Chaos Between Stochasticity and Periodicity in the Prisoner's Dilemma Game

We study the transition from stochasticity to determinism in the three-strategy pair-approximated prisoner's dilemma game. We show that the stochastic solution converges to the deterministic limit cycle attractor as the number of participating players increases. Importantly though, between the stochastic and periodic solutions, we reveal a broad range of population sizes for which the system exhibits deterministic behavior, yet fails to settle onto the limit cycle attractor. We show that these states are characterized by chaos via a rigorous treatment. Results are discussed in view of their sociological importance.