Concept
In decision theory and game theory, Wald's maximin model is a non-probabilistic decision-making model according to which decisions are ranked on the basis of their worst-case outcomes – the optimal decision is one with the least bad worst outcome. It is one of the most important models in robust decision making in general and robust optimization in particular. It is also known by a variety of other titles, such as Wald's maximin rule, Wald's maximin principle, Wald's maximin paradigm, and Wald's maximin criterion. Often 'minimax' is used instead of 'maximin'. This model represents a 2-person game in which the player plays first. In response, the second player selects the worst state in , namely a state in that minimizes the payoff over in . In many applications the second player represents uncertainty. However, there are maximin models that are completely deterministic. The above model is the classic format of Wald's maximin model. There is an equivalent mathematical programming (MP) format: where denotes the real line. As in game theory, the worst payoff associated with decision , namely is called the security level of decision . The minimax version of the model is obtained by exchanging the positions of the and operations in the classic format: The equivalent MP format is as follows: Inspired by game theory, Abraham Wald developed this model as an approach to scenarios in which there is only one player (the decision maker). Player 2 showcases a gloomy approach to uncertainty. In Wald's maximin model, player 1 (the player) plays first and player 2 (the player) knows player 1's decision when he selects his decision. This is a major simplification of the classic 2-person zero-sum game in which the two players choose their strategies without knowing the other player's choice. The game of Wald's maximin model is also a 2-person zero-sum game, but the players choose sequentially. With the establishment of modern decision theory in the 1950s, the model became a key ingredient in the formulation of non-probabilistic decision-making models in the face of severe uncertainty.