We all know that lotteries are a really terrible investment strategy. There are various theories for why people buy tickets anyway; the two main ones are

  • People are stupid and can’t do math.
  • People are not buying an investment, they’re buying a few moments’ thrill and excitement.

Both these theories seem pretty good to me, but I’ve got another which I’ve never seen suggested. I wonder if the usual calculation, “Pay 1 dollar, average return 75 cents” (or whatever it is) gets the expected utility return, as opposed to the dollar return, wrong. If you lose in the lottery, you’re out a dollar which, presumably, you could afford to lose. But if you win, you get (say) a million dollars, all in one chunk. It seems to me that this million might have a very high utility, so that the proper calculation is (1 dollar*1utility/$) to get (epsilon chance of 1 million dollars (=75 cents) times 2M utility/M$), for a total expected return of 1.5. With a million dollars, most people (at least in the lottery-buying public) could solve a whole host of day-to-day problems: Fix the car, move to a better neighbourhood, pay down their mortgage.

Suppose I offer you this deal: You pay me a dollar, and I will pay you back 125 cents. A great investment, right? In dollars it’s certainly a lot better than your average lottery. In utility, though, how much do you really care about an extra quarter? Even if I were stupid enough to give you this deal for a full thousand dollars, and you actually had a thousand lying around to invest, how many problems is a free 250 dollars going to solve?

I think the usual calculation doesn’t take into account the benefit of getting a really huge chunk of money all at once, and out of proportion to your usual income. Most people’s problems are scaled to their income, so getting ten years’ money in a day means, in effect, blowing all your problems out of the water – short, at least, of a really bad drug or gambling habit.

Me, I don’t do lotteries; they’re sort of boring. But if you do, perhaps you’re not irrational after all. 🙂



Filed under Economics

8 responses to “Lotteries

  1. Richard Campbell

    Yeah, if you bet a reasonably small amount (say, less than $5), to gain an amount sufficient to keep you from working again (say, more than $2 million, which yields an inflation-adjusted $100,000 annually if conditions stay anything close to normal), you are effectively betting 0 money against infinite money, which changes the odds significantly.

  2. kingofmen

    A point against this theory, though, is the data on what actually happens to lottery winners. A lot of them are just as poor three years later, and bitter with it.

  3. Carillon

    But that is because they blow it all on stupid crap, not because your original point was necessarily invalid.

  4. kingofmen

    Well, no. My original point does rather rely on the underlying statement “I won’t blow the million on stupid crap”, otherwise there is no 2 utility per million dollars. If anything, that makes the dollar calculation *over-estimate* the value.

  5. Nogbad

    I quite agree. If there is a lottery with a huge expected payout and a fairly low ticket price, it makes sense to buy one ticket. Mathematically I know what my expected payout is. But somebody will win, and if that somebody is me, then I can afford to laugh at all mathematical textbooks.

    But there is one thing to remember, of course: It makes sense to buy 1 ticket, it does not make 1000 times more sense to buy 1000 tickets. Dreams scale badly.

  6. kingofmen

    True about the scaling. There’s measure in all things made. 🙂

    Ah well, perhaps my thought wasn’t as original as all that, nobody seems to be arguing with me. Still, it’s odd that I haven’t seen this pointed out in formal economic analysis. Perhaps the economists tend to forget their own lesson from 101, that dollars do not equal utility.

  7. corp

    AAR reader here. If you are disappointed that nobody is arguing with you, I can help you out with that.

    You forecast the amount of utility one gets per dollar return to go *up* as the amount of dollars increases. This seems counter-intuitive, one would expect it to go down. A thousand dollars is worth a lot more to someone who has ten thousand dollars than to someone who has a million dollars.

    When you think about it, why on earth would a single dollar hold more utility value to a millionaire than to a poor man?

    Similarly, when you offer the opposite analysis I think you get it wrong. To someone who already has a million dollars sitting in the bank, maybe the 250 dollars on a thousand is indeed as insignificant as you find it. Talk to someone who isn’t already rich, however, and I suspect they will not give you the result that you came up with.

    I know as a poor college student I’d take that in a heartbeat. 250 dollars in pocket right now would pay for a months worth of booze right there, and there is plenty of utility in that :p

  8. kingofmen

    I’m saying that the utility of having all your (monetary) problems solved is larger than the utility of a month of free booze, no matter how nice that might be. And I am not arguing that a dollar is worth more to a millionaire; rather I am arguing that a million dollars in one chunk is worth more to a poor guy than a linear extrapolation from the worth of one dollar would indicate. Once he has that million, certainly, the value of the second million goes down quite a bit.

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