Beating Hamilton’s Rule

I’m taking some space here to tout some research of my own that just came out, a paper called “Socially Enforced Nepotism: How Norms and Reputation Can Amplify Kin Altruism”. It’s about one of the ways that human kinship is different from kinship in other living species. (We’re not sure when this happened. Today’s date on Logarithmic History – corresponding to 1.27 to 1.20 million years ago – may be too early for “beating Hamilton’s rule,” but the truth is we don’t know.)

Anyway, there’s a fair amount of math in the paper, but its initial inspiration is non-mathematical. Specifically, generations of anthropologists have observed that altruism is central to human kinship: you’re nice to your kin, even when there’s no chance of their reciprocating, just because they’re your kin.

Kindred and kindness go together – two words whose common derivation expresses … one of the main principles of social life.

Edward Tylor 1881

This sounds like what evolutionary biologists say about kinship: organisms show one-way, unreciprocated altruism toward their genetic relatives because this gets more of their genes into future generations. But anthropologists have often noted something special about kinship in our species. In human societies, a lot of behavior toward kin is socially enforced and follows socially transmitted rules: you sometimes treat your kin (especially distant kin) the way you do not because you’re especially attached to them, but because following the rules is a righteous thing to do, and affects your moral standing in the community.

So the paper develops a mathematical model of “socially enforced nepotism”, where members of a kin group collectively enforce an ethic of altruistic sharing, or “generalized reciprocity”, which benefits the less able group members at the expense of the more able. A central insight here is that altruism toward kin is what economists call a “public good”. When you help one of your kin at some expense to yourself, you are paying the price of helping, but also providing a free genetic benefit to anyone else who is kin to the beneficiary. If you and these other guys can get together and coordinate your assistance, it may pay evolutionarily to be more altruistic than if you work separately.

I developed this insight earlier with a simple game I call the Brothers Karamazov game, where two brothers have a chance to help a third. (See here for the math, and here for a math-lite exposition.) It goes like this: There’s a famous rule in evolutionary theory called Hamilton’s rule, which says that you can get more of your genes into future generations if you help your kin whenever

r b > c

where b is the benefit to your kin (measured in fitness units), c is the fitness cost to you, and r is the fraction of your genes you expect to share with the beneficiaries. For brothers, r is equal to ½. But analyzing the Brothers Karamazov game, I showed that if two brothers act together rather than separately in helping a third, then they may do better to be more altruistic than Hamilton’s rule predicts. The “effective” r for each can go all the way up to 7/10.

In the paper just published I extend this to a more general case involving lots of players of varying abilities. This is more complicated than the Brothers Karamazov game. I let players adopt norms, and treat other players according to their reputations for following the norms, where the reputation of each player depends on how much they help other players and, recursively, on those other players’ reputations. (This is similar to how the Google ranking algorithm assigns scores to a webpage based on the scores of the pages that link to it.) We have to think about about “incentive compatibility” (to use the economists’ term) here. An incentive compatible, stable norm has more able players being altruistic toward weaker players, but not so altruistic that players lose the incentive to signal how able they are. (Those with an economics background will recognize this a version of “optimal taxation.”)

A lot of human social life is like a stage show, where people play assigned roles – and part of playing your assigned role is rewarding other players for sticking to their roles. This makes modeling and understanding human sociality, including kinship, especially difficult. But it may also contribute to making kinship uniquely elaborate and important in our species.

A final caution: The model here doesn’t scale up indefinitely.

[I] implicitly adopt a weak selection assumption: [I] ignore the effects of selection between the time a gene is present in an ancestor and the time it acts in her descendants, which can make genealogical relatedness a poor indicator of identity by descent for genes under strong selection. This assumption is probably roughly correct for groups on the scale of local groups, lineages, and clans, but not on the scale of tribes and ethnic groups.

So socially enforced nepotism is not directly involved in ethnocentrism. It could be indirectly involved insofar as ethnocentrism hijacks some of the psychology of kinship norms.

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6 thoughts on “Beating Hamilton’s Rule

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