Long-run payoff of reactive strategies ($x_{\ast }=[q_1+(p_1-q_1)q_2]/[1-(p_1-q_1)(p_2-q_2)]$)

With initially independent actions, same-round independence is preserved by reactive updates: each next action uses the other independent previous action and fresh independent randomization. Marginal cooperation follows $x^{\prime }=q_1+(p_1-q_1)y$, $y^{\prime }=q_2+(p_2-q_2)x$. If the product of response slopes has modulus below one, these recurrences converge and the joint distribution is their product. Averaging $R,S,T,P$ over that distribution gives the long-time payoffs. At deterministic response slopes of modulus one, initial-memory-specific cycles must instead be averaged directly.