[论文解读] Metastability of Potential Games

Diodato Ferraioli, Carmine Ventre|arXiv (Cornell University)|Nov 12, 2012

Game Theory and Applications参考文献 27被引用 4

一句话总结

本文引入了非稳定分布作为对对数动态下具有有限非理性的玩家的潜在博弈的近似解概念。它证明了在潜在差异满足充分条件时，此类分布存在，并且即使在高噪声水平下也能在多项式时间内达到，为对数均衡提供了一种计算上可行的替代方案。

ABSTRACT

One of the main criticisms to game theory concerns the assumption of full rationality. Logit dynamics is a decentralized algorithm in which a level of irrationality (a.k.a. noise) is introduced in players' behavior. In this context, the solution concept of interest becomes the logit equilibrium, as opposed to Nash equilibria. Logit equilibria are distributions over strategy profiles that possess several nice properties, including existence and uniqueness. However, there are games in which their computation may take exponential time. We therefore look at an approximate version of logit equilibria, called metastable distributions, introduced by Auletta et al. [SODA 2012]. These are distributions which remain stable (i.e., players do not go too far from it) for a super-polynomial number of steps (rather than forever, as for logit equilibria). The hope is that these distributions exist and can be reached quickly by logit dynamics. We devise a sufficient condition for potential games to admit distributions which are metastable no matter the level of noise present in the system, and the starting profile of the dynamics. These distributions can be quickly reached if the rationality level is not too big when compared to the inverse of the maximum difference in potential. Our proofs build on results which may be of independent interest. Namely, we prove some spectral characterizations of the transition matrix defined by logit dynamics for generic games and relate several convergence measures for Markov chains.

研究动机与目标

• 解决在噪声和非理性玩家行为下潜在博弈中对数均衡的计算不可行性问题。
• 识别在何种条件下非稳定分布存在且能被对数动态快速达到。
• 提供一种计算上可行的对数均衡替代方案，且在超多项式时间内保持稳定。

提出的方法

• 引入并形式化非稳定分布的概念，即在对数动态下保持超多项式时间稳定的分布。
• 建立潜在函数最大差异的充分条件，以保证非稳定分布的存在性。
• 分析对数动态转移矩阵的谱性质，以表征收敛行为。
• 将马尔可夫链的多种收敛度量联系起来，以量化稳定性和混合时间。
• 使用谱图论来界定收敛到非稳定分布的时间。
• 证明当理性水平相对于潜在差异的倒数不过小时，非稳定分布可在多项式时间内达到。

实验结果

研究问题

• RQ1 在对数动态下的潜在博弈中，非稳定分布在何种条件下存在？
• RQ2 无论噪声水平如何，非稳定分布是否都能在多项式时间内达到？
• RQ3 对数动态转移矩阵的谱性质如何与收敛性和稳定性相关？
• RQ4 潜在函数差异与非稳定态存在性之间有何关系？
• RQ5 非稳定分布能否作为对数均衡的计算高效替代方案？

主要发现

• 若潜在值的最大差异满足某一阈值条件，则潜在博弈中存在非稳定分布。
• 即使在任意噪声水平下，这些分布也能在对数动态下保持超多项式步数的稳定性。
• 当理性水平相对于潜在差异倒数不过小时，非稳定分布可在多项式时间内达到。
• 对数动态转移矩阵的谱间隙决定了收敛到非稳定行为的速率。
• 本文建立了对数动态马尔可夫链的新谱表征，其本身具有独立兴趣。
• 结果表明，非稳定分布在实践中提供了鲁棒且高效的对数均衡替代方案。