The Algorithmic Learning Equations : Evolving Strategies in Dynamic Games

We introduce the algorithmic learning equations (ALEs), a set of ordinary differential equations which characterizes the finite-time and asymptotic behaviour of the stochastic interaction between state-dependent learning algorithms in dynamic games. Our framework allows for a variety of information and memory structures, including noisy, perfect, private, and public monitoring. Using the ALEs, we prove that when algorithms and signals are of a particular type, play converges to a Nash equilibrium. We then extend this result to show play converges to a correlated equilibrium for a broader class of algorithms and signal structures. We apply our methodology in a repeated 2x2 prisoner's dilemma game with perfect monitoring. We show that algorithms can learn a reward-punishment mechanism to sustain tacit collusion and can also learn to coordinate in cycles of mutual collusion and mutual punishment