The general version of Hamilton's rule

Abstract

The generality of Hamiltons rule1,2 is much debated3-14. In this paper, I show that this debate can be resolved by constructing a general version of Hamiltons rule, which allows for a large variety of ways in which the fitness of an individual can depend on the social behaviour of oneself and of others. For this, I first derive the Generalized Price equation, which reconnects the Price equation15 with the statistics it borrows its terminology from. The Generalized Price equation moreover shows that there is not just one Price equation, but there is a Price-like equation for every possible true model. This implies that there are also multiple, nested rules to describe selection. The simplest rule is the rule for selection of non-social traits with linear fitness effects. This rule is nested in the classical version of Hamiltons rule, for which there is consensus that it works for social traits with linear, independent fitness effects. The classical version of Hamiltons rule, in turn, is nested in more general rules, that for instance allow for nonlinear and/or interdependent fitness effects, like Quellers rule16. The general version of Hamiltons rule therefore is a constructive solution, that allows us to accurately describe when costly cooperation evolves in a wide variety of circumstances. As a byproduct, we also find a hierarchy of nested rules for selection of non-social traits.