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Auction Theory. Class 7 – Common Values, Winner’s curse and Interdependent Values. Outline. Winner’s curse Common values in second-price auctions Interdependent values The single-crossing condition. An efficient auction. Correlated values Cremer & Mclean mechanism. Common Values.

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auction theory

Auction Theory

Class 7 – Common Values, Winner’s curse and Interdependent Values.

outline
Outline
  • Winner’s curse
  • Common values
    • in second-price auctions
  • Interdependent values
    • The single-crossing condition.
    • An efficient auction.
  • Correlated values
    • Cremer & Mclean mechanism
common values
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.
winner s curse
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.
modeling common values
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.
winner s curse some comments
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.
    • 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
a useful notation v x y
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.
a useful notation x i
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,…,xnwhere the i’th coordinate is replaced with z.
second price auctions
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.
second price auctions1
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.
second price auctions2
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.
second price auctions3
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.
second price auctions4
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)
second price auctions5
Second-price auctions

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)

second price auctions example
Second price auctions: example
  • Example: v ~ U[0,1] xi ~ U[0,2v] n = 3
  • Equilibrium strategy:
  • See Krishna’s book for the details.
symmetric valuations
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
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?

information aggregation
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.
outline1
Outline
  • Winner’s curse
  • Common values
    • in second-price auctions
  • Interdependent values
    • The single-crossing condition.
    • An efficient auction.
  • Correlated values
    • Cremer & Mclean mechanism
interdependent values
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 values1
Interdependent values
  • Example:v1(x1, x2,x3) = 5x1 + 3x2 + x3 v2(x1, x2,x3) = 2x1 + 9x2 + (x3)3v2(x1, x2,x3) = 2x1x2 + (x3)2
efficient auctions
Efficient auctions
  • Can we design an efficient auction for settings with interdependent values?
  • No.

Claim: no efficient mechanism exists forv1(x1, x2) = x1

v2(x1, x2) = (x1)2 Where x1 is drawn from [0,2]

efficient auctions1
Efficient auctions

Claim: no efficient mechanism exists forv1(x1, x2) = x1v2(x1, x2) = (x1)2 Where x1 is drawn from [0,2]

  • Proof:
    • What is the efficient allocation?
      • 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.

Together: y1 ≥ z1 contradiction.

y1

z1

1

When 1’s true value is y1:

y1-p1(y1) ≥ 0-p1(z1)

When 1’s true value is z1:0 - p1(z1)≥ z1 – p(y1) (efficiency + truthfulness)

single crossing condition
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
single crossing condition1
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, i’s valuations grows more rapidly with xi than j’s valuation.
single crossing examples
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.

single crossing examples1
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 + x3v3(x1, x2,x3) = 3x1 + 2x2 + 2x3are single crossing

y1

z1

1

x1

an efficient auction
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 )whereyi(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.
an efficient auction1
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)

an efficient auction2
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)

x1

y1(x-1)

an efficient auction3
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

ex post equilibrium
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.
an efficient auction proof
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).
  • Bidderipaysvi(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.

an efficient auction proof1
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).
    • xi< yi(x-i)
    • Payoff of zero
  • To win, I must report zi>yi(x-i).
    • 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.
weakness
Weakness

Weakness of the efficient auction: seller needs to know the valuation functions of the bidders

  • Does not know the signals, of course.
outline2
Outline
  • Winner’s curse
  • Common values
    • in second-price auctions
  • Interdependent values
    • The single-crossing condition.
    • An efficient auction.
  • Correlated values
    • Cremer & Mclean mechanism
revenue
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.
discrete values
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)
correlated values
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

x1

x1

revenue1
Revenue
  • Example: let’s consider the joint distribution
  • Let’s consider 2nd-price auctions:
    • 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 gain information from other bidders’ values)
revenue example
Revenue: example
  • 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) + 0.25*(-2)
    • u1(b1=1) = 0.25*(0.5*2+1/2) +0.5*(0) + 0.25*(-2) = - 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

=0

revenue example1
Revenue: example
  • 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.
revenue2
Revenue
  • Conclusions from the previous example:
    • An incentive compatible, efficient mechanism that gains more revenue than the 2nd-price auction
      • Revenue equivalence theorem doesn’t 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.
revenue3
Revenue
  • The Cremer-Mclean Condition: the conditional correlation matrix has a full rank for every bidder.
    • That is, some minimal level of correlation exists.
the correlation matrix
The correlation matrix

Pr(x1,…,xn)

Pr(x-i | xi)

x-i

Correlated

Full rank (3)

xi

Rank 1

independent

revenue4
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.
revenue5
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).
reminder generalized vcg
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 )whereyi(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.
    • Mi*=vi( yi(x-i) , x-i ) + ci(x-i)
the trick
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:

That’s the conditional probability for which the Cremer-Mclean condition applies

the trick cont
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)

the trick cont1
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 algebra:If the matrix Pr(x-i|xi) has full rank: yes!
  • Economic interpretation of full rank: signals must be “correlated enough”
the cremer mclean mechanism
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
      • 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.

our example
Our example

Payments in a 2nd price auction

Cremer-Mclean payments

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)

summary
Summary
  • Private values is a strong assumption.
    • Many times the item for sale has a common value.
  • Still, bidders have privately known signals.
    • But would know better if knew other signals.
  • Interdependent values:
    • 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