A Pair-Approximation Method for Modelling the Dynamics of Multi-Agent Stochastic Games

Developing a dynamical model for learning in games has attracted much recent interest. In stochastic games, agents need to make decisions in multiple states, and transitions between states, in turn, influence the dynamics of strategies. While previous works typically focus either on 2-agent stochastic games or on normal form games under an infinite-agent setting, we aim at formally modelling the learning dynamics in stochastic games under the infinite-agent setting. With a novel use of pair-approximation method, we develop a formal model for myopic Q-learning in stochastic games with symmetric state transition. We verify the descriptive power of our model (a partial differential equation) across various games through comparisons with agent-based simulation results. Based on our proposed model, we can gain qualitative and quantitative insights into the influence of transition probabilities on the dynamics of strategies. In particular, we illustrate that a careful design of transition probabilities can help players overcome the social dilemmas and promote cooperation, even if agents are myopic learners.