Tuesday, May 13, 2014

More fun with infinite fair lotteries

Imagine an infinite sequence of games such that you are nearly certain to win each one, but you're also certain to lose all but finitely many of them. This seems really absurd. But given an infinite fair lottery, it can be easily arranged. Suppose a secret natural number N is chosen in our infinite fair lottery. Let Gn be the following game, for n a natural number:

  • You win if N>n and you lose otherwise.
Your probability of winning Gn is P(N>n). But P(Nn) is zero or infinitesimal for each finite n. So, your probability of winning is within an infinitesimal of one. But it is guaranteed that you will win at most N−1 games. So, indeed, for each game, you're nearly certain to win it, and you're certain to lose all but finitely many.

Imagine placing bets on this game. If it costs a penny to play and the payoff is a dollar, you'll think it's a great deal: after all, you're nearly certain you will win. But if you play all the games, you will make only finitely many dollars, and lose infinitely many pennies.

Conclusion? I suppose the best one is that infinite fair lotteries are impossible.


Alexander R Pruss said...

Of course, it's crucial here that you don't find out whether you won or lost until all the games are committed to.

Alexander R Pruss said...

Here's a reformulation (with thanks to Josh Rasmussen for making me think about guessing).

For each n you're asked to guess whether N>n. You get a dollar if you guess right and pay a dollar if you guess wrong. You're always nearly certain that the answer is "yes", but nonetheless you know you will do better if you always guess "no".

Joshua Rasmussen said...

Hence: the surest way to maximize true beliefs is to believe, in every case, what is sure to be false.

More simply: certainly false beliefs are certainly nearly all true.