Lecture 2 Auction Design

1
Lecture 2Auction Design
This lecture derives bidding rules for some
auctions where there is incomplete information,
and discusses the the virtues and shortfalls of
alternative auction mechanisms. First we
explore the concept of revenue equivalence, which
is weaker than congruence, and applies to private
value auctions where the bidders are risk
neutral. Then we relax the conditions for
revenue equivalence to apply, seeking to show the
effects on bidding behavior and auction
revenue. Finally we discuss the role of
collusion and entry in auctions.
2
Review of Lecture 1 Congruence revisited
Using the concept of congruence we derived two
rules for bidding Rule 1 Pick the same
reservation price in a Dutch auction that you
would submit in a first price sealed bid
auction. Rule 2 In private value auctions, or
if there are only two bidders, choose the
reservation price for an English or a Japanese
auction, that you would submit in a second price
sealed bid auction.
3
Review of Lecture 1Second price auctions

• We also proved that the bidding strategy in
  sealed bid second price auctions (and ascending
  auctions) is very straightforward if you know
  your own valuation.
• Rule 3 In a second price sealed bid auction,
  bid your valuation if you know it.

4
Relaxing congruence

• In congruent auctions, the revenue to the
  auctioneer and the payoffs to each bidder are
  identical for every game history generated by a
  solution strategy profile.
• This is a very strong form of equivalence, and
  often not met. The bidders and the auctioneer may
  be indifferent between two auctions that are not
  congruent to each other.
• For example, suppose the auctioneer and the
  bidders only care about their expected utility
  from respectively conducting and participating in
  an auction, and did not care about whether each
  individual game history has the same outcome.
• Can we show that such players might be
  indifferent to certain non-congruent auctions
  (where there is incomplete information)?

5
Revenue Equivalence Defined

• The concept of revenue equivalence provides a
  useful tool for exploring this question.
• Two auction mechanisms are revenue equivalent if,
  given a set of players their valuations, and
  their information sets, the expected surplus to
  each bidder and the expected revenue to the
  auctioneer is the same.
• Therefore revenue equivalence is a less stringent
  condition than congruence.
• Thus two congruent auctions are invariably
  revenue equivalent, but not all revenue
  equivalent auctions are congruent.

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Why study revenue equivalence ?

• If the auctioneer and the bidders are risk
  neutral, studying revenue equivalence yields
  conditions under which the players are
  indifferent between auctions that are not
  necessarily congruent.
• Exploiting the principle of revenue equivalence
  can sometimes give bidders a straightforward way
  of deriving their solution bid strategies.

7
Preferences and Expected Payoffs
Let U(vn) denote the expected value of the nth
bidder with valuation vn bidding according to
his equilibrium strategy when everyone else does
too. P(vn) denote the probability the nth
bidder will win the auction when all players bid
according to their equilibrium strategy. C(vn)
denote the expected costs (including any fees to
enter the auction, and payments in the case of
submitting a winning bid).
8
An Additivity Assumption

• We suppose preferences are additive, symmetric
  and private, meaning
• U(v) P(v) v - C(v)
• So the expected value of participating in the
  auction is additive in the expected benefits of
  winning the auction and the expected costs
  incurred.

9
A revealed preference argument

• Suppose the valuation of n is vn and the
  valuation of j is vj.
• The surplus from n bidding as if his valuation
  is vj is U(vj), the value from participating if
  his valuation is vj, plus the difference in how
  he values the expected winnings compared to to a
  bidder with valuation vj, or (vn vj)P(vj).
• In equilibrium the value of n following his
  solution strategy is at least as profitable as
  deviating from it by pretending his valuation is
  vj. Therefore
• U(vn) gt U(vj) (vn vj)P(vj)

10
Revealed preference continued

• For convenience, we rewrite the last slide on
  the previous page as
• U(vn) - U(vj) gt (vn vj)P(vj)
• Now viewing the problem from the jth bidders
  perspective we see that by symmetry
• U(vj) gt U(vn) (vj vn)P(vn)
• which can be expressed as
• (vn vj)P(vn) gt U(vn) - U(vj)

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A fundamental equality

• Say vn gt vj.which then implies P(vn)gt P(vj).
• Putting the two inequalities together, we
  obtain
• (vn vj) P(vn)gt U(vn) - U(vj) gt (vn vj) P(vj)
• Writing
• vn vj dv
• yields
• which, upon integration, yields

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Revenue equivalence

• This equality shows that in private value
  auctions, the expected surplus to each bidder
  does not depend on the auction mechanism itself
  providing two conditions are satisfied
• 1. In equilibrium the auction rules award the bid
  to the bidder with highest valuation.
• 2. The expected value to the lowest possible
  valuation is the same (for example zero).
• Note that if all the bidders obtain the same
  expected surplus, the auctioneer must obtain the
  same expected revenue.

13
A theorem

• Assume each bidder
• - is a risk-neutral demander for the auctioned
  object
• - draws a signal independently from a common,
  strictly increasing, cumulative continuous
  distribution function.
• Consider auction mechanisms where
• - the buyer with the highest signal always wins
• - the bidder with the lowest feasible signal
  expects zero surplus.
• Then the same expected revenue will be
  generated by the auctions, and each bidder will
  make the same expected payment as a function of
  her signal.

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First price sealed bid private value action for
wireless licenses
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Second price sealed bid private value action for
wireless licenses
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All pay private value auction
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The Revenue from Private Value Auctions
Since any auction satisfying the conditions for
the theorem can be used to calculate the expected
revenue, we select the second price, or English
auction, to accomplish this task.
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Steps for deriving expected revenue

• The expected revenue from any auction satisfying
  the conditions of the theorem, is the expected
  value of the second highest bidder.
• To obtain this quantity, we proceed in two
  steps
• 1. derive and analyze the probability
  distribution of the highest valuation,
• 2. and then derive the probability distribution
  of the second highest bidder.

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The unconditional probability of winning

• From the perspective of the auctioneer, or an
  outsider who does not know the valuation of any
  player, each player has an equal chance of
  winning the auction.
• Given N bidders, each of whom has the same
  chance of winning, the probability of bidder n
  winning the auction is 1/N.
• Each player knows his own valuation vn and
  consequently has more information than the
  auctioneer.
• In the solution to a second price sealed bid
  auction, each bidder submits her own valuation to
  the auctioneer, and therefore the winning bidder
  is the player with the highest valuation.

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The probability of winning an auction conditional
on your own valuation

• That is the probability of winning an auction in
  equilibrium is just the probability of being
  endowed with the highest valuation.
• If valuations are identically and independently
  distributed with cumulative probability
  distribution function F(v), the probability that
  v1, for example, is the highest of the N
  valuations equals
• Pr(v2 ? v1) ? (v3? v1) ?. . . ? (vN ? v1)
• Prv2 ? v1x Pr(v3? v1) x . . . x PrvN ? v1
•   F (v1) x F (v1) x . . . x F (v1) (N-1
  times)
• F (v1)N-1

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The probability of the median valuation winning
an auction

• Conditioning on your own valuation, as N
  increases the probability of winning the auction
  declines at a much faster rate than 1/N does.
• For example if v1 is the median of the
  distribution, the probability of winning the
  auction when N2 is 0.5.
• But the probability of winning the auction when
  N10 is
• 0.59 1. 953110-3
• which is orders of magnitude less than 1/10.

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The probability of winning an auction with a
median valuation as a function of the number of
bidders
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Expected revenue in a private value auction

• The expected revenue to the auctioneer is the
  expected value of the second highest valuation.
  This can be calculated as
• Hence the expected revenue for a private values
  auction satisfying the conditions of the theorem
  is this formula.

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Bidding Rules for Private Value Auctions
Armed with the formula on the previous slide, we
can also derive the solution bidding strategies
for auctions that are revenue equivalent to the
second price sealed bid auction.
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Bidding function in a first price sealed bid
auction

• Consider, for example a first price sealed bid
  auctions with independent and identically
  distributed valuations.
• In a symmetric equilibrium to first price sealed
  bid auction, we can show that a bidder with
  valuation vn bids

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An example the uniform distribution

• Valuations are uniformly distributed within the
  closed interval . In this case
• which implies

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Bidding function with the uniform distribution

• Thus in the case of the uniform distribution the
  equilibrium bid of the player with valuation v is
  to bid a weighted average of the lowest possible
  valuation and his own, where the weights are
  respectively 1/N and (N-1)/N

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Comparison of bidding strategies

• The bidding strategies in the first and second
  price auctions markedly differ.
• In a second price auction bidders should submit
  their valuation regardless of the number of
  players bidding on the object.
• In the first price auction bidders should shave
  their valuations, by an amount depending on the
  number of bidders.

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All pay sealed bid auction with private values

• The revenue equivalence theorem implies that the
  amount bidders expect to pay in an all-pay
  auction as in all other auctions satisfying the
  conditions of the theorem.
• In contrast to a first or second price sealed
  bid auctions where only the winner bidder pays
  his bid or the second highest bid in an all pay
  auction losers also pays their bids.
• The amount paid by the nth bidder is certain,
  and not paid with the probability of winning the
  auction, that is F(vn)N-1.
• By the revenue equivalence theorem the amount
  each bidder expect to pay in the first two
  auctions, upon seeing their valuation, equals the
  amount the bidder actually does pay in all pay
  auction.

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Expected revenue in all pay auction

• In an all pay auction the expected revenue from
  a bidder with vn bids F(vn)N-1 multiplied by the
  amount he would bid in a first price auction.
• This is

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When does Revenue Equivalence Fail?

• Bidders might be risk averse, not risk neutral.
• The private valuations of bidders might be drawn
  from the probability distribution that are not
  identical.
• The theorem does not apply when bidders receive
  signals about the value of the object to them
  that are correlated with each other.
• Collusion and entry deterrence are also
  considerations that auctioneers should account
  for.

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Attitudes towards risk in second price sealed bid
auctions with private values

• It remains a weakly dominant strategy for each
  player to bid his or her valuation.
• The optimal bidding strategy for the second
  price sealed bid auction (and also the Japanese
  and English auctions) is independent of a
  bidder's attitude towards risk and uncertainty
  when private values are drawn from a common
  probability distribution.

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Attitudes towards risk in first price sealed bid
auctions with private values

• A strategy of bidding your valuation guarantees
  exactly zero surplus.
• If you place a lower bid than your valuation
  your expected surplus initially increases until
  it reaches the maximum for a risk neutral bidder,
  and then falls, but the variance of the surplus
  increases as well.
• A risk averse gambler is willing to trade a
  lower expected value to reduce the amount of
  uncertainty, he accordingly bids higher than a
  risk neutral bidder.

34
Comparing first and second price sealed bid
auctions

• Revenue generated by a second price auction is
  independent of the bidders' preferences over
  uncertainty, since bidding is unaffected.
• The revenue generated by the first price auction
  is the same as the revenue generated by a second
  price auction when bidders are risk neutral.
• Therefore risk averse bidders generate more
  revenue in a first price auction than they would
  in a second price auction, and they generate more
  revenue in a first price auction than do risk
  neutral bidders.

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Asymmetric valuations

• In many auctions where there are private
  valuations, the bidders have different uses for
  the auctioned object, and this may be common
  knowledge to all the bidders.
• Bidder knows the probability distributions that
  each of the other valuations are drawn from, he
  will typically use that information when making
  his own bid.
• This affects the revenue equivalence theorem,
  and also the auctioneer's preferences towards
  different types of auctions.

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An example of asymmetry

• Instead of assuming that all bidders appear the
  same to the seller and to each other, suppose
  that bidders fall into two recognizably different
  classes.
• Instead of there being a single distribution
  F(v) from which the bidders draw their
  valuations, there are two cumulative
  distributions, F1(v) and F2(v).
• Bidders of type i?1,2 draw their valuations
  independently from the distribution Fi(v).
• Let fi(x) denote the probability density
  function of Fi(x).

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Asymmetry in a first price auction with only two
bidders

• The private valuation of the first bidder is
  drawn from a probability distribution F1(v) that
  stochastically dominates the probability
  distribution for the other probability
  distribution F2(v).
• In fact we make a stronger assumption, that for
  all v
• Then b1(v)lt b2(v). The intuition is to bid
  aggressively from weakness and vice versa.

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Asymmetric first price sealed bid auction
39
Base experiment for asymmetry with only one type
of players
40
Incomplete information about the type of the
bidder

• Suppose each bidder sees his valuation, but does
  not immediately learn whether he comes from the
  high or low probability distribution.
• At that point the bidding strategy cannot depend
  on which probability distribution the valuation
  comes from.
• Then each bidder is told which probability
  distribution his bid is drawn from.
• How should he revise his bid? The second (first)
  bidder learns that the first (second) bidder is
  more likely to draw a higher (lower) valuation
  than himself, realizes the probability of winning
  falls (rises), so adjusts his upwards
  (downwards).

41
Auction Design
In this part of the lecture we relax the
independence assumption for revenue equivalence
to apply, discuss the winners curse, and show
the effects on bidding behavior and auction
revenue. Finally we discuss the role of collusion
and entry in auctions.
42
Relaxing independence

• The revenue equivalence theorem applies to
  situations in which the valuation of each is
  bidder is independently distributed, and this is
  is what we have been focusing on in the first
  part of this lecture.
• This is not always a useful assumption, because
  in many situations a bidder would be informed if
  he had information about the object on the
  auction block that another bidder had, and would
  use the information in a similar way.
• What happens when the signals that bidders get
  about the value of the auctioned item are
  positively correlated?

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Symmetric Valuations

• What happens when the signals that bidders get
  about the value of the auctioned item are
  positively correlated?
• We relax independence and consider the class of
  symmetric valuations, which have two defining
  features
• 1. All bidders have the same utility function.
• 2. Each bidder only cares about the collection
  of signals received by the other bidders, not
  who received them.
• Thus we may write the valuation of bidder n as

44
An example Value of the object not known to
bidders

• Suppose the value of the object to each bidder
  is the same, but this value is unknown to each
  bidder. The nth bidder receives a signal sn which
  is distributed about the common value v, and
  write
• sn v ?n where ?n ? EvInformation of n
  v
• where ?n is independently distributed.
• More generally, each bidder might place more
  significance on their own draw, but still attach
  some value to the assessments of others.

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Summary so far

• When comparing two (or more) auctions, we should
  consider the following questions
• Are the auctions congruent, or if not, revenue
  equivalent?
• Are the bidders risk neutral?
• In private valuation auctions, are bidders
  drawing from the same probability distribution?
• In common valuation auctions, are bidders drawing
  from the same distribution of signals?

46
English auction with common value
47
Dutch auction with common value
48
The winner's curse

• When other bidders have information that you
  lack about the value of the object for sale,
  winning the auction may cause you to decrease
  your conditionally expected value of the object.
• Failure to take into account the bad news about
  others' signals that comes with any victory is
  called the winner's curse.
• The winner's curse describes the fact that
  winning an auction may convey new and unfavorable
  information about the item.
• Because all other bids are less than the winning
  bid, the expected value of the item to the
  winning bidder might fall when the outcome of the
  auction is announced.

49
The expected value of the item upon winning the
auction

• If the nth bidder wins the auction, he will
  realize his signal exceeded the signals of
  everybody else, that is
• sn maxs1,,sN
• so he will condition the expected value of the
  item on this new information.
• His expected value is now the expected value of
  vn conditional upon observing the maximum signal
• Evn sn maxs1,,sN
• This is the value that the bidder should use in
  the auction, not Evnsn, because he should
  recognize that unless his signal is the maximum
  he will receive a payoff of zero.

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Defining the Winners Curse

• The winner's curse can be defined as
• W(sn) Evnsn - Evn sn maxs1,,sN
• Since the max operator is a convex increasing
  function of its arguments
• it follows that W(sn) is a negative function.
• Although bidders should take the winner's curse
  into account, there is widespread evidence that
  novice bidders do not take this extra information
  into account when placing a bid.

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Revenue Comparisons forSymmetric Auctions

• We can rank the expected revenue generated in
  symmetric equilibrium for auctions where
  valuations are also symmetric.
• There are two basic results. In a symmetric
  auction
• 1. The expected revenue from a Japanese auction
  is higher than what an English auction yields.
• 2. The expected revenue from an English auction
  exceeds a first price sealed bid auction.

52
Differential information

• We have discussed several types of information
  structures in auctions
• - perfect foresight
• - independently distributed private valuations
• - symmetric valuations.
• But one important case we have not touched yet,
  when some bidders know more about the common
  value of the object than other bidders do.

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Bidding with differential information

• An extreme form of dependent signals occurs when
  one bidder know the signal and the others do not.
• How should an informed player bid?
• What about an uninformed player?

54
An English common value auction experiment on
asymmetrically informed bidders
55
A sealed bid common value auction on
asymmetrically informed bidders
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Perspective of the less informed bidder

• Suppose the uninformed bidder always makes the
  same positive bid, denoted b. This is an example
  of a pure strategy.
• Is this pure strategy part of a Nash
  equilibrium?
• The best response of the informed bidder is to
  bid a little more than b when the value of the
  object vi is worth more than b, and less than b
  otherwise.
• Therefore the uninformed bidder makes an
  expected loss by playing a pure strategy in this
  auction. A better strategy would be to bid
  nothing.

57
A theorem

• The argument in the previous slide shows that
  the uninformed bidder plays a mixed strategy in
  this game.
• One can show that in equilibrium the informed
  bidder bids according to the strategy
• ?(x) EVV ? x
• and that the uninformed bidder chooses a bid at
  random from the interval 0, EV according to
  the probability distribution H defined by
• H(b) Prob?(V) ? b

58
The field of bidders

• The last part of this lecture discusses two
  issues outside the bidding process itself that
  nevertheless may affect the outcome of the
  auction.
• They are
• Collusion amongst bidders
• Determining the number of bidders
• The degree of collusion and entry deterrence may
  affect how the auctioneer and the bidders rank
  different types of auctions that are revenue
  equivalent, or even strategically equivalent.

59
Collusion

• First we discuss the scope for collusive
  behavior under different auction mechanisms.
• Collusion between bidders is only possible if
  there is a mechanism for determining within a
  designated bidding ring which bidder has the
  highest valuation, and then ensuring members of
  the ring do not break the collusive agreement in
  the bidding.
• We focus on the second point. Sometimes
  determining which member has the highest
  valuation is trivial (for example in common value
  auctions), and sometimes it can be resolved
  through an auction within the ring for the right
  to present the only serious bid to the
  auctioneer.

60
Evidence about collusion in auctions

• Section 1 of the Sherman Act pertains to trusts
  and illegal constraints to trade.
• Over three quarters of the criminal cases filed
  in the 1980s under its provisions were in
  auctions markets.
• Given the difficulty in finding sufficient
  grounds to prosecute this illegal activity, one
  can safely conclude that collusive behavior in
  auctions is often a serious concern for the
  seller.

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What are the gains from collusive behavior?

• Consider a bidding ring of R members out of a
  total of N bidders.
• The goal of the ring is to internally solve
  which of the players have the highest valuation,
  and then make one (serious) bid instead of R
  bids.
• With fewer effective bidders in the auction,
  the expected price of the object is lower.
• If all the bidders are able to collude, meaning
  N R, the auction reduces to a game of bilateral
  bargaining between the auctioneer and the single
  bidder.

62
Bidding rings in second price sealed bid auctions

• For example, suppose the player with the
  highest valuation in the ring bids it, the other
  bidders in the ring submit the auctioneers
  reservation price, and the bidders outside the
  ring respond optimally but individually by
  bidding their true valuations.
• Then the ring benefits from colluding if
• 1. The high bid from the ring wins the auction
• 2. The second highest valuation of bidders in
  the ring exceeds the highest valuation of all
  the bidders outside the ring.

63
Enforcing collusion in first and second price
sealed bid auctions

• One reason why second price auctions are rarely
  used in practice is because they are more
  susceptible to collusion.
• In a first price sealed bid auction, a member
  of the bidding ring must submit a bid over the
  rings low price to win the auction. Providing
  his valuation exceeds the rings bid, there is an
  opportunity to make profits by deviating from the
  rings decision. This makes collusive bidding
  harder to enforce.
• Contrast this with a second price auction. In
  order to win the auction a bidder in the ring who
  breaks the collusive agreement must pay the
  reported value of the ring. The agreement by the
  ring is self enforcing!

64
Ascending auctions encourage communication
amongst the bidders

• A key issue for firms attempting to collude is
  agreeing how to share the spoils.
• In a multiunit ascending auction, bidders can
  use the early stages when prices are still low to
  signal their views about who should win which
  object, and then when consensus has been reached,
  tacitly agree to stop pushing prices up.
• By contrast, bidders cannot easily achieve the
  same coordination in simultaneous sealed-bid or
  descending auctions, in which each player
  simultaneously makes at most a single best and
  final offer to each object.

65
Low reservation prices encourage collusion

• Reducing the reserve price increase the
  potential gains from joint-bidding or colluding,
  because the gains from colluding are greater.
• Therefore the auctioneer should set a
  reservation price that is higher than the
  opportunity cost of failing to sell the auctioned
  item if it reduces the probability that bidders
  will collude.

66
Entry and the provision of information about the
auctioned object

• Should the auctioneer encourage more players to
  enter the auction?
• Potential bidders could, for example, be
  encouraged to bid in an auction, by providing
  them with services that help them to value the
  object for sale.
• Note that simply paying people to participate in
  the auction would not achieve any useful purpose,
  because anyone could accept the payment and then
  make a very low bid.

67
Encouraging entry in private valuation auctions

• If another bidder enters a private valuation
  auction, the the level of the highest or the
  second highest valuation might increase.
• In either case, the revenue from the auction
  would increase.
• Therefore the expected revenue from holding a
  private valuation auction increases with the
  number of bidders.

68
Encouraging entry in common value auctions

• The argument for encouraging participation is
  less definitive.
• In an example discussed earlier we found that
  the auctioneer gains from having a (second)
  uninformed bidder enter the auction even though
  the uninformed bidder is indifferent between
  bidding or not.
• However one can construct examples of common
  value auctions in which the winning bid falls as
  the number of bidders increases.

69
Encouraging entry to avoid collusion

• The lower the number of bidders, the easier it
  is for them to reach a collusive agreement.
• Thus another reason to encourage entry is to
  reduce the probability for facing a bidding
  cartel.

70
Summary and Synthesis

• When comparing two (or more) auctions, we should
  consider the following questions
• Are the auctions congruent, or if not, revenue
  equivalent?
• Are the bidders risk neutral?
• In private valuation auctions, are bidders
  drawing from the same probability distribution?
• In common valuation auctions, are bidders drawing
  from the same distribution of signals?
• Are the auctions symmetric?
• Is collusion between bidders a possibility?
• Should information about the auction be given to
  bidders?
• Not all these questions have easy answers, but
  that is where the value of experiments is
  especially evident.