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.
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.
Bid-shading in this example
• In a first-price sealed bid private value
auction, where each believes that the other’s
value is uniformly distributed on the interval
[0,100], there is a Bayes-Nash equilibrium in
which each bidder bids half of her value.
Back to the oil fields
Single bidder version
• A new oil field has come up for lease.
• The current owner has done no exploration of
the oilfield. All that the current owner knows is
that the field has two halves such that each half is
worth either $0 or $3 million dollars with
independent probabilities of ½ for each value for
each half.
• It is common knowledge that there is only one
possible buyer and that buyer has learned the
value of one half of the field but not the other.
Expected Values
• What is the expected value of the whole
oilfield to the bidder if he knows that his own
half is worth $3 million?
A) $1.5 million
B) $3 million
C) $4 million
D) $4.5 million
E) $6 million
Expected Values
• What is the expected value of the whole
oilfield to the bidder if he knows that his own
half is worth $0 ?
A) $0
B) $1 million
C) $1.5 million
D) $2 million
E) $3 million
Seller posts a price
• The seller posts a price for the oilfield.
• The buyer either buys the field at the posted
price or refuses to buy.
• If the buyer pays the posted price, her profits
are the actual value of the oilfield minus the
price she paid. The seller’s profits are the price
received. If buyer doesn’t buy the field, buyer
and seller both get 0 payoff.
Bayes-Nash equilibrium
• Strategy for Seller--Posts a price.
• Strategy for Buyer—Rule that specifies whether she
will buy or not at each possible price if the side she
knows is worth $0 and if it is worth $3 million.
• Buyer tries to maximize her expected profit.
• Buyer will buy if and only if the price is less than or
equal to her expected value, given her information.
• When seller posts his price, he doesn’t know whether
the side the buyer explored is worth 0 or $3 million.
Clicker question
If the seller posts a price of $4.5 million, what is
her expected profit?
A) $1 million
B) $1.5 million
C) $2 million
D) $2.25 million
E) $3 million
Clicker question
If the seller posts a price of $1.5 million, what is
her expected profit?
A) $.5 million
B) $.75 million
C) $1 million
D) $1.25 million
E) $1.5 million
In a Bayes-Nash Equilibrium
A) Seller will post a price of $4.5 million. Buyer
will buy if buyer’s side is worth $3 million and will
reject if her side is worth $0.
B) Seller will post a price of $1.5 million. Buyer
will buy whether her side is worth $3 million or 0.
C) Seller will post a price of $1.5 million. Buyer
will buy only if her side is worth $3 million.
In Bayes-Nash equilibrium for this game, the
expected revenue of the seller is
A) $1 million
B) $2 million
C) $2.25 million
D) $3 million
E) $4.5 million
What if there are two bidders?
• Each has explored a different half of the oil
field and knows the value of the half she
explored.
• The value of each side is either $3 million or 0,
which depended on the flip of a fair coin.
• Total value of field is the sum of the two sides
• Each bidder knows what her side is worth,
but not the other bidder’s side.
Posted Price
• The seller posts a price.
• If one bidder accepts and the other declines,
oilfield is sold to that bidder at posted price.
• If both bidders accept, the seller tosses a fair
coin to decide which bidder to sell to.
• If neither bidder accepts, oilfield remains
undeveloped and seller and both buyers
receive 0 payoff.
Symmetric Bayes-Nash equilibrium?
Is there a symmetric equilibrium where the
seller asks $4.5 million and both buyers play the
strategy: Don’t buy if you see 0 and buy if you
see $3 million?
A) Yes
B) No
Buyer’s calculation
• If I see $3 million on my side, then the whole field is
worth $3 million with probability ½ and $6 million with
probability ½. But if it is worth $6 million, then the
other buyer’s side is worth $3 million.
• Suppose that her strategy is to offer to buy if her side
is worth $3 million and not offer if her side is worth 0.
• Then if I accept the offer when my side is worth $3
million, I will be sure to get the field if the field is
worth only $3 million and I will get it with probability ½
if it is worth $6 million.
Buyer calculates
• If my side is worth and I pay $4.5 million
Then with probability ½ oilfield is worth $3
million and I pay $4.5 million. With probabilty
¼, oilfield is worth $6 million and I get it for
$4.5 million. With probability ¼ oilfield is worth
$6 million and the other buyer gets it.
So my expected payoff would be
½(3-4.5)+¼ (6-4.5)+¼0=-3/4+3/8=-3/8.
Winner’s curse in common value
auctions
If you win something in an auction or other
bidding contest, makes it more likely that the
object is of low value to other bidders.
If low value to other bidders implies lower value
to you, then it is a mistake to bid as much as
your expected value given only your own
information.
What is it worth if you win it?
In this example, if you get the oilfield for $4.5
million, the probability is 2/3= .5/(.5+.25) that it
is worth $3 million and 1/3 = .25/(.5+.25) that it
is worth $6 million.
So in symmetric solution, the expected value
conditional on your getting it is
2/3x3+1/3x6=$4 million.
Bayes-Nash equilibrium with two
buyers
• There would be a Bayes-Nash equilibrium in
which seller sets price at $4 million and buyers
will offer to buy if their side is worth $3
million and will not offer to buy if their side is
worth 0.
• Expected revenue of seller would then be
(1/4)x0+(3/4)x4 million= $3 million
Wyatt Earp and the Gun Slinger
A Bayesian gunslinger game
The gunfight game when the stranger is (a) a gunslinger
or (b) a cowpoke
What are the strategies?
• Earp
– Draw
– Wait
• Stranger
–
–
–
–
Draw if Gunslinger, Draw if Cowpoke
Draw if Gunslinger, Wait if Cowpoke
Wait if Gunslinger, Draw if Cowpoke
Wait if Gunslinger, Wait if Cowpoke
Things to notice about SPNE
• If stranger is a gunslinger, he will always draw.
• If stranger is a cowboy, he will prefer to draw if
Earp draws and wait if Earp waits.
One Bayes Nash equilibrium
• Suppose that Earp waits and the other guy draws
if he is a gunslinger, waits if he is a cowpoke.
–
–
–
–
Stranger in either case is doing a best response.
If stranger follows this rule, is waiting best for Earp?
Earp’s Payoff from waiting is 3/4x1+1/4x8=2.75
Earp’s Payoff from drawing, given these strategies for
the other guys is (¾)2+(1/4) 4=2.5
• So this is a Bayes Nash equilibrium
There is another equilibrium
• Lets see if there is an equilibrium where
everybody draws.
• If Earp always draws, both cowpoke and
gunslinger are better off drawing.
• Let p be probability stranger is gunslinger.
• If both types always draw, payoff to Earp from
draw is 2p+5(1-p)=5-3p and payoff to Earp from
wait is p+6(1-p)=6-5p
• Now 5-3p>6-5p if p>1/2.
• If Earp always draws, best response for
stranger of either type is to draw.
• If stranger always draws, best response for
Earp is to always , whenever he thinks stranger
is a gunslinger with p>1/2.
• Note that this is so, even though if he knew
stranger was a cowpoke, it would be dominant
strategy to wait.
Problem 10.2
Curly
Bat
FD
FW
SD
SW
FD
20,20
30,-40
30,-40
30,-40
FW
-40,30
50,50
20,20
50,50
SD
-40,30
20,20
20,20
40,-30
SW
-40,30
50,50
-40,30
50,50
Table shows payoffs to Bat and Curly conditional on their actions and type.
Note: This is not a strategic form representation. A strategy takes the form
x/y where x is what you do if you are fast and y is what you do if you are slow.
Is D/D for both players a SPNE?
Need to show that if Curly always draws, Bat will prefer to draw if he is fast
and also if he is slow.
Also need to show that if Bat always draws, Curly will prefer to draw if he is fast
also if he is slow.
Suppose Curly always draws
If Bat draws when he is fast his expected payoff is .6 x 20 +.4 x
30=24.
If Bat waits when he is fast, his expected payoff is .5x(40)+.4x20=-12
So if Bat is fast, Bat’s best response is draw.
If Bat draws when he is slow, his expected payoff is .6x(40)+.4x20=-16.
If Bat waits when he is slow, his expected payoff is .6 x(40)+.4x(-40)=-40.
So if Bat is slow, his best response is draw.
Thus if Curly always draws, Bat’s best response is to always
draw.
Suppose Bat always draws
• Need to show that Curly is better off drawing
than waiting whether he is fast or slow.
• Similar calculation to previous one except that
probability that Bat is fast is .65
Problem 10.5 Find Bayes-Nash equilibrium
Note that if Player 1 is a Low type, he will prefer y to x, whether
Player 2 plays a or b.
So in any SPNE, if Player 1 is a low type, he will choose y.
Problem 10.5
Strategic form (leaving out dominated strategies for Player 1)
a
b
x/y
3, 1
1,3p
y/y
2, 1
5, 2p
• Note that ( x/y, a) is a Bayes- Nash equilibrium if
and only if 3p≤ 1, equivalently, p≤1/3
• Also, (y/y, b) is a Nash equilibrium if and only if
2p≥1, equivalently, p≥1/2.
• If p≤1/3, the only Bayes-Nash equilibrium has
Player 1 goes x if he’s high, and y if he’s low, while
Player 2 goes a.
• If p≥1/2, the only Bayes-Nash has Player 1 always
goes y and Player 2 goes b.
Second Midterm, next Tuesday