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| In [[decision theory]] and [[game theory]], '''[[Abraham Wald|Wald's]] [[Minimax#Maximin|maximin]] model''' is a non-probabilistic decision-making model according to which decisions are ranked on the basis of their worst-case outcomes. That is, the best (optimal) decision is one whose worst outcome is at least as good as the worst outcome of any other decisions. It is one of the most important models in [[robust decision making]] in general and [[robust optimization]] in particular.
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| | | <a href=http://www.unitedfcbermuda.com/>unitedfcbermuda</a> |
| 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'.
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| | | <a href=http://www.lavyan-index.com/>lavyan-index</a> |
| ==Definition== | | <a href=http://narvoo.com/>narvoo</a> |
| Wald's generic maximin model is as follows:
| | <a href=http://www.vestavaluation.com/>vestavaluation</a> |
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| :<math>v^{*}:= \max_{d\in D}\min_{s\in S(d)}f(d,s)</math>
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| where <math>D</math> denotes the decision space; <math>S(d)</math> denotes the set of states associated with decision <math>d</math> and <math>f(d,s)</math> denotes the payoff (outcome) associated with decision <math>d</math> and state <math>s</math>.
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| This model represents a 2-person game in which the <math>\max</math> player plays first. In response, the second player selects the worst state in <math>S(d)</math>, namely a state in <math>S(d)</math> that minimizes the payoff <math>f(d,s)</math> over <math>s</math> in <math>S(d)</math>. In many applications the second player represents uncertainty. However, there are maximin models that are completely deterministic.
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| The above model is the ''classic'' format of Wald's maximin model. There is an equivalent [[Optimization (mathematics)|mathematical programming]] (MP) format:
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| :<math>v^{*}:= \max_{d\in D,\,z\in \mathbb{R}} \{z: z \le f(d,s),\forall s\in S(d)\}</math>
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| where <math>\mathbb{R}</math> denotes the real line.
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| As in [[game theory]], the worst payoff associated with decision <math>d</math>, namely
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| : <math>v(d):= \min_{s\in S(d)} f(d,s)\ , \ d \in D</math> | |
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| is called ''the security level'' of decision <math>d</math>.
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| The minimax version of the model is obtained by exchanging the positions of the <math>\max</math> and <math>\min</math> operations in the classic format:
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| :<math>v^{\circ}:= \min_{d\in D}\max_{s\in S(d)}f(d,s).</math>
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| The equivalent MP format is as follows:
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| :<math>v^{\circ}:= \min_{d\in D,\,z\in \mathbb{R}} \{z: z \ge f(d,s),\forall s\in S(d)\}</math>
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| ==History==
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| Inspired by maximin models of game theory, [[Abraham Wald]] developed this model in the early 1940s <ref name="wald39">Wald, A. (1939). Contributions to the theory of statistical estimation and testing hypotheses. ''The Annals of Mathematics,'' 10(4), 299-326.</ref><ref name="wald45">Wald, A. (1945). Statistical decision functions which minimize the maximum risk. ''The Annals of Mathematics,'' 46(2), 265-280.</ref><ref name="wald50">Wald, A. (1950). ''Statistical Decision Functions,'' John Wiley, NY.</ref> as an approach to situations in which there is only one player (the decision maker). The second player represents a pessimistic (worst case) approach to uncertainty. In Wald's maximin model, player 1 (the <math>\max</math> player) plays first and player 2 (the <math>\min</math> player) knows player 1's decision when he selects his decision. This is a major simplification of the [[Two-person zero-sum game|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.
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| 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.<ref name="resnik">Resnik, M.D. (1987). ''Choices: an Introduction to Decision Theory,'' University of Minnesota Press, Minneapolis.</ref><ref name="french">French, S. (1986). ''Decision Theory: An Introduction to the Mathematics of Rationality,'' Ellis Horwood, Chichester.</ref> It is widely used in diverse fields such as [[decision theory]], [[control theory]], [[economics]], [[statistics]], [[robust optimization]], [[operations research]], [[Maximin (philosophy)|philosophy]], etc.<ref name="ms07">Sniedovich, M. (2007). The art and science of modeling decision-making under severe uncertainty. ''Decision Making in Manufacturing and Services,'' 1(1-2), 111-136.</ref><ref name="ms08">Sniedovich, M. (2008). Wald's maximin model: a treasure in disguise! ''Journal of Risk Finance,'' 9(3), 287-91.</ref>
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| ==Example==
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| One of the most famous examples of a Maximin/Minimax model is
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| : <math>\min_{x\in \mathbb{R}} \max_{y\in \mathbb{R}}\ \{x^{2} - y^{2}\}</math> | |
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| where <math>\mathbb{R}</math> denotes the real line. Formally we can set <math>D=S(d)=\mathbb{R}</math> and <math>f(d,s)=d^{2}-s^{2}</math>. The picture is this
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| [[Image:Saddle point.png|400px]] | |
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| The optimal solution is the (red) [[saddle point]] <math>(x,y)=(0,0)</math>.
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| ==Decision tables==
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| 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.
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| ===Example===
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| 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.
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| {| class="wikitable" align="left"
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| !
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| ! Sun
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| ! Rain
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| |-
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| ! No umbrella
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| | <center>5</center>
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| | <center>−9</center>
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| |-
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| ! Umbrella
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| | <center>1</center>
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| | <center>−5</center>
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| |}
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| {{Clear}}
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| Appending a ''Worst Payoff'' column and a ''Best Worst Payoff'' column to the payoff table, we obtain
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| {| class="wikitable" align="left"
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| !
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| ! Sun
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| ! Rain
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| ! Worst Payoff
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| ! Best Worst Payoff
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| |-
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| ! No umbrella
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| | <center>5</center>
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| | <center>−9</center>
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| | <center>−9</center>
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| | <center></center>
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| |-
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| ! Umbrella
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| | <center>1</center>
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| | <center>−5</center>
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| | <center>−5</center>
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| | <center>−5</center>
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| |}
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| {{Clear}}
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| 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.
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| ==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.<ref name="resnik" /><ref name="french" /><ref name="kouvelis">Kouvelis P, and Yu G. (1997). ''Robust Discrete Optimization and Its Applications,'' Kluwer, Boston.</ref><ref name="ben-tal09">Ben-Tal, A, El Gaoui, L, Nemirovski, A. (2009). ''Robust Optimization.'' Princeton University Press, Princeton.</ref><ref name="sim04">Bertsimas D, and Sim, M. (2004). The price of robustness. ''Operations Research,'' 52(1), 35-53.</ref> For example,
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| ===Savage's minimax regret===
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| [[Leonard Jimmie Savage|Savage's]] [[Minimax regret|minimax regret model]]<ref name="savage51">Savage, L. (1951). The theory of statistical decision. ''Journal of the American Statistical Association,'' 46, 55–67.</ref> is an application of Wald's minimax model to the 'regrets' associated with the payoffs. It can be formulated as follows:
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| : <math>\min_{d\in D}\max_{s\in S} r(d,s)</math>
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| where
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| : <math>r(d,s):= \max_{d\,'\in D} f(d\,',s) - f(d,s)</math> | |
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| is the regret of payoff <math>f(d,s)</math> associated with the (decision,state) pair <math>(d,s)</math>.
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| ==Deterministic models==
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| The sets of states <math>S(d),d\in D,</math> need not represent uncertainty. They can represent (deterministic) variations in the value of a parameter.
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| ===Example===
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| Let <math>D</math> be a finite set representing possible locations of an 'undesirable' public facility (e.g. garbage dump), and let <math>S</math> denote a finite set of locations in the neighborhood of the planned facility, representing existing dwellings.
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| 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:
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| : <math>\max_{d\in D}\min_{s\in S} dist(d,s)</math> | |
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| where <math>dist(d,s)</math> denotes the distance of <math>s</math> from <math>d</math>. Note that in this problem <math>S(d)</math> does not vary with <math>d</math>.
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| 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:
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| : <math>\min_{d\in D}\max_{s\in S} dist(d,s)</math>
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| These are generic [[facility location]] problems.
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| ==Maximin models in disguise==
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| 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.
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| ===Example===
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| Consider the following problem:
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| <blockquote>
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| Given a finite set <math>X</math> and a real valued function <math>g</math> on <math>X</math>, find the largest subset of <math>X</math> such that <math>g(x) \le 0</math> for every <math>x</math> in this subset.
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| </blockquote>
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| The maximin formulation of this problem, in the MP format, is as follows:
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| :<math>\max_{Y\subseteq X} \ \{|Y|: g(x)\le 0,\forall x\in Y\}.</math>
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| Generic problems of this type appear in robustness analysis.<ref name="moffitt08">L. Joe Moffitt, John K. Stranlund, and Craig D. Osteen (2008). Robust detection protocols for uncertain introductions of invasive species. ''Journal of Environmental Management,'' 89(4), 293–299.</ref><ref name="rosenhead72">Jonathan Rosenhead, Martin Elton, Shiv K. Gupta. (1972). Robustness and Optimality as Criteria for Strategic Decisions. ''Operational Research Quarterly,'' 23(4), 413-431.</ref>
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| It has been shown that the [[Stability radius|radius of stability]] model and [[info-gap decision theory|info-gap's robustness]] model are simple instances of Wald's maximin model.<ref name="MS10">Sniedovich, M. (2010). A bird's view of info-gap decision theory. ''Journal of Risk Finance,'' 11(3), 268-283.</ref>
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| ==Constrained maximin models==
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| Constraints can be incorporated explicitly in the maximin models. For instance, the following is a constrained maximin problem stated in the classic format
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| :<math>v^{*}:= \max_{d\in D}\min_{s\in S(d)}\ \{f(d,s): g(d,s) \le 0, \forall s\in S(d)\}.</math>
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| Its equivalent MP format is as follows:
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| :<math>v^{*}:= \max_{d\in D,\,z\in \mathbb{R}} \{z: z \le f(d,s), g(d,s) \le 0, \forall s\in S(d)\}.</math>
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| Such models are very useful in [[robust optimization]].
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| ==The price of robustness==
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| One of the 'weaknesses' of the Maximin model is that the robustness that it provides comes with a ''price''.<ref name="sim04" /> 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.
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| ===Example===
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| Consider the simple case where there are two decisions, d' and d", and where S(d')=S(d")=[a,b]. The Maximin model is then as follows:
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| : <math>\max_{d\in D}\min_{s\in S(d)} f(d,s) = \max_{d\,',d\,''}\ \min_{a\le s \le b}f(d,s) = \max\ \{\min_{a\le s \le b}f(d\,',s),\min_{a\le s\le b}f(d\,'',s)\}</math>
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| Now consider the instance shown by
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| [[Image:Maximin price.png|600px]]
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| Note that although the payoff associated with decision d' is larger than the payoff associated with decision d" over most of the state space S=[a,b], the best worst case according to Wald's model is provided by decision d". Hence, according to Wald's model decision d" is better than decision d'.
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| ==Algorithms==
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| There are no general-purpose algorithms for the solution of maximin problems. Some problems are very simple to solve, others are very difficult.<ref name="reemstem98">Reemstem, R. and R\"{u}ckmann, J. (1998). ''Semi-Infinite Programming,'' Kluwer, Boston.</ref><ref name="rustem02">Rustem, B. and Howe, M. (2002). ''Algorithms for Worst-case Design and Applications to Risk Management,'' Princeton University Press, Princeton.</ref><ref name="ben-tal09" /><ref name="sim04" />
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| ===Example===
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| Consider the case where the state variable is an "index", for instance let <math>S(d)=\{1,2,\dots,k\}</math> for all <math>d \in D</math>. The associated maximin problem is then as follows:
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| : <math>\begin{align}\max_{d\in D}\min_{s\in S(d)} f(d,s) &= \max_{d\in D}\min_{1\le s \le k} \{f_{1}(d),\dots,f_{k}(d)\}\\
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| & = \max_{d\in D, z\in \mathbb{R}} \{z: z\le f_{s}(d),\forall s=1,2,\dots,k\}\end{align}</math>
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| where <math>f_{s}(d) \equiv f(d,s)</math>.
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| If <math>d\in \mathbb{R}^{n}</math>, all the functions <math>f_{s}, s=1,2,\dots,k,</math> are [[linear]], and <math>d\in D</math> is specified by a system of [[linear]] constraints on <math>d</math>, then this problem is a [[linear programming]] problem that can be solved by [[linear programming]] algorithms such as the [[simplex algorithm]].
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| ==References==
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| <references />
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| {{DEFAULTSORT:Wald's Maximin Model}}
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| [[Category:Mathematical optimization]]
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| [[Category:Decision theory]]
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