Question

Homer (H) and Lenny (L) are negotiating over how to divide a giant box of 100 donuts. H will engage L in up to three rounds of negotiations. The order of events is:

FIRST ROUND: H makes L an initial offer. L accepts or rejects. If L accepts, the game ends and H and L each get their donuts. If L rejects, 20 donuts get eaten by mice (because of the delay) and the game continues with 80 donuts to be divided.

SECOND ROUND: L makes an offer. H accepts or rejects. If H accepts, the game ends and H and L get their donuts. If H rejects, 20 donuts get eaten by mice (because of the delay) and the game continues with 60 donuts to be divided.

THIRD ROUND: H makes a final offer. L accepts or rejects. If L accepts, then H and L each get their donuts. If L rejects, the game ends and neither H nor L get any donuts (the mice eat the rest). Note: You should assume that both H and L wish to maximize their donut consumption. You should also apply a tie-breaking rule that both H and L will reject an offer of zero donuts.

(a) If we reach the third round of the game, how many donuts would Homer offer Lenny?

(b) Given that Homer and Lenny know the answer to (a) in round 2, how many donuts would Lenny offer Homer?

(c) Given that Homer and Lenny know the answer to (b) in round 1, how many donuts would Homer offer Lenny?

(d) What is Homer’s strategy (sH) in the first round of the game? What is Lenny’s strategy (sL) in the first round of the game? What is the equilibrium outcome of the game (“outcome”)?

Answer 1

(a) In the third round:

Homer and Lenny know that if the offer is rejected then they will get zero donuts, which is also the tie-breaking point. So Homer will offer at least one donut to lenny and keep 59 donuts.

If Lenny rejects the offer she will get zero. Hence she will accept the offer.

Equilibrium outcome will be Homer 59 donuts and Lenny 1 donut.

(b) Second round;

Since both of them are aware of the outcome in the third round. Also in this round, Lenny has to offer. So anything less than 59 donuts will not be acceptable by Homer. And at 59 donuts Homer will be indifferent.

Hence, equilibrium quantity will be Homer 59 donuts and Lenny 21 donuts.

(c) In the first round:

Now in this round, Homer has to offer to Lenny and both are aware of the second round outcomes.

So in this situation, anything less than 21 donuts will not be acceptable to Lenny. And Lenny will be indifferent at 21 donuts.

Hence, equilibrium quantity will be Homer 79 donuts. and Lenny 21 donuts.

(d) Homer strategy will be to offer Lenny the amount which Lenny could not refuse, so that game does not move to second round.

Lenny strategy will be to that game should not move to the third round.

Equilibrium outcome is Homer 79 donuts and Lenny 21 donuts.