A clever argument for buying lottery tickets

by RolfAndreassen2 min read4th Nov 201135 comments


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I use the phrase 'clever argument' deliberately: I have reached a conclusion that contradicts the usual wisdom around here, and want to check that I didn't make an elementary mistake somewhere.

Consider a lottery ticket that costs $100 for a one-in-ten-thousand chance of winning a million dollars, expected value, $100. I can take this deal or leave it, and of course a realistic ticket actually costs 100+epsilon where epsilon covers the arranger's profit, which is a bad deal.

But now consider this deal in terms of time. Suppose I've got a well-paid job in which it takes me an hour to earn that $100. Suppose further that I work 40 hours a week, 50 weeks a year, and that my living expenses are a modest $40k a year, making my yearly savings $160k. Then, with 4% interest on my $160k yearly, it would take me about 5.5 years to accumulate that million dollars, or 11000 hours. Also note that with these assumptions, once I have my million I don't need to work any more.

It seems to me that, given the assumptions above, I could view the lottery deal as paying one hour of my life for a one-in-ten-thousand chance to win 11000 hours, expected value, 1.1 hours. (Note that leisure hours when young are probably worth more, since you'll be in better health to enjoy it; but this is not necessary to the argument.)

Of course it is possible to adjust the numbers. For example, I could scrimp and save during my working years, and make my living expenses only 20k; in that case it would take me less than 5 years to accumulate the million, and the ticket goes back to being a bad deal. Alternatively, if I spend more than 40k a year, it takes longer to accumulate the million; in this case my standard of living drops when I retire to live off my 4% interest, but the lottery ticket becomes increasingly attractive in terms of hours of life.

I think, and I could be mistaken, that the reason this works is that the rate at which I'm indifferent between money and time changes with my stock of money. Since I work for 8 hours a day at $100 an hour, we can reasonably conclude that I'm *roughly* indifferent between an hour and $100 at my current wealth. But I'm obviously not indifferent to the point that I'd work 24 hours a day for $2400, nor 0 hours a day for $0. Further, once I have accumulated my million dollars (or more generally, enough money to live off the interest), my indifference level becomes much higher - you'd have to offer me way more money per hour to get me to work. Notice that in this case I'm postulating a very sharp dropoff, in that I'm happy to work for $100 an hour until the moment my savings account hits seven digits, and then I am no longer willing to work at all; it seems possible that the argument no longer works if you allow a more gradual change in indifference, but on the other hand "save to X dollars and then retire" doesn't seem like a psychologically unrealistic plan either.

Am I making any obvious mistakes? Of course it may well be the case that the actual lottery tickets for sale in the real world do not match the wages-and-savings situations of real people in such a way that they have positive expected value; that's an empirical question. But it does seem in-principle possible for an epsilon chance at one-over-epsilon dollars paid out right away to be of positive expected value after converting to expected hours of life, even though it's neutral in expected dollars. Am I mistaken?


Edit: Wei Dai found the problem. Briefly, the 100 dollars added to my savings would cut more than 1.1 hours off the time I had to work at the end of the 5.5 years. 

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