Go here for the original puzzle
There is no finite fair amount.
I know it's counterintuitive in the extreme, but it's true.
Look at it this way. You have one chance in two of losing the first flip and earning $1; you can expect to win $.50 in this fashion.
If you win the first flip, on the other hand, you have a 50% chance, or a total chance of 1/4, of losing the second and gaining $2; again, you can expect to gain fifty cents.
The same can be said for each subsequent flip. Of course, realistically, my financial resources will not survive even to the thirteenth flip, since I am a college student, and those of any person in the world could not survive a win on the thirty-seventh.
Also, even if I had infinite dough, you would be a fool to bet tens of thousands of dollars on each game, since you would be extremely likely to go bankrupt before making it big. However, the reasoning is still valid in theory, as with many other things.
P.S. Please don't hardlink to this node, but to the original one, so that others can have the same joy of discovery you did.
And, beable, given that I will be paying you in any event, how is you initially paying me nothing of benefit? (Beable said in a writeup below this node that paying zero would be a fair amount)