# Nash Essay

19934 Words Jan 8th, 2015 80 Pages
Draft chapter from An introduction to game theory by Martin J. Osborne. Version: 2002/7/23. Martin.Osborne@utoronto.ca http://www.economics.utoronto.ca/osborne Copyright © 1995–2002 by Martin J. Osborne. All rights reserved. No part of this book may be reproduced by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from Oxford University Press, except that one copy of up to six chapters may be made by any individual for private study.

2

Nash Equilibrium: Theory

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10

Strategic games 11 Example: the Prisoner’s Dilemma 12 Example: Bach or Stravinsky? 16 Example: Matching Pennies 17 Example: the Stag Hunt 18
For example, the players may be ﬁrms, the actions prices, and the preferences a reﬂection of the ﬁrms’ proﬁts. Or the players may be candidates for political ofﬁce, the actions campaign expenditures, and the preferences a reﬂection of the candidates’ probabilities of winning. Or the players may be animals ﬁghting over some prey, the actions concession times, and the preferences a reﬂection of whether an animal wins or loses. In this chapter I describe some simple games designed to capture fundamental conﬂicts present in a variety of situations. The next chapter is devoted to more detailed applications to speciﬁc phenomena. As in the model of rational choice by a single decision-maker (Section 1.2), it is frequently convenient to specify the players’ preferences by giving payoff functions that represent them. Bear in mind that these payoffs have only ordinal signiﬁcance. If a player’s payoffs to the action proﬁles a, b, and c are 1, 2, and 10, for example, the only conclusion we can draw is that the player prefers c to b and b to a; the numbers do not imply that the player’s preference between c and b is stronger than her preference between a and b. Time is absent from the model. The idea is that each player chooses her action once and for all, and the players choose their actions “simultaneously” in the sense that no player is informed, when she chooses her action, of the action chosen by any other player. (For this reason, a

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