Decentralized Dynamics for Finite Opinion Games

TL;DR

This paper investigates decentralized algorithms for reaching Nash equilibria in opinion formation games on social networks, analyzing convergence rates of best-response and noisy logit dynamics using Markov chain techniques.

Contribution

It provides new bounds on convergence times for best-response and logit dynamics in opinion games, advancing understanding of decentralized equilibrium computation.

Findings

Bounds on convergence time for best-response dynamics.

Analysis of logit dynamics' convergence rate with varying noise.

Application of Markov chain techniques to opinion game dynamics.

Abstract

Game theory studies situations in which strategic players can modify the state of a given system, due to the absence of a central authority. Solution concepts, such as Nash equilibrium, are defined to predict the outcome of such situations. In multi-player settings, it has been pointed out that to be realistic, a solution concept should be obtainable via processes that are decentralized and reasonably simple. Accordingly we look at the computation of solution concepts by means of decentralized dynamics. These are algorithms in which players move in turns to improve their own utility and the hope is that the system reaches an "equilibrium" quickly. We study these dynamics for the class of opinion games, recently introduced by Bindel et al. [Bindel et al., FOCS2011]. These are games, important in economics and sociology, that model the formation of an opinion in a social network. We study best-response dynamics and show upper and lower bounds on the convergence to Nash equilibria. We also study a noisy version of best-response dynamics, called logit dynamics, and prove a host of results about its convergence rate as the noise in the system varies. To get these results, we use a variety of techniques developed to bound the mixing time of Markov chains, including coupling, spectral characterizations and bottleneck ratio.