COS 444 Internet Auctions: Theory and Practice

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COS 444 Internet AuctionsTheory and Practice
Spring 2009 Ken Steiglitz
ken_at_cs.princeton.edu
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the bigger picture, all single item
Myerson 1981 optimal, not efficient asymmetric
bidders
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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
  asymmetric (IPV) case. Why?
• First-price auctions do not

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Inefficiency in FP with asymmetric bidders
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New setup Myerson 81, also BR 89
wins Nobel prize for this and related work, 2007

• 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
  mechanism can be replaced by an
  incentive-compatible direct 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. ?
• This principle is very general and includes
  any sort of negotiation!

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Asymmetric bidders

• We can therefore restrict attention to
  incentive-compatible direct mechanisms!
• Note In the asymmetric case, expected 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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A yet more general RET

• 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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yet more general RE

• Integrate
• Or, using S vQ P ,
• ?In equilibrium, 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

?no longer a common F
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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 They 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!
• For each vector of vs, 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 Qi 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
• This is incentive compatible that is, bidders
  bid truthfully! Why?

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Vickrey 61 yet again
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Wrinkle

• For this argument to work, MR must be an
  increasing function. We call F s with increasing
  MRs regular. (Uniform is regular)
• 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 also that this asks a lot of bidders 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.
• Or, think of MRi(vi) as is bid
• As usual in game-theoretic settings,
  distributions are common knowledge---at least the
  hypothetical auctioneer must know them.

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In the symmetric caseArs are optimal mechanisms!

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

Includes any sort of negotiation whatsoever!
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Efficiency

• Second-price auctions are efficient --- i.e.,
  they allocate the item to the buyer who values it
  the most. (Even in asymm. case, truthful is
  dominant.)
• Weve seen that optimal (revenue-maximizing)
  auctions in the asymmetric case are in general
  inefficient.
• It turns out that second-price auctions are
  optimal in the class of efficient auctions. They
  generalize in the multi-item case to the
  Vickrey-Clark-Groves (VCG) mechanisms. More
  later.

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Laboratory Evidence

• Generally, there are three kinds of empirical
  methodologies
• Field observations
• Field experiments
• Laboratory experiments
• ?Problem people may not behave the same way in
  the lab as in the world
• ?Problem people differ in behavior
• ?Problem people learn from experience

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Laboratory Evidence

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

For more detail, see J. H. Kagel, "Auctions A
Survey of Experimental Research", in The Handbook
of Experimental Economics, J. Kagel and A. Roth
(eds.), Princeton Univ. Press, 1995.
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Best revenue-ranking results for IPV model

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

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• A violation of theory is the
  scientists best news!
• Lets discuss some of the violations
• Second-Price gt English. These are (weakly)
  strategically equivalent. But
• English ? truthful Nash.
• What hint towards an explanation does the
  weakly give us?

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• First-Price gt Dutch. These are strongly
  strategically equivalent. But recall
  Lucking-Reileys pre-eBay internet test with
  Magic cards, where Dutch gt FP by 30!
• Whats going on here?

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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 for a while on a widely accepted
  point prediction
• One explanation, as weve seen, is
• risk aversion
• But is here is an alternative explanation

First-price gt Nash
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Spite MSR 03 MS 03

• Suppose bidders care about the surplus of other
  bidders as well their own.
• Simple example Two bidders, second-price,
  values iid unifom on 0,1. Suppose bidder 2 bids
  truthfully, and suppose bidder 1s utility is not
  her own surplus, but the difference ? between
  hers and her rivals.

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Spite

• Now bidder 1 wants to choose her bid b1 to
  maximize the expectation of
• where I is the indicator function, 1 when
  true, 0 else.
• Taking expectation over v2

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Spite

• Maximizing wrt b1 yields best response to
  truthful bidding
• Intuition?

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Spite

• Maximizing wrt b1 yields best response to
  truthful bidding
• Intuition by overbidding, 1 loses surplus when
  2s bid is between v1 and her bid. But, this is
  more than offset by forcing 2 to pay more when he
  wins.
• Notice that bidder 2 still cannot increase his
  absolute surplus. (Why not?) He must take a hit
  to compete in a pairwise knockout tournament.

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Spite

• Some results from MSR 03 take the case when
  bidders want to maximize the difference between
  their own surplus and that of their rivals.
  Values distributed as F, n bidders. Then
• ? FP equilibrium is the same as in the
  risk-averse CRRA case with ? ½ (utility is t1/2
  ). Thus there is overbidding.
• ? SP equilibrium is to overbid according to

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Spite

• ? Revenue ranking is SP gt FP.
• (Not a trivial proof. Is there a simpler
  one?)
• Thus, this revenue ranking is the opposite of the
  prediction in the risk-averse case, where there
  is overbidding in FP but not in SP. (Testable
  prediction.)
• This explains overbidding in both first- and
  second-price auctions, while risk-aversion
  explains only the first. (Testable prediction.)
• Raises a question do you think people bid
  differently against machines than against people?

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Spiteful behavior in biology

• This model can also explain spiteful behavior in
  biological contexts, where individuals fight for
  survival one-on-one MS 03. Example
• This is a hawk-dove game.
• Winner type replaces loser type.
• In a large population where the success of an
  individual is determined by average individual
  payoff, there is an evolutionarily stable
  solution that is 50/50 hawks and doves.
• If winners are determined by relative payoff in
  each 1-1 contest, the hawks drive out the doves.
• Thus, there is an Invasion of the Spiteful
  Mutants!

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Invasion of the spiteful mutants

• To see this, suppose in the large population
  there is a fraction ? of Hs and (1-? ) of Ds.
• The average payoff to an H in a contest is
• and to a D
• The first is greater than the second iff ?lt1/2. A
  50/50 mixture is an equilibrium.
• But if the winner of a contest is determined by
  who has the greater payoff, an H always replaces
  a D!