Maximin

The mixed extrema

$$\sup_{x\in X}\inf_{y\in Y}F(x,y),\quad\max_{x\in X}\min_{y\in Y}F(x,y),\quad\text{etc.}\label{*}\tag{*}$$

A maximin can be interpreted (for example, in decision theory, operations research or game theory, cf. Games, theory of) as the greatest gain among those that can be attained by decision making under the worst conditions; it is, thereby, a guaranteed gain. Therefore, decision making oriented on a maximin may reasonably be regarded as optimal.

The value of a maximin does not exceed the value of the corresponding minimax. Conditions for their equality are very important in game theory (see Minimax principle). Such conditions are, for example, the presence of a linear structure in $X$, the convexity of $X$ and the concavity of the function $F$ relative to $x\in X$ for each $y\in Y$ (or, the linearity of $X$ and convexity of $Y$, and convexity of $F$ relative to $y\in Y$ for each $x\in X$).

Finding a maximin as a mathematical operation formally consists in the successive calculation of extrema, that is, in the solution of standard ("single-criterion") optimal programming problems, and, therefore, involves no conceptual complications. However, even when $Y$ is "well arranged" and the function $F$ is uniformly continuous on $X$, the function which associates to $x$ the value of $y$ at which the extremum $\inf_{y\in Y}F(x,y)$ is attained (or "almost attained") can turn out to be "badly arranged" and, in particular, can be a discontinuous function of $x$. In these cases the calculation of a maximum \eqref{*} analytically is difficult and it must be found by numerical methods (see Maximin, numerical methods). The above also applies to finding the minimax.

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See also (references

and

to) Maximin, numerical methods.