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Mertens stability is a solution concept used to predict the outcome of a non-cooperative game. A tentative definition of stability was proposed by Elon Kohlberg and Jean-François Mertens for games with finite numbers of players and strategies. Later, Mertens〔Mertens, Jean-François, 1989, and 1991. "Stable Equilibria - A Reformulation," Mathematics of Operations Research, 14: 575-625 and 16: 694-753. ()〕 proposed a stronger definition that was elaborated further by Srihari Govindan and Mertens.〔Govindan, Srihari, and Jean-François Mertens, 2004. "An Equivalent Definition of Stable Equilibria," International Journal of Game Theory, 32(3): 339-357. () ()〕 This solution concept is now called Mertens stability, or just stability. Like other refinements of Nash equilibrium〔Govindan, Srihari & Robert Wilson, 2008. "Refinements of Nash Equilibrium," The New Palgrave Dictionary of Economics, 2nd edition. ()〕 used in game theory stability selects subsets of the set of Nash equilibria that have desirable properties. Stability invokes stronger criteria than other refinements, and thereby ensures that more desirable properties are satisfied. ==Desirable Properties of a Refinement== Refinements have often been motivated by arguments for admissibility, backward induction, and forward induction. In a two-player game, an admissible decision rule for a player is one that does not use any strategy that is weakly dominated by another (see Strategic dominance). Backward induction posits that a player's optimal action in any event anticipates that his and others' subsequent actions are optimal. The refinement called subgame perfect equilibrium implements a weak version of backward induction, and increasingly stronger versions are sequential equilibrium, perfect equilibrium, quasi-perfect equilibrium, and proper equilibrium. Forward induction posits that a player's optimal action in any event presumes the optimality of others' past actions whenever that is consistent with his observations. Forward induction〔Govindan, Srihari, and Robert Wilson, 2009. "On Forward Induction," Econometrica, 77(1): 1-28. () ()〕 is satisfied by a sequential equilibrium for which a player's belief at an information set assigns probability only to others' optimal strategies that enable that information to be reached. Kohlberg and Mertens emphasized further that a solution concept should satisfy the ''invariance'' principle that it not depend on which among the many equivalent representations of the strategic situation as an extensive-form game is used. Thus it should depend only on the reduced normal-form game obtained after elimination of pure strategies that are redundant because their payoffs for all players can be replicated by a mixture of other pure strategies. Mertens〔Mertens, Jean-François, 2003. "Ordinality in Non Cooperative Games," International Journal of Game Theory, 32: 387–430. ()〕〔Mertens, Jean-François, 1992. "The Small Worlds Axiom for Stable Equilibria," Games and Economic Behavior, 4: 553-564. ()〕 emphasized also the importance of the ''small worlds'' principle that a solution concept should depend only on the ordinal properties of players' preferences, and should not depend on whether the game includes extraneous players whose actions have no effect on the original players' feasible strategies and payoffs. Kohlberg and Mertens demonstrated via examples that not all of these properties can be obtained from a solution concept that selects single Nash equilibria. Therefore, they proposed that a solution concept should select closed connected subsets of the set of Nash equilibria.〔The requirement that the set is connected excludes the trivial refinement that selects all equilibria. If only a single (possibly unconnected) subset is selected then only the trivial refinement satisfies the conditions invoked by H. Norde, J. Potters, H. Reijnierse, and D. Vermeulen (1996): ``Equilibrium Selection and Consistency,'' Games and Economic Behavior, 12: 219-225.〕 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「Mertens-stable equilibrium」の詳細全文を読む スポンサード リンク
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