Exercises on Auctions

Transcription

1 Exercises on Auctions 1) Consider sealed bid first price private value auctions where there are two bidders. Each player knows his own valuation and knows possible valuations of the other player and their respective probabilities. Namely the valuations of the two players, t1 and t2 are independent random variables, uniformly distributed between 0 and 100. What are the equilibrium bidding functions a * 1 (t 1 ) =? sealed bid private value auction? a 2 * (t 2 ) =? for the first price 2) Assume you are one of the two bidders in a private value first price auction where a grilled cheese sandwich, which purportedly bore a portrait 1 of the Virgin Mary, is being auctioned. The item is worth $10,000 to you. There is only one other rival bidder at the auction. You know that his valuation is uniformly distributed between 0 and $40,000. Assume that your rival is going to use his equilibrium bidding strategy. a) 1 The grilled cheese sandwich, which purportedly bore a portrait of the Virgin Mary. was sold for $28,000 in The winner of the auction was an online casino GoldenPalace.com. The Florida woman who had made it 10 years earlier said it never went moldy.

2 b) Assume that you learned that the grilled cheese sandwich is worth $14,000 to the second buyer. The second buyer won t be aware that you have this info and therefore would stick to his Bayes-Nash equilibrium strategy. c) Assume that you learned that the grilled cheese sandwich is worth $22,000 to the second buyer. The second buyer won t be aware that you have this info and therefore would stick to his Bayes-Nash equilibrium strategy. d) Assume the auction rules have changed. The highest bid still wins, but the winner has to pay only 75% of his bid. (Your rival is not mathematically sophisticated and would stick to his Bayes-Nash equilibrium strategy) The item is worth $10,000 to you. You know that your rival s valuation is uniformly distributed between 0 and $40,000.

3 3) Assume you are considering auctioning your beloved 2009 Honda Civic, as you near graduation. There are two potential bidders. One of them is a close friend and you know that he values your car at $2,000, the second bidder s valuation is uncertain with a uniform distribution between 0 and $5,000. The two bidders don t know each other s valuation exactly and they each guess that it is a random value with uniform distribution between 0 and $5,000 and they will use Bayes-Nash Equilibrium strategies in this auction. a) Calculate your expected revenue if you use a first-price auction to sell your car. b) Calculate your expected revenue if you use a second-price auction to sell your car. c) Which auction rule would you prefer, first price or second price?

4 4) a) What are the equilibrium bidding functions a 1 * (t 1 ) =? a 2 * (t 2 ) =? for a sealed bid private value second price auction? b) Assume you are one of the two bidders in a second price private value auction where the foam latex Spock ears 2 worn by Leonard Nimoy in Star Trek V is being auctioned. Suppose the Spock ears worth $8,000 to you. There is only one other rival bidder at the auction. You know that his valuation is uniformly distributed between 0 and $12, The foam latex ears with remnants of adhesive on the back were sold for $10,000. Leonard Nimoy wore the ears in Star Trek V which is considered by many one of the worse Star Trek movies ever.

5 5) Suppose you are the owner of a pristine copy of Action Comic No.1, the first comic book to feature Superman3. You would like to sell this item in a closed bid auction. There are two bidders who are interested in this comic book. a) Assume the bidders don t know each other s valuation exactly and they each guess that it is a random value with uniform distribution between 0 and $5 million. As a seller would you prefer a first price or second price auction? (You don t know the buyer valuations, but you know that they are uniformly distributed between 0 and 5 million) Explain. b) Now assume that you learned that the copy of Action Comic No.1 is worth $4 million to the first buyer and 3 million to the second buyer. The two players won t be aware that you have this info and they would stick to their Bayes-Nash equilibrium strategies. Calculate your expected revenue for a first price auction. Calculate your expected revenue for a second price auction. A pristine copy of Action Comic No.1, the first comic book to feature Superman, became the most expensive comic book when it sold for $3.2 million in

6 6) Consider sealed bid first price common value auctions where there are two bidders. The item has a common value, which is partly observed by the bidders. Each player knows his own observation and knows possible observations of the other player and their respective probabilities. Namely the observation of the two players, t1 and t2 are independent random variables, uniformly distributed between 0 and 100. The common value of the item is given by Common value = player 1 s observation + player 2 s observation. i.e. v = t1 + t2 What are the equilibrium bidding functions a * 1 (t 1 ) =? game? a 2 * (t 2 ) =? of this auction 7) FCC is selling rights to the new 10GHz band for long-range microwave communication using a common value auction. The common value of the item is equal to v = t1 + t2, where t1 and t2 are the observations of each bidder. The buyers don t know each other s observation and believe that it is distributed uniformly between 0 and 100. a) Assume you are a consultant to FCC. You learned that the observation of the first buyer is t1=40 and the observation of the second buyer is t2=60. Consider a first price common value auction. Calculate Government s revenue and the bids of each player in the following two cases. i) You secretly tell the first buyer, the observation of the second buyer. (second buyer is unaware of this and will stick to his Bayes-Nash equilibrium strategy) ii) You secretly tell the second buyer, the observation of the first buyer. (first buyer is unaware of this and will stick to his Bayes-Nash equilibrium strategy) i) Buyer 1 bids: Buyer 2 bids: Revenue from the auction: ii) Buyer 1 bids: Buyer 2 bids: Revenue from the auction:

7 b) Consider a first price common value auction again.the buyers don t know each other s observation and believe that it is distributed uniformly between 0 and 100. Assume the observation of the first buyer is t 1 = 40. What would be his optimal bid and his corresponding expected payoff?