COS 444 Internet Auctions: Theory and Practice

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COS 444 Internet AuctionsTheory and Practice
Spring 2008 Ken Steiglitz
ken_at_cs.princeton.edu
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?
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Moving to asymmetric bidders

• Efficiency item goes to bidder with highest
  value
• Very important in some situations!
• Second-price auctions remain efficient in
  asymmetic (IPV) case
• First-price auctions do not

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Inefficiency in FP with asymmetric bidders
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New setup Myerson 81

• Vector of values v
• Allocation function Q (v )
• Qi (v ) is prob. i wins item
• Payment function P (v )
• Pi (v ) is expected payment of i
• Subsumes Ars easily (check SP, FP)
• The pair (Q , P ) is called a
• Direct Mechanism

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New setup Myerson 81

• Definition When agents who participate in a
  mechanism have no incentive to lie about their
  values, we say the mechanism is incentive
  compatible.
• The Revelation Principle In so far as
  equilibrium behavior is concerned, any auction
  can be replaced by an incentive-compatible
  mechanism.

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Revelation Principle

• Proof Replace the bid-taker with a direct
  mechanism that computes equilibrium values for
  the bidders. Then a bidder can bid equilibrium
  simply by being truthful, and there is never an
  incentive to lie.

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Asymmetric bidders

• We can therefore restrict attention to incentive
  compatible direct mechanisms!
• In the asymmetric case, surplus is no longer vi
  F(z) n-1 - P(z)
• (bidding as if value z )
• Next we write expected surplus in the
  asymmetric case

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Asymmetric bidders

• Notation v-i vector v with the i th
• Value omitted. Then the prob. that i wins is
• Where V-i is the space of all vs except vi and
• F (v-i ) is the corresponding distribution

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Asymmetric bidders

• Similarly for the expected payment of bidder i
• Expected surplus is then

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Asymmetric bidders yet more
general RE

• Differentiate wrt z and set to zero when z vi
  
• as usual
• But now take the total derivative wrt vi when z
  vi
• And so

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Asymmetric bidders yet more
general RE

• Integrate
• or
• (S vQ - P )
• ?Expected payment of every bidder depends only on
  allocation function Q !

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Optimal allocation

• Average over vi and proceed as in RS81
• where

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Optimal allocation, cont

• The total expected revenue is
• For participation, Pi (0 ) 0, and seller
  chooses Pi (0) 0 to max surplus. Therefore

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Optimal allocation, cont
When Pi (0 ) 0 we say bidders are individually
rational The dont participate in auctions if
the expected payment with zero value is positive.
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Optimal allocation

• The optimal allocation can now be seen by
  inspection!
• Look for the maximum value of MRi (vi ). Say it
  occurs at i i , and denote it by MR .
• If MR gt 0, then choose that Q i to be 1 and all
  the other Qs to be 0 (bidder i gets the item)
• If MR 0, then hold on to the item (seller
  retains item)

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Optimal allocation (inefficient!)
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Payment rule

• Hint must reduce to second-price when bidders
  are symmetric
• Therefore Pay the least you can while still
  maintaining the highest MR
• Verify This is incentive compatible that is,
  bidders bid truthfully!

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Wrinkle

• For this argument to work, MR must be an
  increasing function. We F s with increasing MRs
  regular. (Uniform OK)
• Its sufficient for the inverse hazard rate (1
  F ) /f to be decreasing.
• Can be fixed See Myerson 81 (ironing)
• Assume MR is regular in what follows

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• Notice that this shows that all the auctions in
  Ars in the symmetric case are optimal auctions.
  (SP is, and the rest are revenue equivalent.)
• Notice also that this asks a lot in the
  asymmetric case. In the direct mechanism the
  bidders must understand enough to be truthful,
  and accept the fact that the highest value
  doesnt always win.

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Ars are optimal mechanisms!

• By the revelation principle, we can restrict
  attention to direct mechanisms
• All direct mechanisms with the same allocation
  rule have the same revenue
• An optimal mechanism in the symmetric case awards
  item to highest-value bidder, and so does any
  auction in Ars
• Therefore any auction in Ars has the same
  allocation rule, and hence revenue, as an optimal
  (general!) mechanism

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Laboratory Evidence

• Generally, there are three kinds of empirical
  methodologies
• Field observations
• Field experiments
• Laboratory experiments
• ?Problem do people behave the same way in the
  lab as in the world?
• ?Problem people differ in experience people
  learn

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Laboratory Evidence

• Conclusions fall into two general categories
• Revenue ranking
• Point predictions (usually revenue relative to
  Nash equilibrium)

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Best results for IPV model

• First-Price gt Dutch Coppinger et al. (80)
• First-Price gt Nash Dyer et al. (89)
• Second-Price gt English Kagel et al. (87)
• English ? truthfulNash Kagel et al. (87)
• First-Price ? Second-Price
• Thus, generally,
• sealed versions gt open versions!

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• See also Kagel Levin 93 for experiments with
  3rd price auctions that test IPV theory
• More about experimental results for common-value
  auctions later
• We next focus a while on
• First-price gt Nash
• One explanation risk aversion
• But is there another explanation