Teoria De Los Juegos

Solution Manual for

A Course in Game Theory

Solution Manual for

A Course in Game Theory

by Martin J. Osborne and Ariel Rubinstein

Martin J. Osborne
Ariel Rubinstein
with the assistance of Wulong Gu

The MIT Press
Cambridge, Massachusetts
London, England

This manual was typeset by the authors, who are greatly indebted to Donald Knuth (the
A
creator of TEX), LeslieLamport (the creator of L TEX), and Eberhard Mattes (the creator
of emTEX) for generously putting superlative software in the public domain, and to Ed
Sznyter for providing critical help with the macros we use to execute our numbering
scheme.

Version 1.1, 97/4/25

Contents

Preface xi
2 Nash Equilibrium 1

Exercise 18.2 (First price auction ) 1
Exercise 18.3 (Second price auction ) 2Exercise 18.5 (War of attrition ) 2
Exercise 19.1 (Location game ) 2
Exercise 20.2 (Necessity of conditions in Kakutani's theorem ) 4
Exercise 20.4 (Symmetric games ) 4
Exercise 24.1 (Increasing payo s in strictly competitive game ) 4
Exercise 27.2 (BoS with imperfect information ) 5
Exercise 28.1 (Exchange game ) 5
Exercise 28.2 (More information may hurt ) 6

3 Mixed, Correlated, andEvolutionary Equilibrium 7
Exercise 35.1 (Guess the average ) 7
Exercise 35.2 (Investment race ) 7
Exercise 36.1 (Guessing right ) 9
Exercise 36.2 (Air strike ) 9
Exercise 36.3 (Technical result on convex sets ) 10
Exercise 42.1 (Examples of Harsanyi's puri cation ) 10
Exercise 48.1 (Example of correlated equilibrium ) 11
Exercise 51.1 (Existence of ESS in 2 2 game ) 12

4 Rationalizabilityand Iterated Elimination of Dominated
Actions 13
Exercise 56.3 (Example of rationalizable actions ) 13
Exercise 56.4 (Cournot duopoly ) 13

vi

Contents

Exercise 56.5 (Guess the average ) 13
Exercise 57.1 (Modi ed rationalizability in location game ) 14
Exercise 63.1 (Iterated elimination in location game ) 14
Exercise 63.2 (Dominance solvability ) 14
Exercise 64.1 (Announcingnumbers ) 15
Exercise 64.2 (Non-weakly dominated action as best response ) 15

5 Knowledge and Equilibrium 17

Exercise 69.1 (Example of information function ) 17
Exercise 69.2 (Remembering numbers ) 17
Exercise 71.1 (Information functions and knowledge functions ) 17
Exercise 71.2 (Decisions and information ) 17
Exercise 76.1 (Common knowledge and di erent beliefs ) 18
Exercise 76.2 (Commonknowledge and beliefs about lotteries ) 18
Exercise 81.1 (Knowledge and correlated equilibrium ) 19

6 Extensive Games with Perfect Information 21

Exercise 94.2 (Extensive games with 2 2 strategic forms ) 21
Exercise 98.1 (SPE of Stackelberg game ) 21
Exercise 99.1 (Necessity of nite horizon for one deviation property ) 21
Exercise 100.1 (Necessity of niteness for Kuhn's theorem ) 22Exercise 100.2 (SPE of games satisfying no indi erence condition ) 22
Exercise 101.1 (SPE and unreached subgames ) 23
Exercise 101.2 (SPE and unchosen actions ) 23
Exercise 101.3 (Armies ) 23
Exercise 102.1 (ODP and Kuhn's theorem with chance moves ) 24
Exercise 103.1 (Three players sharing pie ) 24
Exercise 103.2 (Naming numbers ) 25
Exercise 103.3 (ODP and Kuhn's theorem with simultaneousmoves ) 25
Exercise 108.1 ( -equilibrium of centipede game ) 26
Exercise 114.1 (Variant of the game Burning money) 26
Exercise 114.2 (Variant of the game Burning money) 27

7 A Model of Bargaining 29

Exercise 123.1 (One deviation property for bargaining game ) 29
Exercise 125.2 (Constant cost of bargaining ) 29
Exercise 127.1 (One-sided o ers ) 30
Exercise 128.1 (Finite grid of possibleo ers ) 30
Exercise 129.1 (Outside options ) 32

Contents

vii

Exercise 130.2 (Risk of breakdown ) 33
Exercise 131.1 (Three-player bargaining ) 33

8 Repeated Games 35

Exercise 139.1 (Discount factors that di er ) 35
Exercise 143.1 (Strategies and nite machines ) 35
Exercise 144.2 (Machine that guarantees vi) 35
Exercise 145.1 (Machine for Nash folk theorem ) 36
Exercise 146.1...