Exercise 3.13 —*** 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 Cд the monetary equivalent of getting the ring for Anna (that is, getting the ring is as good, in Anna's mind, as getting $CA) and CB the monetary equivalent of getting the ring for Bess. Not getting the ring is considered by both to be 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: B1 and B2. Write the payoffs to Anna and Bess next to each terminal node.
Exercise 3.13 —*** 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 Cд the monetary equivalent of getting the ring for Anna (that is, getting the ring is as good, in Anna's mind, as getting $CA) and CB the monetary equivalent of getting the ring for Bess. Not getting the ring is considered by both to be 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: B1 and B2. Write the payoffs to Anna and Bess next to each terminal node.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:Exercise 3.13 —*** 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 Cд the monetary equivalent of getting the ring for Anna (that is, getting the
ring is as good, in Anna's mind, as getting $CA) and CB the monetary equivalent of
getting the ring for Bess. Not getting the ring is considered by both to be 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: B1 and B2. Write the payoffs to
Anna and Bess next to each terminal node.
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