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 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
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?