Continuous Strategy Replicator Dynamics for Multi--Agent Learning

TL;DR

This paper extends replicator dynamics to continuous strategy spaces in multi-agent learning, providing a new framework for analyzing adaptive behaviors of Q-learning agents with continuous actions.

Contribution

It introduces a generalized replicator framework using integral-differential equations for continuous strategies, advancing the analysis of multi-agent adaptive dynamics.

Findings

Derived steady-state functional equations for continuous strategies

Analyzed solutions for two-player games

Validated results through simulations

Abstract

The problem of multi-agent learning and adaptation has attracted a great deal of attention in recent years. It has been suggested that the dynamics of multi agent learning can be studied using replicator equations from population biology. Most existing studies so far have been limited to discrete strategy spaces with a small number of available actions. In many cases, however, the choices available to agents are better characterized by continuous spectra. This paper suggests a generalization of the replicator framework that allows to study the adaptive dynamics of Q-learning agents with continuous strategy spaces. Instead of probability vectors, agents strategies are now characterized by probability measures over continuous variables. As a result, the ordinary differential equations for the discrete case are replaced by a system of coupled integral--differential replicator equations that describe the mutual evolution of individual agent strategies. We derive a set of functional equations describing the steady state of the replicator dynamics, examine their solutions for several two-player games, and confirm our analytical results using simulations.