Coin tossing game

This is a question posted in the Wilmott forum some time ago and has attracted considerable attention. It looks like a simple problem (but I don't think it is).

In a fair coin tossing game, your payoff is the number of heads divided by the number of tosses (i.e., m heads of N tosses give you $m/N). You can quit anytime. What is the fair price of playing this game? (fair means that the house breaks even in the long run).

Answer to problem posted in previous blog (22/05/07):
In period 2, player B proposes. If A rejects, both get 0. So B proposes 1 for himself (hence 0 for A), since either way A gets 0 (A is indifferent between the two). Discounted to period 1, B's proposal is d.
In anticipation of B's proposal, in period 1, A proposes 1-d since he has no incentive to give B more than d.
Hence optimal solution is (A,B) = (1-d,d)