Concurrent Imitation Dynamics in Congestion Games

TL;DR

This paper analyzes how concurrent imitation dynamics in atomic congestion games lead to rapid convergence to approximate equilibria, with stable states having near-optimal social costs and a logarithmic dependence on the number of agents.

Contribution

It introduces a potential function-based analysis of imitation dynamics, proving convergence properties and efficiency bounds in congestion games.

Findings

Convergence to stable states is monotonic and rapid.

Approximate equilibria are reached with small latency deviations.

Social costs at stable states are close to optimal in singleton games.

Abstract

Imitating successful behavior is a natural and frequently applied approach to trust in when facing scenarios for which we have little or no experience upon which we can base our decision. In this paper, we consider such behavior in atomic congestion games. We propose to study concurrent imitation dynamics that emerge when each player samples another player and possibly imitates this agents' strategy if the anticipated latency gain is sufficiently large. Our main focus is on convergence properties. Using a potential function argument, we show that our dynamics converge in a monotonic fashion to stable states. In such a state none of the players can improve its latency by imitating somebody else. As our main result, we show rapid convergence to approximate equilibria. At an approximate equilibrium only a small fraction of agents sustains a latency significantly above or below average. In particular, imitation dynamics behave like fully polynomial time approximation schemes (FPTAS). Fixing all other parameters, the convergence time depends only in a logarithmic fashion on the number of agents. Since imitation processes are not innovative they cannot discover unused strategies. Furthermore, strategies may become extinct with non-zero probability. For the case of singleton games, we show that the probability of this event occurring is negligible. Additionally, we prove that the social cost of a stable state reached by our dynamics is not much worse than an optimal state in singleton congestion games with linear latency function. Finally, we discuss how the protocol can be extended such that, in the long run, dynamics converge to a Nash equilibrium.