No Strategy is Evolutionarily Stable in the Repeated Prisoner's Dilemma

Abstract

Following the influential work of Axelrod, the repeated Prisoner's Dilemma game has become the theoretical gold standard for understanding the evolution of co-operative behavior among unrelated individuals. Using the game, several authors have found that a reciprocal strategy known as Tit for Tat ( TFT) has done quite well in a wide range of environments. TFT strategists start out co-operating and then do what the other player did on the previous move. Despite the success of TFT and similar strategies in experimental studies of the game, Boyd & Lorberbaum (1987, Nature, Lond. 327, 58) have shown that no pure strategy, including TFT, is evolutionarily stable in the sense that each can be invaded by the joint effect of two invading strategies when long-term interaction occurs in the repeated game and future moves are discounted. Farrell & Ware (1989, Theor. Popul. Biol. 36 , 161) have since extended these results to include finite mixes of pure strategies as well. Here, it is proven that no strategy is evolutionarily stable when long-term relationships are maintained in the repeated Prisoner's Dilemma and future moves are discounted. Namely, it is shown each completely probabilistic strategy (i.e. one that both co-operates and defects with positive probability after every sequence of behavior) may be invaded by a single deviant strategy. This completes the proof started by Boyd and Lorberbaum and extended by Farrell and Ware. This paper goes on to prove that no reactive strategy with a memory restricted to the opponent's preceding move is evolutionarily stable when there is no discounting of future moves. This is true despite the success of a more forgiving variant of TFT called GTFT in a recent tournament among reactive strategies conducted by Nowak & Sigmund (1992, Nature 355 , 250) where future moves were not discounted. GTFT, for example, may be invaded by a pair of reactive mutants. Since no strategy is evolutionarily stable when future moves are discounted in the repeated game, the restriction of strategy types to those actually maintained by mutation and phenotypic and environmental variability in natural populations may be the key to understanding the evolution of co-operation. However, the result presented here that the somewhat realistic reactive strategies are also not evolutionarily stable at least in the non-discounted game suggests something else may be going on. For one, the proof that no reactive strategy is evolutionarily stable ironically shows the robustness of TFT -like strategies. Also, a significant invasion of strategies like GTFT (among reactive strategies) and TFT which can only be invaded by the joint effect of two deviant strategies relies on the maintenance of at least one of the mutants by recurrent mutation or other process. This weakens this paper's results somewhat. Further, it may be possible that some completely probabilistic strategies which are close to their pure counterparts may only be invadible by strategies with less "errors" in them: thus, a population may tend toward a pure strategy. Such findings could help explain the persistence of certain strategy types in the repeated PD when long-term interaction is possible.