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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 Potential game ： ウィキペディア英語版 Potential game In game theory, a game is said to be a potential game if the incentive of all players to change their strategy can be expressed using a single global function called the potential function. Robert W. Rosenthal created the concept of a congestion game in 1973. Dov Monderer and Lloyd Shapley 〔 〕 created the concept of a potential game and proved that every congestion game is a potential game. The properties of several types of potential games have since been studied. Games can be either ''ordinal'' or ''cardinal'' potential games. In cardinal games, the difference in individual payoffs for each player from individually changing one's strategy ''ceteris paribus'' has to have the same value as the difference in values for the potential function. In ordinal games, only the signs of the differences have to be the same. The potential function is a useful tool to analyze equilibrium properties of games, since the incentives of all players are mapped into one function, and the set of pure Nash equilibria can be found by locating the local optima of the potential function. Convergence and finite-time convergence of an iterated game towards a Nash equilibrium can also be understood by studying the potential function. ==Definition== We will define some notation required for the definition. Let $N$ be the number of players, $A$ the set of action profiles over the action sets $A\_$ of each player and $u$ be the payoff function. A game $G=(N,A=A\_\backslash times\backslash ldots\backslash times\; A\_,\; u:\; A\; \backslash rightarrow\; \backslash reals^N)$ is: * an exact potential game if there is a function $\backslash Phi:\; A\; \backslash rightarrow\; \backslash reals$ such that $\backslash forall\; \},\backslash \; \backslash forall\; \backslash in\; A\_\}$, ::$\backslash Phi(a\text{'}\_,a\_)-\backslash Phi(a\text{'}\text{'}\_,a\_)\; =\; u\_(a\text{'}\_,a\_)-u\_(a\text{'}\text{'}\_,a\_)$ ::That is: when player $i$ switches from action $a\text{'}$ to action $a\text{'}\text{'}$, the change in the potential equals the change in the utility of that player. * a weighted potential game if there is a function $\backslash Phi:\; A\; \backslash rightarrow\; \backslash reals$ and a vector $w\; \backslash in\; \backslash reals\_^N$ such that $\backslash forall\; \},\backslash \; \backslash forall\; \backslash in\; A\_\}$, ::$\backslash Phi(a\text{'}\_,a\_)-\backslash Phi(a\text{'}\text{'}\_,a\_)\; =\; w\_(u\_(a\text{'}\_,a\_)-u\_(a\text{'}\text{'}\_,a\_))$ * an ordinal potential game if there is a function $\backslash Phi:\; A\; \backslash rightarrow\; \backslash reals$ such that $\backslash forall\; \},\backslash \; \backslash forall\; \backslash in\; A\_\}$, ::$u\_(a\text{'}\_,a\_)-u\_(a\text{'}\text{'}\_,a\_)>0\; \backslash Leftrightarrow$ \Phi(a'_,a_)-\Phi(a''_,a_)>0 * a generalized ordinal potential game if there is a function $\backslash Phi:\; A\; \backslash rightarrow\; \backslash reals$ such that $\backslash forall\; \},\backslash \; \backslash forall\; \backslash in\; A\_\}$, ::$u\_(a\text{'}\_,a\_)-u\_(a\text{'}\text{'}\_,a\_)>0\; \backslash Rightarrow$ \Phi(a'_,a_)-\Phi(a''_,a_) >0 *a best-response potential game if there is a function $\backslash Phi:\; A\; \backslash rightarrow\; \backslash reals$ such that $\backslash forall\; i\backslash in\; N,\backslash \; \backslash forall\; \}$, ::$b\_i(a\_)=\backslash arg\backslash max\_\; \backslash Phi(a\_i,a\_)$ where $b\_i(a\_)$ is the best payoff for player $i$ given $a\_$. 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「Potential 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.