The general version of Hamilton’s rule

The generality of Hamilton's rule is much debated. In this paper, I show that this debate can be resolved by constructing a general version of Hamilton's rule, which allows for a large variety of ways in which the fitness of an individual can depend on the social behavior of oneself and of others. For this, I first derive the Generalized Price equation, which reconnects the Price equation 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 Hamilton's rule, for which there is consensus that it works for social traits with linear, independent fitness effects. The classical version of Hamilton's rule, in turn, is nested in more general rules that, for instance, allow for nonlinear and/or interdependent fitness effects, like Queller's rule. The general version of Hamilton's 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.