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COS 444 COS 444 Internet Auctions:Internet Auctions:Theory and PracticeTheory and Practice

Spring 2008

Ken Steiglitz ken@cs.princeton.edu

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Moving to Moving to asymmetric asymmetric biddersbiddersEfficiency:Efficiency: item goes to bidder with item goes to bidder with

highest valuehighest value

• Very important in some Very important in some situations!situations!

• Second-price auctions remain Second-price auctions remain efficient in asymmetic (IPV) caseefficient in asymmetic (IPV) case

• First-price auctions First-price auctions do not do not ……

Inefficiency in FP with asymmetric Inefficiency in FP with asymmetric biddersbidders

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New setup: Myerson New setup: Myerson 8181• Vector of values Vector of values vv• Allocation functionAllocation function Q Q ((v v ) ) ::

QQii ((v v )) is is prob. prob. ii wins item wins item• Payment functionPayment function P P ((v v ) ) ::

PPii ((v v )) is is expected payment of expected payment of ii

• Subsumes ASubsumes Arsrs easily (check SP, FP) easily (check SP, FP)• The pair The pair ((QQ , , P P ) ) is called ais called a

Direct MechanismDirect Mechanism

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New setup: Myerson New setup: Myerson 8181

• Definition: When agents who Definition: When agents who participate in a mechanism have no participate in a mechanism have no incentive to lie about their values, we incentive to lie about their values, we say the mechanism is say the mechanism is incentive incentive compatiblecompatible. .

• The The Revelation PrincipleRevelation Principle: In so far as : In so far as equilibrium behavior is concerned, any equilibrium behavior is concerned, any auction can be replaced by an auction can be replaced by an incentive-compatible mechanism.incentive-compatible mechanism.

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Revelation PrincipleRevelation Principle

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

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Asymmetric biddersAsymmetric bidders We can therefore restrict We can therefore restrict

attention to attention to incentive compatible incentive compatible direct mechanismsdirect mechanisms!!

In the In the asymmetricasymmetric case, surplus is case, surplus is no longer no longer vvi i F(z) F(z) n-1 n-1 − P(z) − P(z)

(bidding as if value = (bidding as if value = z z ))

Next we write expected surplus in Next we write expected surplus in the the asymmetricasymmetric case … case …

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Asymmetric biddersAsymmetric bidders

Notation:Notation: vv−i −i == vectorvector v v with thewith the i – i – thth

Value omitted. Then the Value omitted. Then the prob. that i winsprob. that i wins is is

Where Where VV-i -i is the space of allis the space of all v’ v’s except s except vvii andand

F F ((vv-i -i )) is the corresponding distributionis the corresponding distribution

)(),()( -iiV ii vFdvzQzQi

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Asymmetric biddersAsymmetric bidders

Similarly for the expected payment of Similarly for the expected payment of bidder bidder i i ::

Expected surplus is thenExpected surplus is then

)(),()( iiV ii vdFvzPzPi

)()()( zPzQvzS iiii

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Asymmetric bidders: Asymmetric bidders: yet more yet more general REgeneral REDifferentiate wrt Differentiate wrt z z and set to zero when and set to zero when z z = =

vvi i

as usual:as usual:

But now take the But now take the total derivativetotal derivative wrt v wrt vii when when z z == v vi i ::

And so And so

0)()( '' iiiii vPvQv

)()()()( '''iiiiiiiii vPvQvQvvS

)()(' iiii vQvS

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Asymmetric bidders: Asymmetric bidders: yet more yet more general REgeneral REIntegrate:Integrate:

oror

((SS = = vQ − PvQ − P ) )

Expected payment of every bidder depends Expected payment of every bidder depends only on allocation function Q !only on allocation function Q !

iv

iii dxxQSvS0

)()0()(

dxxQvQvPvPiv

iiiiiii )()()0()(0

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Optimal allocationOptimal allocationAverage over Average over vvi i and proceed as in RS81:and proceed as in RS81:

where where

)()()()0(])(E[ vdFvQvMRPvP ii

V

iii

)(

)(1)(

ii

iiiii vf

vFvvMR

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Optimal allocation, con’tOptimal allocation, con’t

The total expected revenue isThe total expected revenue is

For participation, For participation, PPi i ((00 ) ≤ 0, and seller ) ≤ 0, and seller chooses Pchooses Pi i (0) = 0 to max surplus. (0) = 0 to max surplus. ThereforeTherefore

)()()()0(R vdFvQvMRP ii

V ii

ii

)()()(R vdFvQvMR i

V iii

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Optimal allocation, con’tOptimal allocation, con’t

When PPii (0 (0 ) ≤ 0 we say bidders ) ≤ 0 we say bidders are are individually rational individually rational : The : The don’t participate in auctions if the don’t participate in auctions if the expected payment with zero value expected payment with zero value is positive.is positive.

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Optimal allocationOptimal allocationThe optimal allocation can now be seen by The optimal allocation can now be seen by

inspection!inspection!

Look for the maximum value of Look for the maximum value of MRMRi i ((vvii ). Say it ). Say it occurs at occurs at ii = = i*i* , and denote it by , and denote it by MR*MR* . .

• If If MR*MR* > 0, then choose that Q > 0, then choose that Q i*i* to be 1 and all to be 1 and all the other Q’s to be 0 (bidder the other Q’s to be 0 (bidder i*i* gets the item) gets the item)

• If If MR*MR* ≤ 0, then hold on to the item (seller ≤ 0, then hold on to the item (seller retains item)retains item)

)()()(R vdFvQvMR ii

V ii

Optimal allocation Optimal allocation (inefficient!)(inefficient!)

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PaymentPayment rule rule

Hint: must reduce to second-price Hint: must reduce to second-price when bidders are symmetricwhen bidders are symmetric

Therefore: Therefore: Pay the least you can Pay the least you can while still maintaining the highest while still maintaining the highest MRMR

Verify: This is incentive compatible; Verify: This is incentive compatible; that is, bidders bid truthfully!that is, bidders bid truthfully!

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WrinkleWrinkle

• For this argument to work, MR must For this argument to work, MR must be an increasing function. We be an increasing function. We FF ’s ’s with increasing MR’s with increasing MR’s regularregular. . (Uniform OK)(Uniform OK)

• It’s sufficient for the inverse hazard It’s sufficient for the inverse hazard rate (1 –rate (1 – F F ) / ) /ff to be decreasing. to be decreasing.

• Can be fixed: See Myerson 81 Can be fixed: See Myerson 81 (“ironing”)(“ironing”)

• Assume MR is regular in what followsAssume MR is regular in what follows

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Notice that this shows that all the Notice that this shows that all the auctions in Aauctions in Arsrs in the symmetric in the symmetric case are case are optimal auctionsoptimal auctions. (SP is, . (SP is, and the rest are revenue and the rest are revenue equivalent.)equivalent.)

Notice also that this asks a lot in Notice also that this asks a lot in the asymmetric case. In the direct the asymmetric case. In the direct mechanism the bidders must mechanism the bidders must understand enough to be truthful, understand enough to be truthful, and accept the fact that the and accept the fact that the highest value doesn’t always win.highest value doesn’t always win.

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AArsrs are optimal are optimal mechanismsmechanisms!! By the revelation principle, we can By the revelation principle, we can

restrict attention to direct mechanismsrestrict attention to direct mechanisms All direct mechanisms with the same All direct mechanisms with the same

allocation rule have the same revenueallocation rule have the same revenue An optimal mechanism in the symmetric An optimal mechanism in the symmetric

case awards item to highest-value case awards item to highest-value bidder, and so does any auction in Abidder, and so does any auction in Arsrs

Therefore any auction in ATherefore any auction in Arsrs has the has the same allocation rule, and hence revenue, same allocation rule, and hence revenue, as an optimal (general!) mechanismas an optimal (general!) mechanism

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LaboratoryLaboratory EvidenceEvidenceGenerally, there are three kinds of Generally, there are three kinds of

empirical methodologies:empirical methodologies:• Field observationsField observations• Field experimentsField experiments• Laboratory experimentsLaboratory experiments

Problem: do people behave the same Problem: do people behave the same way in the lab as in the world?way in the lab as in the world?

Problem: people differ in experience; Problem: people differ in experience; people learnpeople learn

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LaboratoryLaboratory EvidenceEvidence

Conclusions fall into two general Conclusions fall into two general categories:categories:

• Revenue rankingRevenue ranking• Point predictions (usually revenue Point predictions (usually revenue

relative to Nash equilibrium)relative to Nash equilibrium)

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Best results for IPV Best results for IPV modelmodel

• First-Price > Dutch First-Price > Dutch Coppinger et al. (80) Coppinger et al. (80)

• First-Price > Nash First-Price > Nash Dyer et al. (89)Dyer et al. (89) • Second-Price > English Second-Price > English Kagel et al. (87)Kagel et al. (87)

• English English truthful=Nash truthful=Nash Kagel et al. (87)Kagel et al. (87)

• First-Price ? Second-PriceFirst-Price ? Second-Price

Thus, generally,Thus, generally, sealed versions > open versions!sealed versions > open versions!

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See also See also Kagel & Levin 93Kagel & Levin 93 for experiments for experiments with 3with 3rdrd price auctions that test IPV price auctions that test IPV theorytheory

More about experimental results for More about experimental results for common-value auctions latercommon-value auctions later

We next focus a while on We next focus a while on

First-price > Nash First-price > Nash

One explanation: One explanation: risk aversionrisk aversion

But is there another explanation…But is there another explanation…