Some Further Thoughts on the Minimax Principle

In a recent paper Aumann and Maschler [Aumann, R. J., M. Maschler. 1972. Some thoughts on the minimax principle. Management Sci.18 (5, Part 2, January) P-54.] (we will subsequently refer to them as A-M) raised some questions about certain matters that had long been considered settled. In their words,

“It is generally agreed that the minimax solution to a two-person zero-sum matrix game is intuitively satisfactory. Now in many applications of game theory a game is not described in matrix (or ‘normal’ or ‘strategic’) form, but rather in extensive form, i.e., by its rules. A game described in such a way may be reduced to a matrix game by means of the concept of ‘strategy.’ If, moreover, it is of perfect recall, then all mixed strategies, and in particular the optimal strategies of each player, are equivalent to behavior strategies The usual conclusion from these considerations is that for 2-person 0-sum games in extensive form, the minimax solution is intuitively satisfactory, and that in games of perfect recall, in particular, the players would do well to play in accordance with optimal (minimax) behavior strategies. In this paper we shall discuss some examples that, we believe, cast doubt on these conclusions.”