 A 2IPD game between memory-one agents (and indeed any 2-player, 2-move game between memory-one agents) can be represented in a particularly perspicuous way. Let \(\bO_1, \bO_2, \bO_3, \bO_4\) be the four outcomes \(\bC\bC, \bC\bD, \bD\bC\) and \(\bD\bD\). The memory-one strategies (as noted in the discussion of evolution above) are the strategies \(\bS(p_1,p_2,p_3,p_4)\) of cooperating with probability \(p_1, p_2, p_3, p_4\) after outcomes \(\bO_1, \bO_2, \bO_3, \bO_4\). (If we assume that the game is repeated infinitely many times and that \(0 \lt p_i \lt 1\) for \(i=1,2,3,4\), the initial move can be ignored.). Let \(\bS(p_1,p_2,p_3,p_4)\) and \(\bS(q_1,q_2,q_3,q_4)\) be the strategies of players One and Two. (The subscripts are switched for Player Two, so that \(p_2\) and \(q_2\) both give the probability of cooperation after receiving the sucker payoff and \(p_3\) and \(q_3\) give the probability of cooperation after receiving temptation.) Let \(p'_i = 1 - p_i\) and \(q'_i=1 - q_i\) (for \(i=1,2\)) (so that \(p'_i\) and \(q'_i\) are odds of defection). Then we can represent the 2IPD between One and Two as a “Markov transition matrix,” that displays the odds of moving from any state to any state.