Inclusive fitness is something you've heard of before no doubt. J. B. S. Haldane, one of the greatest evolutionary geneticists of the 20th century, once quipped that he would "...lay down [his] life for two brothers or eight cousins," a succinct expression of the subset of this framework which is bracketed under kin selection. The logic is pretty self-evident, but in the 1960s a lonely graduate student in England, William D. Hamilton, toiled away attempting to formally explain the mystery of altruism. The fruit of his labors were two papers, The genetical evolution of social behaviour - I and The genetical evolution of social behaviour - II, published in 1964 in the The Journal of Theoretical Biology. In Narrow Roads of Gene Land, Volume 1: Evolution of Social Behaviour he bemoans in a biographical preface to these papers his incredible self-doubts about the worth of the questions he was asking, the ubiquitous hostility or incomprehension that Hamilton was subject to from all his potentional mentors. During this period the explanation of altruism was self-evident to most, and the biologist V. C. Wynne-Edwards expressed the reigning consensus about the power of group level effects in shaping social dynamics. In short, it was "for the good of the species." Like Charles Darwin and his hero R. A. Fisher Hamilton thought there was more there than met the eye, and that a genetic explanation from first principles was necessary. In the prologue to his preface to The genetical evolution of social behaviour - I he asks rhetorically why after all are there amicable relations between organisms which are not mates or the parent-offspring dyad? In this paper he attempts to mathematically "prove" why a gene selectionist viewpoint is sufficient as an explanation for altruism. Hamilton admits that his formalism was clumsy, he would later enthusiastically admit the elegance and greater generality of George Price's solution toward the problem of altruism. But until then these two papers were essential as markers of an inflection point in a new direction in evolutionary biology and social theory. The genetical evolution of social behaviour - I is not a hallmark of clear exposition; rather, The genetical evolution of social behaviour - II performs the task of illustrating the biological relevance of the somewhat opaque formalisms which Hamilton introduces. In reality it is actually one argument, but the presentation and level of abstraction differs to such a great extent that one may imagine many simply dove into the second with barely a glance at the first. My aim in his post is to go over the major points in homage to Hamilton, who ruefully admitted that though the first part of the paper had copious citations it seemed likely that very few actually read it (1,700+ citations according to Google Scholar right now). The notation used is somewhat peculiar, and I'm going to make a few compromises because of the difficulties of rendering mathematical equations in HMTL (for purposes of accessibility I'm not going to make recourse to a specialty markup language).
The fitness of an individual is the sum of his basic unit of his personal genotype and the total of effects on due to neighbors which will depend on their genotypes:
