I've been going through a problem solving book, and I'm a little stumped on the following question:
At each round, draw a number 1-100 out of a hat (and replace the number after you draw). You can play as many rounds as you want, and the last number you draw is the number of dollars you win, but each round costs an extra $1. What is a fair value to charge for entering this game?
One thought I had was to suppose I only have N rounds, instead of an unlimited number. (I'd then let N approach infinity.) Then my expected payoff at the Nth round is (Expected number I draw - N) = 50.5 - N. So if I draw a number d at the (N-1)th round, my current payoff would be d - (N-1), so I should redraw if d - (N-1) < 50.5 - N, i.e., if d < 49.5. So my expected payoff at the (N-1)th round is 49(50.5-N) + 1/100*[(50 - (N-1)) + (51 - (N-1)) + ... + (100 - (N-1))] = 62.995 - N (if I did my calculations correctly), and so on.
The problem is that this gets messy, so I think I'm doing something wrong. Any hints/suggestions to the right approach?