Transcript Slide 1

Auction Theory
Class 7 – Common Values, Winner’s curse
and Interdependent Values.
1
Outline
•
Winner’s curse
•
Common values
–
•
Interdependent values
–
–
•
in second-price auctions
The single-crossing condition.
An efficient auction.
Correlated values
–
Cremer & Mclean mechanism
2
Common Values
•
Last time in class we played 2 games:
1. 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.
2. 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.
3
Winner’s curse
•
These phenomena demonstrate the Winner’s 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 item’s value.
•
Note: Winner’s curse does not happen in
equilibrium. Bidders account for that in their
strategies.
4
Modeling common values
•
First model: Each bidder has an estimate ei=v + xi
–
–
–
•
v is some common value
ei is an unbiased estimator (E[xi]=0)
Errors xi are independent random variables.
Winner’s 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 Winner’s curse.
5
Winner’s curse: some comments
•
The winner’s curse grows with the market size:
if my signal is greater than lots of my competitors,
over-estimation is probably higher.
–
•
With common values: English auctions and Vickrey
auctions are no longer equivalent.
–
•
The highest-order statistic is not an unbiased estimator.
Bidders update beliefs after other bidders drop out.
Two cases where the two auctions are equivalent:
–
–
2 bidders (why?)
Private values
6
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) = E[v1 | x1=x and max{y2,…,yn}=y ]
•
We will assume that v(x,y) is increasing in both
coordinates and that v(0,0)=0.
7
A useful notation: x-i
• We will sometime use x=x1,…,xn
• Given a bidder i, let x-i denote the signals of the
other bidders: x-i=x1,…,xi-1,xi+1,…,xn
• x=(xi,x-i)
• (z,x-i) is the vector x1,…,xn where the i’th
coordinate is replaced with z.
Second-price auctions
•
With common values, how should bidder bid?
•
Naïve approach:
bid according to the estimate you have: v+xi
–
•
Problem: does not take into account the winner’s curse.
Bidders will thus shade their bids below the
estimates they currently have.
9
Second-price auctions
In the common value setting:
• Theorem: bidding according to β(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 competition:
I bid as if I know that all other bidders have signals
below my signal (and the highest equals my signal)
–
With small competition, no winner’s curse effect.
10
Second-price auctions
In the common value setting:
• Theorem: bidding according to β(xi)=v(xi,xi) is a
Nash equilibrium in a second-price auction.
•
Equilibrium concept:
Unlike 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.
11
Second-price auctions
In the common value setting:
• Theorem: bidding according to β(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 ε in my bid:
since the other bidder bids with b(), if his bid is far from b(xi) then an
ε 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.
12
Second-price auctions
In the common value setting:
• Theorem: bidding according to β(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 β is
–
•
•
•
Where y=max{x-i}
g[y|x] is the density of y given x.
Bidder i wins when all other signals are less than b-1(β)
 1 ( b )
ui (b, x) 
 v( x, y)  v( y, y) g[ y | x]dy
0
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Second-price auctions
 1 ( b )
ui (b, x) 
 v( x, y)  v( y, y) g[ y | x]dy
0
Let’s 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= β(x)= v(x,x)
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Second price auctions: example
•
Example:
v ~ U[0,1]
xi ~ U[0,2v]
n=3
2x
 ( x) 
2 x
•
Equilibrium strategy:
•
See Krishna’s book for the details.
15
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.
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?
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Information Aggregation
Common-value auctions are mechanisms for
aggregating information.
• “The wisdom of the crowds” and Galton’s ox.
• In our model, the average is a good estimation
–
•
E[ei] = E[v+xi] = E[v] + E[xi] = v+E[xi] ≈ 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.
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Outline
•
Winner’s curse
•
Common values
–
•
Interdependent values
–
–
•
in second-price auctions
The single-crossing condition.
An efficient auction.
Correlated values
–
Cremer & Mclean mechanism
19
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)=E[vi | x1,…,xn ]
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
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]
Efficient auctions
Claim: no efficient mechanism exists for
v1(x1, x2) = x1 v2(x1, x2) = (x1)2
Where x1 is drawn from [0,2]
• Proof:
– What is the efficient allocation?
y1 1
z1
• give the item to 1 when x1<1, otherwise give it to 2.
– Let p be a payment rule of an efficient mechanism.
– Let y1<1<z1 be two types of player 1.
When 1’s true value is z1:
0 - p1(z1)≥ z1 – p(y1)
(efficiency + truthfulness)
Together: y1 ≥ z1  contradiction.
When 1’s true value is y1:
y1-p1(y1) ≥ 0-p1(z1)
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) = 2x1+5x2,
v2(x1, x2) = 4x1+2x2
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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,
v j
vi
( x) 
( x)
xi
xi
•
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, i’s valuations grows
more rapidly with xi than j’s valuation.
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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.
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Single crossing: examples
• v1(x1, x2) = x1 , v2(x1, x2) = (x1)2 are not single crossing.
y1 1
• v1(x1, x2,x3) = 5x1 + 3x2 + x3
v2(x1, x2,x3) = 2x1 + 9x2 + x3
v3(x1, x2,x3) = 3x1 + 2x2 + 2x3
are single crossing
z1
x1
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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.
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An Efficient Auction
What is the payment of bidder 1 when he wins with a
signal x1* ?
v1(x1, x-i)
v2(x1, x-i)
v3(x1, x-i)
M*(i)
y1(x-1)
x1*
x1
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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
winner’s bid.
v1(x1, x-i)
v2(x1, x-i)
v3(x1, x-i)
M*(i)
*
1
y1(x-1) x
x1
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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
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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.
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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) < vi(xi ,x-i)

non-negative surplus.
Due to single crossing:
–
–
For any bid zi>yi(x-i), his value will remain maximal, and he
will still win (paying the same amount).
For any bid zi≤yi(x-i), he will lose and pay zero.
 No profitable deviation for a winner.
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An Efficient Auction:proof
Proof (cont.):
• Suppose i loses for the reports x1,…,xn ,
that is, vi(xi,x-i) < maxj≠i vj(xi,x-i).
–
–
•
To win, I must report zi>yi(x-i).
–
•
xi< yi(x-i)
Payoff of zero
Still losing when bidding lower (single crossing).
Then payment will be:
M*(i) = vi( yi(x-i) , x-i ) > vi(xi, x-i )
generating a negative payoff.
34
Weakness
Weakness of the efficient auction:
seller needs to know the valuation functions of the
bidders
–
Does not know the signals, of course.
35
Outline
•
Winner’s curse
•
Common values
–
•
Interdependent values
–
–
•
in second-price auctions
The single-crossing condition.
An efficient auction.
Correlated values
–
Cremer & Mclean mechanism
36
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.
37
Discrete values
•
We will assume now that signals are discrete
–
–
drawn from a distribution on Xi={Δ, 2Δ, 3Δ,….,TiΔ}
(For simplicity, let Xi={1, 2, 3,….,Ti} )
think about Δ 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, Δ+x-i) - vi(xi,x-i)≥ vk(xi, Δ + x-i) - vk(xi,x-i)
38
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 distributions:
f1(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
1
x1 2
1/12
1/6
1/12
1/8
1/4
1/8
3
1
2
3
1/6
1/12
1/12
x1 2
1/12
1/6
1/12
3
1/12
1/12
1/6
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Revenue
•
•
Example: let’s consider the joint distribution
2
3
1
1/6
1/12
1/12
2
1/12
1/6
1/12
3
1/12
1/12
1/6
Let’s consider 2nd-price auctions:
–
–
–
•
1
Expected welfare: 14/6
Expected revenue for the seller: 10/6
Expected revenue with optimal reserve price (R=2): 11/6
Can the seller do better?
–
Intuitively, information rent should be smaller (seller can
40
gain information from other bidders’ values)
Revenue: example
•
Prob 1
2
3
1
1/6
1/12 1/12
2
1/12 1/6
3
1/12 1/12 1/6
1/12
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: the auction is truthful
–
Example: when x1=2, assume bidder 2 is truthful.
–
–
u1(b1=2)= 0.25*(2-0)
+ 0.5*(0.5*2-1)
u1(b1=1) = 0.25*(0.5*2+1/2) +0.5*(0)
–
–
+ 0.25*(-2)
+ 0.25*(-2)
=0
= - 0.125
Note: although bidder 1 bids 1, the true probabilities are according to x1=2.
u1(b1=3) = 0.25*(2-0)
+ 0.5*(2-2)
+ 0.25*( 0.5*2 –3.5 ) = -0.125
41
Revenue: example
•
1
2
3
1
1/6
1/12 1/12
2
1/12 1/6
3
1/12 1/12 1/6
1/12
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: E[seller’s revenue]=14/6
–
–
Equals the expected social welfare
Easy way to see: the expected surplus of each bidder is 0.
42
Revenue
•
Conclusions from the previous example:
1. An incentive compatible, efficient mechanism that gains
more revenue than the 2nd-price auction

Revenue equivalence theorem doesn’t hold with correlated values.
2. 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.
43
Revenue
•
The Cremer-Mclean Condition: the conditional
correlation matrix has a full rank for every bidder.
–
That is, some minimal level of correlation exists.
44
The correlation matrix
•
Pr(x1,…,xn)
Pr(x-i | xi)
x-i
Correlated
1
1
2
1/6
1/12 1/12
2
1/12 1/6
3
1/12 1/12 1/6
1
independent
3
2
1/12
1
2
3
½
¼
¼
2
¼
½
¼
3
¼
¼
½
1
2
3
½
½
¼
¼
½
¼
1
xi
3
1
1/24 1/12 1/24
1
2
1/12 1/6
1/12
2
¼
¼
3
1/8
1/8
3
¼
1/4
Full
rank (3)
Rank 1
45
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, seller’s profit = the maximal social welfare
The expected surplus of each bidder is zero.
46
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).
47
Reminder:”Generalized VCG”
–
–
Bidders report their signals x1,…,xn
The winner: the bidder with the highest value (given the
reported signals).
– Payments:
the winner pays Mi*=vi( yi(x-i) , x-i ) + ci(x-i)
where
yi(x-i) = min{ zi | vi(zi,x-i) ≥ maxj≠i vj(zi,x-i) }
A general observation: adding to the
payment of bidder any term which is
independent of her bid will not change her
behavior.
• Mi*=vi( yi(x-i) , x-i ) + ci(x-i)
48
The trick
•
The expected surplus of each bidder:

Ui* ( xi )   Pr(xi xi ) Qi ( xi , xi )vi ( xi , xi )  M i* ( xi )

xi
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:
Ui* ( xi )   Pr(xi xi )  ci ( xi )
xi
That’s the conditional probability for which the
Cremer-Mclean condition applies
49
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:


Pr(xi xi ) Qi ( xi , xi )vi ( xi , xi )  M i* ( xi )  ci ( xi )

xi

  Pr( xi xi ) Qi ( xi , xi )vi ( xi , xi )  M i* ( xi )

xi
  Pr(xi xi )ci ( xi )
xi
=Ui* by
definition
=Ui* due to the
choice of ci(x-i)
0
50
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:
Ui* ( xi )   Pr(xi xi )  ci ( xi )
xi
Is there a solution?
• From linear algebra:
If the matrix Pr(x-i|xi) has full rank: yes!
• Economic interpretation of full rank:
signals must be “correlated enough”
U i*  P  ci
51
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 MiCM=vi( yi(x-i) , x-i )+ci(x-i)
where
1. yi(x-i) = min{ zi | vi(zi,x-i) ≥ maxj≠i vj(zi,x-i) }
2. ci(x-i) are the solution to the system of equations
(Ui*(xi) is the expected surplus without the ci(x-i) term):
Ui* ( xi )   Pr(xi xi )  ci ( xi )
xi
Under the Cremer-Mclean condition:
it is truthful, efficient and leaves bidders with a 0 surplus.
52
Our
example
Payments in a 2
nd
price auction
1
2
3
1
1/6
1/12 1/12
2
1/12 1/6
3
1/12 1/12 1/6
1/12
Pay 1
Cremer-Mclean
payments
2
3
Pay 1
2
3
1
0.5
0
0
1
-0.5 0
2
2
1
1
0
2
0
1
2
3
1
2
1.5
3
0
2
3.5
U(x1=1) = 0.5*(½*1-0.5) + 0.25*(0)
+ 0.25*(0)
=0
U(x1=2) = 0.25*(2-1)
+ 0.5*(½*2-1) + 0.25*(0)
=¼
U(x1=3) = 0.25*(3-1)
+ 0.25*(3-2)
+ 0.5*(½*3-1.5) = ¾
We would like to find c1,c2,c3 such that:
0.5*c1 + 0.25*c2 + 0.25*c3
= U(x1=1) = 0
0.25*c1 + 0.5*c2 + 0.25*c3
= U(x1=2) = ¼
0.25*c1 + 0.25*c2 + 0.5*c3
= U(x1=3) = ¾
Solution: (c1,c2,c3) = (-1,0,2)
53
Summary
•
Private values is a strong assumption.
–
•
Still, bidders have privately known signals.
–
•
But would know better if knew other signals.
Interdependent values:
–
–
•
•
Many times the item for sale has a common value.
We saw how bidders account for the winner’s curse in
second-price auctions
We saw an efficient auction (under the “single-crossing”).
New equilibrium concept: ex-post Nash.
Correlated values:
seller can extract the whole surplus
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