Conditional cooperation with longer memory

Direct reciprocity is a wide-spread mechanism for 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 co-player's previous move. Here we extend reactive strategies to longer memories. A reactive-n strategy takes into account the sequence of the last n moves of the co-player. A reactive-n counting strategy records how often the co-player has cooperated during the last n rounds. We derive an algorithm to identify all partner strategies among reactive-n strategies. We give explicit conditions for all partner strategies among reactive-2, reactive-3 strategies, and reactive-n counting strategies. Partner strategies are those that ensure mutual cooperation without exploitation. We perform evolutionary simulations and find that longer memory increases the average cooperation rate for reactive-n strategies but not for reactive counting strategies. Paying attention to the sequence of moves is necessary for reaping the advantages of longer memory.