The Emergence and Stability of Cooperation

We establish conditions under which cooperative strategies will stabilize in conflict situations modelled as tournaments of iterated Prisoner's Dilemma (IPD) games. Our results resolve the controversy over the issue of stability of cooperative strategies in tournaments no pure strategy is evolutionarily stable in the tournaments of IPD. More recently Farrell and Ware (1989) generalized this result to mixed strategies, concluding that the concept of evolutionary stability is "too strong" to be used in such analyses. We show that the form of stability required by Boyd and Lorberbaum is indeed much too strong: it is unattainable, not only in torunaments of IPD, but also in tournaments of all nontrivial games. Yet a slight and basically inconsequential generalization of the standard Maynard Smith notion of evolutionary stability solves this problem. Evolutionarily stable strategies in this sense (ESS) not only exist but turn out to support the whole spectrum of stable state from purely defective to purely cooperative. This phenomenon is similar to the result of the standard folk theorems. In addition to the proliferation of ESS's, we show that evolutionary stability is not easily related to any desirable properties of strategies (e.g, properties pointed out by Axelrod (1984)) which makes it essentially impossible to infer why some ESS's like TIT FOR TAT (TFT) are in any sense superior to others. We propose a solution to this problem which looks at the minimal frequency an ESS needs to stabilze in the population. ESS's that attain the minimal stabilizing frequencies are the most robust ones. We prove that all nice (never the first to defect) and retaliatory (defect immediately in response to partner's defection) strategies (TFT being one of them) are indeed the most robust ESS's when the 'shadow of the future' is sufficiently long. Axelrod's simulation findings are fully supported in that all nice and retaliatory strategies prove to be evolutionarily stable once in the majority, leading to the unique stable state of the population--universal cooperation.