The second problem with using the law of large numbers to justify EUM has to do with a mathematical theorem known as gambler’s ruin. Imagine that you and I flip a fair coin, and I pay you $1 every time it comes up heads and you pay me $1 every time it comes up tails. We both start with $100. If we flip the coin enough times, one of us will face a situation in which the sequence of heads or tails is longer than we can afford. If a long-enough sequence of heads comes up, I’ll run out of $1 bills with which to pay you. If a long-enough sequence of tails comes up, you won’t be able to pay me. So in this situation, the law of large numbers guarantees that you will be better off in the long run by maximizing expected utility only if you start the game with an infinite amount of money (so that you never go broke), which is an unrealistic assumption. (For technical convenience, assume utility increases linearly with money. But the basic point holds without this assumption.)
I don’t understand. The final result is 50% for $0 and 50% for $200. Expected money is $100, same as if I didn’t play. What’s the problem?
I don’t understand. The final result is 50% for $0 and 50% for $200. Expected money is $100, same as if I didn’t play. What’s the problem?