# A History of Game Theory - Vol. 1

By Mary Ann Dimand; Robert W. Dimand | Go to book overview

7

THE MINIMAX APPROACH TO NON-COOPERATIVE STRATEGIC GAMES FROM WALDEGRAVE TO BOREL

As previous chapters have shown, writers before von Neumann and Morgenstern’s epochal 1944 publication did a good deal of work on the analysis of strategic interdependence—the stuff of game theory. While economists such as Stackelberg (see Chapter 3) gave considerable thought to the nature of equilibrium in such situations, relatively little formal work on the definition and existence of equilibrium for games in general was written before 1944, among the most important of which was von Neumann (1928a, b). Most writers employed a notion of equilibrium ‘natural’ to the situation modelled. Strategic games, whose outcome depends on the skill of the participants in choosing a strategy of play, received widespread attention among mathematicians and economists only with the publication in 1944 of the first edition of von Neumann and Morgenstern’s Theory of Games and Economic Behavior. The book marked an important advance, but it built upon an existing literature on strategic games, to which both its authors had contributed.

This chapter examines the pre-1944 literature on formalizing equilibrium conditions and on the minimax theorem, which holds that two-person zero-sum games with finitely many pure strategies (or a continuum of pure strategies and continuous convex payoffs) have solutions (equilibria) under maximin and minimax strategies which are identical. An agent A whose choices are governed by a maximin criterion looks at each strategy she might follow and, in each case, considers the lowest payoff she can receive by following this strategy. She then chooses to play the strategy whose minimum payoff is the highest. This is an extremely conservative and pessimistic approach: it assumes that B’s ability to deliver to A her lowest payoff possible, given her choice of strategy, is the paramount element in As choice of strategy. Player A ensures her minimal payoff by taking this approach. A player C taking the minimax approach, on the other hand, looks at the payoffs her opponent D can achieve given each strategy of C.C then chooses to play the strategy which will give D the lowest payoff, if D would always play so as to maximize his payoff subject to C’s strategy. While the maximin approach presumes a player who wishes to guarantee her own minimum payoff, minimax conjectures a player who wants to guarantee her opponents

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A History of Game Theory - Vol. 1

• Title Page iii
• Contents vii
• Preface ix
• 1 - Introduction 1
• 2 - Strategic Interdependence 18
• 3 - Strategic Interdependence 35
• 4 - Strategic Interdependence 54
• 5 - Lewis Carroll and the Game of Politics 84
• 6 - Early Mathematical Models of Conflict 104
• 7 - The Minimax Approach to Non-Cooperative Strategic Games from Waldegrave to Borel 120
• 8 - From Games of Pure Chance to Strategic Games 131
• 9 142
• Notes 158
• Bibliography 170
• Index 184
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