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 CA 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 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 B₂. Write
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 CA 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 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 B₂. Write
Chapter1: Making Economics Decisions
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Transcribed Image Text: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 CA 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 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 B₂. Write
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