Minimax

A mixed extremum

$$\inf_{y\in Y}\sup_{x\in X}F(x,y),\quad\min_{y\in Y}\max_{x\in X}F(x,y),$$

etc. (see also Maximin); it can be interpreted (for example, in decision theory, operations research or statistics) as the least of the losses which cannot be prevented by decision making under the given circumstances.

#### Comments[edit]

Cf. Minimax statistical procedure for an interpretation in statistics, [a1] for minimaxima in game theory, and [a2], [a3] for a discussion of minimaxima and maximinima in decision theory. Minimax (and maximin) considerations also occur in other parts of mathematics, for instance in approximation theory, [a4].

#### References[edit]

[a1] |  J. Szép, F. Forgó, "Introduction to the theory of games" , Reidel (1985) pp. Sect. 9.1

|

[a2] |  R.J. Thierauf, R.A. Grosse, "Decision making through operations research" , Wiley (1970) pp. Chapt. 3
[a3] |  J.K. Sengupta, "Stochastic optimization and economic models" , Reidel (1986) pp. Chapt. IV
[a4] |  T.J. Rivlin, "An introduction to the approximation of functions" , Dover, reprint (1981)