A Puzzle of Infinitely Many Mutually Independent Bets

Abstract

I present a new puzzle where a player is offered infinitely many mutually independent bets each of which has a positive expected return but which collectively lead to a net loss almost surely with probability 1. This puzzle is similar to the one presented by Vann McGee (1999) where there are infinitely many mutually dependent bets each of which has a positive expected return but whose cumulative outcome is a sure loss. Both puzzles pose a serious threat (while my new puzzle, because of the mutual independence of the bets, poses a more serious threat) to the principle of expected utility maximization because the principle seems to recommend us to take each and every one of the bets whereas it does not seem rational to take all of them. I then argue that the two puzzles can be solved in a uniform way: It is impossible to take all of the bets if you have only a finite amount of money and it is not irrational to take them all if you have an infinite amount of money.