Editing Game theory (section)

===Stochastic outcomes (and relation to other fields)===
Individual decision problems with stochastic outcomes are sometimes considered "one-player games". They may be modeled using similar tools within the related disciplines of [[decision theory]], [[operations research]], and areas of [[artificial intelligence]], particularly [[AI planning]] (with uncertainty) and [[multi-agent system]]. Although these fields may have different motivators, the mathematics involved are substantially the same, e.g. using [[Markov decision process]]es (MDP).<ref>{{cite book |last1=Lozovanu |first1=D |last2=Pickl |first2=S |title=A Game-Theoretical Approach to Markov Decision Processes, Stochastic Positional Games and Multicriteria Control Models |date=2015 |publisher=Springer, Cham |isbn=978-3-319-11832-1 }}</ref>

Stochastic outcomes can also be modeled in terms of game theory by adding a randomly acting player who makes "chance moves" ("[[Move by nature|moves by nature]]").{{sfnp|Osborne|Rubinstein|1994}} This player is not typically considered a third player in what is otherwise a two-player game, but merely serves to provide a roll of the dice where required by the game.

For some problems, different approaches to modeling stochastic outcomes may lead to different solutions. For example, the difference in approach between MDPs and the [[Minimax|minimax solution]] is that the latter considers the worst-case over a set of adversarial moves, rather than reasoning in expectation about these moves given a fixed probability distribution. The minimax approach may be advantageous where stochastic models of uncertainty are not available, but may also be overestimating extremely unlikely (but costly) events, dramatically swaying the strategy in such scenarios if it is assumed that an adversary can force such an event to happen.<ref name="McMahan">{{cite thesis |first=Hugh Brendan |last=McMahan |date=2006 |title=Robust Planning in Domains with Stochastic Outcomes, Adversaries, and Partial Observability |type=PhD dissertation |publisher=Carnegie Mellon University |url=https://www.cs.cmu.edu/~mcmahan/research/mcmahan_thesis.pdf |archive-url=https://web.archive.org/web/20110401124804/http://www.cs.cmu.edu/~mcmahan/research/mcmahan_thesis.pdf |archive-date=2011-04-01 |url-status=live |pages=3–4}}</ref> (See [[Black swan theory]] for more discussion on this kind of modeling issue, particularly as it relates to predicting and limiting losses in investment banking.)

General models that include all elements of stochastic outcomes, adversaries, and partial or noisy observability (of moves by other players) have also been studied. The "[[gold standard]]" is considered to be partially observable [[stochastic game]] (POSG), but few realistic problems are computationally feasible in POSG representation.<ref name="McMahan" />