Question $3$: Abe and Beth

When Abe graduated from LMU, he bought a diamond ring with a lion insignia to impress his high school friends. The person most impressed was his old sweetheart, Be he $.$ They grew so fond of each other that Abe asked Beth to wear his ring. Then the quarreling began. The relationship deteriorated until, in a fit of anger, Beth enrolled as a student at Pepperdine. Abe, naturally, broke off the relationship and asked for the return of his ring. Beth replied that it was a gift to her and she was not about to return it$.$ He answered that he had only loaned it to her to wear for the duration of their friendship.

Abe and Beth are now involved in a legal dispute over the ring. Beth possesses the ring and she declares that she intends to sell it$.$ Abe threatens to sue her. If Abe wins at trial, the court will require Beth to return the ring to him. The ring is worth $$1,000$ to Abe. If Beth wins at trial, the court will allow her to keep the ring. She would then sell the ring for $$400$ to an acquaintance. $($The acquaintance does not know Abe and will not resell the ring to him.$)$ In the event of a trial, each one expects to win with probability $0\mathrm{.}5.$ A trial will cost Abe $$250$ and it will also cost Beth $$250.$

Abe and Beth start negotiating to reach a settlement and avoid a trial. The costs of settling out of court are nil.

Questions:

A$)$ What is Abe's threat value?

B$)$ What is Beth's threat value?

C$)$ What is the total value of the noncooperative solution?

D$)$ What is the total value of cooperation?

E$)$ What is the surplus from cooperation?

F$)$ The "reasonable" solution would be for Beth to return the ring to Abe and for Abe to pay Beth how much?

G$)$ If the judge were to decide the case by applying the "normative Hobbes Theorem," who would win the case and get the ring? Abe or Beth?