The Rotten Kid Theorem

Consider a situation with one altruist ("parent") and two beneficiaries ("kids"). One of them is a rotten kid who would enjoy kicking his little sister. The analysis I have just described implies that if the dollar value to the rotten kid of kicking his sister (the number of dollars worth of consumption he would, if necessary, give up in order to do so) is less than the dollar cost to the sister of being kicked, the rotten kid is better off not kicking her. After the parent has adjusted his expenditure on the kids in response to the increased utility of the kid and the decreased utility of the kicked sister, the rotten kid will have lost more than he has gained. Here again, the argument does not depend on the parent observing the kick but only on his observing how happy the two kids are.
This result--that a rotten kid, properly allowing for the effects of parental altruism, will find it in his self-interest to kick his sister only if it is efficient to do so--is known as the Rotten Kid Theorem. There is a sense in which the altruist in such a situation functions, unintentionally, as a stand-in for the bureaucrat-god, at least as far as the tiny society made up of altruist and beneficiaries is concerned. Because of the altruist's peculiar utility function--which contains the beneficiaries' utilities among its arguments--both altruist and beneficiaries find it in their private interest to maximize Marshall efficiency, to make decisions according to whether the net effect on altruist and beneficiaries is or is not a Marshall improvement.

--David Friedman