Logit Dynamics with Concurrent Updates for Local-Interaction Games

TL;DR

This paper investigates the properties of all-logit dynamics in local interaction games, characterizing when they are reversible and comparing their stationary behavior and mixing times to one-logit dynamics.

Contribution

It characterizes local interaction games for which all-logit dynamics are reversible and compares their stationary expectations and mixing times to one-logit dynamics.

Findings

All-logit dynamics are reversible in local interaction games.

Expected values of decomposable observables are equal for bipartite social graphs.

Mixing time behavior varies with game type and rationality level beta.

Abstract

Logit choice dynamics are a family of randomized best response dynamics based on the logit choice function [McFadden, 1974], used for modeling players with limited rationality and knowledge. In this paper we study the all-logit dynamics, where at each time step all players concurrently update their strategies according to the logit choice function. In the well studied one-logit dynamics [Blume, 1993] instead at each step only one randomly chosen player is allowed to update. We study properties of the all-logit dynamics in the context of local interaction games, a class of games that has been used to model complex social phenomena and physical systems. In a local interaction game, players are the vertices of a social graph whose edges are two-player potential games. Each player picks one strategy to be played for all the games she is involved in and the payoff of the player is the sum of the payoffs from each of the games. We prove that local interaction games characterize the class of games for which the all-logit dynamics is reversible. We then compare the stationary behavior of one-logit and all-logit dynamics. Specifically, we look at the expected value of a notable class of observables, that we call decomposable observables. We prove that the difference between the expected values of the observables at stationarity for the two dynamics depends only on the rationality level beta and on the distance of the social graph from a bipartite graph. In particular, if the social graph is bipartite then decomposable observables have the same expected value. Finally, we show that the mixing time of the all-logit dynamics has the same twofold behavior that has been highlighted in the case of the one-logit: for some games it exponentially depends on the rationality level beta, whereas for other games it can be upper bounded by a function independent from beta.