Auction Theory

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Auction Theory

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Title: Auction Theory

1
Auction Theory

Class 7 Common Values, Winners curse and
Interdependent Values.

2
Outline

Winners curse
Common values
in second-price auctions
Interdependent values
The single-crossing condition.
An efficient auction.
Correlated values
Cremer Mclean mechanism

3
Common Values

Last time in class we played 2 games
Each student had a private knowledge of xi, and
the goal was to guess the average.
Students with high signals tended to have higher
guesses.
Students were asked to guess the total value of a
bag of coins.
We should have gotten some bidders overestimate.
Today we will model environments when there is a
common value, but bidders have different pieces
of information about it.

4
Winners curse

These phenomena demonstrate the Winners Curse
Winning means that everyone else was more
pessimistic than you? the winner should update
her beliefs after winning.
Winning is bad news
Winners typically over-estimate the items value.
Note Winners curse does not happen in
equilibrium. Bidders account for that in their
strategies.

5
Modeling common values

First model Each bidder has an estimate eiv
xi
v is some common value
ei is an unbiased estimator (Exi0)
Errors xi are independent random variables.
Winners curse consider a symmetric equilibrium
strategy in a 1st-price auction.
Winning means all the other had a lower signal ?
my estimate should decrease.
Failing to foresee this leads to the Winners
curse.

6
Winners curse some comments

The winners curse grows with the market sizeif
my signal is greater than lots of my competitors,
over-estimation is probably higher.
The highest-order statistic is not an unbiased
estimator.
With common values English auctions and Vickrey
auctions are no longer equivalent.
Bidders update beliefs after other bidders drop
out.
Two cases where the two auctions are equivalent
2 bidders (why?)
Private values

7
A useful notation v(x,y)

What is my expected value for the item if
My signal is x.
I know that the highest bid of the other bidders
is y?v(x,y) Ev1 x1x and maxy2,,yny
We will assume that v(x,y) is increasing in both
coordinates and that v(0,0)0.

8
A useful notation x-i

We will sometime use xx1,,xn
Given a bidder i, let x-i denote the signals of
the other bidders x-ix1,,xi-1,xi1,,xn
x(xi,x-i)
(z,x-i) is the vector x1,,xn where the ith
coordinate is replaced with z.

9
Second-price auctions

With common values, how should bidder bid?
Naïve approach bid according to the estimate
you have vxi
Problem does not take into account the winners
curse.
Bidders will thus shade their bids below the
estimates they currently have.

10
Second-price auctions

In the common value setting
Theorem bidding according to ß(xi)v(xi,xi) is
a Nash equilibrium in a second-price auction.

That is, each bidder bids as if he knew that the
highest signal of the others equals his own
signal.
Bid shading increases with competitionI bid as
if I know that all other bidders have signals
below my signal (and the highest equals my
signal)
With small competition, no winners curse effect.

11
Second-price auctions

In the common value setting
Theorem bidding according to ß(xi)v(xi,xi) is
a Nash equilibrium in a second-price auction.

Equilibrium conceptUnlike the case of private
values, equilibrium in the 2nd-price auction is
Bayes-Nash and not dominant strategies.
Bidder need to take distributions into account.

12
Second-price auctions

In the common value setting
Theorem bidding according to ß(xi)v(xi,xi) is
a Nash equilibrium in a second-price auction.

Intuition (assume 2 bidders)
b() is a symmetric equilibrium strategy.
Consider a small change of e in my bid since
the other bidder bids with b(), if his bid is far
from b(xi) then an e change will not matter.
A small change in my bid will matter only if the
bids are close.
I might win and figure out that the other signal
was very close to mine.
I might lose and figure out the same thing.
I should be indifferent between winning and pay
b(x), and losing.

13
Second-price auctions

In the common value setting
Theorem bidding according to ß(xi)v(xi,xi) is
a Nash equilibrium in a second-price auction.

Proof
Assume that the other bidders bid according to
b(xi)v(xi,xi).
The expected utility of bidder i with signal x
that bids ß is
Where ymaxx-i
gyx is the density of y given x.
Bidder i wins when all other signals are less
than b-1(ß)

14
Second-price auctions

Lets plot v(x,y)-v(y,y)
Recall v(x,y) increasing in x (for all x,y)
y
x
? Utility is maximized when bidding b ß(x)
v(x,x)
15
Second price auctions example

Example v U0,1 xi U0,2v n 3
Equilibrium strategy
See Krishnas book for the details.

16
Symmetric valuations

The exact theorem and proof actually works for a
more general model symmetric valuations.
That is, there is some function u such that for
all i
vi(x1,.,xn)u(xi,x-i)
Generalizes private values vi(x1,.,xn)u(xi)
It also works for joint distributions, as long
they are symmetric.

17
Game of Trivia

Question 1 What is the distance between Paris
and Moscow?
Question 2 What is the year of birth of David
Ben-Gurion?

18
Information Aggregation

Common-value auctions are mechanisms for
aggregating information.
The wisdom of the crowds and Galtons ox.
In our model, the average is a good estimation
Eei Evxi Ev Exi vExi v
One can show if bidders compete in a 1st-price
or a 2nd-price auctions, the sale price is a good
estimate for the common value.
Some conditions apply.
Intuition Thinking that the largest value of the
others is equal to mine is almost true with many
bidders.

19
Outline

Winners curse
Common values
in second-price auctions
Interdependent values
The single-crossing condition.
An efficient auction.
Correlated values
Cremer Mclean mechanism

20
Interdependent values

We now consider a more general model
interdependent values
the valuations are not necessarily symmetric.
The value of a bidder is a functions of the
signals of all bidders vi(x1,,xn)
We assume vi is non decreasing in all variables,
strictly increasing in xi.
Again, private values are a special case
vi(x1,,xn)vi(xi)
There might still be more uncertainty then,
vi(x1,,xn) is the expected value over the
remaining uncertainty.
vi(x1,,xn)Evi x1,,xn

21
Interdependent values

Example v1(x1, x2,x3) 5x1 3x2 x3 v2(x1,
x2,x3) 2x1 9x2 (x3)3 v2(x1, x2,x3) 2x1x2
(x3)2

22
Efficient auctions

Can we design an efficient auction for settings
with interdependent values?
No.

Claim no efficient mechanism exists for v1(x1,
x2) x1 v2(x1, x2) (x1)2 Where x1 is
drawn from 0,2
23
Efficient auctions
Claim no efficient mechanism exists for v1(x1,
x2) x1 v2(x1, x2) (x1)2 Where x1 is drawn
from 0,2
y1
z1
1

Proof
What is the efficient allocation?
give the item to 1 when x1lt1, otherwise give it
to 2.
Let p be a payment rule of an efficient
mechanism.
Let y1lt1ltz1 be two types of player 1.
Together y1 z1 ? contradiction.

When 1s true value is y1 y1-p1(y1)
0-p1(z1)
When 1s true value is z1 0 - p1(z1) z1
p(y1) (efficiency truthfulness)
24
Single-crossing condition

Conclusion For designing an efficient auction we
will need an additional technical condition.
Intuitively for every bidder, the effect of her
own signal on her valuations is stronger than the
effect of the other signals.
v1(x1, x2) x1, v2(x1, x2) (x1)2
v1(x1, x2) 2x15x2, v2(x1, x2) 4x12x2

25
Single-crossing condition

Definition Valuations v1,,vn satisfy the
single-crossing condition if for every pair of
bidders i,j we have for all x,
Actually, a weaker condition is often sufficient
Inequality holds only when vi(x)vi(y) and both
are maximal.
Single crossing fixing the other signals, is
valuations grows more rapidly with xi than js
valuation.

26
Single crossing examples

For example when we plot v1(x1, x2,x3) and
v2(x1, x2,x3) as a function of x1 (fixing x2 and
x3)

v1(x1, x2,x3)
v2(x1, x2,x3)
x1
For every x, the slope of v1(x1, x2,x3) is
greater.
27
Single crossing examples

v1(x1, x2) x1 , v2(x1, x2) (x1)2 are not
single crossing.
v1(x1, x2,x3) 5x1 3x2 x3 v2(x1, x2,x3)
2x1 9x2 x3 v3(x1, x2,x3) 3x1 2x2
2x3are single crossing

y1
z1
1
x1
28
An Efficient Auction

Consider the following direct-revelation auction
Bidders report their signals x1,,xn
The winner the bidder with the highest value
(given the reported signals).
Argmax vi(x1,,xn)
Payments the winner pays M(i)vi( yi(x-i) ,
x-i )where yi(x-i) min zi vi(zi,x-i)
maxj?i vj(zi,x-i)
In other words, yi(x-i) is the lowest signal for
which i wins in the efficient outcome (given the
signals x-i of the other bidders)
Losers pay zero.

29
An Efficient Auction

What is the payment of bidder 1 when he wins with
a signal ?

v1(x1, x-i)
v2(x1, x-i)
v3(x1, x-i)
M(i)
x1
y1(x-1)
30
An Efficient Auction

What is the problem with the standard
second-price payment (given the reported
signals)?
i.e., 1 should pay v2(x1, x-i)?
In the proposed payments, like 2nd-price auctions
with private value, price is independent of the
winners bid.

31
An Efficient Auction
Theorem when the valuations satisfy the
single-crossing condition, truth-telling is an
efficient equilibrium of the above auction.

Equilibrium concept stronger than Nash (but
weaker than dominant strategies) ex-post Nash

32
Ex-post equilibrium

Given that the other bidders are truthful,
truthful bidding is optimal for every profile of
signals.
No bidder, nor the seller, need to have any
distributional assumptions.
A strong equilibrium concept.
Truthfulness is not a dominant strategy in this
auction.
Why?
My declared value depends on the declarations
of the others.If some crazy bidder reports a
very high false signal, I may win and pay more
than my value.

33
An Efficient Auction proof

Proof
Suppose i wins for the reports x1,,xn, that is,
vi(xi,x-i) maxj?i vj(xi,x-i).
Bidder i pays vi(yi(x-i) ,x-i), where yi(x-i) is
its minimal signal for which his value is greater
than all others.
vi(yi(x-i) ,x-i) lt vi(xi ,x-i) ?
non-negative surplus.
Due to single crossing
For any bid zigtyi(x-i), his value will remain
maximal, and he will still win (paying the same
amount).
For any bid ziyi(x-i), he will lose and pay
zero.
? No profitable deviation for a winner.

34
An Efficient Auctionproof

Proof (cont.)
Suppose i loses for the reports x1,,xn ,that
is, vi(xi,x-i) lt maxj?i vj(xi,x-i).
xilt yi(x-i)
Payoff of zero
To win, I must report zigtyi(x-i).
Still losing when bidding lower (single
crossing).
Then payment will be M(i) vi( yi(x-i) ,
x-i ) gt vi(xi, x-i )generating a negative payoff.

35
Weakness

Weakness of the efficient auction seller needs
to know the valuation functions of the bidders
Does not know the signals, of course.

36
Outline

Winners curse
Common values
in second-price auctions
Interdependent values
The single-crossing condition.
An efficient auction.
Correlated values
Cremer Mclean mechanism

37
Revenue

In the first few classes we saw with private,
independent values, bidders have an information
rent that leaves them some of the social
surplus.
No way to make bidders pay their values in
equilibrium.
We will now consider revenue maximization with
statistically correlated types.

38
Discrete values

We will assume now that signals are discrete
drawn from a distribution on Xi?, 2?,
3?,.,Ti?(For simplicity, let Xi1, 2,
3,.,Ti )
think about ? as 1 cent
The analysis of the continuous case is harder.
We still require single-crossing valuations, with
the discrete analogue for all i and k, and
every xi, vi(xi, ?x-i) - vi(xi,x-i)
vk(xi, ? x-i) - vk(xi,x-i)

39
Correlated values

For the Generalized-VCG auction to work, signals
are not necessarily statistically independent
correlation is allowed.
Which one is not a product of independent
distributions?

Independent distributionsf1(1)1/6, f1(2)1/3,
f1(3)1/2 f2(1)1/4, f2(2)1/2, f2(3)1/4
A joint distribution
x2
x2
1 2 3
1 1/24 1/12 1/24
2 1/12 1/6 1/12
3 1/8 1/4 1/8
1 2 3
1 1/6 1/12 1/12
2 1/12 1/6 1/12
3 1/12 1/12 1/6
x1
x1
40
Revenue

Example lets consider the joint
distribution
Lets consider 2nd-price auctions
Expected welfare 14/6
Expected revenue for the seller 10/6
Expected revenue with optimal reserve price
(R2) 11/6
Can the seller do better?
Intuitively, information rent should be smaller
(seller can gain information from other bidders
values)

1 2 3
1 1/6 1/12 1/12
2 1/12 1/6 1/12
3 1/12 1/12 1/6
41
Revenue example
Pay 1 2 3
1 -0.5 0 2
2 0 1 2
3 0 2 3.5
Prob 1 2 3
1 1/6 1/12 1/12
2 1/12 1/6 1/12
3 1/12 1/12 1/6

Consider the following auction
Efficient allocation (given the bids), ties
randomly broken.
Payments see table for payment for bidder 1
Claim the auction is truthful
Example when x12, assume bidder 2 is truthful.
u1(b12) 0.25(2-0) 0.5(0.52-1)
0.25(-2)
u1(b11) 0.25(0.521/2) 0.5(0)
0.25(-2) - 0.125
Note although bidder 1 bids 1, the true
probabilities are according to x12.
u1(b13) 0.25(2-0) 0.5(2-2) 0.25(
0.52 3.5 ) -0.125

0
42
Revenue example
1 2 3
1 1/6 1/12 1/12
2 1/12 1/6 1/12
3 1/12 1/12 1/6
Pay 1 2 3
1 -0.5 0 2
2 0 1 2
3 0 2 3.5

Consider the following auction
Efficient allocation (given the bids), ties
randomly broken.
Payments see table for payment for bidder 1
Claim Esellers revenue14/6
Equals the expected social welfare
Easy way to see the expected surplus of each
bidder is 0.

43
Revenue

Conclusions from the previous example
An incentive compatible, efficient mechanism that
gains more revenue than the 2nd-price auction
Revenue equivalence theorem doesnt hold with
correlated values.
The expected surplus of each bidder is 0
Seller takes all surplus. No information rent.
Is this a general phenomenon?
Surprisingly with correlated types, the seller
can get all surplus leaving bidders with 0
surplus.
Even with slight correlation.

44
Revenue

The Cremer-Mclean Condition the conditional
correlation matrix has a full rank for every
bidder.
That is, some minimal level of correlation exists.

45
The correlation matrix

Pr(x-i xi)
Pr(x1,,xn)
x-i
1 2 3
1 1/6 1/12 1/12
2 1/12 1/6 1/12
3 1/12 1/12 1/6
1 2 3
1 ½ ¼ ¼
2 ¼ ½ ¼
3 ¼ ¼ ½
Correlated
Full rank (3)
xi
1 2 3
1 1/24 1/12 1/24
2 1/12 1/6 1/12
3 1/8 1/4 1/8
1 2 3
1 ¼ ½ ¼
2 ¼ ½ ¼
3 ¼ ½ ¼
Rank 1
independent
46
Revenue

The Cremer-Mclean Condition the conditional
correlation matrix has a full rank for every
bidder.
That is, some minimal level of correlation exists.

Theorem (Cremer Mclean, 1988)Under the
Cremer-Mclean condition, then there exists an
efficient, truthful mechanism that extracts the
whole surplus from the bidders.
That is, sellers profit the maximal social
welfare
The expected surplus of each bidder is zero.

47
Revenue

We will now construct the Cremer-Mclean auction.
Idea modify the truthful auction (generalized
VCG) that we saw earlier.
Remark The Cremer-Mclean auction is
not ex-post individually rational
(sometimes bidders pay more than their actual
value)
Interim individually rational
Given the bidder value, he will gain zero surplus
in expectation (over the values of the others).

48
ReminderGeneralized VCG

Bidders report their signals x1,,xn
The winner the bidder with the highest value
(given the reported signals).
Payments the winner pays Mivi( yi(x-i) , x-i
)where yi(x-i) min zi vi(zi,x-i) maxj?i
vj(zi,x-i)

ci(x-i)

A general observation adding to the payment of
bidder any term which is independent of her bid
will not change her behavior.
Mivi( yi(x-i) , x-i ) ci(x-i)

49
The trick

The expected surplus of each bidder

As before, Qi(x1,,xn) is the probability that
bidder i wins.

For every i, we would like now to find values
ci(x-i) such that and for every xi

Thats the conditional probability for which the
Cremer-Mclean condition applies
50
The trick (cont.)

If we could find such values ci(x-i), we will add
it to the bidders payments.
As observed, it will not change the incentives.
The expected surplus of bidder i is now

Ui by definition
Ui due to the choice of ci(x-i)
51
The trick (cont.)

Can we find such values ci(x-i)?
For each bidder i, and every signal xi, we would
like to solve the following system of equations
Is there a solution?
From linear algebraIf the matrix Pr(x-ixi) has
full rank yes!
Economic interpretation of full rank signals
must be correlated enough

52
The Cremer-Mclean mechanism

Bidders report their signals x1,,xn
The winner the bidder with the highest value
(given the reported signals).
Payments the winner pays MiCMvi( yi(x-i) , x-i
)ci(x-i)where
yi(x-i) min zi vi(zi,x-i) maxj?i
vj(zi,x-i)
ci(x-i) are the solution to the system of
equations (Ui(xi) is the expected surplus
without the ci(x-i) term)

Under the Cremer-Mclean condition it is
truthful, efficient and leaves bidders with a 0
surplus.
53
Our example
Payments in a 2nd price auction
Cremer-Mclean payments
1 2 3
1 1/6 1/12 1/12
2 1/12 1/6 1/12
3 1/12 1/12 1/6
Pay 1 2 3
1 0.5 0 0
2 1 1 0
3 1 2 1.5
Pay 1 2 3
1 -0.5 0 2
2 0 1 2
3 0 2 3.5