Question

My sons (“Boy 1” and “Boy 2”) are negotiating over how to divide a pile of 20 chocolates. Boy 1 will engage Boy 2 in up to three rounds of negotiations. The order of events is:

FIRST ROUND: Boy 1 makes Boy 2 an initial offer. Boy 2 accepts or rejects. If he accepts, the game ends and the two boys get their chocolates. If Boy 2 rejects, I punish them for not working together by eating 5 chocolates myself. The game then continues with 15 chocolates to be divided.

SECOND ROUND: Boy 2 makes an offer. Boy 1 accepts or rejects. If he accepts, the game ends and the two boys get their chocolates. If Boy 1 rejects, I eat 5 chocolates myself and the game continues with 10 chocolates to be divided.

THIRD ROUND: Boy 1 makes a final offer. Boy 2 accepts or rejects. If he accepts, the two boys get their chocolates. If Boy 2 rejects, the game ends and I eat all the remaining chocolates. Note that I am expecting you to make an assumption of spiteful players, all else equal. Said another way, I assume that “reject” will break the indifference of getting zero either way...

(a) (5) If we reach the third round of the game, what would be Boy 1’s offer?

(b) (5) Given that my sons know (a) in round 2, what would be Boy 2’s 2nd Round offer?

(c) (5) Given that my sons know (b) in round 1, what would be Boy 1’s 1st Round offer?

(d) (10) What is Boy 1’s first-round strategy? What is Boy 2’s first-round strategy? What is the equilibrium outcome of the game?

Answer 1