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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 Shapley value ： ウィキペディア英語版 Shapley value In game theory, the Shapley value, named in honour of Lloyd Shapley, who introduced it in 1953, is a solution concept in cooperative game theory.〔Lloyd S. Shapley. "A Value for ''n''-person Games". In ''Contributions to the Theory of Games'', volume II, by H.W. Kuhn and A.W. Tucker, editors. ''Annals of Mathematical Studies'' v. 28, pp. 307–317. Princeton University Press, 1953. 〕〔Alvin E. Roth (editor). ''The Shapley value, essays in honor of Lloyd S. Shapley''. Cambridge University Press, Cambridge, 1988.〕 To each cooperative game it assigns a unique distribution (among the players) of a total surplus generated by the coalition of all players. The Shapley value is characterized by a collection of desirable properties. Hart (1989) provides a survey of the subject.〔Sergiu Hart, ''Shapley Value'', The New Palgrave: Game Theory, J. Eatwell, M. Milgate and P. Newman (Editors), Norton, pp. 210–216, 1989.〕〔''A Bibliography of Cooperative Games: Value Theory'' by Sergiu Hart()〕 The setup is as follows: a coalition of players cooperates, and obtains a certain overall gain from that cooperation. Since some players may contribute more to the coalition than others or may possess different bargaining power (for example threatening to destroy the whole surplus), what final distribution of generated surplus among the players should arise in any particular game? Or phrased differently: how important is each player to the overall cooperation, and what payoff can he or she reasonably expect? The Shapley value provides one possible answer to this question. == Formal definition == Formally, a coalitional game is defined as: There is a set ''N'' (of ''n'' players) and a function $v$ that maps subsets of players to the real numbers: $v\; \backslash ;\; :\; \backslash ;\; 2^N\; \backslash to\; \backslash mathbb$, with $v(\backslash emptyset)=0$, where $\backslash emptyset$ denotes the empty set. The function $v$ is called a characteristic function. The function $v$ has the following meaning: if ''S'' is a coalition of players, then ''$v$''(''S''), called the worth of coalition ''S'', describes the total expected sum of payoffs the members of $S$ can obtain by cooperation. The Shapley value is one way to distribute the total gains to the players, assuming that they all collaborate. It is a "fair" distribution in the sense that it is the only distribution with certain desirable properties listed below. According to the Shapley value, the amount that player ''i'' gets given a coalitional game $(\; v,\; N)$ is :$\backslash phi\_i(v)=\backslash sum\_(v(S\backslash cup\backslash )-v(S))$ where ''n'' is the total number of players and the sum extends over all subsets ''S'' of ''N'' not containing player ''i''. The formula can be interpreted as follows: imagine the coalition being formed one actor at a time, with each actor demanding their contribution ''v''(''S''∪) − ''v''(''S'') as a fair compensation, and then for each actor take the average of this contribution over the possible different permutations in which the coalition can be formed. An alternative equivalent formula for the Shapley value is: :$\backslash phi\_i(v)=\; \backslash frac\backslash sum\_R\backslash left\; (v(P\_i^R\; \backslash cup\; \backslash left\; \backslash )\; -\; v(P\_i^R)\; \backslash right)\backslash ,\backslash !$ where the sum ranges over all $|N|!$ orders $R\backslash ,\backslash !$ of the players and $P\_i^R\backslash ,\backslash !$ is the set of players in $N\backslash ,\backslash !$ which precede $i\backslash ,\backslash !$ in the order $R\backslash ,\backslash !$. 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「Shapley value」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.