Population Games With Erlang Clocks: Convergence to Nash Equilibria For Pairwise Comparison Dynamics

TL;DR

This paper introduces a new framework for analyzing population games with Erlang-distributed revision times, extending existing models to account for sub-strategies and proving convergence to Nash equilibria under certain conditions.

Contribution

It develops a methodology for population games with Erlang clocks and sub-strategies, providing convergence proofs for potential and contractive games.

Findings

Proves convergence to Nash equilibria in potential games.

Establishes conditions for convergence in contractive games.

Models Erlang-distributed revision intervals in population dynamics.

Abstract

The prevailing methodology for analyzing population games and evolutionary dynamics in the large population limit assumes that a Poisson process (or clock) inherent to each agent determines when the agent can revise its strategy. Hence, such an approach presupposes exponentially distributed inter-revision intervals, and is inadequate for cases where each strategy entails a sequence of sub-tasks (sub-strategies) that must be completed before a new revision time occurs. This article proposes a methodology for such cases under the premise that a sub-strategy's duration is exponentially-distributed, leading to Erlang distributed inter-revision intervals. We assume that a so-called pairwise-comparison protocol captures the agents' revision preferences to render our analysis concrete. The presence of sub-strategies brings on additional dynamics that is incompatible with existing models and results. Our main contributions are twofold, both derived for a deterministic approximation valid for large populations. We prove convergence of the population's state to the Nash equilibrium set when a potential game generates a payoff for the strategies. We use system-theoretic passivity to determine conditions under which this convergence is guaranteed for contractive games.