Editing Minimax (section)

== Minimax for individual decisions ==

=== Minimax in the face of uncertainty ===
Minimax theory has been extended to decisions where there is no other player, but where the consequences of decisions depend on unknown facts. For example, deciding to prospect for minerals entails a cost, which will be wasted if the minerals are not present, but will bring major rewards if they are. One approach is to treat this as a game against ''nature'' (see [[move by nature]]), and using a similar mindset as [[Murphy's law]] or [[resistentialism]], take an approach which minimizes the maximum expected loss, using the same techniques as in the two-person zero-sum games.

In addition, [[expectiminimax tree]]s have been developed, for two-player games in which chance (for example, dice) is a factor.

=== Minimax criterion in statistical decision theory ===
{{Main article|Minimax estimator}}
In classical statistical [[decision theory]], we have an [[estimator]] <math>\ \delta\ </math> that is used to estimate a [[parameter]] <math>\ \theta \in \Theta\ .</math> We also assume a [[risk function]] <math>\ R(\theta,\delta)\ .</math> usually specified as the integral of a [[loss function]]. In this framework, <math>\ \tilde{\delta}\ </math> is called '''minimax''' if it satisfies

: <Math>\sup_\theta R(\theta,\tilde{\delta}) = \inf_\delta\ \sup_\theta\ R(\theta,\delta)\ .</math>

An alternative criterion in the decision theoretic framework is the [[Bayes estimator]] in the presence of a [[prior distribution]] <math>\Pi\ .</math> An estimator is Bayes if it minimizes the ''[[average]]'' risk

: <Math>\int_\Theta R(\theta,\delta) \ \operatorname{d} \Pi(\theta)\ .</math>

=== Non-probabilistic decision theory ===
A key feature of minimax decision making is being non-probabilistic: in contrast to decisions using [[expected value]] or [[expected utility]], it makes no assumptions about the probabilities of various outcomes, just [[scenario analysis]] of what the possible outcomes are. It is thus [[:wikt:robust|robust]] to changes in the assumptions, in contrast to these other decision techniques. Various extensions of this non-probabilistic approach exist, notably [[minimax regret]] and [[Info-gap decision theory]].

Further, minimax only requires [[ordinal measurement]] (that outcomes be compared and ranked), not ''interval'' measurements (that outcomes include "how much better or worse"), and returns ordinal data, using only the modeled outcomes: the conclusion of a minimax analysis is: "this strategy is minimax, as the worst case is (outcome), which is less bad than any other strategy". Compare to expected value analysis, whose conclusion is of the form: "This strategy yields {{nobr| {{math|ℰ}}({{mvar|X}}) {{=}} {{mvar|n}} ."}} Minimax thus can be used on ordinal data, and can be more transparent.