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**Unformatted text preview: **Publicly-available solutions for AN INTRODUCTION TO G AME T HEORY Publicly-available solutions for AN INTRODUCTION TO G AME T HEORY M ARTIN J. O SBORNE University of Toronto Copyright © 2004 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 (T E X), Leslie Lamport (L A T E X), Diego Puga (mathpazo), Christian Schenk (MiKT E X), Ed Sznyter (ppctr), Timothy van Zandt (PSTricks), and others, for generously making superlative software freely available. The main font is 10pt Palatino. Version 2: 2004-4-27 Contents Preface xi 1 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 Prisoners 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 3 Nash Equilibrium: Illustrations 11 Exercise 58.1 ( Cournots duopoly game with linear inverse demand and different unit costs ) 11 Exercise 60.2 ( Nash equilibrium of Cournots duopoly game and the collusive outcome ) 12 Exercise 63.1 ( Interaction among resource-users ) 12 Exercise 67.1 ( Bertrands duopoly game with constant unit cost ) 13 Exercise 68.1 ( Bertrands oligopoly game ) 13 Exercise 68.2 ( Bertrands 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 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...