  We initiate the study of game dynamics in the population protocol model: $n$
agents each maintain a current local strategy and interact in pairs uniformly
at random. Upon each interaction, the agents play a two-person game and receive
a payoff from an underlying utility function, and they can subsequently update
their strategies according to a fixed local algorithm. In this setting, we ask
how the distribution over agent strategies evolves over a sequence of
interactions, and we introduce a new distributional equilibrium concept to
quantify the quality of such distributions. As an initial example, we study a
class of repeated prisoner's dilemma games, and we consider a family of simple
local update algorithms that yield non-trivial dynamics over the distribution
of agent strategies. We show that these dynamics are related to a new class of
high-dimensional Ehrenfest random walks, and we derive exact characterizations
of their stationary distributions, bounds on their mixing times, and prove
their convergence to approximate distributional equilibria. Our results
highlight trade-offs between the local state space of each agent, and the
convergence rate and approximation factor of the underlying dynamics. Our
approach opens the door towards the further characterization of equilibrium
computation for other classes of games and dynamics in the population setting.
