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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク continuous game ： ウィキペディア英語版 continuous game A continuous game is a mathematical generalization, used in game theory. It extends the notion of a discrete game, where the players choose from a finite set of pure strategies. The continuous game concepts allows games to include more general sets of pure strategies, which may be uncountably infinite. In general, a game with uncountably infinite strategy sets will not necessarily have a Nash equilibrium solution. If, however, the strategy sets are required to be compact and the utility functions continuous, then a Nash equilibrium will be guaranteed; this is by Glicksberg's generalization of the Kakutani fixed point theorem. The class of continuous games is for this reason usually defined and studied as a subset of the larger class of infinite games (i.e. games with infinite strategy sets) in which the strategy sets are compact and the utility functions continuous. ==Formal definition== Define the ''n''-player continuous game $G\; =\; (P,\; \backslash mathbf,\; \backslash mathbf)$ where :: $P\; =$ is the set of $n\backslash ,$ players, :: $\backslash mathbf=\; (C\_1,\; C\_2,\; \backslash ldots,\; C\_n)$ where each $C\_i\backslash ,$ is a compact metric space corresponding to the $i\backslash ,$ ''th'' player's set of pure strategies, :: $\backslash mathbf=\; (u\_1,\; u\_2,\; \backslash ldots,\; u\_n)$ where $u\_i:\backslash mathbf\backslash to\; \backslash R$ is the utility function of player $i\backslash ,$ : We define $\backslash Delta\_i\backslash ,$ to be the set of Borel probability measures on $C\_i\backslash ,$, giving us the mixed strategy space of player ''i''. : Define the strategy profile $\backslash boldsymbol\; =\; (\backslash sigma\_1,\; \backslash sigma\_2,\; \backslash ldots,\; \backslash sigma\_n)$ where $\backslash sigma\_i\; \backslash in\; \backslash Delta\_i\backslash ,$ Let $\backslash boldsymbol\_$ be a strategy profile of all players except for player $i$. As with discrete games, we can define a best response correspondence for player $i\backslash ,$, $b\_i\backslash$. $b\_i\backslash ,$ is a relation from the set of all probability distributions over opponent player profiles to a set of player $i$'s strategies, such that each element of :$b\_i(\backslash sigma\_)\backslash ,$ is a best response to $\backslash sigma\_$. Define :$\backslash mathbf(\backslash boldsymbol)\; =\; b\_1(\backslash sigma\_)\; \backslash times\; b\_2(\backslash sigma\_)\; \backslash times\; \backslash cdots\; \backslash times\; b\_n(\backslash sigma\_)$. A strategy profile $\backslash boldsymbol$ * is a Nash equilibrium if and only if $\backslash boldsymbol$ * \in \mathbf(\boldsymbol *) The existence of a Nash equilibrium for any continuous game with continuous utility functions can been proven using Irving Glicksberg's generalization of the Kakutani fixed point theorem.〔I.L. Glicksberg. A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points. Proceedings of the American Mathematical Society, 3(1):170–174, February 1952.〕 In general, there may not be a solution if we allow strategy spaces, $C\_i\backslash ,$'s which are not compact, or if we allow non-continuous utility functions. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「continuous game」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.