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Private Information and Auctions

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Auction Situations Private Value – Everybody knows their own value for the object – Nobody knows other peoples values. Common Value – The object has some ``true value that it would be worth to anybody – Nobody is quite sure what it is worth. Different bidders get independent hints.

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Auction types English Auction – Bidders see each others bids, bid sequentially. Bidding continues until nobody will raise bid. – Object is sold to highest bidder. Sealed bid first price auction – Bidders each submit a single bid. – Object is sold to highest bidder at high bidders bid Sealed bid second price auction – Bidders submit a single bid – Object is sold to high bidder at second highest bidders bid price

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English Auction Suppose bidding goes up by increments of $1. – What is a sensible strategy in this auction? – Is there a weakly dominant strategy? What does Nash equilibrium look like? – Who gets the object in Nash equilibrium? – How much does the buyer pay?

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Sealed bid, second price auction Suppose that your value for the object is V. Claim: Bidding V is a weakly dominant strategy. Suppose you bid more X where X<V. When would the result be the same as if you bid V? When would it be different? Could you be better off bidding X than V? Could you be worse off? Suppose you bid X where X>V – Same questions

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Dominant strategy equilibrium For the sealed bid, second price auction, what is the only Bayes-Nash equilibrium if you think that it is possible that other bids could be any number? Who gets the object in thise equilibrium? How much does the winner pay? How does this outcome compare with that of the English auction?

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First Price Sealed Bid Auction Suppose that everyone knows their own value V for an object, but all you know is that each other bidder has a value that is equally likely to be any number between 1 and 100. A strategy is an instruction for what you will do with each possible value. Lets look for a symmetric Nash equilibrium.

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Case of two bidders. Lets see if there is an equilibrium where everyone bids some fraction a of their values. Lets see what that fraction would be. Suppose that you believe that if the other guys value is X, he will bid aX. If you bid B, the probability that you will be the high bidder is the probability that B>aX. The probability that B>aX is the probability that X<B/a.

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Two bidder case We have assumed that the probability distribution of the other guys value is uniform on the interval [0,100]. For number X between 0 and 100, the probability that his value is less than X is just X/100. The probability that X<B/a is therefore equal to B/(100 a). This is the probability that you win the object if you bid B.

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So whats the best bid? If you bid B, you win with probability B/(100a). Your profit is V-B if you win and 0 if you lose. So your expected profit if you bid B is (V-B) times B/(100a)=(1/100a)(VB-B 2 ). To maximize expected profit, set derivative equal to zero. We have V-2B=0 or B=V/2. This means that if the other guy bids proportionally to his value, you will too, and your proportion will be a=1/2.

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What if there are n bidders? Suppose that the other bidders each bid the same fraction a of their values. If you bid B, you will be high bidder if each of them bids less than B. If others bid aX when there values are X, the probability that you outbid any selected bidder is the probability that aX<B, which is B/(100a).

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Winning the object You get the object only if you outbid all other bidders. The probability that with bid B you outbid all n-1 other guys is (B/100a) n-1. If you bid B and get the object, you win V-B. So your expected winnings if you bid B are (V-B) (B/100a) n-1 =(1/100a) n-1 (V B n-1 -B n ) To maximize expected winnings set derivative with respect to B equal to 0.

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Equilbrium bid-shading Derivative of (1/100a) n-1 (V B n-1 -B n ) is equal to zero if (n-1)VB n-2 -nB n-1 =0 This implies that (n-1)V=nB and hence B= V(n-1)/n Therefore if everybody bids a fraction a of their true value, it will be in the interest of everybody to bid the fraction n-1/n of their true value.

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A Common Value Auction

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Classroom Exercise Form groups of 3. One is auctioneer, two are oil field bidders. Each bidder explores half the oilfield and determines what his half is worth. (Either $3 million or 0) Neither will know what other half is worth. Total value is sum of the values of the two halves.

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Implementation Auctioneer flips a coin. – If the coin is heads, Player As side is worth 3 million. – If the coin is tails this Player As side is worth zero. – Auctioneer writes result down and shows it to A but not to B. Next auctioneer does this for B. Next auctioneer conducts a sealed bid second price auction for the oilfield. Auctioneer records coin toss results, bids, auction winner and profit or loss.

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The lemons market

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Classroom experiment Value of cars to owners was uniformly distributed 1 to 1000. Value to buyer of any car is 1.5 times its value to current owner. What happened? – Most people who bought lost money. – After a few rounds few cars were sold.

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Why was that? Suppose there were a single price P>0 for cars. Which cars would be available? What would be the average value to its owner of an available car? What would be the expected value of a used car be to a buyer? How many cars would you expect buyers to buy at this price?

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Another lemons example Just two kinds of cars, good ones and lemons Good cars are worth $700 to their owners and $1200 to potential buyers. Lemons are worth $200 to their owners and $400 to buyers. There are 150 lemons and 50 good cars in town.

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Beliefs Suppose that there are more than 200 buyers, who believe that all used cars will come on the market. Then average used car is worth (3/4)400+(1/4)1200=$600 to a buyer. This would be the price. Which cars would be available?

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Self-confirming belief? No. Belief that all used cars come to market results in only lemons reaching market.

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Another belief Suppose buyers believe that only lemons will reach market. Then used cars are worth $400 to buyers. Price will be $400. Only lemons will be sold. This belief is confirmed.

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The paradox Even though it would efficient for all cars to be sold, (since buyers value them more than sellers) the market for good used cars vanishes.

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Another version Story is as before, but now there are 100 good cars and 100 lemons in town. If buyers believe that all cars will come to market, average car is worth (1/2)1200+(1/2)400=$800. At this price, even good car owners will sell their cars. Belief that all cars are good is confirmed.

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A second equilibrium Again suppose there are 100 good cars and 100 lemons in town But suppose buyers believe that only lemons will come on the market. Price of a used car will be $400. Only lemons come on the market.

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Two distinct equilibria When there are 100 good cars and 100 lemons available, there are two equilibria with self-confirming beliefs. All believe that all used cars come to market. With this belief they are priced at $800. All believe that only lemons come to market. With this belief they are priced at $200. In each case, beliefs are supported by outcome.

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Would you buy a used car from this man?

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Optimal auction design Roger Myerson Mathematics of Operations research 1981.

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