auction theory class 7 – common values, winner’s curse and interdependent values. 1
TRANSCRIPT
Outline• Winner’s curse
• Common values– in second-price auctions
• Interdependent values– The single-crossing condition.– An efficient auction.
• Correlated values– Cremer & Mclean mechanism
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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.
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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.
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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.
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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
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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.
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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.
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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.
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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.
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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.
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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)
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)(
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
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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.
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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?
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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
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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
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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.
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)()( 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)
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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
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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.
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An Efficient AuctionWhat is the payment of bidder 1 when he wins with a
signal ?
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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.
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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
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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.
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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.
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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.
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Weakness
Weakness of the efficient auction: seller needs to know the valuation functions of the bidders
– Does not know the signals, of course.
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Outline• Winner’s curse
• Common values– in second-price auctions
• Interdependent values– The single-crossing condition.– An efficient auction.
• Correlated values– Cremer & Mclean mechanism
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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.
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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?:
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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.
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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.
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Revenue• The Cremer-Mclean Condition: the conditional
correlation matrix has a full rank for every bidder.– That is, some minimal level of correlation exists.
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The correlation matrix•
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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.
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• 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).
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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) }
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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:
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)(),(),()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.)
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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”
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)()Pr()(*iiii
xii xcxxxU
i
ii cPU *
The Cremer-Mclean mechanism
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– 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