Approachability in Population Games

TL;DR

This paper extends approachability theory to population games using PDE models, analyzing equilibrium stability and applying it to regret dynamics for convergence to Bayesian equilibrium.

Contribution

It introduces a novel PDE framework for population games, combining approachability with mean-field game theory and analyzing equilibrium properties.

Findings

Established conditions for 1st-moment approachability.

Developed coupled PDE models for population dynamics.

Proved convergence to Bayesian equilibrium under certain conditions.

Abstract

This paper reframes approachability theory within the context of population games. Thus, whilst one player aims at driving her average payoff to a predefined set, her opponent is not malevolent but rather extracted randomly from a population of individuals with given distribution on actions. First, convergence conditions are revisited based on the common prior on the population distribution, and we define the notion of \emph{1st-moment approachability}. Second, we develop a model of two coupled partial differential equations (PDEs) in the spirit of mean-field game theory: one describing the best-response of every player given the population distribution (this is a \emph{Hamilton-Jacobi-Bellman equation}), the other capturing the macroscopic evolution of average payoffs if every player plays its best response (this is an \emph{advection equation}). Third, we provide a detailed analysis of existence, nonuniqueness, and stability of equilibria (fixed points of the two PDEs). Fourth, we apply the model to regret-based dynamics, and use it to establish convergence to Bayesian equilibrium under incomplete information.