S and J are trying to split 100 dollars. In bargaining round 1, S makes an offer at cost 0, proposing to keep S1 for himself and J either accepts (ending the game) or rejects. In round 2, J makes an offer at cost 10 of S2 for S and S either accepts or rejects. In round 3, S makes an offer of S3 at cost c, and J either accepts or rejects. If no offer is ever accepted, the 100 dollars goes to a third player, D.

Question

S and J are trying to split 100 dollars. In bargaining round 1, S makes an offer at cost 0, proposing to keep S1 for himself and J either accepts (ending the game) or rejects. In round 2, J makes an offer at cost 10 of S2 for S and S either accepts or rejects. In round 3, S makes an offer of S3 at cost c, and J either accepts or rejects. If no offer is ever accepted, the 100 dollars goes to a third player, D.

a) if c=0, what is the equilibrium outcome? what if c=80? c=10?
b) what happens if c=0, but Jones is very emotional and would spit S's face and throw the 100 dollars to D if S proposes S=100? Assume here that S knows J's personality perfectly.

Answer 1

a) To find the equilibrium outcomes for different values of c, let's analyze the bargaining process step by step for each scenario:

1. c = 0:
- In round 1, S proposes S1 = $100 for himself and offers J $0. J will accept this offer since it is more favorable than receiving nothing from D.
- Therefore, the equilibrium outcome is S1 = $100 for S and $0 for J.

2. c = 80:
- In round 1, S proposes S1 = $100 for himself and offers J $0. J rejects this offer since it is not beneficial for him.
- In round 2, J proposes S2 = $90 for S and offers himself $10. S will accept this offer since receiving $90 is better than nothing from D.
- Therefore, the equilibrium outcome is S2 = $90 for J and $10 for S.

3. c = 10:
- In round 1, S proposes S1 = $100 for himself and offers J $0. J rejects this offer.
- In round 2, J proposes S2 = $90 for S and offers himself $10. S rejects this offer.
- In round 3, S proposes S3 = $90 for himself and offers J $10. J will accept this offer since receiving $10 is better than nothing.
- Therefore, the equilibrium outcome is S3 = $90 for S and $10 for J.

b) In this scenario, if S proposes S = $100, and J would spit S's face and throw the $100 to D, we need to evaluate the impact of J's emotional response. Assuming S knows J's personality perfectly, the bargaining process changes as follows:

1. In round 1, since S is aware of J's response, he offers S1 = $0 for himself and offers J $100.
2. J reacts emotionally, spits S's face, and throws the $100 to D, resulting in J receiving $0 and S receiving $0 as well.
3. Therefore, the equilibrium outcome is $0 for both S and J, while D receives the $100.

Note: This analysis assumes perfect knowledge of J's personality by S.