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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 Bayesian game ： ウィキペディア英語版 Bayesian game In game theory, a Bayesian game is one in which information about characteristics of the other players (i.e. payoffs) is incomplete. Following John C. Harsanyi's framework,〔Harsanyi, John C., 1967/1968. "Games with Incomplete Information Played by Bayesian Players, I-III." Management Science 14 (3): 159-183 (Part I), 14 (5): 320-334 (Part II), 14 (7): 486-502 (Part III).〕 a Bayesian game can be modelled by introducing Nature as a player in a game. Nature assigns a random variable to each player which could take values of ''types'' for each player and associating probabilities or a probability density function with those types (in the course of the game, ''nature'' randomly ''chooses'' a type for each player according to the probability distribution across each player's type space). Harsanyi's approach to modelling a Bayesian game in such a way allows games of incomplete information to become games of imperfect information (in which the history of the game is not available to all players). The type of a player determines that player's payoff function. The probability associated with a type is the probability that the player, for whom the type is specified, is that type. In a Bayesian game, the incompleteness of information means that at least one player is unsure of the type (and so the payoff function) of another player. Such games are called ''Bayesian'' because of the probabilistic analysis inherent in the game. Players have initial beliefs about the type of each player (where a belief is a probability distribution over the possible types for a player) and can update their beliefs according to Bayes' Rule as play takes place in the game, i.e. the belief a player holds about another player's type might change on the basis of the actions they have played. The lack of information held by players and modelling of beliefs mean that such games are also used to analyse imperfect information scenarios. ==Specification of games== The normal form representation of a non-Bayesian game with perfect information is a specification of the strategy spaces and payoff functions of players. A strategy for a player is a complete plan of action that covers ''every contingency of the game'', even if that contingency can never arise. The strategy space of a player is thus the set of all strategies available to a player. A payoff function is a function from the set of strategy profiles to the set of payoffs (normally the set of real numbers), where a strategy profile is a vector specifying a strategy for every player. In a Bayesian game, one has to specify strategy spaces, type spaces, payoff functions and beliefs for every player. A strategy for a player is a complete plan of action that covers every contingency that might arise for every type that player might be. A strategy must not only specify the actions of the player given the type that he is, but must specify the actions that he would take if he were of another type. Strategy spaces are defined as above. A type space for a player is just the set of all possible ''types'' of that player. The beliefs of a player describe the uncertainty of that player about the types of the other players. Each belief is the probability of the other players having particular types, given the type of the player with that belief (i.e. the belief is $p(\backslash mbox|\backslash mbox)$). A payoff function is a 2-place function of strategy profiles and types. If a player has payoff function $U(x,y)$ and he has type t, the payoff he receives is $U(x^$ *,t), where $x^$ * is the strategy profile played in the game (i.e. the vector of strategies played). One of the formal definitions of such a game looks like the following: The game is defined as: $G\; =\; \backslash langle\; N,\; \backslash Omega,\; \backslash langle\; A\_i,u\_i,T\_i,\backslash tau\_i,p\_i,C\_i\; \backslash rangle\_\; \backslash rangle$, where # $N$ is the set of players. # $\backslash Omega$ is the set of states of nature. For instance, in a card game, it can be any order of the cards. # $A\_i$ is the set of actions for player $i$. Let $A=A\_1\backslash times\; A\_2\backslash times\backslash dotsb\backslash times\; A\_N$. # $T\_i$ is the type of player $i$, decided by the function $\backslash tau\_i\backslash colon\; \backslash Omega\; \backslash rightarrow\; T\_i$. So for each state of the nature, the game will have different types of players. The outcome of the players is what determines its type. Players with the same outcome belong to the same type. # $C\_i\; \backslash subseteq\; A\_i\; \backslash times\; T\_i$ defines the available actions for player $i$ of some type in $T\_i$. # $u\_i\backslash colon\; \backslash Omega\; \backslash times\; A\; \backslash rightarrow\; R$ is the payoff function for player $i$. More formally, let $L=\backslash$, and $u\_i\backslash colon\; L\; \backslash rightarrow\; R$. # $p\_i$ is the probability distribution over $\backslash Omega$ for each player $i$, that is to say, each player has different views of the probability distribution over the states of the nature. In the game, they never know the exact state of the nature. The pure strategy $s\_i\backslash colon\; T\_i\; \backslash rightarrow\; A\_i$ should satisfy $(s\_i(t\_i),t\_i)\; \backslash in\; C\_i$ for all $t\_i$. So the strategy for each player only depends on his type, since he may not have any knowledge about other players' types. And the expected payoff to player $i$ for such a strategy profile is $u\_i(S)=E\_(\backslash omega\; ,s\_1(\backslash tau\_1(\; \backslash omega\; )),\backslash dotsc,s\_N(\backslash tau\_N(\; \backslash omega\; ))))$. Let $S\_i$ be the set of pure strategies, $S\_i\; =\; \backslash .$ A Bayesian Equilibrium of the game $G$ is defined to be a (possibly mixed strategy) Nash equilibrium of the game $\backslash hat\; =\; \backslash langle\; N,\backslash hat=S\_1\backslash times\; S\_2\backslash times\backslash dotsb\backslash times\; S\_N,\; \backslash hat\; =u\; \backslash rangle$. So for any finite game $G$, Bayesian Equilibria always exist. 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「Bayesian 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.