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In game theory, a subgame perfect equilibrium (or subgame perfect Nash equilibrium) is a refinement of a Nash equilibrium used in dynamic games. A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every subgame of the original game. Informally, this means that if (1) the players played any smaller game that consisted of only one part of the larger game and (2) their behavior represents a Nash equilibrium of that smaller game, then their behavior is a subgame perfect equilibrium of the larger game. Every finite extensive game has a subgame perfect equilibrium. A common method for determining subgame perfect equilibria in the case of a finite game is backward induction. Here one first considers the last actions of the game and determines which actions the final mover should take in each possible circumstance to maximize his/her utility. One then supposes that the last actor will do these actions, and considers the second to last actions, again choosing those that maximize that actor's utility. This process continues until one reaches the first move of the game. The strategies which remain are the set of all subgame perfect equilibria for finite-horizon extensive games of perfect information.〔 However, backward induction cannot be applied to games of imperfect or incomplete information because this entails cutting through non-singleton information sets. A subgame perfect equilibria necessarily satisfies the One-Shot deviation principle. The set of subgame perfect equilibria for a given game is always a subset of the set of Nash equilibria for that game. In some cases the sets can be identical. The Ultimatum game provides an intuitive example of a game with fewer subgame perfect equilibria than Nash equilibria. ==Example== An example for a game possessing an ordinary Nash equilibrium and a subgame perfect equilibrium is shown in Figure 1. The strategies for player 1 are given by whereas player 2 has the choice between as his choice to be kind or unkind to player 1 might depend on the choice previously made by player 1. The payoff matrix of the game is shown in Table 1. Observe that there are two different equilibria, which are also shown in Figure 1. Consider the equilibrium given by the strategy profile (shown in the middle). Observe that while the profile is obviously a Nash equilibrium the behaviour of player 2 is rather hard to justify when we look at his choice at the node : By choosing strategy instead of player 2 would increase his profit if node would actually be reached during the progress of the game. More formally, the equilibrium is not an equilibrium with respect to the subgame induced by node . It is likely that in real life player 2 would choose the strategy instead which would in turn inspire player 1 to change his strategy to . The resulting profile (shown on the right) is not only a Nash equilibrium but it is also an equilibrium in all subgames (induced by the nodes resp ). It is therefore a subgame perfect equilibrium. 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「subgame perfect equilibrium」の詳細全文を読む スポンサード リンク
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