The evolution of cooperation under imperfect phenotypic recognition

Abstract

Phenotypic similarity is a classical mechanism for the evolution of cooperation. Most existing models assume a binary rule in which individuals cooperate only with others of the same phenotype. This assumption is biologically restrictive, since recognition and discrimination are often gradual rather than all-or-nothing. In this paper, we extend the multidimensional phenotype-space model of cooperation by allowing the probability of helping to decline with phenotypic distance. In a large population under weak selection with mutation in both strategy and phenotype, we derive a generalized threshold for selection to favor the abundance of cooperation and express it using a new Laplace-type transform. We show that this threshold decreases strictly as discrimination becomes sharper, meaning that exact phenotype matching is the most favorable limit within this family of recognition rules. We also show that the threshold decreases strictly with phenotype-space dimension, meaning that higher-dimensional phenotype spaces promote cooperation even when recognition is imperfect. Our asymptotic analysis further shows that low phenotype mutations strongly inhibit cooperation, whereas sufficiently high phenotype mutations drive the threshold toward its minimal value. However, high strategy mutation makes cooperation harder to maintain.