[bobman0330]

NE isn't ideally suited to this game. Firstly, if it's a finitely repeated game, the only NE is for B to get only 1 every round. He might play some head games, but none of his threats are strictly credible.

If this game is played for a randomly-determined, unknown to the players length of time, then a different concept comes into play, what is called an "Evolutionarily Stable Strategy" (ESS). The principle behind this is that player B will refuse to accept certain bargains. His threat is given credibility because there is no final round where he has to revert to the accepting 1 strategy.

What makes B willing to reject on any given round, an immediately -EV move, is his expectation that he will get a better deal on future rounds. Future rounds have to be discounted according to the probability that they will not occur. For ease of calculation, assume the rule is that each round there's a K% chance that the game terminates. B's future benefit for refusing to agree to anything less than N, assume the offer is X is (N-X)(1-K) + (N-X)(1-K)^2 + ... = (N-X)(1-K)/K. His loss is X + X(1-K) + X (1-k)^2 = X / K.

Now, the downside of B's strategy is that he refuses to accept any payoff less than N, so he has to have a strong enough future incentive to refuse an immediate payoff of X=N-1 in order for his strategy to be ESS. (otherwise A could offer X=N-1 and "break his will.")

So, (N-X)(1-K)/K >= X/K when X=N-1.

(N-N+1)(1-K)/K = (N-1)/K
1-K = N-1
N < 2-K
N=1.

Now, that seems like it can't possibly be correct, but I can't find the problem in my math/logic. Anyone?