(Pitman 3.4.9) Suppose we play the following game based on tosses of a fair coin. You pay me $10, and I agree to pay you $n 2 if heads comes up first on the nth toss. If we play this game repeatedly, how much money do you expect to win or lose per game over the long run?

Let's make a couple of assumptions to clarify the situation.  First, the coin flipping is fair, that is, each flip is independent of all the others and for each flip, the probabilities of heads and tails are both 1/2.  Second, you have enough money to pay me no matter how many tails are flipped before the first head.

Under those assumptions, the expected amount of money I will win in infinite.  

In decision theory, utility is often used to make decisions rather than money. If my utility is proportional to expected monitory payoff, I should pay whatever I can scrape up, my total assets.  For some reason, economists often assume utility functions have deminishing returns, and eventually flatten out.

In that case, the expected amount of utility payoff will be lower than the maximum utility.  What does that mean for this game?  It means it won't matter to me whether I get some large quantity of money like a trillion dollars, or any larger quantity of money, like a quadrillion dollars.  All my needs are met by a trillion dollars.  That's 240 dollars.  So I certainly shouldn't pay more than $40 to play the game.  As the utility function starts to flatten out earlier, perhaps $30 would come out to be a fair payment.