Optimality under noise: higher memory strategies for the alternating prisoner's dilemma

The Alternating Prisoner's Dilemma is a variant of the iterated Prisoner's Dilemma in which the players alternate in the roles of actor and recipient. We searched for strategies which are "optimal" in the Alternating Prisoner's Dilemma with noise (a non-zero probability that a player's decision will be transmitted incorrectly). In order to achieve success against a variety of other strategies, a strategy must be "self-cooperating" (able to achieve mutual cooperation with its clone), "C-exploiting" (able to exploit unconditional cooperators), and "D-unexploitable" (able to resist exploitation by defectors). It must also have high evolutionary "dominance", a general measure of evolutionary performance which considers both resistance to invasion and the ability to invade other strategies. A strategy which meets these optimality criteria can evolve cooperation by invading a population of defectors and establishing a stable cooperative society. Most of the strategies commonly discussed in the Alternating Prisoner's Dilemma literature are low-memory strategies such as Tit For Tat, Pavlov, and Firm But Fair, but none of these strategies can simultaneously meet all of the optimality criteria. However, we discovered a class of higher memory "Firm Pavlov" strategies, which not only meet our stringent optimality criteria, but also achieve remarkable success in round-robin tournaments and evolutionary interactions. These higher memory strategies are friendly enough to cooperate with their clone, pragmatic enough to exploit unconditional cooperators, and wary enough to resist exploitation by defectors: they are truly "optimal under noise" in the Alternating Prisoner's Dilemma.