# 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 if the equality

(*) |

holds, that is, if there are a value of the game, equal to , 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 (*) is equivalent to the inequalities (see Saddle point in game theory):

for all , , where and are the strategies on which the external extrema in (*) 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 (respectively, ). 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 [1]).

#### References

[1] | 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 MR0154750 Zbl 0279.90044 |

#### 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

[a1] | J. von Neumann, O. Morgenstern, "Theory of games and economic behavior" , Princeton Univ. Press (1947) MR21298 Zbl 1112.91002 Zbl 1109.91001 Zbl 0452.90092 Zbl 0205.23401 Zbl 0097.14804 Zbl 0053.09303 Zbl 0063.05930 |

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Minimax principle.

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