  This paper addresses a mathematically tractable model of the Prisoner's
Dilemma using the framework of active inference. In this work, we design pairs
of Bayesian agents that are tracking the joint game state of their and their
opponent's choices in an Iterated Prisoner's Dilemma game. The specification of
the agents' belief architecture in the form of a partially-observed Markov
decision process allows careful and rigourous investigation into the dynamics
of two-player gameplay, including the derivation of optimal conditions for
phase transitions that are required to achieve certain game-theoretic steady
states. We show that the critical time points governing the phase transition
are linearly related to each other as a function of learning rate and the
reward function. We then investigate the patterns that emerge when varying the
agents' learning rates, as well as the relationship between the stochastic and
deterministic solutions to the two-agent system.
