协调对策Logit动力学的亚稳定性
Metastability of Logit Dynamics for Coordination Games
摘要：
Logit动力学[Blume，Games and Economic Behavior，1993]是策略博弈的随机最佳反应动力学：在每一个时间步，一个参与者被随机地均匀选择，她根据一个偏向于有更高收益的策略的概率分布选择一个新策略。这个过程定义了一个遍历马尔可夫链，在一组策略配置文件的游戏，其唯一的平稳分布是长期均衡的游戏概念。但是，当链的混合时间较长时（例如，当参与者的数量呈指数增长时，平稳分布失去了作为均衡概念的吸引力，马尔可夫链的瞬态阶段变得重要起来。可能发生的是，链是“亚稳的”，即，在比混合时间短的时间尺度上，它保持接近状态空间上的某个概率分布，而在混合时间的时间尺度倍数中，它从一个分布跳到另一个分布。本文给出了马尔可夫链的“亚稳定概率分布”的定量定义，并研究了几类协调对策的logit动力学的亚稳定性。我们首先考虑一个纯粹的$n$球员协调游戏，突出了我们的亚稳定性概念的分布的基础上的显着特点。然后，我们研究了没有风险主导策略的集团上的协调博弈（这相当于Curie-Weiss模型的著名的Glauber动力学）和环上的协调博弈（有和没有风险主导策略）。
英文摘要：
Logit Dynamics [Blume, Games and Economic Behavior, 1993] are randomized best response dynamics for strategic games: at every time step a player is selected uniformly at random and she chooses a new strategy according to a probability distribution biased toward strategies promising higher payoffs. This process defines an ergodic Markov chain, over the set of strategy profiles of the game, whose unique stationary distribution is the long-term equilibrium concept for the game. However, when the mixing time of the chain is large (e.g., exponential in the number of players), the stationary distribution loses its appeal as equilibrium concept, and the transient phase of the Markov chain becomes important. It can happen that the chain is "metastable", i.e., on a time-scale shorter than the mixing time, it stays close to some probability distribution over the state space, while in a time-scale multiple of the mixing time it jumps from one distribution to another. In this paper we give a quantitative definition of "metastable probability distributions" for a Markov chain and we study the metastability of the logit dynamics for some classes of coordination games. We first consider a pure $n$-player coordination game that highlights the distinctive features of our metastability notion based on distributions. Then, we study coordination games on the clique without a risk-dominant strategy (which are equivalent to the well-known Glauber dynamics for the Curie-Weiss model) and coordination games on a ring (both with and without risk-dominant strategy).
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