Hamilton's Rule

Predict when an allele for a costly social behaviour spreads by relocating the accounting from organism to gene: it is favoured whenever rB > C — the relatedness-weighted benefit to relatives exceeds the cost to the actor.

Version

v4 · 2026-09-28 · History · 13 corrections

Domain-specific #

484

Domain group

Natural Sciences

Origin domain

Biology & Ecology

Subdomain

Evolutionary Biology and Social Behavior → Biology & Ecology

Core Idea

Hamilton's rule (1964) is the condition under which an allele encoding a costly social behaviour is favoured by selection: an altruistic act imposing fitness cost C on the actor and delivering benefit B to a recipient spreads when rB > C, where r is the coefficient of relatedness — the excess probability that the recipient carries the same allele by descent. It resolves the paradox of altruism by shifting the unit of accounting from organism to allele: the organism is altruistic, but the allele, in the gene's-eye accounting, is selfish.

How would you explain it like I'm…

The Family Helping Rule

Sometimes an animal helps its family even when it costs the helper something. Family members share many of the same instructions inside their bodies. So helping a close relative can mean more copies of the 'helping' instructions end up in the world, as long as the help is big enough and the family is close enough. Hamilton's Rule is the math for when that works.

When Helping Family Pays

Hamilton's Rule explains how helping others can evolve even when the helper pays a price. It says a helping behaviour spreads when r × B is bigger than C. C is the cost to the helper, B is the benefit to the one being helped, and r is how related they are — how likely they are to share the same gene for helping. Siblings are more related than cousins, so helping a sibling is more likely to pay off. The animals don't do math; natural selection does the 'counting' over many generations.

Relatedness-Weighted Altruism Rule

Hamilton's Rule, formalized by W. D. Hamilton in 1964, states that a gene for a costly helping behaviour is favoured by natural selection when rB > C. C is the fitness cost to the actor, B the benefit to the recipient, and r the relatedness: how likely, above the population average, the recipient carries a copy of the same gene inherited from a common ancestor. This solves a puzzle: how can a trait that lowers an individual's own reproduction spread? The answer is to track the gene, not the individual — a gene can gain copies by helping relatives who carry it. Full siblings have r = 0.5 and cousins 0.125. In bees, wasps and ants, sisters can be related by 0.75, more than to their own offspring, which may help explain sterile worker castes.

 

Hamilton's Rule (Hamilton, 1964) gives the condition rB > C for an allele causing a costly social behaviour to increase in frequency, where C is the actor's fitness cost, B the recipient's fitness benefit, and r the coefficient of relatedness — the probability, above the population average, that the recipient carries an allele identical by descent to the actor's. It follows from gene-level accounting of allele-frequency change and resolves the paradox of altruism within standard population genetics: the organism may be altruistic while the allele is, in accounting terms, selfish. In diploids r is 0.5 for full sibs, 0.25 for half-sibs and 0.125 for first cousins. Haplodiploidy in Hymenoptera makes full sisters r = 0.75 versus 0.5 to offspring, a potential explanation for repeated evolution of worker sterility and eusociality. Predictions include kin-directed cooperation, thresholds rising with cost, and more cooperation in kin-structured populations — supported by Sherman's work on ground-squirrel alarm calls and by bacterial public-goods systems. It is the core of inclusive fitness theory, which classifies altruism, mutualism, selfishness and spite by the signs of actor cost and recipient benefit.

Scope of Application

Hamilton's rule lives across the social-evolution subfields of evolutionary biology, ranging over every population, sexual or clonal, where relatedness can be defined.

• Behavioural ecology — cooperation, alarm calls, food-sharing, allocare, helping-at-the-nest, with relatedness-targeted helping the testable signature.
• Evolution of eusociality — worker sterility in ants, bees, wasps; the haplodiploid r = 0.75 sister asymmetry is Hamilton's headline application.
• Conflict theory — parent-offspring conflict, sibling rivalry, intragenomic conflict, each the same inequality with different r.
• Microbial social evolution — secreted public goods and cheating, with r from clonal patch structure approaching one.
• Non-kin cooperation theory — reciprocity and enforcement framed as departures requiring non-r mechanisms.

Clarity

Hamilton's rule resolves the sharpest standing paradox in Darwinian theory — how selection could favour a behaviour that lowers the actor's own reproduction — through a change in the unit of accounting: the allele gains net copies whenever rB exceeds C. It converts "the evolution of altruism" from a verbal puzzle into a quantitative inequality whose three terms can be measured in the field, and it organises the whole space of social behaviour, with r made a measurable quantity rather than an intuition.

Manages Complexity

Deciding whether a social-behaviour allele will spread is, literally, a forbidding population-genetic bookkeeping problem. Hamilton's rule collapses it onto three measurable scalars and one inequality — r, B, C, with rB > C the spread condition — so the analyst predicts cooperation's fate by estimating three quantities rather than re-deriving allele dynamics. It further organises social behaviour into a four-cell table (altruism, mutualism, selfishness, spite), and makes special cases single substitutions: haplodiploidy reads off worker-sterility accessibility; cooperation at r ≈ 0 flags a non-kin mechanism.

Abstract Reasoning

The rule licenses a diagnostic move (reading targeted helping as a signature of relatedness, and cooperation among near-strangers as a pointer to a non-kin mechanism), an interventionist move (raising r or shifting B and C to move the outcome, as a kin-removal experiment would test), a boundary-drawing move (separating kin-selected from non-kin regimes, and the haplodiploid sterility regime), and predictive/order-of-events moves (forecasting which systems evolve cooperation and building parent-offspring conflict into the r arithmetic).

Knowledge Transfer

Within evolutionary biology Hamilton's rule transfers as mechanism across every population, sexual or clonal, with definable relatedness, because the cargo is one inequality grounded in allele-frequency accounting; the three scalars, four-cell table, and near-stranger diagnostic carry without translation across behavioural ecology, eusociality, conflict theory, and microbial cooperation — different content domains of one substrate. Beyond replicators-under-selection the named rule does not travel: cultural or computational analogies strain the genetics, since there is no allele or identity-by-descent. What genuinely recurs is the deeper move — multilevel accounting of a replicator's success weighted by copy-probability — captured more generally by the Price-equation decomposition, from which Hamilton's rule can be derived as one special case.

Relationships to Other Abstractions

Current abstraction Hamilton's Rule Domain-specific

Parents (2) — outgoing typed relations

• Hamilton's Rule presupposes Inclusive Fitness Domain-specific

  Hamilton’s rule presupposes the inclusive-fitness quantity whose marginal change its inequality tests.

• Hamilton's Rule includes Threshold Prime

  Hamilton’s rule contains a threshold boundary at rB equals C between favored and disfavored alleles.

Children (1) — abstractions linked here by typed relations

• Kin selection Domain-specific includes Hamilton's Rule

  Kin selection contains Hamilton’s rule as its operational spread criterion.

Hierarchy paths (2) — routes to 2 parentless roots

• Hamilton's Rule → presupposes Inclusive Fitness → presupposes Natural Selection → is a kind of Selection

Show alternative path (1)

Neighborhood in Abstraction Space

Hamilton's Rule sits in a moderately populated region (52^nd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Population Genetics & Selection (11 abstractions)

Nearest neighbors

• Kin selection — 0.96
• Inclusive Fitness — 0.91
• Fisher's Principle (Sex-Ratio Equilibrium) — 0.85
• Haldane's Sieve — 0.84
• Fisher's Fundamental Theorem of Natural Selection — 0.83

Computed from structural-signature embeddings · 2026-10-25