the second hundred · metaphor 199
Whom would you sacrifice for, and how much? Beneath loyalty that feels like the least calculating thing about us, there may be a cold little sum — one that has quietly decided, for a billion years, exactly where a creature's self-sacrifice pays.
We tell ourselves that love doesn't count the cost. And in the moment, it doesn't — the parent runs into the fire, the sister gives the kidney, the soldier covers the grenade, no ledger in sight. Yet look across a whole species, or a whole family, and a pattern surfaces that no one intended: sacrifice flows thickest toward the closest kin, thins with distance, and all but vanishes among strangers. As if something were keeping score.
Something is — not in any mind, but in the arithmetic of which genes survive. A gene that makes you help can still spread, even if helping costs you, so long as it lands often enough on others who carry the same gene. Your relatives are exactly the people likely to carry it. So the question “is this worth it?” has, hidden inside it, a second question the heart never asks aloud: how much of me is also in you? Hamilton's rule is that second question, made exact.
The arithmetic
Hamilton's insight was to stop asking whether an act helps the individual and ask whether it helps the gene that causes it. That gene sits in the helper, who pays a cost c to their own reproduction. But it very likely also sits in the one being helped, who gains a benefit b — and the chance they carry the same gene is exactly the relatedness r. So the gene's net return on the act is r·b − c: what it gains through copies in the relative, minus what it loses in the self.
Set that return above zero and you have the whole rule: help is favoured when r·b > c. It draws a single line through the space of possible acts — a frontier whose slope is the relatedness. Steepen r and the line tilts up, sweeping more acts into the “worth it” region: closer kin justify costlier sacrifice. Flatten r toward a stranger and the frontier collapses onto the vertical axis — almost nothing is favoured, because almost none of you is in them. The heart's warmth toward family and coolness toward strangers is this frontier, felt from the inside.
The lower panel closes the loop. A rule about which genes pay is a prediction about which genes spread: put a rare helping gene into a population and let selection run. When r·b − c is positive it climbs to fixation — everyone, eventually, is a helper; when negative it vanishes. The frontier isn't a metaphor for evolution's bookkeeping. It is the bookkeeping.
What to try
Haldane is said to have quipped he would lay down his life for two brothers or eight cousins — and you can watch the sum behind the joke. Take a sacrifice that costs a full life (c = 1). For a brother (r = ½) the act pays only once the benefit tops two lives saved: ½·b > 1 needs b > 2. For a first cousin (r = ⅛) you'd need to save more than eight. Load those presets and watch the dot cross the frontier and the gene's fate flip in the panel below.
Then just drive one slider at a time. Raise the cost until a favoured act tips over the line — the gene that was spreading goes into decline. Or hold cost and benefit fixed and walk the kinship diagram outward, twin → sibling → cousin → stranger: the same helpful act is favoured for close kin and abandoned for the distant, purely because r shrank. Park the dot exactly on the line (rb = c) and the population goes flat — neither spreading nor dying, evolution's perfect indifference. That knife-edge is the rule stated as an equation.
The mapping
The rule was built to solve a puzzle Darwin feared: sterile workers. A honeybee gives up her own reproduction entirely — an infinite cost, seemingly impossible to evolve — yet workers are so closely related to the sisters they raise that r·b towers over c, and self-sacrifice becomes the winning move. Alarm calls: a ground squirrel that shrieks to warn the colony draws the hawk's eye to itself, and calls most when close kin are near. Human echoes: we leave more, on average, to children than nieces, to nieces than strangers; we feel the tug to help tilt with the family tree even when no gene is doing the thinking.
What the mapping offers is not a licence to reduce love to genetics, but a lens on why loyalty has the shape it does. The pull toward kin isn't arbitrary sentiment layered on top of a selfish creature; it's the visible surface of an old accounting in which helping those who share your genes was, often enough, indistinguishable from helping yourself. When you feel the difference between what you'd give a child and what you'd give a stranger, you are feeling a frontier that predates you — the same rb − c that sorts which sacrifices a lineage can afford to keep making.
Read as life lessons
How much you'll give isn't a constant — it scales with relatedness. The frontier tilts up for close kin and flattens for strangers, and that tilt is felt as the graded warmth of loyalty.
An act that loses for the individual can still win for the gene that causes it, if it lands on others who carry it. Fitness is counted in copies, not in bodies.
The pull toward family isn't sentiment bolted onto a cold machine — it's the surface of an ancient sum, rb − c, that shaped whom a lineage could afford to love.
The mapping, exactly
| Mathematics | Life |
|---|---|
| relatedness r | How much of you is also in them — the fraction of genes shared, and the reach of your loyalty. |
| benefit b | What the sacrifice buys for the one helped — the life saved, the hunger fed, the danger averted. |
| cost c | What it costs you — the risk taken, the resource given up, the reproduction forgone. |
| r·b − c | The gene's-eye return on the act — the cold sum beneath the warm impulse to help. |
| the line rb = c | The frontier of worthwhile sacrifice — where a costlier act needs closer kin to justify it. |
| the gene going to fixation | Loyalty becoming the norm — a disposition to help kin spreading through a lineage until it's simply who they are. |
The honest model
Three dials — relatedness r, benefit b, cost c — feed one inclusive-fitness effect, Δ = r·b − c. The phase panel plots your act as a point in the b–c plane and draws the frontier c = r·b: below it Δ > 0 (favoured), above it Δ < 0. The kinship diagram just sets r to the textbook coefficients — identical twin 1, sibling or parent 1⁄2, half-sib or grandchild 1⁄4, first cousin 1⁄8 — the fraction of genes two relatives share by descent.
The lower panel is a live simulation, not a drawing of the rule. A rare helping allele starts at frequency 0.04 and each generation updates by a replicator step, p ← p + s·p·(1−p) with selection s ∝ (rb − c) — the standard result that an allele's advantage is proportional to its inclusive-fitness effect. The curve is integrated fresh from the current dials: positive Δ drives it to 1, negative to 0, and Δ = 0 leaves it flat. Every chip, the frontier's slope, and the trajectory's fate are computed on the spot; nothing is scripted to a preset answer.
Where the metaphor tears
Nothing in you computes rb − c. The rule describes the statistical pressure that shaped inherited dispositions over deep time — not a sum any creature performs, and not a motive you feel. Read as a theory of feeling it collapses into caricature: people love, grieve, and give for reasons that have long since floated free of reproductive accounting. The arithmetic explains the distribution of a tendency across a species, not the content of a single heart.
Hamilton's rule covers help routed by relatedness, but humans cooperate massively with strangers — through reciprocity, reputation, punishment, shared norms, and institutions that reward the helpful and shun the free rider. Those channels can favour altruism where r is essentially zero, exactly where this frontier says give nothing. A model tuned to blood is silent about the friendships, contracts, and moral codes that carry most of human generosity.
Real benefits and costs are uncertain, delayed, and context-dependent; relatedness through a diploid pedigree is a clean average that hides mutation, inbreeding, and whole different genetic systems (a worker bee's math isn't a mammal's). The tidy line rb = c holds under assumptions — weak selection, additive effects — that the living world honours only roughly. It's the right first equation, not the last word.