Analytical and Finite-Population Properties of Defector-Responsive Strategy Switching

Defector-responsive switching from unconditional cooperation to Tit-for-Tat has been proposed as a way to rescue cooperation in the repeated Prisoner's Dilemma. The response rule itself was presented by Pal and coauthors; here we study analytical and finite-population consequences that were not characterized there. We first compare constant switching with switching proportional to the frequency of unconditional defectors. When maintaining the responsive mechanism reduces the payoff of unconditional cooperators, we derive an exact break-even cost for general benefit and cost parameters and give the conditions under which the relevant boundary equilibria resist invasion by defectors. We then implement a finite-population Moran process in which continuous switching intensities are mapped to valid per-update probabilities and cooperation is measured for a fixed period after rescue. Constant switching rapidly creates the retaliatory strategy but continues to remove unconditional cooperators after defectors disappear. Responsive switching can rescue more slowly at low intensity, but it shuts off after rescue and therefore preserves unconditional cooperation. These results identify a quantitative cost constraint and a finite-population tradeoff for an existing feedback mechanism, while also clarifying the biological assumptions required to interpret strategy switching as a phenotypic response.