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'.)
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'.)
Managerial Economics: A Problem Solving Approach
5th Edition
ISBN:9781337106665
Author:Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor
Publisher:Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor
Chapter16: Bargaining
Section: Chapter Questions
Problem 16.4IP
Related questions
Question
Solve only if all subparts
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step 1: Defining the problem:
VIEWStep 2: a) To draw an extensive game and write the payoffs of Anna and Bess:
VIEWStep 3: (b)To find backward-induction solution:
VIEWStep 4: (c) To find the backward-induction solution case 1:
VIEWStep 5: (d) To find the backward-induction solution for case 2:
VIEWStep 6: (e) Money made by Saibo, Anna and Bess in equilibrium solution:
VIEWSolution
VIEWStep by step
Solved in 7 steps with 1 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, economics and related others by exploring similar questions and additional content below.Recommended textbooks for you
Managerial Economics: A Problem Solving Approach
Economics
ISBN:
9781337106665
Author:
Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor
Publisher:
Cengage Learning
Principles of Economics 2e
Economics
ISBN:
9781947172364
Author:
Steven A. Greenlaw; David Shapiro
Publisher:
OpenStax
Managerial Economics: A Problem Solving Approach
Economics
ISBN:
9781337106665
Author:
Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor
Publisher:
Cengage Learning
Principles of Economics 2e
Economics
ISBN:
9781947172364
Author:
Steven A. Greenlaw; David Shapiro
Publisher:
OpenStax