Symmetric equilibrium of multi-agent reinforcement learning in repeated prisoner's dilemma

We investigate the repeated prisoner's dilemma game where both players alternately use reinforcement learning to obtain their optimal memory-one strategies. We theoretically solve the joint Bellman optimum equations of reinforcement learning. We find that the Win-stay Lose-shift strategy, the Grim strategy, and the strategy which always defects can form symmetric equilibrium of the mutual reinforcement learning process amongst sixteen deterministic strategies.