Question

Problem 1. When Abe graduated from Texas A&M, he bought a diamond ring with a golden insignia to impress his high school friends. The person most impressed was his old sweetheart, Beth. 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 University of Texas. 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 $600 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 0.5. 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. Assume that these numbers are common knowledge for all parties.

Abe's expected value from going to trial is:

Beth's expected value from going to trial is:

The bargaining surplus over which the two are negotiating is:

If Abe and Beth were to split the bargaining surplus evenly, what kind of exchange (ring and currency) should take place?

A) Abe should let Beth keep the ring in exchange of her giving him $200.

B) Beth should give Abe the ring in exchange of him giving her $600.

C) Beth should give Abe the ring in exchange of him giving her $400.

D) Beth should give Abe the ring in exchange of him giving her $250.

E) Abe should let Beth keep the ring in exchange of her giving him $1000.

Answer 1

For Abe

Ring worth = $1000

Trial fee = $250

If Abe going the trail, he will get a total amount worth (1000 - 250 = ) $750 if win and will cost -$250 if lose

The expected value is obtained using the formula

X	P(X=x)	X*P(X=x)
750	0.5	375
-250	0.5	-125
	Sum	250

For Beth

Ring worth = $600

Trial fee = $250

If Beth going the trail, she will get a total amount worth (600 - 250 = ) $350 if won and will cost -$250 if lose

X	P(X=x)	X*P(X=x)
350	0.5	175
-250	0.5	-125
	Sum	50

B) Beth should give Abe the ring in exchange in him giving her $600.

Explanation:

A) In this case Abe will get ( = - $1000 + $ 200) = - $800 and Beth will get ( = $600 - $ 200) = $400

B) In this case Abe will get ( = $1000 - $ 600) = $400 and Beth will get ( = - $600 + $ 600) = $0

C) In this case Abe will get ( = $1000 - $ 400) = $600 and Beth will get ( = - $600 + $ 400) = - $200

D) In this case Abe will get ( = $1000 - $ 250) = $750 and Beth will get ( = - $600 + $ 250) = - $350

E) In this case Abe will get ( = - $1000 + $ 1000) = $0 and Beth will get ( = $600 - $ 1000) = - $400

In case B) there is no loss for each of them