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