Repeated Game Dynamics in Population Protocols

We initiate the study of repeated game dynamics in the population model, in which we are given a population of $n$ nodes, each with its local strategy, which interact uniformly at random by playing multi-round, two-player games. After each game, the two participants receive rewards according to a given payoff matrix, and may update their local strategies depending on this outcome. In this setting, we ask how the distribution of player strategies evolves with respect to the number of node interactions (time complexity), as well as the number of possible player states (space complexity), determining the stationary properties of such game dynamics. Our main technical results analyze the behavior of a family of Repeated Prisoner's Dilemma dynamics in this model, for which we provide an exact characterization of the stationary distribution, and give bounds on convergence time and on the optimality gap of its expected rewards. Our results follow from a new connection between Repeated Prisoner's Dilemma dynamics in a population, and a class of high-dimensional, weighted Ehrenfest random walks, which we analyze for the first time. The results highlight non-trivial trade-offs between the state complexity of each node's strategy, the convergence of the process, and the expected average reward of nodes in the population. Our approach opens the door towards the characterization of other natural evolutionary game dynamics in the population model.