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

a) within the third round:

Boy one and Boy a pair of apprehend that if the provide is rejected then they'll get zero donuts, that is additionally the tie-breaking purpose. therefore Boy one can provide a minimum of one friedcake to Boy a pair of and keep fifty nine donuts.

If Boy a pair of rejects the provide she is going to get zero. therefore she is going to settle for the provide.

Equilibrium outcome are Boy one fifty nine friedcakes and Boy a pair of one donut.

(b) Second round;

Since each of them square measure alert to the end result within the third spherical. conjointly during this spherical, Boy a pair of needs to provide. therefore something but fifty nine donuts won't be acceptable by Boy one. And at fifty nine donuts Boy one are indifferent.

Hence, equilibrium amount are Boy one fifty nine donuts and Boy a pair of twenty one donuts.

(c) within the 1st round:

Now during this spherical, Boy one needs to provide to Boy a pair of and each square measure alert to the second spherical outcomes.

So during this scenario, something but twenty one donuts won't be acceptable to Boy a pair of. And Boy a pair of are indifferent at twenty one donuts.

Hence, equilibrium amount are Boy one seventy nine donuts. and Boy a pair of twenty one donuts.

(d) Boy one strategy are to supply Boy a pair of the quantity that Boy a pair of couldn't refuse, so game doesn't move to second spherical.

Boy a pair of strategy are to it game shouldn't move to the third spherical.

Equilibrium outcome is Boy one seventy nine donuts and Boy a pair of twenty one donuts.