|Game Theory and Business Strategy.|
Chapter 15 Problem 4 (1st Edition)
Chapter 16 Problem 6 (2nd Edition)
(a) If you offer $3,000, the current owner will sell to you only if the car's true value to him is less than $3,000. Since all values between $1,000 and $3,000 are equally likely, a car that you'll buy has an expected value of $2,000. By increasing its value by a third, on average, such a car will be worth (4/3) x $2,000 = $2,667 to you. If you offer $3,000, therefore, you can expect the transaction to cause you to lose $333 in value.
(b) Let o indicate the amount offered. Since your offer will only be accepted if the car is worth between $1000 and o the expected value of the car is:
(1000 + o) / 2Considering your repair skills, the car will be worth an additional 1/3 to you. Hence, the expected value of the car if you win is:
(4/3) × (1000 + o) / 2 = (2/3) × (1000 + o)To not lose money on the deal, your offer, o, has to be less than this expected value:
(2/3) x (1000+o) > oHence, any offer above 2000 will result in an expected loss. However, any offer below 2000 will result in an expected gain. This is the winner's curse adjustment. Offering the expected value of the car results in a loss, hence the bid needs to be revised downward to reflect the expected value of the car if you win it!
≡ (2/3) 1000 > (1/3) o
≡ 2000 > o