Are there really no evolutionarily stable strategies in the iterated prisoner's dilemma?

The evolutionary form of the iterated prisoner's dilemma (IPD) is a repeated game where players strategically choose whether to cooperate with or exploit opponents and reproduce in proportion to game success. It has been widely used to study the evolution of cooperation among selfish agents. In the past 15 years, researchers proved over a series of papers that there is no evolutionarily stable strategy (ESS) in the IPD when players maintain long-term relationships. This makes it difficult to make predictions about what strategies can actually persist as prevalent in a population over time. Here, we show that this no ESS finding may be a mathematical technicality, relying on implausible players who are "too perfect" in that their probability of cooperating on any move is arbitrarily close to either 0 or 1. Specifically, in the no ESS proof, all strategies were allowed, meaning that after a strategy X experiences any history H, X cooperates with an unrestricted probability p (X, H) where 0< or =p (X, H)< or =1. Here, we restrict strategies to the set S in which X is a member of S [corrected] if after any H, X cooperates with a restricted probability p (X, H) where e< or =p (X, H)< or =1-e and 0<e<1/2. The variables e and 1-e may be thought of as the biological limits to how perfect (pure) a strategy can be. In S, we first prove that an ESS must be a Nash equilibrium and boundary strategy of S where a boundary strategy has allowable probabilities of cooperating of e or 1-e after any history. We then prove that X is an ESS in S if and only if X is a Nash equilibrium in the set consisting of only S's boundary strategies. Thus, in searching for ESSs in S, we only need to consider what happens when boundary strategies interact with one another. This greatly simplifies the search for ESSs. Using these results, we finally show that when e is sufficiently small, exactly three one-move memory ESSs exist in S: [1] Pavlov-e which generally cooperates (i.e. cooperates with maximum probability 1-e) after both players either simultaneously cooperated or simultaneously defected; otherwise, Pavlov-e generally defects (i.e. cooperates with minimum probability e). [2] Grudge-e which generally cooperates after both players simultaneously cooperated; otherwise, Grudge-e generally defects. [3] ALLD-e which generally defects after all histories.