# Maximin

 Wald‘s (1902-1950) Maximin model (circa 1940) is one of the most popular tools for dealing with severe uncertainty. Informally, it can be stated as follows: Rank decisions according to their worst-case outcome. Hence, select the decision whose worst-case outcome is the best. More formally, in the context of constrained optimization problems, it can be stated as follows: $\displaystyle \max_{x\in X}\ \min_{s\in S(x)} \left\{f(x,s): \mathrm{constraints}(x,s),\forall s\in S(x) \right\}$ where $X$= decision space: set of all available decisions. $S(x)$ = state space associated with decision $x\in X$. $f$ = objective function: $f(x,s)$ denotes the payoff generated by $(x,s)$. $\mathrm{constraints}(x,s)$ = list of constraints imposed on decision $x$, which may depend on the state $s$. In this framework, the decision ($x$) is controlled by the decision maker and the state variable ($s$) is controlled by Nature (uncertainty). This formulation represents a game with two players: the decision maker (represented by the outer $\max$) and Nature (represented by the inner $\min_{\,}$), where the decision maker plays first. Note that Nature knows the value of $x$ when She conducts the $\displaystyle \min_{s\in S(x)}$ operation. Maximin models dominate the scene in decision-making under severe uncertainty in general, and in robust optimization in particular. They are used to model both local and global robustness.