1. Class 6 – Common Values, Winner’s curse and Interdependent Values.
Auction Theory
Class 6 – Common Values, Winner’s curse and Interdependent Values.

2. Outline Winner’s curse Common values Interdependent values
in second-price auctions
Interdependent values
The single-crossing condition.
An efficient auction.

3. 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.

4. 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.

5. 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.

6. 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

7. 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.

8. 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.

9. 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.

10. 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.

11. 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.

12. 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.

13. 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)

14. 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)

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

16. 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.

17. 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?
2495 km, 1886

18. 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.
From wikipedia: In 1906 Galton visited a livestock fair and stumbled upon an intriguing contest. An ox was on display, and the villagers were invited to guess the animal's weight after it was slaughtered and dressed. Nearly 800 gave it a go and, not surprisingly, not one hit the exact mark: 1,198 pounds. Astonishingly, however, the mean of those 800 guesses came close — very close indeed. It was 1,197 pounds.

19. Outline Winner’s curse Common values Interdependent values
in second-price auctions
Interdependent values
The single-crossing condition.
An efficient auction.

20. 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 ]

21. 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

22. Efficient auctions
Can we design an efficient auction for settings with interdependent values?
No.
Claim: no efficient mechanism exists for v1(x1, x2) = x1
v2(x1, x2) = (x1)2 Where x1 is drawn from [0,2]

23. Efficient auctions
Claim: no efficient mechanism exists for v1(x1, x2) = x1 v2(x1, x2) = (x1)2 Where x1 is drawn from [0,2] y1 z1 1
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.
When 1’s true value is z1: p1(z1)≥ z1 – p(y1) (efficiency + truthfulness)
When 1’s true value is y1:
y1-p1(y1) ≥ 0-p1(z1)

24. 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

25. 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.

26. 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.

27. 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 y1 z1 1 x1

28. 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 ) 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.

29. 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)

30. 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. x1
v1(x1, x-i)
v2(x1, x-i)
y1(x-1)
v3(x1, x-i)
M*(i)

31. 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

32. An Efficient Auction:proof
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: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.

34. 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 some very high false signal, I may win and pay more than my value.

35. Weakness
Weakness of the efficient auction: seller needs to know the valuation functions of the bidders
Does not know the signals, of course.

36. Summary Private values is a strong assumptions.
Many times the item for sale has a common value.
Still, bidders have privately known signals.
But would know better if knew other signals.
We analyzed auctions with interdependent values: the value depends on the signals of all players.
We saw how bidders account for the winner’s curse in second-price auctions
We saw an efficient auction (under the single-crossing assumptions)
New equilibrium concept: ex-post Nash.a