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Lotteries and Expectation Values

Expectation values are nifty things, and are useful for many purposes … but not, I think, for all purposes. In particular, there is a class of games, similar to lotteries, for which expectation values are misleading.

The problem is easiest to see in games that are played only once. Suppose I offer you the following deal: you pay me a hundred dollars, and in return, I give you a one-in-a-million chance of winning a billion dollars. Suppose, too, that the deal is verifiable … that I have a billion dollars, that I'll put it in escrow, that the chance is random and fair, and so on. Should you play the game?

The expectation value of the game is $900, nice and positive, so according to conventional wisdom, you should. I'm not convinced, though. It seems to me that the odds of winning are just too low—if you play, you are just giving away your money.

What about games that are played more than once? Well, it turns out that any such game can be converted into a game that is played only once, as follows. First, pick an approximate number of times you would consider playing, or perhaps could play over the course of your life. Then, roll up that many copies of the game: imagine playing that many games all at once, look at the various possible outcomes and their probabilities, and consider that to be a single game.

Then, of course, if the odds of winning that single game are too low, you shouldn't play.

In case that explanation of rolled-up games isn't clear, here's a quick example. Suppose the game is that you flip a coin and call it, and I pay you a dollar if you call it correctly; and suppose we play the game six times. Here are the outcomes and probabilities for the rolled-up game.

1615201561/ 64

For games that are similar to lotteries it's actually quite easy to compute the odds of winning the rolled-up game. Small probabilities are essentially the same as risks, and so can be added. As a result, the probability of winning the rolled-up game is about the same as the probability of winning a single game times the number of times the game is played. I could, for example, let you play the billion-dollar game a hundred times, and the odds of winning would still only be about one in ten thousand.

Actually, speaking of rolled-up games, here's an interesting thing: there is a mathematical basis for the conventional wisdom that games with positive expectation values are worth playing! That basis is known as the law of large numbers, and what it tells you, among other things, is that by playing such a game enough times, you can make the odds of winning the rolled-up game as high as you like. The problem with games that are similar to lotteries is simply that the number of times you play is not large enough.

You have to be careful what claim you make, though. If the result for a single game has mean m and standard deviation s, then the average result per round for the rolled-up game has mean m and standard deviation s/(root n). So, if you make n large enough, the standard deviation will be small, and the result will almost certainly be positive. You might, for example, require that the standard deviation be less than the mean, which translates to

n > (s/m)2.

The total result for the rolled-up game, however, has mean mn and standard deviation s × (root n). So, the standard deviation does not become small, and the result is not guaranteed to converge to mn.

So, that's my theory about lotteries and expectation values. Just for fun, here are some other kinds of things that can be rolled up: decisions (Accumulated Notes) and strategies in iterated games (The Shape of Strategy Space).


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@ July (2002)