论可持续均衡
On Sustainable Equilibria
摘要：
Hofbauer（2000）在Myerson（1996）的基础上，将有限对策的纳什均衡定义为可持续的，如果通过删除/添加劣对策的策略子集，可以使有限对策的纳什均衡成为唯一的，本文证明了关于可持续均衡的两个结果。第一个涉及霍夫鲍尔-迈尔森关于均衡可持续性与其指数之间关系的猜想：对于一类一般博弈，均衡是可持续的，当且仅当其指数为$+1$。Von Schemde和von Stengel（2008）证明了双矩阵博弈的这个猜想;我们证明了这个猜想对所有有限博弈都是正确的。更确切地说，我们证明了一个孤立的均衡有指数+1当且仅当它可以在一个更大的游戏中唯一，通过添加许多策略，是劣对策的均衡。我们的第二个结果给出了可持续性的公理化扩展到所有的游戏，并表明，只有纳什组件的积极指数可以持续。
英文摘要：
Following the ideas laid out in Myerson (1996), Hofbauer (2000) defined a Nash equilibrium of a finite game as sustainable if it can be made the unique Nash equilibrium of a game obtained by deleting/adding a subset of the strategies that are inferior replies to it. This paper proves two results about sustainable equilibria. The first concerns the Hofbauer-Myerson conjecture about the relationship between the sustainability of an equilibrium and its index: for a generic class of games, an equilibrium is sustainable iff its index is $+1$. Von Schemde and von Stengel (2008) proved this conjecture for bimatrix games; we show that the conjecture is true for all finite games. More precisely, we prove that an isolated equilibrium has index +1 if and only if it can be made unique in a larger game obtained by adding finitely many strategies that are inferior replies to that equilibrium. Our second result gives an axiomatic extension of sustainability to all games and shows that only the Nash components with positive index can be sustainable.
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