Wednesday, September 1, 2010

Hamiltonian Altruism

So I'm reading Steven Pinker's The Blank Slate right now, which is a fantastic book, but has one rather glaring (or at least, misleading, if you interpret the analogy less literally) mistake in the discussion of kin selection, which William Hamilton proposed is the evolutionary explanation for altruism. Even if it's not a widespread mistake, it's an interesting example of the confusion between consumption and utility, utility and marginal utility, and marginal utility and overall motive.

So a bit of background first... the idea of kin selection is that since relatives share genes, and evolution acts on the level of genes (individual genes that are good at self-replication will survive better than those that aren't), genes that cause their host to assist relative who very likely also contain that gene will be selected for. That is, altruism towards family members, and more altruism towards closer family members, can be evolutionarily advantageous.

Hamilton's rule is the calculation that individual genes might be making when considering a costly action that benefits a relative. Since parents share about half of their genes with any given child, and siblings share half their genes with each other, and cousins share one quarter of their genes with each other, etc, this means that if a parents can do something to help their children more than twice the amount it costs them, they should. And if someone can help their cousin four times more than the cost of the action, it's worth it. That is, you should care twice as much about yourself as your child or sibling, four times as much as your cousins or grandchildren, etc.

Steven Pinker considers this principle in the context of sibling rivalry over how to split a pie. He says that since siblings care twice as much about themselves as each other, each of two siblings should desire 2/3 of the pie. Parents, on the other hand, care about their children equally and want a 50-50 split of the pie. Hence, sibling rivalry, and children thinking their siblings got a better deal than they did despite parents' adamant claim of fairness.

But pie isn't the same thing as utility (economist-speak for whatever benefit something provides to someone, in any sense, in sum). If I know that my brother really loves pie and I'm pretty ambivalent about it, I might want him to get much more than 1/3 of it.

Beyond that, at any given moment, marginal utility is more relevant than absolute utility. I don't want to divide a whole pie with my brother, I want to divide each individual bite separately. Say my brother loves every bite of pie equally, but I get sick of it quickly. The first few bites, I like pie at least half as much as my brother, so I keep every bite for myself. But soon I get sick of it, and I let my brother eat the rest. It's unclear, then, how much of the total pie I want my brother to have.

But, note that if utility is concave and the same for each child, it is true that parents want to split the pie evenly between their children. That's because if the allocation is ever uneven, it helps the pie-poor child to get one more bite more than it hurts the pie-rich child to get a bite less. On the other hand, concavity gets us nowhere in predicting how much each child wants to share. If utility is linear, every child wants to eat the entire pie. If utility is concave, the child may still want to eat the entire pie, none at all, or until they dislike continuing to eat. Or any other amount.

Even this is an incorrect way to view Hamilton's rule of pie. Evolution doesn't select for happiness, it selects for reproductive success. And what is the chance that an extra bite of pie will increase the number of my brother's offspring more than twice as much as it increases mine? Pretty much nil, unless my brother is on the brink of starvation. No matter how much I know my brother loves pie compared to me, I will want to eat the pie until I'm thoroughly sick of it.

Steven Pinker is brilliant and, what makes me love him especially, precise. Might as well get this detail right too. Maybe kin selection led to altruism in an environment where everyone was on the brink of death and every kind act pulled the recipient back from the edge, but that's hardly true in today's society, even if that ingrained impulse generalizes to giving change to the homeless guy on the corner so he can get on the bus or buy a beer.

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