Partial Imitation Hinders Emergence of Cooperation in the Iterated Prisoner’s Dilemma with Direct Reciprocity

Abstract

The evolutionary time scales for various strategies in the iterated Prisoner’s Dilemma on a fully connected network are investigated for players with finite memory, using two different kinds of imitation rules: the (commonly used) traditional imitation rule where the entire meta-strategy of the role model is copied, and the partial imitation rule where only the observed subset of moves is copied. If the players can memorize the last round of the game, a sufficiently large random initial population eventually reaches a cooperative equilibrium, even in an environment with bounded rationality (noise) and high temptation. With the traditional imitation rule the time scale to cooperation increases linearly with decreasing intensity of selection (or increasing noise) in the weak selection regime, whereas partial imitation results in an exponential dependence. Populations with finite lifetimes are therefore unlikely to ever reach a cooperative state in this setting. Instead, numerical experiments show the emergence and long persistence of a phase characterized by the dominance of always defecting strategies.