Game Dynamics and Equilibrium Computation in the Population Protocol Model

TL;DR

This paper explores how populations of agents playing repeated games evolve over time under local update rules, introducing a new equilibrium concept and analyzing the dynamics through high-dimensional Ehrenfest random walks.

Contribution

It introduces a novel distributional equilibrium concept and links population game dynamics to high-dimensional Ehrenfest walks, providing exact stationary distributions and convergence bounds.

Findings

Dynamics relate to high-dimensional Ehrenfest random walks.

Derived exact stationary distributions and mixing time bounds.

Showed convergence to approximate distributional equilibria.

Abstract

We initiate the study of game dynamics in the population protocol model: $n$ agents each maintain a current local strategy and interact in pairs uniformly at random. Upon each interaction, the agents play a two-person game and receive a payoff from an underlying utility function, and they can subsequently update their strategies according to a fixed local algorithm. In this setting, we ask how the distribution over agent strategies evolves over a sequence of interactions, and we introduce a new distributional equilibrium concept to quantify the quality of such distributions. As an initial example, we study a class of repeated prisoner's dilemma games, and we consider a family of simple local update algorithms that yield non-trivial dynamics over the distribution of agent strategies. We show that these dynamics are related to a new class of high-dimensional Ehrenfest random walks, and we derive exact characterizations of their stationary distributions, bounds on their mixing times, and prove their convergence to approximate distributional equilibria. Our results highlight trade-offs between the local state space of each agent, and the convergence rate and approximation factor of the underlying dynamics. Our approach opens the door towards the further characterization of equilibrium computation for other classes of games and dynamics in the population setting.