Hamilton׳s Rule in finite populations with synergistic interactions

Much debate has appeared in the literature over the generality of the inclusive fitness approach in the modeling of evolutionary behavior. Here I focus on the capacity of the inclusive fitness approach to effectively handle non-additive or synergistic interactions. I work with a binary interaction with the matrix game [abcd] and I restrict attention to transitive (homogeneous) populations with weak selective effects. First of all I observe that the construction of "higher-order" relatedness coefficients permits these synergistic interactions to be analyzed with an inclusive fitness analysis. These coefficients are an immediate generalization of Hamilton׳s original coefficient and can be calculated with exactly the same type of recursive equations. Secondly I observe that for models in which the population is not too large and local genetic renewal is rare (e,g, rare mutation), these higher order coefficients are not needed even with non-additive interactions; in fact the synergistic interaction is entirely equivalent to a closely-related additive one. The overall conclusion is that in the study of synergistic binary social interactions (2-player games) in a finite homogeneous population with weak selection and rare genetic renewal, a standard inclusive-fitness analysis is able to predict the direction of allele-frequency change. I apply this result to analyze a recent model of Allen and Nowak (2015).