EFFICIENT AUCTIONS

1
EFFICIENT AUCTIONS

• PARTHA DASGUPTA AND ERIC MASKIN
• QUARTERLY JOURNAL OF ECONOMICS
• May 2000

2
INTRODUCTION

• Efficient auctions auctions that put goods into
  the hands of the buyers who value them the most.
• Most of the theoretical literature on auctions
  primarily concentrates on revenue-maximization.

3
Motivation

• A leading rationale for the widespread
  privatization of state-owned assets in recent
  years is to enhance efficiency.

4
The Case of many buyers

• If there are a sufficiently large number of
  potential buyers, competition will render
  virtually any kind of auction approximately
  efficient.
• In practice, the number of serious bidders is
  often severely limited.
• For many properties sold in the FCC spectrum
  auctions, the number of bidders submitting
  realistic bids was as low as two or three.

5
Common values

• Common values - where one buyers valuation can
  depend on the private information of another
  buyer.
• Example several wildcatters are bidding for the
  right to drill for oil on a given tract of land.

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The principal contribution of this paper

• Show that the Vickrey auction can be generalized
  so as to attain efficiency even when there are
  common values.
• Show that this auction remains efficient
  regardless of the number of goods being sold, and
  of the nature of those goods, e.g., whether they
  are substitutes or complements.

7
Formulation

• Suppose that there is a single unit of a good
  available for auction.
• There are n risk-neutral buyers.
• Buyer i observes a private real-valued Signal si.
• Let vi (s1,..., sn) be buyer is expected
  valuation for the good, conditional on all the
  signals (s1,..., sn).

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Formulation (Cont.)

• If buyer i is awarded the good and pays price p,
  his net payoff is
• vi (s1 , ... , sn ) p
• Assume that, for all i, vi() is continuously
  differentiable in its arguments and that a higher
  signal value si corresponds to a higher valuation

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Example Definition

• If vi(s1 , ... , sn ) si, then this is an
  auction of private values.
• We call an auction efficient if, for all signal
  values (s1,..., sn), the winner in equilibrium is
  buyer i such that
• vi(s1 , ... , sn ) vj(s1 , ... , sn ) for all
  j.

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Familiar auction types

• High-bid auction The buyers submit sealed bids,
  the winner is the high bidder.
• Second-price (or Vickrey) auction Has the same
  rules as the high-bid, except that the winner
  pays only the second-highest bid.
• English auction The buyers call out bids
  publicly. The winner is the last buyer to bid,
  and he pays his bid.

11
High-bid auction

• Even with private values, the high-bid auction is
  not, in general, efficient.
• Example s1 is drawn from a continuous
  distribution on 0,1 whereas s2 is drawn
  (independently) from a continuous distribution on
  0,10.
• The equilibrium bid functions (b1(),b2())
  satisfy b1(1) b2(10), where b2() is strictly
  increasing at s210.

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Second-price auction

• Is efficient in the case of private values.
• Not efficient in the case of common values !!

13
Second-price auction - Example

• Suppose that there are three buyers, whose
  valuations are
• v1(s1 ,s2 ,s3 ) s1 ½ s2 ¼ s3
• v2(s1 ,s2 ,s3 ) s2 ¼ s1 ½ s3
• v3(s1 ,s2 ,s3 ) s3
• In a neighborhood of (s1,s2,s3) (1,1,1),
  efficient allocation of the good between buyers 1
  and 2 depends on the value of s3.

14
Direct revelation mechanisms

• Each buyer i reports a signal value si.
• The good is awarded to the buyer i for whom
  vi(s1,..., sn) maxi?j vj(s1,..., sn).
• In equilibrium, si equals the true value si.

15
Direct revelation mechanisms - Problems

• It would require the mechanism designer (or
  auctioneer) to know the physical signal spaces
  S1, S2, and S3 and the functional forms of the
  valuation functions v1(), v2(), and v3(). A
  strong assumption.

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Condition on valuations

• for all i and j?i,
• At any point where
• vi(s1,...,sn) vj(s1,...,sn) maxk
  vk(s1,...,sn).
• The condition says that (if buyers i and j have
  equal and maximal valuations) buyer is signal
  must have a greater marginal effect on his own
  valuation than on that of buyer j.

17
Example to establish condition

• Two wildcatters are competing for the right to
  drill for oil on a given tract of land.
• Wildcatter 1 has a fixed cost of 1 and a marginal
  cost of 2.
• Wildcatter 2s fixed cost is 2 and marginal cost
  is 1.
• Oil can be sold at a price of 4.
• Only wildcatter 1 performs a test, and discovers
  that the expected size of the oil reserve is s1
  units.

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Example (cont.)

• The value functions are
• v1(s1 ,s2) (4-2)s1 - 1 2s1 - 1
• v2(s1 ,s2) (4-1)s1 - 2 3s1 2
• Notice that
• Efficiency dictates that wildcatter 1 get the
  drilling rights if ½ lt s1 lt 1 and that wildcatter
  2 get the drilling rights if s1 gt 1.

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Example (cont. 2)

• Suppose that wildcatter 1 is given a reward R(s1)
  if he claims that there are s1 units of oil.
• If s1 gt 1 gt s1 gt ½ , incentive compatibility and
  efficiency demand that
• R (s1 ) 2s1 1 R(s1)
• 2s1 1 R(s1) R(s1)
• And we get 2(s1 - s1) 0 contradiction.

20
Auctions with Two Buyers

• Instead of a single bid, we will have each buyer
  i report a bid function,
• where j ? i.
• We can interpret as buyer is bid if
  the other buyers valuation turns out to be vj.

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Find a fixed point

• Given the bid functions let us
  look for a fixed point, i.e., a pair (v1,v2)
  such that
• Buyer i is the winner

22
Bidding truthfully

• To see that this allocation rule is the right
  one, consider what happens when buyers bid
  truthfully.
• That is, if buyer 1s signal value is s1, the
  truthful bid function is b1() such that
• b1(v2(s1 ,s2)) v1(s1 ,s2) for all s2.
• Similarly,
• b2(v1(s1 ,s2)) v2(s1 ,s2) for all s1.

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Bidding truthfully (cont.)

• Observe that
• (v1,v2) (v1(s1 ,s2) , v2(s1 ,s2))
• is a fixed point of the mapping
• (v1,v2) ? (b1(v2) , b2(v1))
• This means that, if buyers bid truthfully, our
  allocation rule ensures that buyer 1 wins if and
  only if v1(s1,s2) gt v2(s1,s2).

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Example of two fixed points

• v1(s1,s2) s1² s1s2 s2² s1 - 2s2 24
• v2(s1,s2) s2² s1s2 s1² - 9s1 13
• If (s1,s2) (2,3), then one fixed point is
• (v1(2,3) , v1(2,3)) (21,6)
• However, for these signal values,
• (v1(2,4) , v2(1,3)) (14,15) also constitutes a
• fixed point, because vi(2,4) vi(1,3), and so
• v2(1,3) b2(v1(1,3)), and v1(2,4)
  b1(v2(2,4)).

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Stronger conditions to ensure unique fixed point

• for all i and j?i,
• At any point.
• Note that
• And so, we obtain for
  all v1
• and vice versa.

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Winnings buyer payment

• It remains to establish that there exists a
  payment scheme that induces truthful bidding.
• The way that the Vickrey auction induces
  truthfulness in the private-values case is to
  make a winning buyers payment equal to the
  lowest bid that he could have made for which he
  would still have won the auction.

27
Winnings buyer payment (cont.)

• Try to adhere to this principle means that, if
  buyer 1 is the winner, then he should pay
  
• where
• This is because if buyer 1 were restricted to
  constant bids, v1 would be the lowest such bid
  for which buyer 1 would still win the auction.

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Incentive to bid truthfully

• If buyer 1 wins, his payoff is
• v1(s1,s2) b2(v1) where v1 b2(v1).
• To see that buyer 1 has an incentive to bid
  truthfully in equilibrium, it suffices to show
  that if buyer 1 sets then he wins if
  and only if his payoff is positive.

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Proof of equilibrium

• () The payoff is positive iff for any v1
• From the intermediate value theorem, there exists
  a value of v1 such that

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Proof of equilibrium (cont.)

• Hence, () holds if and only if
• v1(s1,s2) - v1 gt b2(v1(s1,s2)) - b2(v1).
• But v1 b2(v1) ,
• and b2(v1(s1,s2)) v2(s1,s2)
• () Hence, () holds if and only if
• v1(s1,s2) gt v2(s1,s2).
• But, when he is truthful, buyer 1 wins if and
  only if () holds. Hence, if buyer 1 bids
  truthfully, () is indeed positive if and only if
  buyer 1 wins.

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To summarize

• Consider the two-buyer auction in which,
• for i 1,2,
• buyer i reports i ? j
• and a contingent bid function
• that satisfy
• a fixed point (v1,v2) is taken, and the winner
  is determined.
• Winner i pays

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To summarize (cont.)

• This auction is efficient
• It is an equilibrium for each buyer i to bid
  truthfully.
• If both buyers do so, the auction results in an
  efficient outcome.

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Remark 1

• It may seem very demanding to insist that a buyer
  make his bid a function of the other buyers
  valuation.
• Indeed, suppose that buyer 1 knew nothing about
  the nature of v2(), He could, make an
  uncontingent bid b1() ? b1.
• In this sense, having buyers report contingent
  bids should be viewed as giving them an
  opportunity to express their interdependencies.

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Remark 2

• Some degree of common knowledge about valuation
  functions is needed to ensure that players can
  calculate equilibrium.
• Why we do not go all the way and have each
  buyer i report a pair of valuation functions
  (v1(),v2()) and then
• (i) use a direct revelation mechanism, in
  which each buyer reports his signal value and
  these are then plugged into the reported
  valuation functions. (ii) or punish buyers in
  some way if their reports disagree.

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Remark 2 (cont.)

• There is a difficulty, however, with having buyer
  1 report v2(), namely, he may not even know
  what buyer 2s physical signal space.
• Notice that there is no contradiction in
  supposing that buyer 1 does not know v2() but
  does know v2.
• The ability of buyer i to calculate vj can be
  thought of as the weakest hypothesis that ensures
  efficiency in equilibrium.

36
Auctions with More than Two Buyers

• Each buyer i (i 1,..., n) submits a bid
  correspondence
• where
• A fixed point (v1,..., vn) is calculated so
  that for all i.
• if vi maxj?i vj , the good is awarded to
  buyer i.

37
Auctions with More than Two Buyers (cont.)

• If buyer i is the winner, he makes a payment
  maxj?ivj , where (v1,, ivn) is a vector such
  that vj maxj?ivj
• and for all k?i.
• What if there are multiple fixed points ?
• What if there are multiple payment points ?

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Conditions to ensure efficiency equilibrium

• (i) For all i, for all s-i?Si , there exists
• s'i?Si such that vi(s'i,s-i) gt maxj?iv(s'i,s-i).
  
• (ii) for all m1,,n

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Proposition

• Assume that, for all i 1,...,n, buyer is
  valuation function satisfies (i), and that
  buyers valuation functions collectively satisfy
  (ii), then it is an equilibrium for each buyer i
  to bid truthfully.
• Moreover, if buyers are truthful, the auction is
  efficient.

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Example with three users

• (i) v1(s1 ,s2 ,s3 ) s1 ½ s2 ¼ s3
• (ii) v2(s1 ,s2 ,s3 ) s2 ¼ s1 ½ s3
• (iii) v3(s1 ,s2 ,s3 ) s3
• Buyer 3s valuation does not depend on s1 and s2
  and so, given s3, his truthful bid function
  b3(v1,v2) s3.
• b1(v2,v3) s1 ½ (v2 ¼ s1 ½ v3) ¼ v3
• 7/8 s1 ½ v2.
• b2(v1,v3) 7/8 s2 ¼ v1 7/16 v3

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Example (cont.)

• Suppose for example, that s1 s2 1 and that s3
  is either slightly less than 1. Then buyer 1 is
  the winner, and he should pay v1 b2(v1,v3)
  
• 7/8 ¼ v1 7/16v3
• i.e., v1 7/6 7/12 s3
• Hence, buyer 1s net payoff is
• (1 ½ ¼ s3 ) (7/6 7/12 s3 )

42
Multidimensional Signals

• Example There are two wildcatters competing for
  the right to drill for oil on a tract of land
  consisting of an eastern and western region.
• Wildcatter 1 has a (fixed) cost of drilling c1,
  which is private information. She also performs a
  private test that tells her that the expected
  quantity of oil in the eastern region is q1.
  Wildcatter 2 has fixed cost c2 and expected
  quantity q2 in the western region.

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Multidimensional Signals(cont)

• wildcatter 1s information can be summarized,
  from her own standpoint, by the one-dimensional
  signal
• t1 q1- c1.
• However, t1 is not an adequate summary of 1s
  information from wildcatter 2s standpoint.

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Proposition

• If there exist signal values s?i , s?i and
• s?-i such that vi(s?i,) vi(s?i ,), but
• arg maxj vj(s?i,,s?-i) ? arg maxj vj(s?i,,s?-i)
• then there is no efficient auction with regular
  equilibria.

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Multiple goods

• Each buyer has valuation for every possible set
  of goods.
• For each subset of goods, the buyer report bid
  function.
• Again we will look for fixed points.

46
Problem with multiple goods

• Vickrey auctions for multiple goods are sometimes
  criticized as demanding too much information of a
  buyer he is asked to submit a bid for each
  possible combination of goods.
• Further-more, in our common-values setting, these
  bids must be made contingent on all other buyers
  valuations.

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An answer

• In our view, these criticisms are overblown.
• A buyer could be permitted to submit bids only on
  those combinations of goods he is potentially
  interested in.
• Furthermore, he could choose to make his bids
  contingent only on those other buyers valuations
  that, he believes, share a significant common
  component with his own valuation.

48
An open question

• there are at least two important advantages that
  an English auction could have over a generalized
  Vickrey auction
• at any instant, a buyer in an English auction
  need make only a binary decision whether or not
  to drop out.
• Back to the 3 buyers example In the English
  auction, buyer 3s true signal value can be
  inferred even though he does not win. In the
  generalized Vickrey auction, by contrast, buyer 3
  must truthfully bid b3(v1,v2) in order for s3 to
  be revealed.

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An open question (cont.)

• Thus, on both counts, we regard finding an
  appropriate English auction (i.e., a dynamic
  auction with binary decisions at each instant)
  counterpart to our Vickrey auction with multiple
  goods as a leading topic for further research.