Mechanism Design with Correlated Information
Sushil Bikhchandani
Workshop on Mechanism Design
I.S.I. Delhi
August 4, 2015
Auctions
August 4, 2015
1/9
Optimal mechanism design
Auctions
August 4, 2015
2/9
Optimal mechanism design
When bidder information is correlated
There exists a mechanism that is efficient and leaves bidders with
zero expected surplus
Full-surplus extraction
Auctions
August 4, 2015
2/9
Symmetric Model with Correlated Information
Auctions
August 4, 2015
3/9
Symmetric Model with Correlated Information
Single indivisible object
n risk-neutral buyers or bidders, i = 1, 2, . . . , n
Auctions
August 4, 2015
3/9
Symmetric Model with Correlated Information
Single indivisible object
n risk-neutral buyers or bidders, i = 1, 2, . . . , n
For simplicity, assume n = 2 and that bidders are symmetric
Bidder i’s valuation is Vi ,
Auctions
August 4, 2015
3/9
Symmetric Model with Correlated Information
Single indivisible object
n risk-neutral buyers or bidders, i = 1, 2, . . . , n
For simplicity, assume n = 2 and that bidders are symmetric
Bidder i’s valuation is Vi , information signal is Xi
Auctions
August 4, 2015
3/9
Symmetric Model with Correlated Information
Single indivisible object
n risk-neutral buyers or bidders, i = 1, 2, . . . , n
For simplicity, assume n = 2 and that bidders are symmetric
Bidder i’s valuation is Vi , information signal is Xi
Each Xi takes values in a finite number X, |X| = m
Auctions
August 4, 2015
3/9
Symmetric Model with Correlated Information
Single indivisible object
n risk-neutral buyers or bidders, i = 1, 2, . . . , n
For simplicity, assume n = 2 and that bidders are symmetric
Bidder i’s valuation is Vi , information signal is Xi
Each Xi takes values in a finite number X, |X| = m
i’s exp. valuation, vi (Xi , Xj ), is a function of signals Xi , Xj
Auctions
August 4, 2015
3/9
Symmetric Model with Correlated Information
Single indivisible object
n risk-neutral buyers or bidders, i = 1, 2, . . . , n
For simplicity, assume n = 2 and that bidders are symmetric
Bidder i’s valuation is Vi , information signal is Xi
Each Xi takes values in a finite number X, |X| = m
i’s exp. valuation, vi (Xi , Xj ), is a function of signals Xi , Xj
Joint probability distribution P over (X1 , X2 )
Auctions
August 4, 2015
3/9
Symmetric Model with Correlated Information
Single indivisible object
n risk-neutral buyers or bidders, i = 1, 2, . . . , n
For simplicity, assume n = 2 and that bidders are symmetric
Bidder i’s valuation is Vi , information signal is Xi
Each Xi takes values in a finite number X, |X| = m
i’s exp. valuation, vi (Xi , Xj ), is a function of signals Xi , Xj
Joint probability distribution P over (X1 , X2 )
All this is common knowledge
Auctions
August 4, 2015
3/9
Model (continued)
From P define a matrix of conditional probabilities, P1
P1 is a m × m matrix with elements Pr(x2 |x1 ), x1 , x2 ∈ X
Each row of P1 is a conditional probability distribution of X2 given X1
P2 is similarly defined
Auctions
August 4, 2015
4/9
Model (continued)
From P define a matrix of conditional probabilities, P1
P1 is a m × m matrix with elements Pr(x2 |x1 ), x1 , x2 ∈ X
Each row of P1 is a conditional probability distribution of X2 given X1
P2 is similarly defined
Assumption: P1 and P2 are full-rank matrices
Assumption: Single-crossing condition is satisfied
Auctions
August 4, 2015
4/9
The optimal mechanism
u1 (x1 ) ≡ buyer 1’s expected surplus in second-price auction when X1 = x1
Auctions
August 4, 2015
5/9
The optimal mechanism
u1 (x1 ) ≡ buyer 1’s expected surplus in second-price auction when X1 = x1
Let u1 be an m vector of the u1 (x1 )’s.
Auctions
August 4, 2015
5/9
The optimal mechanism
u1 (x1 ) ≡ buyer 1’s expected surplus in second-price auction when X1 = x1
Let u1 be an m vector of the u1 (x1 )’s.
As P1 has full rank, there exists a m vector c1 such that
P1 c1 = u1
Auctions
August 4, 2015
5/9
The optimal mechanism
u1 (x1 ) ≡ buyer 1’s expected surplus in second-price auction when X1 = x1
Let u1 be an m vector of the u1 (x1 )’s.
As P1 has full rank, there exists a m vector c1 such that
P1 c1 = u1
That is, for all x1 ∈ X,
X
Pr(x2 |x1 )c1 (x2 ) = u1 (x1 )
x2 ∈X
Auctions
August 4, 2015
5/9
The optimal mechanism
u1 (x1 ) ≡ buyer 1’s expected surplus in second-price auction when X1 = x1
Let u1 be an m vector of the u1 (x1 )’s.
As P1 has full rank, there exists a m vector c1 such that
P1 c1 = u1
That is, for all x1 ∈ X,
X
Pr(x2 |x1 )c1 (x2 ) = u1 (x1 )
x2 ∈X
c2 is defined similarly
Auctions
August 4, 2015
5/9
The optimal mechanism
A second-price auction plus the payments ci is an optimal mechanism –
maximizes efficiency and revenue
Auctions
August 4, 2015
6/9
The optimal mechanism
A second-price auction plus the payments ci is an optimal mechanism –
maximizes efficiency and revenue
It is incentive compatible because c1 does not depend on x1 and c2 does
not depend on x2
Auctions
August 4, 2015
6/9
The optimal mechanism
A second-price auction plus the payments ci is an optimal mechanism –
maximizes efficiency and revenue
It is incentive compatible because c1 does not depend on x1 and c2 does
not depend on x2
It is efficient as single-crossing is satisfied
Auctions
August 4, 2015
6/9
The optimal mechanism
A second-price auction plus the payments ci is an optimal mechanism –
maximizes efficiency and revenue
It is incentive compatible because c1 does not depend on x1 and c2 does
not depend on x2
It is efficient as single-crossing is satisfied
It is revenue-maximizing because buyer expected surplus is zero:
Auctions
August 4, 2015
6/9
The optimal mechanism
A second-price auction plus the payments ci is an optimal mechanism –
maximizes efficiency and revenue
It is incentive compatible because c1 does not depend on x1 and c2 does
not depend on x2
It is efficient as single-crossing is satisfied
It is revenue-maximizing because buyer expected surplus is zero:
At X1 = x1 , for any x1 ∈ X, buyer 1’s expected surplus is
u1 (x1 ) −
X
Pr(x2 |x1 )c1 (x2 ) = u1 (x1 ) − u1 (x1 ) = 0
x2 ∈X
Auctions
August 4, 2015
6/9
Example
Two buyers with types X1 , X2 ∈ {0, 1} and values Vi = Xi .
Auctions
August 4, 2015
7/9
Example
Two buyers with types X1 , X2 ∈ {0, 1} and values Vi = Xi .




Pr(0, 0) Pr(0, 1)
0.375 0.125
 = 

P = 
Pr(1, 0) Pr(1, 1)
0.125 0.375
Auctions
August 4, 2015
7/9
Example
Two buyers with types X1 , X2 ∈ {0, 1} and values Vi = Xi .




Pr(0, 0) Pr(0, 1)
0.375 0.125
 = 

P = 
Pr(1, 0) Pr(1, 1)
0.125 0.375




Pr(0 | 0) Pr(0 | 1)
0.75 0.25
 = 

Pi = 
Pr(1 | 0) Pr(1 | 1)
0.25 0.75
Auctions
August 4, 2015
7/9
Example
Efficient allocation rule: Q(X1 , X2 ) = (prob 1 gets it, prob 2 gets it)


(0, 0)
(0, 1)

Q(X1 , X2 ) = 
(1, 0) (0.5, 0.5)
Auctions
August 4, 2015
8/9
Example
Efficient allocation rule: Q(X1 , X2 ) = (prob 1 gets it, prob 2 gets it)


(0, 0)
(0, 1)

Q(X1 , X2 ) = 
(1, 0) (0.5, 0.5)
Buyer’s exp. surplus in second-price auction: ui = (0, 0.25)
Auctions
August 4, 2015
8/9
Example
Efficient allocation rule: Q(X1 , X2 ) = (prob 1 gets it, prob 2 gets it)


(0, 0)
(0, 1)

Q(X1 , X2 ) = 
(1, 0) (0.5, 0.5)
Buyer’s exp. surplus in second-price auction: ui = (0, 0.25)
Select ci = (−0.125, 0.375).
Auctions
August 4, 2015
8/9
Example
Efficient allocation rule: Q(X1 , X2 ) = (prob 1 gets it, prob 2 gets it)


(0, 0)
(0, 1)

Q(X1 , X2 ) = 
(1, 0) (0.5, 0.5)
Buyer’s exp. surplus in second-price auction: ui = (0, 0.25)
Select ci = (−0.125,
 0.375).
 
0
 =  
ui − Pi ci = 
0.25 − [(0.25)(−0.125) + (0.75)(0.375)]
0
0 − [(0.75)(−0.125) + (0.25)(0.375)]
Auctions

August 4, 2015
8/9
Caveats
The optimal mechanism is
Is not detail free
Not ex post individually rational
Does not work with limited liability
Auctions
August 4, 2015
9/9