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bioRxiv · 10.64898/2026.01.16.699891

Successful strategies in the voluntarily repeated Prisoner's Dilemma

Izquierdo, L. R.·Izquierdo, S. S.·Boyd, R. T.
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Abstract

Reciprocity is central to explanations of cooperation among unrelated individuals in societies of humans and other animals. Most mathematical analyses of the evolution of reciprocity are based on the repeated Prisoners Dilemma (RPD) and typically assume that new strategies are rarely introduced by mutation or analogous cultural processes, that behavioral errors are absent or infrequent, and that agents are bound to interact with the same partner. Here we analyze a version of the RPD in which new strategies are frequently introduced, behavioral errors occur at a substantial rate, and actors may have the option to leave their current partner. In this environment, the usual indeterminacy disappears and the mix of strategies and cooperation levels are quite stable. With the option to leave, cooperation persists at a substantially higher level than without the option to leave. Classical strategies such as Grim, Tit-for-Tat, or Win-Stay-Lose-Shift disappear and are replaced by strategies that sanction cheaters by leaving rather than by retaliatory defection. Beyond a threshold, increasing the number of times partners interact decreases the level of cooperation without the option to leave, but increases it when leaving is possible. Significance StatementCooperation is central to human societies, and the repeated Prisoners Dilemma has long been used to explain reciprocity. Most models assume that individuals must interact repeatedly with the same partner. We show that when behavioral errors and variability are substantial, allowing individuals to leave their partner fundamentally changes which strategies succeed and how much cooperation is sustained. Classical strategies such as Grim, Tit-for-Tat, and Win-Stay-Lose-Shift are eliminated. Instead, strategies that sanction cheaters by leaving prevail. Leaving both protects cooperators from exploitation and generates positive assortment, as similar strategies interact more often with each other. These results reveal a simple principle: freedom to leave can promote cooperation at the population level.

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Izquierdo, L. R., Izquierdo, S. S., Boyd, R. T.. 2026-01-20. Successful strategies in the voluntarily repeated Prisoner's Dilemma. https://doi.org/10.64898/2026.01.16.699891

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