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)

Game theory

A simple game-theoretic model can provide insights to a problem. It is another tool used in the analysis of the interaction of players. Everyday life poses game theory problems, e.g. firms deciding whether to enter a particular industry, how companies set prices in view of the actions of competitors, how one responds to a given incentive or disincentive, etc.

My microeconomics exam is on this Friday. I have never liked this subject, and having lecturers like mine didn't further help the cause. Anyway, in preparation for the paper, I was forced to read up and found to my liking, the simple applications of game theory (though Nash equilibrium is an easy concept, the solutions can sometimes be a little illogical at first glance).

A simple problem:
Suppose there are 2 players (A and B) bargaining over a cake of size 1 in 2 periods. There is a common discount factor, d (if you have 1 in period 2, that means it is worth d in period 1).
In period 1, player A proposes a share x for himself (and hence 1-x for player B). If B accepts, the game ends, otherwise we go to period 2.
In period 2, B proposes a share y for himself. The game ends after period 2.
What are the optimal responses for both players?

ps. You should watch "A Beautiful Mind" if you haven't already done so. It's one of my all-time favourite films.