Network Economics -- Lecture 5: Auctions and...

NetworkEconomics--

Lecture5:Auctionsandapplications

PatrickLoiseauEURECOMFall2016

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References• V.Krishna,“AuctionTheory”,Elseiver AP2009(secondedition)– Chapters2,3,5

• P.Milgrom,“Puttingauctiontheorytowork”,CUP2004– Chapter1

• D.EasleyandJ.Kleinberg,“Networks,CrowdsandMarkets”,CUP2010– Chapters9and15

• BenPolak’s onlinecoursehttp://oyc.yale.edu/economics/econ-159– Lecture24

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Outline

1. Generalitiesonauctions2. Privatevalueauctions3. Commonvalueauctions:thewinner’scurse4. Mechanismdesign5. Generalizedsecondpriceauction

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Outline


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Whereareauctions?

• Everywhere!– Ebay– Googlesearchauctions– Spectrumauctions– Artauctions– Etc.

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Whatisanauction?

• Sellersellsoneitemofgoodthroughbidding– Setofbuyers

• Buyerbuysoneitemofgood– Setofsellers– Calledprocurementauction(governments)

• Auctionsareusefulwhenthevaluationofbiddersisunknown

• Morecomplexauctions–Multi-items– Combinatorial

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Standardauction

• Standardauction:thebidderwiththehighestbidwins

• Exampleofnonstandardauction:lottery

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Thetwoextremesettings

• Commonvaluesßà Privatevalues

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Maintypesofauctions

1. Ascendingopenauction(English)

2. Descendingopenauction(Dutch)

3. First-pricesealedbidauction

4. Secondpricesealedbidauction(Vickrey)

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Relationshipsbetweenthedifferenttypesofauctions

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Outline


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Privatevalueauctions:Model

• Oneobjectforsale• Nbuyers• ValuationXi• Xi’si.i.d.distributedon[0,w],cdf F(.)• Bidderi knows– Realizationxi ofhisvalue– ThatotherbiddershavevaluesdistributedaccordingtoF

• Def:symmetric:allbiddershavethesamedistributionofvalue

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Game

• Thegameisdeterminedbytheauctionrules– Gamebetweenthebidders

• Bidder’sstrategy:βi:[0,w]à [0,∞)

• Lookforsymmetricequilibria– 1st priceauction– 2nd priceauction– Compareseller’srevenue

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Second-pricesealed-bidauction

• Proposition:Inasecond-pricesealed-bidauction,biddingitstruevalueisweaklydominant

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First-pricesealed-bidauction

• Biddingtruthfullyisweaklydominated

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First-pricesealed-bidauction(2)

• Whatistheequilibriumstrategy?

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First-pricesealed-bidauction(3)

• Proposition:Symmetricequilibriumstrategiesinafirst-pricesealed-bidauctionaregivenby

whereY1 isthemaximumofN-1independentcopiesofXi

β(x) = E Y1 |Y1 < x[ ]

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Example

• Valuesuniformlydistributedon[0,1]

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Revenuecomparison

• Withindependentlyandidenticallydistributedprivatevalues,theexpectedrevenueinafirst-priceandinasecond-priceauctionarethesame

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Proof

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Warning

• Thisisnottrueforeachrealization• Example:2bidders,uniformvaluesin[0,1]

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Revenueequivalencetheorem• Generalizationofthepreviousresult

• Theorem:Supposethatvaluesareindependentlyandidenticallydistributedandallbiddersareriskneutral.Thenanysymmetricandincreasingequilibriumofanystandardauctionsuchthattheexpectedpaymentofabidderwithvaluezeroiszeroyieldsthesameexpectedrevenuetotheseller.

• Seeanevenmoregeneralresultinthe(beautiful)paperR.Myerson,“OptimalAuctionDesign”,MathematicsofOperationResearch1981– 2007NobelPrize

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Proof

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Reserveprice

• r>0,suchthatthesellerdoesnotsellifthepricedeterminedbytheauctionislower

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Reservepriceinsecond-priceauction

• Nobidderwithvaluex<rcanmakeapositiveprofit

• Biddingtruthfullyisstillweaklydominant• Winnerpaysrifthedeterminedpriceislower• Expectedpayment

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Reservepriceinfirst-priceauction

• Nobidderwithvaluex<rcanmakeapositiveprofit

• Symmetricequilibrium:

• Expectedpayment:

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Effectofreservepriceonrevenue

• Sellerhasvaluationx0 ofthegood• Setsr>x0!

• Optimalreserveprice:

• Increasestheseller’srevenue– Sometimescalledexclusionprinciple

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Remark

• Efficiency:maximizesocialwelfare– Goodendsupintheendofthehighestvalueamongbiddersandseller

• EfficientisNOTthesameasrevenueoptimality• Example– Sellerwithvaluationzero

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Outline


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Playingwithajarofcoins

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Thewinner’scurse

• GoodhasvalueV,sameforallbidders– Example:oilfield

• Eachbidderhasani.i.d.estimatexi=V+ei ofthevalue(E(ei)=0)

• Theyallbid(e.g.,first-priceauction)

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Thewinner’scurse(2)

• Supposebidder1wins• Uponwinning,hefindsouthisestimatewastoolarge!à badnews:winner’scurse

• Bidasifyouknowyouwin!

• Remark:thewinner’scursedoesnotariseatequilibrium,ifyourbidtakesitintoaccount.

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Outline


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Mechanismdesign

• Anauctionisonlyoneofmanywaystosellagood

• Mechanismdesignstudiesthedesignofrulessuchthattheresultinggameyieldsadesiredoutcome

• The2007NobelMemorialPrizeinEconomicScienceswasawardedtoLeonidHurwicz,EricMaskin,andRogerMyerson"forhavinglaidthefoundationsofmechanismdesigntheory"

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Setting

• Buyers• Values• Setofvalues• Distributions• Productset• Jointdensity

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Mechanisms

• Setofmessages(bids)• Allocationrule• Paymentrule

• Example:1st or2nd priceauction

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Directmechanism

• Definition

• Characterization:Pair(Q,M)

• Truthfulequilibrium

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Revelationprinciple

• Givenamechanismandanequilibriumforthatmechanism,thereexistsadirectmechanismsuchthat1. Itisanequilibriumforeachbuyertoreporthis

valuetruthfully2. Theoutcomes(probabilitiesQandexpected

paymentM)arethesameasintheequilibriumoftheoriginalmechanism

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Proof

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Incentivecompatibility(IC)

• AdirectrevelationmechanismisICifitisoptimalforabuyertoreporthisvaluetruthfullywhenallotherbuyersreporttheirvaluetruthfully

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Revenueequivalence

• Ifthedirectmechanism(Q,M)isincentivecompatible,thentheexpectedpaymentis

• Thus,theexpectedpaymentinanytwoincentivecompatiblemechanismswiththesameallocationruleareequivalentuptoaconstant

• Generalizesthepreviousversion

mi (xi ) =mi (0)+ qi (xi )xi − qi (ti )0

xi∫ dti

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Twoquestions

• Howtodesignarevenueoptimalmechanism?

• Howtodesignanefficientmechanism?

• Restrictingto– ICmechanisms– Individuallyrationalmechanisms(i.e.,suchthattheexpectedpayoffofeverybuyerisnonnegative)

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Optimalmechanism

• Definethevirtualvaluation

• Define

• Undersomeregularityconditions,theoptimalmechanismis:allocatetothebuyerwithhighestvirtual valuation(ifitisnonnegative),withexpectedpaymentyi(x-i)

ψi (xi ) = xi −1−Fi (xi )fi (xi )

yi (x−i ) = inf zi :ψi (zi ) ≥ 0 andψi (zi ) ≥ψ j (x j ) for all j{ }

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Symmetriccase

• Wefindthesecondpriceauctionwithreserveprice ψ−1(0)

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Efficientmechanism

• SocialwelfaremaximizedbyQ*

• Ifthereisnotie:allocationtothebuyerwithhighestvalue

• Notation:

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VCGmechanism:definition

• TheVCGmechanismis(Q*,MV),where

• Note:theW’sarecomputedwiththeefficientallocationrule

MiV (x) =W (0, x−i )−W−i (x)

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VCGmechanism:properties• TheVCGmechanismis– Incentivecompatible– truthfulreportingisweaklydominant

– Individuallyrational– Efficient

• i’sequilibriumpayoffisthedifferenceinsocialwelfareinducedbyhistruthfulreportinginsteadof0

• Proposition:Amongallmechanismsforallocatingasinglegoodthatareefficient,ICandIR,theVCGmechanismmaximizestheexpectedpaymentofeachagent 47

Example

• Inthecontextofauctions:VCG=2nd priceauction!

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Outline


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Sponsoredsearch

• Adsinsponsoredbox

• Severalspots:multipleitemsauction

• Payperclickfortheadvertiser

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Generalizedsecondpriceauction(GSP)

• HowdoesGoogledeterminewhichadisshownforagivenkeyword?

• Advertiserssubmitbids• Googleranksadsbybidxexpectednb ofclicks– Adqualityfactor

• Advertiserspaythepricedeterminedbythebidbelow(GSP)

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GSPproperties• GSPisnottruthful• GSPisnotVCG• GSPmayhaveseveralequilibria• GSP’srevenuemaybehigherorlowerthanVCG’srevenue

• B.Edelman,M.Ostrovsky,M.Schwarz,“InternetAdvertisingandtheGeneralizedSecond-PriceAuction:SellingBillionsofDollarsWorthofKeywords”,AmericanEconomicReview2007

• H.Varian,“Positionauctions”,InternationalJournalofIndustrialOrganization2007

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