Source
Hamilton (1964), “The Genetical Evolution of Social Behaviour. I”, LIT-497 (Journal of Theoretical Biology 7: 1–16), read in full in NOTE-401. Some displayed equations lost symbols in extraction; results are taken in the form the prose states.
What was actually shown
The model (§2). One autosomal locus, any number of alleles; organisms reproduce once at the end of a fixed period; random mating; relationship measured by Wright’s coefficient r in a non-inbred population. An individual’s fitness is a basic unit plus its own genotypic effect plus the effects of its neighbours’ genotypes on it.
The decomposition. A genotype’s total effect splits as δT = δR + δS. δR falls on genes identical by descent in relatives and sets the direction of gene-frequency change. δS, the “diluting effect”, changes only the magnitude. The quantity that plays classical fitness’s role is therefore inclusive fitness: own fitness stripped of the social environment’s effects, plus the effects caused on others weighted by r.
The theorem. Mean inclusive fitness increases if δS̄ ≥ 0, which holds when no genotype’s effects on others are negative. For harmful effects the change is only “aimed somewhere in the direction of a local maximum”.
The threshold (§5). With δT = kδa, a trait costly to its bearer (δa < 0) raises inclusive fitness if and only if −k > 1/r. A selfish trait spreads iff −k < 1/r. A pure transfer to a relative (k = −1) is never selected.
Special cases (§3). A benefit placed on relatives of degree r′ spreads r′ times as fast as one kept by the actor. Genes that give to the nearest relatives and take from the most distant are favoured, and “viscous” populations should show the most giving traits.
What could have come out otherwise. The direction of selection could have depended on the diluting part as well. Under the model it does not.
What this does not say
- The rb > c form. That is a later restatement. Nowak (LIT-501) gives it as r > c/b.
- That inclusive fitness is maximized for traits that harm others. Only a direction is shown.
- That r is exact. It is exact only for loci not under selection and for ancestors. Hamilton “has so far failed to reach any definite general conclusions” on the error (§4), and the account is “a good approximation … when selection is slow”.
- That it holds for new mutations. Relatives carry the gene with probability r only after some generations (§4).
- That it holds beyond the model. Overlapping generations and general life histories are deferred to Part II (LIT-506), where the extension is called “unrigorous” (THEORY-655).
- Anything about morality. The model gives a selective reason to favour kin by degree of relationship. That kin-directed behaviour is judged good is a further step this paper does not take.
Connections
- Part II (LIT-506, THEORY-655) generalizes this and applies it.
- Axelrod & Hamilton (LIT-498, THEORY-672) use kinship as the route by which reciprocity can start from universal defection. Nowak (LIT-501, THEORY-567) lists kin selection as the first of five mechanisms.
- Trivers (LIT-510, THEORY-663) builds reciprocal altruism for the cases where kin selection “can be ruled out”.
- Morality-as-cooperation cites “Hamilton, 1964” for its kin domain (LIT-503). Its reference gives Part II’s DOI only (NOTE-401, corrections).
- The later derivation of the rule from the Price equation is not in the record. Luque’s analysis of the Price equation (LIT-205, NOTE-101) has no section on Hamilton’s rule.