auction theory class 7 – common values, winner’s curse and interdependent values. 1

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Auction Theory Class 7 – Common Values, Winner’s curse and Interdependent Values. 1

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Auction Theory

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

1

Outline• Winner’s curse

• Common values– in second-price auctions

• Interdependent values– 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.

– 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

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

• 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

In the common value setting:• Theorem: bidding according to β(xi)=v(xi,xi) is a Nash

equilibrium in a second-price auction.

Second-price auctions

• 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

In the common value setting:• Theorem: bidding according to β(xi)=v(xi,xi) is a Nash

equilibrium in a second-price auction.

Second-price auctions

• 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

In the common value setting:• Theorem: bidding according to β(xi)=v(xi,xi) is a Nash

equilibrium in a second-price auction.

Second-price auctions

• 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)

13

)(

0

1

]|[),(),(),(b

i dyxygyyvyxvxbu

In the common value setting:• Theorem: bidding according to β(xi)=v(xi,xi) is a Nash

equilibrium in a second-price auction.

Second-price auctions

14

x

)(

0

1

]|[),(),(),(b

i dyxygyyvyxvxbu

Let’s plot v(x,y)-v(y,y)

Recall: v(x,y) increasing in x (for all x,y)

Utility is maximized when bidding b= β(x)= v(x,x)

y

Second price auctions: example• Example: v ~ U[0,1]

xi ~ U[0,2v]n = 3

• Equilibrium strategy:

• See Krishna’s book for the details.

15

x

xx

2

2)(

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 TriviaQuestion 1:

What is the distance between Paris and Moscow?

Question 2: What is the year of birth of David Ben-Gurion?

17

Information AggregationCommon-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.18

Outline• Winner’s curse

• Common values– in second-price auctions

• Interdependent values– 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 forv1(x1, x2) = x1

v2(x1, x2) = (x1)2

Where x1 is drawn from [0,2]

Efficient auctions

• 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.

1y1 z1

Claim: no efficient mechanism exists forv1(x1, x2) = x1 v2(x1, x2) = (x1)2

Where x1 is drawn from [0,2]

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

When 1’s true value is y1: y1-p1(y1) ≥ 0-p1(z1)

Single-crossing conditionConclusion: 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

24

Single-crossing conditionDefinition: 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.

25

)()( xx

vx

x

v

i

j

i

i

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)

26

x1

v1(x1, x2,x3)

v2(x1, x2,x3)

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 + x3 v3(x1, x2,x3) = 3x1 + 2x2 + 2x3

are single crossing

27

1y1 z1

x1

An Efficient AuctionConsider 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.

28

An Efficient AuctionWhat is the payment of bidder 1 when he wins with a

signal ?

29

x1

v1(x1, x-i)

v2(x1, x-i)

y1(x-1)

v3(x1, x-i)M*(i)

*1x

*1x

An Efficient AuctionWhat 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.

30

x1

v1(x1, x-i)

v2(x1, x-i)

y1(x-1)

v3(x1, x-i)M*(i)

*1x

An Efficient Auction

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

31

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

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.

32

An Efficient Auction: proofProof: • 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.

33

An Efficient Auction:proofProof (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.

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– in second-price auctions

• Interdependent values– 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 valuesFor the Generalized-VCG auction to work, signals are not

necessarily statistically independent: correlation is allowed.

Which one is not a product of independent distributions?:

39

1 2 3

1 1/24 1/12 1/24

2 1/12 1/6 1/12

3 1/8 1/4 1/8

Independent distributions:f1(1)=1/6, f1(2)=1/3, f1(3)=1/2f2(1)=1/4, f2(2)=1/2, f2(3)=1/4

A joint distribution

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

x2

x1

x2

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) 40

1 2 3

1 1/6 1/12 1/12

2 1/12 1/6 1/12

3 1/12 1/12 1/6

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 41

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

=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.

42

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

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•

45

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

Pr(x-i | xi)x-i

xi

Pr(x1,…,xn)

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 ¼2 ¼ ½ ¼3 ¼ ½ ¼

Full rank (3)

Rank 1

Correlated

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.

46

• 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).

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 )where

yi(x-i) = min{ zi | vi(zi,x-i) ≥ maxj≠i vj(zi,x-i) }

48

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)

+ ci(x-i)

The trick• The expected surplus of each bidder:

49

)(),(),()Pr()( **iiiiiiiiii

xii xMxxvxxQxxxU

i

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

)()Pr()(*iiii

xii xcxxxU

i

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

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

The trick (cont.)

50

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( *iiiiiiiiiiii

x

xcxMxxvxxQxxi

)(),(),()Pr( *iiiiiiiiii

x

xMxxvxxQxxi

)()Pr( iiiix

xcxxi

0

=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”

51

)()Pr()(*iiii

xii xcxxxU

i

ii cPU *

The Cremer-Mclean mechanism

52

– Bidders report their signals x1,…,xn

– The winner: the bidder with the highest value (given the reported signals).

– Payments: the winner pays Mi

CM=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):)()Pr()(*

iiiix

ii xcxxxUi

Under the Cremer-Mclean condition: it is truthful, efficient and leaves bidders with a 0 surplus.

Our example

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

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

Payments in a 2nd price auction

Cremer-Mclean payments

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 surplus54