Convergence of Payoff-Based Higher-Order Replicator Dynamics in Contractive Games

TL;DR

This paper analyzes the convergence of a payoff-based higher-order replicator dynamic in contractive games, using passivity theory to establish local and global convergence to Nash equilibria.

Contribution

It introduces a novel higher-order replicator dynamic framework and applies passivity-based analysis to prove convergence in contractive games.

Findings

Local convergence to Nash equilibrium under strict passivity and stability.

Global convergence in symmetric matrix contractive games.

Extension of passivity-based methods to higher-order learning dynamics.

Abstract

We study the convergence properties of a payoff-based higher-order version of replicator dynamics, a widely studied model in evolutionary dynamics and game-theoretic learning, in contractive games. Recent work has introduced a control-theoretic perspective for analyzing the convergence of learning dynamics through passivity theory, leading to a classification of learning dynamics based on the passivity notion they satisfy, such as \textdelta-passivity, equilibrium-independent passivity, and incremental passivity. We leverage this framework for the study of higher-order replicator dynamics for contractive games, which form the complement of passive learning dynamics. Standard replicator dynamics can be represented as a cascade interconnection between an integrator and the softmax mapping. Payoff-based higher-order replicator dynamics include a linear time-invariant (LTI) system in parallel with the existing integrator. First, we show that if this added system is strictly passive and asymptotically stable, then the resulting learning dynamics converge locally to the Nash equilibrium in contractive games. Second, we establish global convergence properties using incremental stability analysis for the special case of symmetric matrix contractive games.