  We investigate symmetric equilibria of mutual reinforcement learning when
both players alternately learn the optimal memory-two strategies against the
opponent in the repeated prisoners' dilemma game. We provide a necessary
condition for memory-two deterministic strategies to form symmetric equilibria.
We then provide three examples of memory-two deterministic strategies which
form symmetric mutual reinforcement learning equilibria. We also prove that
mutual reinforcement learning equilibria formed by memory-two strategies are
also mutual reinforcement learning equilibria when both players use
reinforcement learning of memory-$n$ strategies with $n>2$.
