Mixing Time and Stationary Expected Social Welfare of Logit Dynamics

TL;DR

This paper analyzes the long-term social welfare and mixing times of logit dynamics in strategic games, providing bounds and insights into how these properties depend on game parameters and structure.

Contribution

It offers new bounds on mixing times and evaluates stationary social welfare for specific games under logit dynamics, highlighting different behaviors based on game types.

Findings

Mixing time can depend exponentially on the noise parameter β in some games.

Stationary social welfare varies across different game types.

Certain games exhibit mixing times independent of β.

Abstract

We study "logit dynamics" [Blume, Games and Economic Behavior, 1993] for strategic games. This dynamics works as follows: at every stage of the game a player is selected uniformly at random and she plays according to a "noisy" best-response where the noise level is tuned by a parameter $\beta$. Such a dynamics defines a family of ergodic Markov chains, indexed by $\beta$, over the set of strategy profiles. We believe that the stationary distribution of these Markov chains gives a meaningful description of the long-term behavior for systems whose agents are not completely rational. Our aim is twofold: On the one hand, we are interested in evaluating the performance of the game at equilibrium, i.e. the expected social welfare when the strategy profiles are random according to the stationary distribution. On the other hand, we want to estimate how long it takes, for a system starting at an arbitrary profile and running the logit dynamics, to get close to its stationary distribution; i.e., the "mixing time" of the chain. In this paper we study the stationary expected social welfare for the 3-player CK game, for 2-player coordination games, and for two simple $n$-player games. For all these games, we also give almost tight upper and lower bounds on the mixing time of logit dynamics. Our results show two different behaviors: in some games the mixing time depends exponentially on $\beta$, while for other games it can be upper bounded by a function independent of $\beta$.