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Case 1: the legal owner is Anna. Let us assume that this implies that C₁ > C. Case 2: the legal owner is Bess. Let us assume that this implies that C₂ > CA (c) Find the backward-induction solution for Case 1 and show that it implies that the ring goes to Anna. (d) Find the backward-induction solution for Case 2 and show that it implies that the ring goes to Bess. (e) How much money does Sabio make in equilibrium? How much money do Ann and Bess end up paying in equilibrium? (By 'equilibrium' we mean 'backward induction solution'.)

Case 1: the legal owner is Anna. Let us assume that this implies that C₁ > C. Case 2: the legal owner is Bess. Let us assume that this implies that C₂ > CA (c) Find the backward-induction solution for Case 1 and show that it implies that the ring goes to Anna. (d) Find the backward-induction solution for Case 2 and show that it implies that the ring goes to Bess. (e) How much money does Sabio make in equilibrium? How much money do Ann and Bess end up paying in equilibrium? (By 'equilibrium' we mean 'backward induction solution'.)

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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000000000000 Exercise 2.13: Challenging Question. Two women, Anna and Bess, claim to be the legal owners of a diamond ring that each claims - has great sentimental value. Neither of them can produce evidence of ownership and nobody else is staking a claim on the ring. Judge Sabio wants the ring to go to the legal owner, but he does not know which of the two women is in fact the legal owner. He decides to proceed as follows. First he announces a fine of $F > 0 and then asks Anna and Bess to play the following game. Move 1: Anna moves first. Either she gives up her claim to the ring (in which case Bess gets the ring, the game ends and nobody pays the fine) or she asserts her claim, in which case the game proceeds to Move 2. Move 2: Bess either accepts Anna's claim (in which case Anna gets the ring, the game ends and nobody pays the fine) or challenges her claim. In the latter case, Bess must put in a bid, call it B, and Anna must pay the fine of $F to Sabio. The game goes on to Move 3. Move 3: Anna now either matches Bess's bid (in which case Anna gets the ring. Anna pays $B to Sabio in addition to the fine that she already paid - and Bess pays the fine of $F to Sabio) or chooses not to match (in which case Bess gets the ring and pays her bid of $B to Sabio and, furthermore, Sabio keeps the fine that Anna already paid). Denote by C, the monetary equivalent of getting the ring for Anna (that is, getting the ring is as good, in Anna's mind, as getting $C) and C, the monetary equivalent of getting the ring for Bess. Not getting the ring is considered by both as good as getting zero dollars. (a) Draw an extensive game with perfect information to represent the above situation, assuming that there are only two possible bids: B, and B. Write the payoffs to Anna and Bess next to each terminal node. (b) Find the backward-induction solution of the game you drew in part (a) for the case where B, > C > C > B > F> 0. Now consider the general case where the bid B can be any non-negative number and assume that both Anna and Bess are very wealthy. Assume also that CA, C and F are positive numbers and that CA and C, are common knowledge between Anna and Bess. We want to show that, at the backward- induction solution of the game, the ring always goes to the legal owner. Since we (like Sabio) don't know who the legal owner is, we must consider two cases. Case 1: the legal owner is Anna. Let us assume that this implies that CA > C Case 2: the legal owner is Bess. Let us assume that this implies that C > CA- (c) Find the backward-induction solution for Case 1 and show that it implies that the ring goes to Anna. (d) Find the backward-induction solution for Case 2 and show that it implies that the ring goes to Bess. (e) How much money does Sabio make in equilibrium? How much money do Ann and Bess end up paying in equilibrium? (By 'equilibrium' we mean 'backward induction solution'.)
Transcribed Image Text:000000000000 Exercise 2.13: Challenging Question. Two women, Anna and Bess, claim to be the legal owners of a diamond ring that each claims - has great sentimental value. Neither of them can produce evidence of ownership and nobody else is staking a claim on the ring. Judge Sabio wants the ring to go to the legal owner, but he does not know which of the two women is in fact the legal owner. He decides to proceed as follows. First he announces a fine of $F > 0 and then asks Anna and Bess to play the following game. Move 1: Anna moves first. Either she gives up her claim to the ring (in which case Bess gets the ring, the game ends and nobody pays the fine) or she asserts her claim, in which case the game proceeds to Move 2. Move 2: Bess either accepts Anna's claim (in which case Anna gets the ring, the game ends and nobody pays the fine) or challenges her claim. In the latter case, Bess must put in a bid, call it B, and Anna must pay the fine of $F to Sabio. The game goes on to Move 3. Move 3: Anna now either matches Bess's bid (in which case Anna gets the ring. Anna pays $B to Sabio in addition to the fine that she already paid - and Bess pays the fine of $F to Sabio) or chooses not to match (in which case Bess gets the ring and pays her bid of $B to Sabio and, furthermore, Sabio keeps the fine that Anna already paid). Denote by C, the monetary equivalent of getting the ring for Anna (that is, getting the ring is as good, in Anna's mind, as getting $C) and C, the monetary equivalent of getting the ring for Bess. Not getting the ring is considered by both as good as getting zero dollars. (a) Draw an extensive game with perfect information to represent the above situation, assuming that there are only two possible bids: B, and B. Write the payoffs to Anna and Bess next to each terminal node. (b) Find the backward-induction solution of the game you drew in part (a) for the case where B, > C > C > B > F> 0. Now consider the general case where the bid B can be any non-negative number and assume that both Anna and Bess are very wealthy. Assume also that CA, C and F are positive numbers and that CA and C, are common knowledge between Anna and Bess. We want to show that, at the backward- induction solution of the game, the ring always goes to the legal owner. Since we (like Sabio) don't know who the legal owner is, we must consider two cases. Case 1: the legal owner is Anna. Let us assume that this implies that CA > C Case 2: the legal owner is Bess. Let us assume that this implies that C > CA- (c) Find the backward-induction solution for Case 1 and show that it implies that the ring goes to Anna. (d) Find the backward-induction solution for Case 2 and show that it implies that the ring goes to Bess. (e) How much money does Sabio make in equilibrium? How much money do Ann and Bess end up paying in equilibrium? (By 'equilibrium' we mean 'backward induction solution'.)
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