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.

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