= Long-run payoff of reactive strategies
{title2=$x_*=[q_1+(p_1-q_1)q_2]/[1-(p_1-q_1)(p_2-q_2)]$}
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 $x'=q_1+(p_1-q_1)y$, $y'=q_2+(p_2-q_2)x$. 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.
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