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This paper investigates selfish gradient ascent learning in iterated prisoner's dilemma using reactive strategies defined by reciprocity and forgiveness. It demonstrates that heterogeneity in learning rates is crucial for achieving cooperative outcomes, contrasting with uniform learning rates that converge to defection. Specifically, variable learning rates enable convergence to generous-tit-for-tat (GTFT), yielding near-optimal collective payoffs and allowing recovery from perturbations, unlike uniform learning.
Selfish learners can converge to persistent cooperation in social dilemmas, but only if their learning rates are heterogeneous.
Learning agents, whether humans or algorithms, adapt to one another rather than to a fixed environment. Selfish learning need not produce a good outcome for the group, especially when learners face a social dilemma. The iterated prisoner's dilemma (IPD) is the classical example, but it is usually studied in populations of agents who imitate successful strategies. Here, we instead study selfish gradient ascent learners on the two-dimensional space of reactive strategies. A reactive strategy is described by two continuous traits: reciprocity, the chance of cooperating after the co-player cooperates, and forgiveness, the chance of cooperating after the co-player defects. Each learner updates its strategy in the direction that most rapidly improves its payoff against all other learners, by an amount governed by its learning rate. We show that heterogeneity in learning rates can qualitatively change the long-term outcome: a population with uniform learning rates converges to always-defect (ALLD), whereas a population with variable learning rates approaches generous-tit-for-tat (GTFT), which produces nearly optimal collective payoffs. In simulations, every population passes through a transient state of minimal forgiveness, in which the distribution of reciprocity determines the outcome. Variable learning rates promote cooperation because slow learners preserve high reciprocity until the population reaches this state. We prove sufficient conditions on the distribution of reciprocity for convergence to ALLD or to tit-for-tat (TFT). Variable learning rates also allow cooperation to recover from perturbation, which is provably impossible under uniform learning. These results suggest that purely selfish and myopic learning can produce persistent cooperation provided the population has intrinsic variability in learning rates.