Exercises on Auctions. What are the equilibrium bidding functions a * 1 ) = ) = t 2 2. ) = t 1 2, a 2(t 2

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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 t are independent random variables, uniformly distributed between 0 and 100. What are the equilibrium bidding functions a * 1 ) =? sealed bid private value auction? a * (t ) =? for the first price Equilibrium bidding functions are: a 1, a (t In first price sealed bid private value auction with two bidders, players bid half of their valuation. ) 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) Since my rival is going to use his equilibrium bidding strategy, he would use a (t Since my valuation is $10,000 (t1 = $10,000), I would bid a 1 ) = With a bid of $5,000, I would win if my bid exceeds my rival s bid: 5,000 > a à 5,000 > t/ > t < 10,000 So I d win if my rival s valuation is less than $10,000. Prob ( t < 10,000)= ¼ My expected payoff = ¼(10,000-5,000) = $1,50 10, 000 = 5, The grilled cheese sandwich, which purportedly bore a portrait of the Virgin Mary. was sold for $8,000 in 004. 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. Since the second buyer won t be aware that I have this info and would stick to his equilibrium strategy and use a (t I would then bid = 7,000 + ε My expected payoff = 10,000 (7,000 + ε) = 3,000 - ε. he d bid a = 14,000/ = $7,000 c) Assume that you learned that the grilled cheese sandwich is worth $,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. Since the second buyer won t be aware that I have this info and would stick to his equilibrium strategy, he d use a (t and bid a=,000/ = $11,000 I would not want to win this auction and therefore, I d bid less than 11,000. (Any bid < 11,000 would work). My expected payoff would be zero. 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. Since my rival d stick to his equilibrium strategy, he d use a (t find what would be our optimal bid, a * 1? we need to When would I win with a bid of a 1? With a bid of a 1 I would win if my bid exceeds my rival s bid: a 1 > a à a 1 > t/ > t < a 1 So I d win if my rival s valuation is less than a 1. Prob ( t < a 1 )= a 1 /100 Note that when I win, I ll be asked to pay 0.75 a 1 according to the auction rules. EP = My expected payoff = a 1 /100(10, a 1 ) = 00 a a 1 /100 dep / da 1 = 00 3a 1 /100 = 0 à a 1 =$6,666.6

3 3) Assume you are considering auctioning your beloved 009 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 $,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. The bidders will use equilibrium strategies given by a 1, a (t You know player 1 s valuation t1 = 000, and therefore a1=1000. Two possible cases: 1) player 1 will win if a < a > t/ < > t < 000, Prob (t < 000) = 0.4 So with probability 0.4 and you will get a1 =1000 as your revenue. ) player will win if a > a > t / > > t > 000, Prob (t > 000) = 0.6 So with probability 0.6 and you will get player s bid a as your revenue. When player wins, t > 000. t~[,000, 5000] then his bids a~[1,000,,500] on the average you will get the average a which is $1750. Therefore your expected revenue will be 0.4 * *1750= $1,450 b) Calculate your expected revenue if you use a second- price auction to sell your car. The bidders will use equilibrium strategies given by a 1, a (t You know player 1 s valuation t1=000, two possible cases: 1) player 1 will win if a < a > t < 000 Prob (t < 000) = 0.4 So with probability 0.4 and you will get player s bid, a as your revenue. a is distributed between [0, 000], so on the average, you will get $1000 as your revenue. ) player will win if a > a > t > 000, Prob (t > 000) = 0.6 So with Probability 0.6 and you will get player 1 s bid, a1=$000 as your revenue. Therefore your expected revenue will be 0.4 * *000= $1600. c) Which auction rule would you prefer, first price or second price? Second price auction since 1,600 > 1,450

4 4) a) What are the equilibrium bidding functions a 1 * ) =? a * (t ) =? for a sealed bid private value second price auction? In a second price sealed bid private value auction, the equilibrium bidding functions are: a 1, a (t which are weakly dominant strategies. b) Assume you are one of the two bidders in a second price private value auction where the foam latex Spock ears 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 $1,000. Since a 1 I would bid a 1 = 8, 000. We know that my rival will bid a (t With a bid of $8,000, I would win if my bid exceeds my rival s bid: 8,000 > a à 8,000 > t > t < 8,000 So I d win if my rival s valuation is less than $8,000. i.e. t~(0, 8,000) Prob( t < 8,000 ) = 8,000/1,000 = /3 When I win I would have to pay my rival s bid since this s a second price auction. We need to estimate the range of his bids (hence the price I pay) when I win. Since I d win if my rival s valuation is less than $8,000, t~(0, 8,000) and a (t So his bids, a~(0,8,000). So my rival s average bid, (hence the average price I will pay) is (0+8,000)/ = 4,000 My expected payoff = /3(8,000-4,000) = $,666.6 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. In a sealed bid auction where the valuation of the bidders are private knowledge, First price auction and Second price auction rules bring the same expected revenue for the seller. Therefore I d be indifferent. 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. In a first price private value auction, the equilibrium bidding functions are: t t 4 3 a1 (t1 ) = 1, a (t ) = So a1 (t1 ) = = and a (t ) = = 1.5 The first bidder would win and pay $m. So my expected revenue would be $ million. In a second price private value auction, the equilibrium bidding functions are: a1 (t1 ) = t1, a (t ) = t which are weakly dominant strategies. So a1 (t1 ) = 4 and a (t ) = 3 The first bidder would win and pay $3m. (the second highest price). So my expected revenue would be $3 million. 3 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. million in 015.

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 t 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 s observation. i.e. v = t1 + t What are the equilibrium bidding functions a * 1 ) =? game? a * (t ) =? of this auction In a first price common value auction the equilibrium bidding functions are: a 1, a (t 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 + t, where t1 and t 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 t=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: a 1 = 60 +ε Buyer bids: a = 60 Revenue from the auction: 60 +ε ii) Buyer 1 bids: a 1 = 40 Buyer bids: a = 40 +ε Revenue from the auction: 40 +ε

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? Since a 1 he would bid a 1 = 40 and we know that player would use a (t With a bid of $40, I would win if my bid exceeds my rival s bid: 40 > a à 40 > t > t < 40 So I d win if my rival s observation is less than $40. i.e. if t is in this range t~(0,40). Now we have to estimate the probability of player 1 winning, as well as the common value of the item when player 1 wins. Probability of player 1 winning is the probability that t < 40 which is given by Prob( t < 40 ) = 0.4 Remember the common value of the item is v = t1 + t We know t1 = 40 and t~(0,40) when player 1 wins. So the common value of the item will be in the range v= t1 + t~(40, 80) The average common value will be = (40+80)/=60 My expected payoff = 0.4(60-40) = $8.