Tuesday, May 22, 2007

Of Sex Pots and Sniper Shots

Eugene links to the fascinating case of a woman who was impregnated by twins. Or rather, she was impregnated by a twin, but she doesn’t know which one because she was sleeping with both! Because the twins are identical, DNA tests cannot sort out the paternity.
The identical Missouri twins [Raymon and Richard Miller] say they were unknowingly having sex with the same woman. And according to the woman's testimony, she had sex with each man on the same day. Within hours of each other.
Are those dueling banjos I hear? I especially enjoyed this passage (specifically, the parenthetical):
The two brothers are not the only ones in an awkward situation. Jean Boyd, the mother of the twins (and the child's grandmother — they're sure she is the grandmother) has felt caught in the middle.
I almost threw this case into the Law & Econ exam I gave today. But I decided to spare my poor students, because I soon realized that finding the efficient rule for a case like this is a surprisingly vexing problem. I still haven’t worked out all the details. This will, therefore, be one of those half-baked posts.

In fact, the problem is not even easy when the two possible fathers are unrelated and DNA can identify the biological father with certainty. (I’ll be focusing on this question exclusively, and hopefully return to the twins later.) The natural answer – and the current legal answer – is to make the biological father pay full child support. The goal of our current child-support system is, for the most part, to make sure somebody pays for the kid so the welfare state doesn’t have to. But that goal could be served as easily by pulling a male name out of a hat and sticking some random schmuck with the bill – which, by the way, is not far from reality in some cases. But if we’re going to dump the bill on somebody, we might as well try to create efficient incentives while we’re at it. We should be trying to get potential fathers to have sex if and only if their perceived marginal benefits exceeds the expected marginal costs.

Taking the potential mother’s behavior as given (a problematic assumption – hey, I said this would be a half-baked post), it turns out that it makes sense to make a father pay less than the full child-support cost if the mother had other partners – even if none of those lucky fellows has to pay.

Here’s the analogy. Say you have two snipers aiming at the same victim. Each sniper has a 20% chance of success, and a single hit is enough to kill. The total probability of the victim getting killed is not 40%, but 36%. This is because of the overlap, as there’s a 4% chance they’ll both hit. And in that case, what is each sniper’s marginal contribution to the likelihood of a kill? Since the probability is 20% for one sniper and 36% for two, the marginal contribution of the second sniper is only 16%. And since there’s no special reason to regard one sniper as “sniper #1” and the other as “sniper #2”, 16% is actually the marginal contribution of both snipers. (If that seems strange, think about it this way: If both snipers are aiming, either one could put down his rifle, and the probability of a kill would drop by 16%.) If we add a third sniper, the total probability of a kill is 48.8%, yielding a marginal contribution of only (48.8% – 36%) = 12.8%. And so on. In general, if there are n snipers with probability p of hitting, the marginal contribution of each sniper is:
p(1 – p)^(n - 1)
which is decreasing in n. Similarly, as we increase the number of men having sex with the same woman, the marginal contribution to the probability of conception declines. The formula given above for snipers works just as well for sex partners, if we reinterpret p as the probability of a lone man impregnating the woman. If the men have different probabilities, due to higher sperm counts or greater frequency of intercourse for instance, then the formula will be more complicated. But it will still be true that each man’s marginal contribution will be less than his solo contribution.

How does this matter to efficiency? Back to the snipers. Suppose you’re the mob boss who wants the victim dead. How much would you be willing to pay for one more sniper? Clearly, each additional sniper is worth less than the last, so the amount you should be willing to pay should also fall. The number of snipers affects willingness to pay for another sniper. And by analogy, the number of sex partners should affect our willingness to punish another sex partner.

Specifically, efficiency dictates that each man should only have sex with the woman if his marginal benefit B exceeds the marginal expected cost of doing so. The marginal expected cost is the increase in probability (as given by the formula above) multiplied by the cost of child support L. It follows that if we charge the full amount in every case where a given man’s DNA matches – an event whose frequency will necessarily exceed his marginal contribution – he will have too great an incentive not to have sex with this woman.

Sound odd? Here’s some intuition: if we wanted to minimize the number of children created while maximizing the number of men who have sex, the obvious way to do it would be to make all the men have sex with just one woman, thereby resulting in at most one pregnancy.

The solution, then, is for family courts to allow a “she was shagging other dudes” defense even when biological paternity is clear. If successful, the father would not be totally off the hook, but his child support payments would be reduced by some fraction to reflect his smaller marginal contribution to the likelihood of pregnancy.

But then who would pick up the rest of the tab? Aye, there’s the rub. To be continued... maybe.

8 comments:

Anonymous said...

Except, as any idiot can tell you, the number of times a woman has sex in one small period of time does not increase her chances of getting her pregnant. The odds are essentially the same whether she has sex with one or ten.

Glen Whitman said...

Anon -- If you're right, that actually strengthens the point. It means the marginal contribution of each partner (beyond the first) is approximately zero. And that would imply that liability for child support should also be reduced to zero.

However, I doubt that's really true. It might be the case if we're talking about 2+ men having sex with a woman on the very same days. But if we're talking about semi-regular liaisons that aren't perfectly in sync, then each new partner increases the likelihood of sex occurring during the woman's most fertile period.

Ran said...

One difference between your two examples is that while one sniper's success in killing the target guarantees another's failure (or at least, guarantees that the other's success is irrelevant, in the case that they both shoot simultaneously), one spermatozoon's success does not necessarily guarantee another's failure, in the case that the woman releases two eggs. (I don't understand the processes involved in this, but it's what makes fraternal twins possible.) Of course, there's still a decrease in marginal probability contribution, firstly in that one spermatozoon's success in fertilizing an egg will decrease others' chance of success (since they now have only one egg to aim for), and secondly in that one father may supply the spermatozoa that fertilize both eggs (which, I hope, is the usual story with fraternal twins). So your overall points are valid, but the math is potentially much more complicated than you're suggesting, and will depend on whether the mother releases two eggs each month.

mich said...

Surely it's all very well if you assume the father has no interest in his child - but if the DNA test means it is his, then it's in his evolutionary interests to give it money, unless it gets money from all the people who'd been sleeping with the mother.
If it gets money from all the men who slept with its mother the cost of having sex for men rises with the potential of no gain at all, so why not stick with the usual system except in exceptional cases like this?

Steve said...

Isn't the marginal contribution 18% not 16%? It's 18% because one of the snipers is #1, one is #2, just we don't knopw which. So you average the 2, 20% and 16%...

Thomas said...

The only problem with this is that only one man (excepting the case of double ovulation and both eggs getting fertilized by different men) can impregnate a woman. There is no need to calculate marginal contributions to the event that she gets pregnant. Only one did it! We're not mob bosses trying to slay enemies in a turf wars game of oneupmanship; we're judges trying to assign the responsibility of supporting a child brought into the world. And that child is here because of the actions of a single man. Never mind that the blokes before and after him couldn't slip one past the goalie. He's the one who impregnated her, and he's the one with the bill.

You never made any discussion of the utility of encouraging such behavior (matching marginal costs with marginal benefits) and you certainly made no discussion of why we (as a society armed with the police power of the courts) should do this. I submit that you ignored this because there is no good reason for encouraging such behavior. What say you?

Glen Whitman said...

Steve: "Isn't the marginal contribution 18% not 16%? It's 18% because one of the snipers is #1, one is #2, just we don't knopw which. So you average the 2, 20% and 16%..."

Nope, because 18% is not the marginal contribution of a sniper. The marginal contribution is simply the difference between the probability with two snipers and the probability with just one. That difference is 16%. Even if you designate one sniper as #1 and the other as #2, either one of them could put down his rifle, and the corresponding reduction in probability would be 16%.

Glen Whitman said...

Thomas: "The only problem with this is that only one man (excepting the case of double ovulation and both eggs getting fertilized by different men) can impregnate a woman. There is no need to calculate marginal contributions to the event that she gets pregnant. Only one did it!"

Yes, only one did it, and (unless they're twins) DNA can tell us which one. I conceded that all along. But that's not the point, unless your only purpose is merely to find a guy to stick with the bill. The point is that, in some fraction of cases, there will be a child regardless. If Joe's sperm hadn't done the job, Bill's would have.

Here's another way to think about it. There's a child that resulted from Joe's sperm. Let's call her Jenny. But if Joe's sperm hadn't been there, Bob's sperm would have resulted in a different child. Let's call her Belinda. Now, if the question is which guy is responsible for the existence of Jenny specifically, DNA tells us it's Joe. But if the question is which guy is responsible for the existence of a child, the answer is not Joe, because a child would have existed whether or not Joe's sperm was present.

Now, you might say the question is obviously about Jenny, not just "a child." But from an ex ante perspective, that's just wrong. Rules about consequences affect people's choices before the fact, so if we're talking about the right rules to apply in the context of child support, we have to think about how they will affect choices before any specific child exists.

Thomas: "You never made any discussion of the utility of encouraging such behavior (matching marginal costs with marginal benefits) and you certainly made no discussion of why we (as a society armed with the police power of the courts) should do this. I submit that you ignored this because there is no good reason for encouraging such behavior. What say you?"

I assume by "such behavior" you mean sex. And in fact, I discussed the marginal costs and benefits of sexual behavior explicitly. I said people should have sex when the marginal benefits exceed the marginal costs. It's not about encouraging more sex or less sex, but about giving people incentives to have the right amount of sex, for them, given the costs and benefits.