt) 131 Exercise 246.1 (Market with one owner and two heterogeneous buyers) 131 Exercise 247.1 (Vote trading) 131 Exercise 248.1 (Core of landowner–worker game) 133 Exercise 249.1 (Unionized workers in landowner–worker game) 133 Exercise 249.2 (Landowner–worker game with increasing marginal products) 134 Exercise 254.1 (Range of prices in horse market) 134 Exercise 256.1 (Horse trading game with single seller) 134 Exercise 256.2 (Horse trading game with large seller) 134 Exercise 258.1 (House assignment with identical preferences) 135 Exercise 260.1 (Emptiness of the strong core when preferences are not strict) 135 Exercise 261.1 (Median voter theorem) 135 Exercise 262.1 (Cores of q-rule games) 135 Exercise 265.1 (Deferred acceptance procedure with proposals by Y’s) 136 Exercise 266.1 (Example of deferred acceptance procedure) 136 Exercise 267.1 (Strategic behavior under the deferred acceptance procedure) 136 Exercise 267.2 (Empty core in roommate problem) 137 Exercise 267.3 (Spatial preferences in roommate problem) 137

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Contents

Bayesian Games 139 Exercise 276.1 (Equilibria of a variant of BoS with imperfect information) 139 Exercise 277.1 (Expected payoffs in a variant of BoS with imperfect information) 139 Exercise 282.1 (Fighting an opponent of unknown strength) 140 Exercise 282.2 (An exchange game) 141 Exercise 282.3 (Adverse selection) 141 Exercise 284.1 (Infection argument) 142 Exercise 287.1 (Cournot’s duopoly game with imperfect information) 143 Exercise 288.1 (Cournot’s duopoly game with imperfect information) 143 Exercise 290.1 (Nash equilibria of game of contributing to a public good) 145 Exercise 291.1 (Reporting a crime with an unknown number of witnesses) 146 Exercise 294.1 (Weak domination in second-price sealed-bid action) 147 Exercise 294.2 (Nash equilibria of a second-price sealed-bid auction) 148 Exercise 296.1 (Auctions with risk-averse bidders) 148 Exercise 299.1 (Asymmetric Nash equilibria of second-price sealed-bid common value auctions) 149 Exercise 299.2 (First-price sealed-bid auction with common valuations) 150 Exercise 306.1 (Signal-independent equilibria in a model of a jury) 150 Exercise 307.1 (Swing voter’s curse) 151 Exercise 309.2 (Property of the bidding function in a first-price sealed-bid auction) 152 Exercise 309.3 (Example of Nash equilibrium in a first-price auction) 153 Exercise 310.2 (Reserve prices in second-price sealed-bid auction) 153

10 Extensive Games with Imperfect Information 155 Exercise 316.1 (Variant of card game) 155 Exercise 318.2 (Strategies in variants of card game and entry game) 155 Exercise 319.3 (Nash equilibrium of card game) 156 Exercise 320.1 (Nash equilibria of variant of card game) 156 Exercise 331.1 (Selten’s horse) 157 Exercise 331.2 (Weak sequential equilibrium and Nash equilibrium in subgames) 158 Exercise 335.1 (Pooling and separating equilibria in a signaling game) 158 Exercise 335.2 (Sir Philip Sydney game) 160 Exercise 340.1 (Pooling equilibria of game in which expenditure signals quality) 161 Exercise 342.1 (Pooling equilibria of game in which education signals ability) 162 Exercise 346.1 (Comparing the receiver’s expected payoff in two equilibria) 162 Exercise 350.1 (Variant of model with piecewise linear payoff functions) 162 Exercise 350.2 (Pooling equilibrium in a general model) 163 11 Strictly Competitive Games and Maxminimization 165 Exercise 363.1 (Maxminimizers in a bargaining game) 165

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Exercise 363.3 (Finding a maxminimizer) 165 Exercise 364.2 (Nash equilibrium payoffs and maxminimized payoffs) 166 Exercise 365.1 (Nash equilibrium payoffs and maxminimized payoffs) 166 Exercise 366.2 (Determining strictly competitiveness) 166 Exercise 369.2 (Equilibrium payoffs in symmetric game) 167 Exercise 370.2 (Maxminimizing in BoS) 167 Exercise 372.1 (Increasing payoffs and eliminating actions) 167 Exercise 372.2 (Equilibrium in strictly competitive game) 168 Exercise 372.3 (Morra) 168 Exercise 372.4 (O’Neill’s game) 169 12 Rationalizability 171 Exercise 379.2 (Best responses to beliefs) 171 Exercise 384.1 (Mixed strategy equilibria of game in Figure 384.1) 171 Exercise 387.2 (Finding rationalizable actions) 172 Exercise 387.3 (Morra) 172 Exercise 387.4 (Guessing two-thirds of the average) 172 Exercise 387.5 (Hotelling’s model of electoral competition) 172 Exercise 388.1 (Contributing to a public good) 173 Exercise 388.2 (Cournot’s duopoly game) 174 Exercise 391.1 (Example of dominance-solvable game) 174 Exercise 391.2 (Dividing money) 175 Exercise 391.3 (Voting) 175 Exercise 392.1 (Bertrand’s duopoly game) 175 Exercise 392.2 (Strictly competitive extensive games with perfect information)

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13 Evolutionary Equilibrium 177 Exercise 400.1 (Evolutionary stability and weak domination) 177 Exercise 400.2 (Example of evolutionarily stable actions) 177 Exercise 402.1 (Mixed strategy ESSs) 177 Exercise 405.1 (Hawk–Dove–Retaliator) 178 Exercise 405.2 (Variant of BoS) 178 Exercise 405.3 (Bargaining) 179 Exercise 408.1 (Equilibria of C and of G) 179 Exercise 409.2 (Variant of BoS) 180 Exercise 414.1 (A coordination game between siblings) 180 Exercise 414.2 (Assortative mating) 180 Exercise 416.1 (Darwin’s theory of the sex ratio) 181 14 Repeated Games: The Prisoner’s Dilemma 183 Exercise 423.1 (Equivalence of payoff functions) 183 Exercise 425.1 (Subgame perfect equilibrium of finitely repeated Prisoner’s Dilemma) 183 Exercise 428.1 (Strategies in an infinitely repeated Prisoner’s Dilemma) 184

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Exercise 429.1 (Grim trigger strategies in a general Prisoner’s Dilemma) 184 Exercise 430.1 (Limited punishment strategies in an infinitely repeated Prisoner’s Dilemma) 184 Exercise 431.1 (Tit-for-tat in an infinitely repeated Prisoner’s Dilemma) 185 Exercise 431.2 (Nash equilibria of an infinitely repeated Prisoner’s Dilemma) 185 Exercise 433.1 (Feasible payoff pairs in a Prisoner’s Dilemma) 186 Exercise 439.1 (Finitely repeated Prisoner’s Dilemma with switching cost) 186 Exercise 442.1 (Deviations from grim trigger strategy) 188 Exercise 443.1 (Delayed modified grim trigger strategies) 189 Exercise 443.2 (Different punishment lengths in subgame perfect equilibrium) 189 Exercise 445.1 (Tit-for-tat as a subgame perfect equilibrium) 190 15 Repeated Games: General Results 193 Exercise 452.3 (Minmax payoffs of some games) 193 Exercise 454.2 (Examples of application of Nash folk theorem) Exercise 454.3 (Repeated Bertrand duopoly) 194 Exercise 459.1 (Costly price changing) 195 Exercise 459.2 (Detection lags) 195 Exercise 459.3 (Alternating moves) 196

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16 Bargaining 197 Exercise 468.1 (Two-period bargaining with constant cost of delay) 197 Exercise 468.2 (Three-period bargaining with constant cost of delay) 197 Exercise 473.1 (One-sided offers) 198 Exercise 473.2 (Alternating offer bargaining with constant cost of delay) 198 Exercise 479.1 (One seller–two buyer game with random matching) 200 Exercise 479.2 (One seller–two buyer game with choice of partner) 200 Exercise 480.1 (One seller–two buyer game with public price announcements) 202 Exercise 486.1 (Implications of PAR, SYM, and I IA) 202 Exercise 488.1 (Bargaining solutions) 203 Exercise 488.2 (Wage bargaining) 203 17 Appendix: Mathematics 205 Exercise 497.1 (Maximizer of quadratic function) Exercise 499.3 (Sums of sequences) 205 Exercise 504.2 (Bayes’ law) 205 References

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Preface

This manual contains the solutions to all the exercises in my book An Introduction to Game Theory (Oxford University Press, 2004). The sources of the problems are given in the section entitled “Notes” at the end of each chapter of the book. Please alert me to errors.

I am grateful to the following people for pointing out errors in earlier versions of this manual: Akihito Asano, Joel Blit, Buddy Chang, Jingyu Chen, Douglas Dion, Xiaohua Fang, Satoshi Fukuda, Martin van Hees, Haifeng Huang, Jochen Konemann, Rebecca Morton, Carolyn Pitchik, William Sandholm, Ahmer Tarar, Levent Ulku, Georg Vanberg, Mingxiao Ye, and Zhanna Zhanabekova.

M ARTIN J. O SBORNE martin.j.osborne@gmail.com

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Introduction

5.3 Altruistic preferences Person 1 is indifferent between (1, 4) and (3, 0), and prefers both of these to (2, 1).

The payoff function u defined by u( x, y) = x + 12 y, where x is person 1’s income and y is person 2’s, represents person 1’s preferences. Any function that is an increasing function of u also represents her preferences. For example, the functions k ( x + 12 y) for any positive number k, and ( x + 12 y)2 , do so.

6.1 Alternative representations of preferences The function v represents the same preferences as does u (because u( a) < u(b) < u(c) and v( a) < v(b) < v(c)), but the function w does not represent the same preferences, because w( a) = w(b) while u( a) < u(b).

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Nash Equilibrium

16.1 Working on a joint project The game in Figure 3.2 models this situation (as does any other game with the same players and actions in which the ordering of the payoffs is the same as the ordering in Figure 3.2).

Work hard Goof off

Work hard 3, 3 2, 0

Goof off 0, 2 1, 1

Figure 3.2 Working on a joint project (alternative version).

17.1 Games equivalent to the Prisoner’s Dilemma The game in the left panel differs from the Prisoner’s Dilemma in both players’ preferences. Player 1 prefers (Y, X ) to ( X, X ) to ( X, Y ) to (Y, Y ), for example, which differs from her preference in the Prisoner’s Dilemma, which is ( F, Q) to ( Q, Q) to ( F, F ) to ( Q, F ), whether we let X = F or X = Q. The game in the right panel is equivalent to the Prisoner’s Dilemma. If we let X = Q and Y = F then player 1 prefers ( F, Q) to ( Q, Q) to ( F, F ) to ( Q, F ) and player 2 prefers ( Q, F ) to ( Q, Q) to ( F, F ) to ( F, Q), as in the Prisoner’s Dilemma.

18.1 Hermaphroditic fish A strategic game that models the situation is shown in Figure 3.3.

Either role Preferred role

1 1 2 ( H + L ), 2 ( H + L )

Either role

Preferred role L, H

H, L

S, S

Figure 3.3 A model of encounters between pairs of hermaphroditic fish whose preferred roles differ.

In order for this game to differ from the Prisoner’s Dilemma only in the names of the players’ actions, there must be a way to associate each action with an action in the Prisoner’s Dilemma so that each player’s preferences over the four outcomes are the same as they are in the Prisoner’s Dilemma. Thus we need L < S < 12 ( H + L).

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Chapter 2. Nash Equilibrium

That is, the probability of a fish’s encountering a potential partner must be large enough that S > L, but small enough that S < 12 ( H + L).

20.1 Games without conflict Any two-player game in which each player has two actions and the players have the same preferences may be represented by a table of the form given in Figure 4.1, where a, b, c, and d are any numbers.

T B

L a, a c, c

R b, b d, d

Figure 4.1 A strategic game in which conflict is absent.

27.1 Variant of Prisoner’s Dilemma with altruistic preferences a. A game that model the situation is given in Figure 4.2.

Quiet Fink

Quiet 4, 4 3, 3

Fink 3, 3 2, 2

Figure 4.2 The payoffs in a variant of the Prisoner’s Dilemma in which the players are altruistic.

This game is not the Prisoner’s Dilemma because one (in fact both) of the players’ preferences are not the same as they are in the Prisoner’s Dilemma. Specifically, player 1 prefers (Quiet, Quiet) to (Fink, Quiet), while in the Prisoner’s Dilemma she prefers (Fink, Quiet) to (Quiet, Quiet). (Alternatively, you may note that player 1 prefers (Quiet, Fink) to (Fink, Fink), while in the Prisoner’s Dilemma she prefers (Fink, Fink) to (Quiet, Fink), or that player 2’s preferences are similarly not the same as they are in the Prisoner’s Dilemma.) b. For an arbitrary value of α the payoffs are given in Figure 5.1. In order that the game be the Prisoner’s Dilemma we need 3 > 2(1 + α) (each player prefers Fink to Quiet when the other player chooses Quiet), 1 + α > 3α (each player prefers Fink to Quiet when the other player choose Fink), and 2(1 + α) > 1 + α (each player prefers (Quiet, Quiet) to (Fink, Fink)). The last condition is satisfied for all nonnegative values of α. The first two conditions are both equivalent to α < 12 . Thus the game is the Prisoner’s Dilemma if and only if α < 12 .

If α = 12 then all four outcomes (Quiet, Quiet), (Quiet, Fink), (Fink, Quiet), and (Fink, Fink) are Nash equilibria; if α > 12 then only (Quiet, Quiet) is a Nash equilibrium.

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