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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|operations research]] or game theory, cf. [[Games, theory of|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|minimax]]. Conditions for their equality are very important in game theory (see [[Minimax principle|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|Maximin, numerical methods]]). The above also applies to finding the minimax.



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and

to) [[Maximin, numerical methods|Maximin, numerical methods]].