Learning Equilibria in Games by Stochastic Distributed Algorithms

TL;DR

This paper introduces stochastic distributed algorithms that learn equilibria in games, proving their convergence to Nash equilibria in certain classes of games using mean-field limits and Lyapunov functions.

Contribution

It establishes the convergence of fully stochastic, distributed algorithms to equilibria in games by linking them to mean-field ODEs and Lyapunov functions, including potential and Lyapunov games.

Findings

Stochastic algorithms converge to Nash equilibria in potential games.

Lyapunov functions serve as super-martingales, providing convergence bounds.

Applicable to load balancing and congestion games.

Abstract

We consider a class of fully stochastic and fully distributed algorithms, that we prove to learn equilibria in games. Indeed, we consider a family of stochastic distributed dynamics that we prove to converge weakly (in the sense of weak convergence for probabilistic processes) towards their mean-field limit, i.e an ordinary differential equation (ODE) in the general case. We focus then on a class of stochastic dynamics where this ODE turns out to be related to multipopulation replicator dynamics. Using facts known about convergence of this ODE, we discuss the convergence of the initial stochastic dynamics: For general games, there might be non-convergence, but when convergence of the ODE holds, considered stochastic algorithms converge towards Nash equilibria. For games admitting Lyapunov functions, that we call Lyapunov games, the stochastic dynamics converge. We prove that any ordinal potential game, and hence any potential game is a Lyapunov game, with a multiaffine Lyapunov function. For Lyapunov games with a multiaffine Lyapunov function, we prove that this Lyapunov function is a super-martingale over the stochastic dynamics. This leads a way to provide bounds on their time of convergence by martingale arguments. This applies in particular for many classes of games that have been considered in literature, including several load balancing game scenarios and congestion games.