Wald's maximin model
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'. Definition. This model represents a 2-person game in which the formula_1 player plays first. In response, the second player selects the worst state in formula_2, namely a state in formula_2 that minimizes the payoff formula_4 over formula_5 in formula_2. 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 formula_7 denotes the real line. As in game theory, the worst payoff associated with decision formula_8, namely is called "the security  level" of decision formula_8. The minimax version of the model is obtained by exchanging the positions of the formula_1 and formula_12 operations in the classic format: The equivalent MP format is as follows: History. 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 formula_1 player) plays first and player 2 (the formula_12 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. It is widely used in diverse fields such as decision theory, control theory, economics, statistics, robust optimization, operations research, philosophy, etc. Example. One of the most famous  examples of a Maximin/Minimax model is where formula_7 denotes the real line. Formally we can set formula_19 and formula_20. The picture is this The optimal solution is the (red) saddle point formula_21. Decision tables. There are many cases where it is convenient to 'organize' the Maximin/Minimax model as a 'table'. The convention is that the rows of the table represent the decisions, and the columns represent the states. Example. Henri is going for a walk. The sun may shine, or it may rain. Should Henri carry an umbrella? Henri does not like carrying an umbrella, but he dislikes  getting wet even more. His "payoff matrix", viewing this as a Maximin game pitting Henri against Nature, is as follows. Appending a "Worst Payoff"  column and a "Best Worst Payoff"  column to the payoff table, we obtain The worst case, if Henri goes out without umbrella, is definitely worse than the (best) worst case when carrying an umbrella. Therefore, Henri takes his umbrella with him. Variations on a theme. Over the years a variety of related models have been developed primarily to moderate the pessimistic approach dictated by the worst-case orientation of the model. For example, Savage's minimax  regret. Savage's minimax regret model is associated with the payoff regrets. Deterministic models. The sets of states formula_22 need not represent uncertainty. They can represent (deterministic) variations in the value of a parameter. Example. Let formula_23 be a finite set representing possible locations of an 'undesirable' public facility (e.g. garbage dump), and let formula_24 denote a finite set of locations in the neighborhood of the planned facility, representing existing dwellings. It might be desirable to build the facility so that its shortest distance from an existing dwelling is as large as possible. The maximin formulation of the problem is  as follows: where formula_26 denotes the distance of formula_5 from formula_8. Note that in this problem formula_2 does not vary with formula_8. In cases where is it desirable to live close to the facility, the objective could be to minimize the maximum distance from the facility. This yields the following minimax problem: These are generic facility location problems. Maximin models in disguise. Experience has shown that the formulation of maximin models can be subtle in the sense that problems that 'do not look like' maximin problems can be formulated as such. Example. Consider the following problem: The maximin formulation of  this problem, in the MP format, is as follows: Generic problems of this type appear in robustness analysis. It has been shown that the radius of stability model and info-gap's robustness model are simple instances of Wald's maximin model. Constrained maximin models. Constraints can be incorporated explicitly in the maximin models. For instance, the following is a constrained maximin problem stated in the classic format Its equivalent MP format is as follows: Such models are very useful in robust optimization. The price of robustness. One of the 'weaknesses' of the Maximin model is that the robustness that  it provides comes with a "price". By playing it safe, the Maximin model tends to generate conservative decisions, whose price can be high. The following example illustrates this important feature of the model. Example. Suppose there are two options, and , and where formula_41. The model is then as follows: Algorithms. There are no general-purpose algorithms for the solution of maximin problems. Some problems are very simple to solve, others are very difficult. Example. Consider the case where the state variable is an "index", for instance let formula_43 for all formula_44. The associated maximin problem is then as follows: where  formula_46. If formula_47, all the functions formula_48 are linear, and formula_49 is specified by a system of linear constraints on formula_8, then this problem is a linear programming problem that can be solved by linear programming algorithms such as the simplex algorithm.