# Minimax principle

An optimality principle for a two-person zero-sum game, expressing the tendency of each player to obtain the largest sure pay-off. The minimax principle holds in such a game $\Gamma=\langle A,B,H\rangle$ if the equality

$$v=\max_{a\in A}\min_{b\in B}H(a,b)=\min_{b\in B}\max_{a\in A}H(a,b)\label{*}\tag{*}$$

holds, that is, if there are a value of the game, equal to $v$, and optimal strategies for both players.

For a matrix game and for certain classes of infinite two-person zero-sum games (see Infinite game) the minimax principle holds if mixed strategies are used. It is known that \eqref{*} is equivalent to the inequalities (see Saddle point in game theory):

$$H(a,b^*)\leq H(a^*,b^*)\leq H(a^*,b)$$

for all $a\in A$, $b\in B$, where $a^*$ and $b^*$ are the strategies on which the external extrema in \eqref{*} are attained. Thus, the minimax principle expresses mathematically the intuitive conception of stability, since it is not profitable for either player to deviate from his optimal strategy $a^*$ (respectively, $b^*$). At the same time the minimax principle guarantees to player I (II) a gain (loss) of not less (not more) than the value of the game. An axiomatic characterization of the minimax principle for matrix games has been given (see ).

How to Cite This Entry:
Minimax principle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minimax_principle&oldid=44728
This article was adapted from an original article by E.B. Yanovskaya (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article