Conditional cooperation with longer memory

Direct reciprocity is a wide-spread mechanism for the evolution of cooperation. In repeated interactions, players can condition their behavior on previous outcomes. A well-known approach is given by reactive strategies, which respond to the coplayer's previous move. Here, we extend reactive strategies to longer memories. A reactive-<i>n</i> strategy takes into account the sequence of the last <i>n</i> moves of the coplayer. A reactive-<i>n</i> counting strategy responds to how often the coplayer cooperated during the last <i>n</i> rounds. We derive an algorithm to identify the partner strategies within these strategy sets. Partner strategies are those that ensure mutual cooperation without exploitation. We give explicit conditions for all partner strategies among reactive-2, reactive-3 strategies, and reactive-<i>n</i> counting strategies. To further explore the role of memory, we perform evolutionary simulations. We vary several key parameters, such as the cost-to-benefit ratio of cooperation, the error rate, and the strength of selection. Within the strategy sets we consider, we find that longer memory tends to promote cooperation. This positive effect of memory is particularly pronounced when individuals take into account the precise sequence of moves.