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.

a) In this scenario, we will determine the equilibrium outcomes for different values of 'c'. The equilibrium outcome refers to the result that both S and J would agree upon if they act rationally and optimize their own utility.

If c = 0:
In round 1, S proposes an offer where S1 = 100 (keeping the entire $100 for himself), and J has to either accept or reject the offer. Since J's utility from accepting the offer is 0, he will reject it and the game proceeds to round 2.

In round 2, J proposes an offer where S2 = 100 (keeping the entire $100 for himself), and S has to either accept or reject the offer. Since S's utility from accepting the offer is 0, he will reject it. As a result, there is no agreement between S and J, and the $100 goes to D.

The equilibrium outcome when c = 0 is that the $100 goes to D.

If c = 80:
In round 1, S proposes an offer where S1 = 100 (keeping the entire $100 for himself), and J has to either accept or reject the offer. Since J's utility from accepting the offer is 0, he will reject it, knowing that he can potentially get a better offer in the future rounds.

In round 2, J proposes an offer where S2 = 90 (giving $90 to S and keeping $10 for himself), and S has to either accept or reject the offer. Since S's utility from accepting the offer is lower than if he waits for the next round, he will reject it.

In round 3, S proposes an offer where S3 = 100 (keeping the entire $100 for himself), and J has to either accept or reject the offer. Since J's utility from accepting the offer is 0, he will reject it, and the $100 goes to D.

The equilibrium outcome when c = 80 is still that the $100 goes to D.

If c = 10:
In round 1, S proposes an offer where S1 = 100 (keeping the entire $100 for himself), and J has to either accept or reject the offer. Since J's utility from accepting the offer is 0, he will reject it, knowing that he can potentially get a better offer in the future rounds.

In round 2, J proposes an offer where S2 = 90 (giving $90 to S and keeping $10 for himself), and S has to either accept or reject the offer. Since S's utility from accepting the offer is higher than if he waits for the next round, he will accept it.

The equilibrium outcome when c = 10 is that S receives $90, and J keeps $10.

b) If c = 0, but J is very emotional and would spit on S's face and throw the $100 to D if S proposes S = 100, then the outcome would be different.

In this scenario, S knows J's personality perfectly and is aware of the extreme emotional response of J. Therefore, S would not propose S = 100 in round 3 to avoid the negative consequences.

The equilibrium outcome would still be that the $100 goes to D since neither S nor J would make an offer that results in an agreement.