fair value of a hat-drawing game 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?

Your expected return if you draw a number on the last round is 49.5 (because it costs a dollar to make the draw).  On round N-1, you should keep what you have if it is greater than 49.5, or take your chances if it is less.  The expected value if N=2 is then $\frac {51}{100}\frac {100+50}{2} -1 + \frac {49}{100}49.5=61.505$ where the first term is the chance that you keep the first draw times the expectation of that draw (assuming you will keep it), the second is the cost of the first draw, and the third is the chance that you will decline the first draw and take your chances on the second times the expectation of the second draw.
Added:  As Yuval makes more explicit, your strategy will be to quit when you get a number at least $X$.  The gain then is $\frac{100+X}{2}-\frac{100}{101-X}$ where the first is the payoff and the second is the cost of the expected number of plays.  As he says, this is maximized at X=87 with value $\frac{1209}{14}=86.3571$.  I'll have to think where I was a bit off.
