AN INTRODUCTION TO GAME THEORY

Publicly-available solutions for

AN INTRODUCTION TO GAME THEORY

M ARTIN J. O SBORNE

University of Toronto

Copyright © 2005 by Martin J. Osborne All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Martin J. Osborne. This manual was typeset by the author, who is greatly indebted to Donald Knuth A (TEX), Leslie Lamport (L TEX), Diego Puga (mathpazo), Christian Schenk (MiKTEX), Ed Sznyter (ppctr), Timothy van Zandt (PSTricks), and others, for generously making superlative software freely available. The main font is 10pt Palatino.

Version 5: 2005-10-7

Contents

Preface 1

xi

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

1

2

Nash Equilibrium 3 Exercise 16.1 (Working on a joint project) 3 Exercise 17.1 (Games equivalent to the Prisoner’s Dilemma) 3 Exercise 20.1 (Games without conﬂict) 3 Exercise 31.1 (Extension of the Stag Hunt) 4 Exercise 34.1 (Guessing two-thirds of the average) 4 Exercise 34.3 (Choosing a route) 5 Exercise 37.1 (Finding Nash equilibria using best response functions) 6 Exercise 38.1 (Constructing best response functions) 6 Exercise 38.2 (Dividing money) 7 Exercise 41.1 (Strict and nonstrict Nash equilibria) 7 Exercise 47.1 (Strict equilibria and dominated actions) 8 Exercise 47.2 (Nash equilibrium and weakly dominated actions) 8 Exercise 50.1 (Other Nash equilibria of the game modeling collective decision-making) 8 Exercise 51.2 (Symmetric strategic games) 9 Exercise 52.2 (Equilibrium for pairwise interactions in a single population) 9 Nash Equilibrium: Illustrations 11 Exercise 58.1 (Cournot’s duopoly game with linear inverse demand and different unit costs) 11 Exercise 60.2 (Nash equilibrium of Cournot’s duopoly game and the collusive outcome) 12 Exercise 63.1 (Interaction among resource-users) 12 Exercise 67.1 (Bertrand’s duopoly game with constant unit cost) 13 Exercise 68.1 (Bertrand’s oligopoly game) 13 Exercise 68.2 (Bertrand’s duopoly game with different unit costs) 13 Exercise 73.1 (Electoral competition with asymmetric voters’ preferences) 14 Exercise 75.3 (Electoral competition for more general preferences) 14 Exercise 76.1 (Competition in product characteristics) 15 Exercise 79.1 (Direct argument for Nash equilibria of War of Attrition) 15 Exercise 85.1 (Second-price sealed-bid auction with two bidders) 16 v

3

vi

Contents

Exercise 86.2 (Nash equilibrium of ﬁrst-price sealed-bid auction) 17 Exercise 87.1 (First-price sealed-bid auction) 17 Exercise 89.1 (All-pay auctions) 18 Exercise 90.1 (Multiunit auctions) 18 Exercise 90.3 (Internet pricing) 19 Exercise 96.2 (Alternative standards of care under negligence with contributory negligence) 19 4 Mixed Strategy Equilibrium 23 Exercise 101.1 (Variant of Matching Pennies) 23 Exercise 106.2 (Extensions of BoS with vNM preferences) 23 Exercise 110.1 (Expected payoffs) 24 Exercise 111.1 (Examples of best responses) 24 Exercise 114.1 (Mixed strategy equilibrium of Hawk–Dove) 25 Exercise 117.2 (Choosing numbers) 26 Exercise 120.2 (Strictly dominating mixed strategies) 26 Exercise 120.3 (Strict domination for mixed strategies) 26 Exercise 127.1 (Equilibrium in the expert diagnosis game) 27 Exercise 130.3 (Bargaining) 27 Exercise 132.2 (Reporting a crime when the witnesses are heterogeneous) 28 Exercise 136.1 (Best response dynamics in Cournot’s duopoly game) 29 Exercise 139.1 (Finding all mixed strategy equilibria of two-player games) 29 Exercise 145.1 (All-pay auction with many bidders) 30 Exercise 147.2 (Preferences over lotteries) 31 Exercise 149.2 (Normalized vNM payoff functions) 31 Extensive Games with Perfect Information: Theory 33 Exercise 163.1 (Nash equilibria of extensive games) 33...