Question

Recall the “Nash Demand Game” from the presentation in which two siblings (A and B) must bargain for an inheritance of $1 million. Instead of the two submitting sealed proposals (Nash’s original story) or the two submitting potentially infinite offers and counteroffers to each other (Rubenstein’s bargaining game), suppose the will specifies the following procedure for splitting the inheritance: Sibling A will go first and submit an offer to Sibling B for how they are to split the $1 million. B can accept or reject this offer. If B accepts A’s offer, A and B are paid accordingly. If B rejects A’s offer, A and B will wait one year. After one year, B will submit an offer to A for how they are to split the $1 million. A can accept or reject this offer. If A accepts B’s offer, A and B are paid accordingly. If A rejects B’s offer, A will receive $200,000 and B will get nothing (i.e., the remaining $800,000 will go to charity). Assume that the siblings have identical discount factors of 0.75 (i.e., “a dollar one year from now is worth 75 cents to me today”). Also assume that both siblings are risk neutral, neither sibling receives any utility if the inheritance goes to charity, and neither sibling is worried about the fairness of the result. What will happen in this negotiation? How much of the inheritance will A receive and how much will B receive? (Hint: What happens in one year if B rejects A’s initial offer? What does that imply about what A’s initial offer must be for B to accept it?) Which sibling has the advantage under these rules?

Answer 1

Backward Induction Method

Period 02:

B proposes an offer to A.

If A accepts the offer then it will split the inheritance accordingly.

If A rejects the offer then A will get 200,000 and 80,000 will go to charity.

So, B will plan an offer which will make A accept the offer otherwise if A rejects then B will get nothing.

B realizes that if he refuses the offer, A will be receiving 200,000. So, B will agree to send A 200,000 and retain 800,000. In this case, A will consider the bid, as in either case he gets 200,000. In this scenario, A refuses the offer, and in the second cycle, A will get 200,000 and B will get 800,000.
A anticipates this situation in the first round and A would like to make an offer in the first round itself that B would accept. One knows that the outcome will be A getting 200,000 and B getting 800,000 in the second period.

The discount rate is 0.75. This is $1 in the second round in the first round, which is worth just $0.75. Therefore, the current inheritance value received by A is 0.75(200,000) = $150,000 and the current inheritance value gained by B is 0.75(800,000) = $600,000, respectively.

Period 01:
So, now A needs to make an offer B accepts. A realizes that if the game moves to the second round and A gets $150,000, B will get the equal of $600,000 in wealth. Therefore, in the first round, A must offer B $600,000 and retain him $400,000. So, B realizes the inheritance he gets in the second round is $600,000 equivalent and takes the bid. A will be better off that way because he is now getting $400,000 which is more than what he is receiving in the second round ($150,000).

A makes an offer to retain $400,000 and send B$600,000 in the first round and B will consider the deal.
Provided inheritance: A=$400,000 B= $600,000. Sibling B has an advantage under those rules.