Private Information and Auctions Auction Situations Private Value

Private Information and Auctions

Auction Situations • Private Value – Everybody knows their own value for the object – Nobody knows other people’s 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.

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 bidder’s bid • Sealed bid second price auction – Bidders submit a single bid – Object is sold to high bidder at second highest bidder’s bid price

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?

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

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?

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. • Let’s look for a symmetric Nash equilibrium.

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

Two bidder case • We have assumed that the probability distribution of the other guy’s 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.

So what’s the best bid? • If you bid B, you win with probability B/(100 a). • 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/(100 a)=(1/100 a)(VB-B 2). To maximize expected profit, set derivative equal to zero. We have V-2 B=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.

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 a. X when there values are X, the probability that you outbid any selected bidder is the probability that a. X<B, which is B/(100 a).

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/100 a)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/100 a)n-1=(1/100 a)n-1(V Bn-1 -Bn) • To maximize expected winnings set derivative with respect to B equal to 0.

Equilbrium bid-shading • Derivative of (1/100 a)n-1(V Bn-1 -Bn) is equal to zero if • (n-1)VBn-2 -n. Bn-1=0 • This implies that (n-1)V=n. B 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.

A Common Value Auction

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.

Implementation • Auctioneer flips a coin. – If the coin is heads, Player A’s side is worth 3 million. – If the coin is tails this Player A’s 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.

The lemons market

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.

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?

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.

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?

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

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.

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.

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.

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.

Two distinct equilibria • When there are 100 good cars and 100 lemons available, there are two equilibria with selfconfirming 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.

Would you buy a used car from this man?