Deterministic limit of temporal difference reinforcement learning for stochastic games

TL;DR

This paper develops a new method to derive deterministic equations for multistate reinforcement learning algorithms, enabling analysis of complex dynamics like chaos in multiagent environments.

Contribution

It introduces a novel approach to obtain deterministic limits of temporal difference learning in multistate settings, extending prior work focused on single-state games.

Findings

Demonstrates convergence to fixed points, limit cycles, and chaos in multiagent environments.

Applies the method to Q, SARSA, and Actor-Critic algorithms.

Reveals diverse dynamical regimes in multistate reinforcement learning.

Abstract

Reinforcement learning in multiagent systems has been studied in the fields of economic game theory, artificial intelligence and statistical physics by developing an analytical understanding of the learning dynamics (often in relation to the replicator dynamics of evolutionary game theory). However, the majority of these analytical studies focuses on repeated normal form games, which only have a single environmental state. Environmental dynamics, i.e., changes in the state of an environment affecting the agents' payoffs has received less attention, lacking a universal method to obtain deterministic equations from established multistate reinforcement learning algorithms. In this work we present a novel methodological extension, separating the interaction from the adaptation time scale, to derive the deterministic limit of a general class of reinforcement learning algorithms, called temporal difference learning. This form of learning is equipped to function in more realistic multistate environments by using the estimated value of future environmental states to adapt the agent's behavior. We demonstrate the potential of our method with the three well established learning algorithms Q learning, SARSA learning and Actor-Critic learning. Illustrations of their dynamics on two multiagent, multistate environments reveal a wide range of different dynamical regimes, such as convergence to fixed points, limit cycles, and even deterministic chaos.