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1 Introduction
Consider the classic maximin paradigm ([9], [10] Wald, 1945, 1950): Equation 1 [Figure omitted. See Article Image.] where D is some set; S (d )⊆S, ∀d ∈D for some set S; and f is a real-valued function on D×S.
This model represents a game played by a decision maker against antagonistic nature. The ingredients of this model are construed as follows:
- D represents the decision space pertaining to the decision maker;
- S represents the state space pertaining to nature;
- S (d ) represents the set of feasible states associated with decision d ; and
- f represents the payoff (reward) function.
The game consists of three stages:
- Act 1. The decision maker selects a decision d ∈D.
- Act 2. In response, nature selects the worst state in S (d ) related to the decision d selected by the decision maker. Call this state s (d ).
- Act 3. The payoff f (d ,s (d )) is awarded to the decision maker.
Clearly, the situation outlined by this model depicts a conflict: the decision maker is striving to maximize her payoff while nature is endeavoring to minimize it.
For our purposes it suffices to note that this simple model is the foremost paradigm used in classical decision theory ([6] Resnik, 1987; [4] French, 1988) and affiliated fields such as robust optimization ([5] Kouvelis and Yu, 1997; [3] Ben-Tal et al. , 2006). It is a specialversion of the maximin paradigm of game theory ([8] von Neumann and Morgenstern, 1944) where nature is assigned the role of one of the players.
2 The art of mathematical modeling
For all its prominence and widespread use in decision theory, the maximin paradigm presents the modeler/analyst with a considerable challenge. This is due to the paradigm's austere simplicity. The maximin's lean structure requires that the analyst/modeller make the most of the model's components, which means making the most of the possibilities that are latent in its structure. To this end, the analyst/modeller can fall back on concepts, techniques, etc. from optimization theory that are commonly used for the purpose of mathematical modeling.
For one thing, it is often convenient to rewrite equation (1)...