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 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.

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/(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-B2). 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.

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).

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 Bn-1-Bn)• To maximize expected winnings set derivative

with respect to B equal to 0.

Equilbrium bid-shading

• Derivative of (1/100a)n-1(V Bn-1-Bn) is equal to zero if

• (n-1)VBn-2-nBn-1=0• This implies that (n-1)V=nB and henceB= 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 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.

Would you buy a used car from this man?