van Veelen Matthijs
Department of Economics, Universiteit van Amsterdam, Amsterdam, Netherlands.
Elife. 2025 Sep 12;14:RP105065. doi: 10.7554/eLife.105065.
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.
汉密尔顿法则的普遍性备受争议。在本文中，我表明通过构建汉密尔顿法则的通用版本可以解决这一争议，该通用版本允许个体的适合度以多种方式依赖于自身及他人的社会行为。为此，我首先推导出广义普莱斯方程，它将普莱斯方程与其所借用术语的统计学重新联系起来。此外，广义普莱斯方程表明并非只有一个普莱斯方程，而是对于每一个可能的真实模型都有一个类似普莱斯的方程。这意味着也有多个嵌套的规则来描述选择。最简单的规则是用于选择具有线性适合度效应的非社会性状的规则。这个规则嵌套在汉密尔顿法则的经典版本中，人们普遍认为该经典版本适用于具有线性、独立适合度效应的社会性状。反过来，汉密尔顿法则的经典版本又嵌套在更一般的规则中，例如像奎勒法则那样允许非线性和/或相互依赖的适合度效应的规则。因此，汉密尔顿法则的通用版本是一个建设性的解决方案，它使我们能够准确描述在各种情况下代价高昂的合作何时会进化。作为一个副产品，我们还发现了一个用于选择非社会性状的嵌套规则层次结构。
