Unifying genetic and game theoretic models of kin selection for continuous traits

A framework is presented for unifying single locus genetic and game theoretic models of continuous traits under frequency-dependent selection when there are interactions among relatives. This framework serves two purposes. First, it is used to determine how "games between relatives" must be modeled to be genetically valid. There are two commonly employed phenotypic approaches used in this setting, and we demonstrate that, although some of their predictions are always genetically valid, others are invalid in general, and this is true for both haploid asexual and diploid sexual organisms. In particular, we show that both approaches obtain the correct equilibrium and convergence stability conditions, but neither obtains the correct condition for evolutionary stability. Unlike earlier results for discrete trait matrix games (Hines & Maynard Smith, 1979), there is no simple correspondence between phenotypic and genetic predictions, and we provide two examples to illustrate this point. It is possible however, to obtain these earlier results within the present setting by restricting attention to a particular class of fitness functions. These results demonstrate that, even when selection is weak, phenotypic models can fail if fitness is frequency-dependent. The second purpose is to determine when population mean inclusive fitness effect provides an adaptive topography in games between relative. Our results show that the fitness function must have a special form for this to be true, and this form differs between haploid and diploid organisms.