Auction Theory

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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

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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
• 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
• 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
• 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
• 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)
• 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
• 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
• 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 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 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 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 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 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
• Example: v ~ U[0,1] xi ~ U[0,2v] n = 3
• Equilibrium strategy:
• See Krishna’s book for the details.
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?

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.
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
• 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)3v2(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 forv1(x1, x2) = x1

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

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

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 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
• 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: 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

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 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 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 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
• 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

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

• Does not know the signals, of course.
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
• 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
• 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

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

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
• 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: 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.
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.
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

Pr(x1,…,xn)

Pr(x-i | xi)

x-i

Correlated

Full rank (3)

xi

Rank 1

independent

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.
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”
• 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 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.)
• 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 (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
• 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

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
• 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