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the following question is a kind of Rubinstein bargaining model:

2 players, A and B, have 100dollars to divide between them. They agree to spend T days negotiating over this division.

The first day, A will make an offer, B either accepts or comes back with a counteroffer the next day, and on T day, B gets to make one final offer. If they cannot reach an agreement in T days, both players get 0 dollar.

Assuming that both A and B are having the same degree of impatience: A and B discount payoff in the future at a rate of r per day.

Finally, we assume that if a player is indifferent between two offers, he wil accept the one that is most preferred by his opponent.

This idea is that the opponent could offer some arbitrary small amount that would make the player strictly prefer one choice and that this assumption allows us to approximate such an"arbitrarily small amount" by zero. It turns out that there is a unique subgame perfect nash equilibrium of this bargaining game.

So the question is that, what is the SPNE in this alternating-offer bargaining game when T is even?

Am i making it much clearer?

Thanks for all the advice that has given on modifying this question, thanks for being patient for my stupid question.

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I think you need to be more specific about the rules of the game $p_1$ and $p_2$ are playing. – beauby Nov 23 '12 at 10:34
beauby i have modified it a bit, hope this could explain more about this question, thanks! – Steve Nov 23 '12 at 10:57
You should tell us what an offer is and what it means to accept an offer. – Michael Greinecker Nov 23 '12 at 11:16
what i only know is that there is an amount of money to be shared by 2 players, – Steve Nov 23 '12 at 11:46
and player 1 would take the first turn to give an offer to player 2(in T=1), what player 2 can do is to accept the offer and the game ends, if this is not the case, player 2 will give an offer to player 1 in the next period (T=2) then this time for player 1 to decide whether to accept it or give player 2 a new offer(in T=3). and the important thing is that, the money being shared today will cost more than that when they end the game in later period and the discounting factor is r. – Steve Nov 23 '12 at 11:51

I am not sure what you mean by "...the opponent could offer some arbitrary small amount." The first mover has advantage in a Rubinstein-type bargaining. So when it's A's turn to offer, the next-period subgame where B becomes first mover gives B a reservation utility that A must meet.

Replace "100 dollars" by division of a pie of size $1$. An offer is of the form $(x,y)$ where $x + y = 1$. $x$ is A's piece of the pie and $y$ for B.

SPNE by backward induction:

In the subgame starting at the last period $T$, B offers the division $(0, 1)$ and A accepts. (This being the last period, A has reservation utility $0$.)

So in period $T-1$, B has reservation utility $r$. A offers $(1-r, r)$.

In period $T-2$, A has reservation utility $r(1-r)$. B offers $(r(1-r), 1 - r(1-r))$.

In period $T-1$, B has reservation utility $r(1 - r(1-r))$. A offers $( 1-r(1-r(1-r)), r(1-r(1-r) )$.

Continue this way by induction gives the SPNE outcome.

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