A much-discussed, quantitative criterion for the spread of an altruistic gene is Hamilton's rule1,2c/b < R (1)where c and b are additive decrement and increment to fitness of altruist and recipient, respectively, and R is a measure of genetic relatedness between the two individuals. When rearranged as -c.1+b.R > 0, the rule can be interpreted as requiring that the gene-caused action increase the 'inclusive fitness' of the actor. Since its introduction, Hamilton's rule and the attendant concept of inclusive fitness have gained increasing acceptance and use among biologists and have become integral in the field now named sociobiology. However, the essentially heuristic reasoning used in deriving these concepts, along with the lack of a complete specification even in the original outbred model2, have led to many investigations into the population genetical underpinnings of Hamilton's rule3-18. On the basis of these considerations, several reports 3,10,12,13,18 have proposed new formulae for R. These formulae have no obvious relation to each other or to the coefficients originally suggested by Hamilton. This proliferation of coefficients is undoubtedly confusing to many and the net effect may be to generate distrust both of the rule and of the notion of inclusive fitness. Our purpose here is to show that these various formulae for R (refs 3,10,12,13,18), although independently derived, are actually the same.
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