Does it make sense to play the lottery?
A very smart friend of mine told me yesterday that he buys a lottery ticket every week. I’ve encountered other smart people who do the same, and it always confused me. If you’re aware (as these people all are) that the odds are stacked against you, then why do you play?
I raised this question on Facebook recently and got some insightful replies. One economist pointed out that money is different from utility (i.e., happiness, or well-being), and that it’s perfectly legitimate to get disproportionately more utility out of a jackpot win than you lose from buying a ticket.
So for example, let’s say a ticket costs $1 and gives you a 1 in 10 million chance of winning $8 million. Then the expected monetary value of the ticket equals $8 million/ 10 million – $1 = negative $0.20. But what if you get 15 million units of utility (utils) from $8 million, and you only sacrifice one util from losing $1? In that case, the expected utility value of the ticket equals 15 million utils / 10 million – 1 util = 0.5 utils. Which means buying the ticket is a smart move, because it’ll increase your utility.
I’m sympathetic to that argument in theory — it’s true that we shouldn’t assume that there’s a one-to-one relationship between utility and money, and that someone could hypothetically have a utility curve that makes it a good deal for them to play the lottery. But in practice, the relationship between money and utility tends to be disproportionate in the opposite direction from the example above, in that the more dollars you have, the less utility you get out of each additional dollar. So the $8 million you would gain if you won the lottery will probably give you less than 8 million times as much utility as the $1 that you’re considering spending on the ticket. Which would make the ticket a bad purchase even in terms of expected utility, not just in terms of expected money.
Setting that theoretical argument aside, the most common actual response I get from smart people who play the lottery is that they’re buying a fantasy aid. Purchasing the ticket allows them to daydream about what it would be like to win $8 million, and the experience of daydreaming itself gives them enough utility to make up for the expected monetary loss. “Why can’t you just daydream without paying the $1?” I always ask. “Because it’s not as satisfying if I know that I have no chance of winning,” they reply. Essentially, they don’t care how small their chance of winning is, they just need to know that they have some non-zero chance at the jackpot in order to be able to daydream about it.
I used to accept that argument. But in talking with my friend yesterday, it occurred to me that it’s not true that your chances of winning a fortune are zero without a lottery ticket. For example, you could suddenly discover that you have a long-lost wealthy aunt who died and bequeathed her mansion to you. Or you could find a diamond ring in the gutter. Or, for that matter, you could find a winning lottery ticket in the gutter. The probability of any of these events happening isn’t very high, of course, but it is non-zero.
So you simply can’t say that the lottery ticket is worth the money because it increases your chances of becoming rich from zero to non-zero. All it’s really doing is increasing your chances of becoming rich from extremely tiny to very tiny. And if all you need to enable your Scrooge McDuck daydreams is the knowledge that they have a non-zero chance of coming true, then you can keep those daydreams and your ticket money too.