View source for Minimax principle

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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|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 [[#References|[1]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Vilkas,   "Axiomatic definition of the value of a matrix game"  ''Theory Probabl. Appl.'' , '''8'''  (1963)  pp. 304–307  ''Teor. Veroyatnost. i Primenen.'' , '''8''' :  3  (1963)  pp. 324–327  {{MR|0154750}} {{ZBL|0279.90044}} </TD></TR></table>



====Comments====
The fact that the minimax principle holds if mixed strategies are allowed is called the minimax theorem; it is due to J. von Neumann.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. von Neumann,   O. Morgenstern,   "Theory of games and economic behavior" , Princeton Univ. Press  (1947)  {{MR|21298}} {{ZBL|1112.91002}} {{ZBL|1109.91001}} {{ZBL|0452.90092}} {{ZBL|0205.23401}} {{ZBL|0097.14804}} {{ZBL|0053.09303}} {{ZBL|0063.05930}} </TD></TR>
</table>

[[Category:Game theory, economics, social and behavioral sciences]]