State lotteries: Part 2
Yesterday I started a multi-post series on why state lotteries are dumb. I tried to show that it was irrational to play the lottery. As I said yesterday, though, I left one potential counterargument to deal with today. This is Posner’s argument about a U-shaped marginal utility curve:
And finally and most interestingly, there are people whose marginal utility of income is U-shaped rather than everywhere declining. Usually we think of it as declining: my second million dollars confers less utility on me than my first million, and that is why I would not pay a million dollars for a lottery ticket that gave me a 50.1 percent or probably even an 80 percent probability of winning $2 million. But maybe I lead a rather drab life, and this might make such a gamble rational even if it were not actuarially fair. Suppose that for a $2 lottery ticket I obtain a one in a million chance of winning $1 million. It is not a fair gamble because the expected value of $1 million discounted by .000001 is $1, not $2. But if having $1 million would transform my life, the expected utility of the gamble may exceed $2, and then it is rationally attractive.
Now, Posner is right in that this is a more interesting argument than the ones I dealt with yesterday, but I also think it’s the least realistic. However, the reasons why will take some explaining.
Marginal utility, for those of you out there who aren’t up on your economics, is the “utility” (think “happiness”) that you get out of an additional dollar. So if I have $700, and you give me a dollar, that dollar will make me more happy, and the amount of additional happiness it gives me is its marginal utility. The marginal utility of each dollar is different, however. If I had $10,000 already, and you gave me a dollar, probably it would add less to my happiness than the dollar that put me up to $701 did.
That idea, that the later dollar, when you’re more wealthy, gives you less happiness, is an example of what economists call “decreasing marginal utility.” Economists assume that in general each dollar is worth less to you than the dollar before it. They have pretty good reasoning behind this. Let’s say I get $10. There are a huge number of ways I can spend that $10. Presumably, I pick whichever of those ways makes me happiest and spend it on that. Then I get another $10. If the thing I spent the first $10 is repeatable — say, buying dinner at a restaurant — then maybe I’ll do it again. Probably it’ll provide less happiness for me now, but at best it provides the same happiness as before. And if it isn’t repeatable — say, seeing a movie — then I pick one of the other options, which I had previously decided gave me less utility. Overall, each additional amount of money gives me less benefit than the one before, so marginal utility is decreasing.
This decreasing marginal utility is the reason why economists expect people to be risk averse. Say you have $100, and you have the option of betting $10 on a coin flip, so that you’ll end up with either $90 or $110 with equal probability. On average, you have $100 whether you take the bet or not, so a risk-neutral person would be indifferent towards taking the bet. However, if you have decreasing marginal utility, the money you could lose (dollars 91 through 100) would give you on average more happiness than those you could win (dollars 101 through 110). Therefore, you would choose not to take the bet, even though the expected value in each situation is the same. Now, for small amounts of money, the difference is not so big, so if you had a 55% chance of winning this bet, you’d probably take it. But for large amounts of money, the differences are huge, so if someone offered you double-or-nothing on your entire life’s savings, you probably wouldn’t take the bet even if you had a 70% chance of winning. The extent of risk-aversion varies between people, but it’s pretty consistently there, and it rests on very rational foundations.
Now, Posner suggests that people might have a U-shaped marginal utility curve, meaning that they have decreasing marginal utility up to some point, but that it then increases again. If that’s true to a large enough extent, it’s conceivable that the average marginal utility of the dollars in your lottery winnings is higher than the marginal utility of the dollar you gave up to buy the ticket, and if it’s enough higher (like, twice as high) it could mean that the bet is worth making even though on average you’ll lose money.
I just don’t find this believable. The only real way for marginal utility to be increasing is if, say, with an additional $20 you can buy something that makes you more than twice as happy as anything you could buy with $10. This is possible for small amounts of money if you really want a particular item and getting only a portion of the money for it doesn’t do you much good. However, if you got a million dollars, you’d buy lots of different stuff. If you got half a million, you’d presumably buy half that stuff, and the half that you most want, meaning that half-million has a higher marginal utility. Now you can imagine things like a huge house that have price tags that make them not possible to split up into parts, but in those cases you could buy one for half the money that is still (at least) half as good, which has the same implications about decreasing marginal utility. It just seems very unlikely that increasing marginal utility really exists.
The other problem with the U-shaped marginal utility curve is that even if it did exist, it would more strongly suggest that the wealthy should play the lottery than the poor. Assume someone is at a level of wealth so that they are currently experiencing decreasing marginal utility (on the left side of the U) but that their marginal utility curve is U-shaped. In order for the lottery to be worth playing, a win has to give them so much money that the marginal utility of those dollars very far to the right brings the average above the marginal utility of the dollar they spend on playing it. However, if that’s true, imagine what happens if we make the person a bit wealthier. The average marginal utility of the lottery winnings goes up, because as we shift the winnings to the right, we get more of those high-utility dollars to the right that are more valuable than the dollar they would have bought the lottery ticket with (and therefore the dollars to its immediate right as well). However, the utility of the dollar needed to buy the lottery ticket is now lower, because we’ve moved right on the left half of the U. This means that the bet is now more worth taking than before. If people had U-shaped marginal utility curves and were rational, the rich would play the lottery more than the poor, which is not what’s happening.
Now, it’s conceivable that the poor tend to have U-shaped utility curves while the wealthy don’t, but this seems simply impossible to me. I see no reason why people of different income levels would have (on average) differently shaped marginal utility curves. Say you take someone making $20k per year and someone making $500k per year and reversed their wealth and income. I see no reason to believe that the person previously making $20k would be any more or less happy with their new $500k/yr status than the person previously making $500k had been (at least, after the initial transitional euphoria died down).
It’s true that the effect of winning would be huge — it might even “transform your life” — but it’s a huge effect because of the huge amount of money, not because the value of any fixed amount of money is greater at greater wealth.
Now, you can imagine that for whatever reason — an unusual U-shaped marginal utility curve, the suspense, the dreaming about winning — someone rationally plays the lottery. But, even if these explanations are theoretically possible, they seem to me to be much more of a stretch than the “people are irrational” explanation. People are notoriously bad at estimating very small probabilities. (For more on that, interesting talk here, hat tip to BoingBoing.)
Hopefully I’ve convinced you that it’s dumb to play the lottery. In the next post I’ll get to why the state shouldn’t actually run a lottery, which is a bit more complicated than just whether it’s smart to play.
Update: The final installment has been posted here.
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[...] this is the third part of a series on the problems I see with state lotteries. In parts one and two, I give what I hope is a convincing argument that playing the lottery is irrational for the vast [...]