Direct fitness or inclusive fitness: how shall we model kin selection?

Two standard mathematical formulations of kin-selection models can be found. Inclusive fitness is an actor-centred approach, which calculates the fitness effect on a number of recipients of the behaviour of a single actor. Direct fitness is a recipient-centred approach, which calculates the fitness effect on the recipient of the behaviour of a number of actors. Inclusive fitness offers us a powerful heuristic, of choosing behaviour to maximize fitness, but direct fitness can be mathematically easier to work with and has recently emerged as the preferred approach of theoreticians. In this paper, we explore the fundamental connection between these two approaches in both homogeneous and class-structured populations, and we show that under simple assumptions (mainly fair meiosis and weak selection) they provide equivalent formulations, which correspond to the predictions of Price's equation for allele frequency change. We use a couple of examples to highlight differences in their conception and formulation, and we briefly discuss a two-species example in which we have a class of 'actor' that is never a 'recipient', which the standard direct fitness method can handle but the usual inclusive fitness cannot.