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PricesandAuctionsinMarketswithComplexConstraints

PaulMilgromStanfordUniversity&AuctionomicsAugust2016

Conferenceon“FrontiersofEconomicsandComputerScience”Becker-FriedmanInstitute

MarketDesignPerspective• Marketdesign– implementingpracticalmarketrules– forceseconomiststodealexplicitlywithsomeissuesthatareusuallyde-emphasized.

• Onesuchissuecluster,emphasizedhere,isproductdefinition,productheterogeneity,andconstraintsonmarketallocations…

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ThreeMarket-DesignQuestions1. Whatisaproduct?Debreudescribedcommoditiesbytheir

physicalcharacteristicsandtheirtime&placeofavailability.• ElectricityinPaloAlto,CAat2:05pmisnotthesamecommodityaselectricityinPaloAltoat2:06pm.…northesameaselectricityinSanJose.

2. Ifheterogeneousproductsarelumpedtogether(astheymustbe!),thentheremaybemoreconstraintsthanjusttotalresourceconstraints.Howcanwehandlethose?• MyreturnflighttoSFOwilluse“time-space”resources.

3. Whencaneconomicmodelsthataredrasticallysimplerthan“realistic”engineeringmodelsbeusefulformarketdesign?• “SFOairporthasacapacityofXpassengers/hour” 5

COMPLEXCONSTRAINTSANDTHEROLESOFPRICES&ORGANIZATION

Examples

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Real-EstateHoldoutExamples

12• Seattle,WA • SanAntonio,TX

Co-ChannelInterferenceAroundOneStation

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ReallocatingRadioSpectrum

About130,000co-channelinterferenceconstraints,andabout2.7millionconstraintsinthefullmodel.

Thegraph-coloringisNP-complete.TheFCCmaysometimesbeunabletodetermine,inreasonabletime,whetheracertainsetofstationscanbeassignedtoagivensetofchannels.

PRICESANDINCENTIVESINTHE“KNAPSACKPROBLEM”

An“In-Between”ModelofO’HareAirport?!?

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KnapsackProblem• Notation• Knapsacksize:�̅�.• Items𝑛 = 1,… ,𝑁• Eachitemhasavalue𝑣* > 0 andasize𝑠* > 0.• Inclusiondecision:𝑥* ∈ {0,1}.

• Knapsackproblem:

𝑉∗ = max7∈ 8,9 :

; 𝑣*𝑥*<

*=9subjectto; 𝑠*𝑥*

<

*=9≤ 𝑆̅.

• Theclassofknapsackproblems(verification)isNP-complete,buttherearefastalgorithmsfor“approximateoptimization.” 20

Dantzig’s “GreedyAlgorithm”• OrdertheitemssothatIJKJ

> ⋯ > I:K:.

• Algorithm:1. 𝑆9 ← �̅�.(Initialize“availablespace”)2. For𝑛 = 1,… ,𝑁3. If𝑠* ≤ 𝑆*, set𝑥* = 1,elseset𝑥* = 0.4. Set𝑆*N9 = 𝑆* − 𝑥*𝑠*.5. Next𝑛.6. End

• Theitemsselectedare𝛼QRSSTU 𝑣; 𝑠 ≝ {𝑛|𝑥* = 1}.• Thisfunctionis“monotonic.”

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“NearlytheSame”AlgorithmCanbeFormulatedasanAuction• Let𝑝 0 > IJ

KJ;𝑆 0 = �̅�;andlabelall𝑛 “Out.”

Discretegreedyalgorithm• For𝑡 = 1,2,…𝑝(0)/𝜀.• Mark“Rejected”any𝑛 forwhom𝑠* + 𝑆 𝑡 − 1 > 𝑆̅• Set𝑝 𝑡 = 𝑝 𝑡 − 1 − 𝜀 (where𝜀 > 0 isthe“biddecrement”)• Ifsome𝑛 whoismarked“Out”has𝑣* > 𝑝 𝑡 𝑠*,markit“Accepted”andset𝑆 𝑡 = 𝑆 𝑡 − 1 − 𝑠*.

• Next𝑡

GeneralizableInsight:• Closeequivalencebetweenvariousclockauctionsandrelatedgreedyalgorithms(Milgrom&Segal).

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TheLOSAuction• Model(Lehman,O’CallaghanandShoham(2002))• Eachitemn isownedbyaseparatebidder.• Sizes𝑠* areobservabletotheauctioneer.

• “LOS”DirectMechanism• Eachbidder𝑛 reportsitsvalue𝑣*.• Allocatespacetothesetofbidders𝛼QRSSTU(𝑣; 𝑠).• Chargeeachbidderits“thresholdprice”,definedby:

𝑝* 𝑣a*; 𝑠 ≝ inf 𝑣*e 𝑛 ∈ 𝛼QRSSTU 𝑣*e , 𝑣a*; 𝑠 .

• Theorem.TheLOSauctionistruthful.

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LOSMechanismCanLeadtoExcessiveInvestments• Consideragameinwhich,beforetheauction,eachbidder𝑛can,byinvesting𝑐,reducethesizeofitsitemto𝑠* − Δ.

• Examples:Supposethereare𝑁 items,eachofsize1andtheknapsackhassize𝑁 − 1,so𝛼QRSSTU 𝑣, 𝑠 = 1,… ,𝑁 − 1 .

• Lossesfromexcessiveinvestment: IfΔ = 9< and𝑐 = 𝑣< − 𝜀,

thenthereisaNashequilibriuminwhichallinvest,eventhoughthecostisnearlyN timesthebenefit.

• Canauniformpricemechanismperformsimilarlywell?

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Uniform-PriceGreedyMechanism• Determinetheallocation

1. OrdertheitemssothatIJKJ> ⋯ > I:

K:.

2. Initialize𝑛 ← 1 andset𝑆9 ← 𝑆.3. If𝑠* > 𝑆*,gotostep54. Incrementnandgotostep35. Set𝛼hij 𝑣, 𝑠 ← {1,… , 𝑛 − 1}.• Setasupportingprice(a“uniform”priceofspace)

6. Define�̂� ≝ 𝑣*/𝑠* (the“pseudo-equilibrium”priceofspace)7. Set𝑝l ← �̂�𝑠l for𝑗 = 1,… , 𝑛 − 1,and𝑝l ← 0 for𝑗 = 𝑛,… ,𝑁.8. End

• Define𝑉hij(𝑣, 𝑠) ≝ ∑ 𝑣*.�*∈pqrs(I,K)

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PackingEfficiencyandTruthfulness

• Theorem.The“Alt”mechanismistruthful.Moreover,

𝑉QRSSTU(𝑣, 𝑠) ≥ 𝑉hij(𝑣, 𝑠) ≥ 𝑉∗(𝑣, 𝑠) − 𝑆 − 𝑆hij �̂�.

• Theboundonefficiencylossisobservable:itisequaltothe“value”oftheunusedspaceintheknapsack.

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InvestmentEfficiency• Supposethateachbiddern can,byexpending𝑐* ∈ 𝐶, determinethesize𝑠*(𝑐*) ofitsitem,where𝐶 isafiniteset.Let𝑛∗ denotetheindexofthefirstbiddernotpackedbytheAltmechanism.

• Notation:𝑉∗∗ ≝ max

𝑥, 𝑐 𝑐*∗ = ⋯ = 𝑐< = 0; (𝑥*𝑣* − 𝑐*)

�

* 𝑠. 𝑡.; 𝑥*𝑠*(𝑐*)

�

*≤ 𝑆

𝑐*∗ ≝ argmaxxy∈z

max7y

𝑣*𝑥* − �̂�𝑠* 𝑐* − 𝑐*

• Theorem.ThecombinedlossfrompackingandinvestmentinAlt is(“observably”)boundedasfollows:

𝑉∗∗ − 𝑉hij(𝑠 𝑐∗ ) ≤ 𝑆 − 𝑆hij 𝑐∗ �̂�

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GreedyAlgorithmsandAuctions• Inanauctionwithsingle-mindedbidders,agreedyalgorithmcanbeusedtosortthebiddersintotwosets,winnersandlosers.• LOSAlgorithmandRelated:Selectwinnersbyagreedyalgorithm;otherbiddersarelosers.

• DAAAlgorithmandRelated:Selectlosersbyagreedyalgorithm;otherbiddersarewinners.

• Thetwocategoriesareeconomically distinctbecause• winnerscollectpayments• losersdonot

• TheFCCdescendingclockauctionisaDAAalgorithm.• OnecouldspecifyanascendingclockforanLOS-stylealgorithm. 28

GREEDYONMATROIDSMoreGreedyAlgorithms

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MatroidTermsDefined• Givenafine“groundset”𝑋,let℘ 𝑋 denoteitspowerset.• Example:𝑋 isthesetofrowsofafinitematrix

• Letℛ ⊆ ℘ 𝑋 benon-emptyandallitselements“independentsets.”• Example:alllinearlyindependentsetsofrowsin𝑋.

• A“basis”isamaximalindependentset.• Example:amaximallinearlyindependentsetofrowsof𝑋.

• Thepair(𝑋,ℛ) (orjustthesetℛ)isamatroid if1. [Freedisposal]If𝑆′ ⊆ 𝑆 ∈ ℛ,then𝑆′ ∈ ℛ .2. [AugmentationProperty]Given𝑆, 𝑆′ ∈ ℛ,if 𝑆 > |𝑆′|,then

thereexists𝑛 ∈ 𝑆 − 𝑆′ suchthat𝑆′ ∪ 𝑛 ∈ ℛ. 30

GreedyAlgorithm• Givenanycollectionofindependentsetsℛ.• Ordertheitemssothat𝑣9 > ⋯ > 𝑣<. (Novolumes)

• Algorithm:1. Initialize𝑆8 ← ∅.2. For𝑛 = 1,… ,𝑁

3. 𝑆* ← �𝑆*a9 ∪ 𝑛 if𝑆*a9 ∪ 𝑛 ∈ ℛ𝑆*a9otherwise

4. Next𝑛5. Output𝑆<.

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OptimizationonMatroids• Forsimplicity,assumeauniqueoptimum.• Theorem.Ifℛ isamatroidand𝑆< isthegreedysolution,then

𝑆< = argmax�∈ℛ

; 𝑣*�

*∈�

• Intuition.Supposethat𝑆∗ = 𝑖9, … , 𝑖� ∈ ℛdoesnotincludethemostvaluableitem,whichisitem1.• Then𝑆∗ isnotoptimal,becausewecanaugment theset{1}usingitemsfrom𝑆∗ tocreateak itemsetthatisstrictlymorevaluable.

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FullProofisbyInduction• Supposethatthesetselectedbythegreedyalgorithmis{𝑔9, …𝑔�} andthatthe𝑔* istheelementwiththelowestindexthatsuchthatfortheoptimalset𝑆,𝑔* ∉ 𝑆.So,𝑆 =𝑔9,… , 𝑔*a9 ∪ 𝑆e andforeachelement𝑠 ∈ 𝑆′,𝑣�y > 𝑣K.

• Bytheaugmentationproperty,itispossibletoaugment{𝑔9, … , 𝑔*} toabasis𝐵 setbyiterativelyaddingelementsfrom𝑆′,whileomittingjustoneelement,say�̂�.

• Bythen𝑆wasnotoptimal,because𝐵 isbetter:

; 𝑣l�

l∈�−; 𝑣l

�

l∈�= 𝑣�y − 𝑣K̂ > 0. ∎

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MatroidsandSubstitutes• Letℛ beanon-emptycollectionofindependentsetssatisfyingfreedisposal.

• Foreachgoodin𝑥 ∈ 𝑋,thereisabuyer𝑣(𝑥) andaprice𝑝(𝑥).Thebuyer’sdemandisdescribedby:

𝑉∗ ℛ, 𝑣 ≝ max�∈ℜ

; (𝑣 𝑥 − 𝑝 𝑥 )�

7∈�

𝑑∗ 𝑝|ℛ, 𝑣 ≝ argmax�∈ℜ

; (𝑣 𝑥 − 𝑝 𝑥 )�

7∈�

• Theorem.Theitemsin𝑋 aresubstitutesifandonlyifℜ isamatroid.

• Intuition:prettymuchthesameasoptimalityofgreedyalgorithm! 34

ProofSketch• Supposethat𝑛 ∈ 𝑑∗ 𝑝|ℛ, 𝑣 andconsideraprice𝑝e 𝑛 > 𝑝(𝑛)suchthat𝑛 ∉ 𝑑∗ 𝑝\𝑝′(𝑛)|ℛ, 𝑣 .Let𝑛e ∉ 𝑑∗ 𝑝|ℛ, 𝑣 bethefirstnewitemchoseninsteadduringthegreedyalgorithmwithprices𝑝\𝑝′(𝑛).Letthestateofthegreedyalgorithmwhenitischosenbe𝑆′ andlet𝑆 = 𝑆e ∪ 𝑛 − {𝑛e}.

• Bytheaugmentationproperty,thethefeasiblenextchoicestoaugment𝑆′ and𝑆 areidentical.Hence,𝑑 𝑝\𝑝e 𝑛 ℛ, 𝑣 =𝑑 𝑝 ℛ, 𝑣 − 𝑛 ∪ {𝑛e},asrequired.

• Conversely,ifℛisnotamatroid,then…

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NecessityofMatroids• Theorem.Ifℛ isanon-emptyfamilythatsatisfiesfreedisposalbutnottheaugmentationproperty,thenthereissomevectorofvalues𝑣 suchthat(thegreedyalgorithm“fails”)𝑆< ∉ argmax

�∈ℛ∑ 𝑣*�*∈� .

• Proof.ℛ doesnothavetheaugmentationproperty,sothereissome𝑆, 𝑆′ ∈ ℛ suchthat 𝑆 > |𝑆′| andthereisno𝑛 ∈ 𝑆 − 𝑆′ suchthat𝑆′ ∪ 𝑛 ∈ ℛ.

• Let𝜖 > 0 besmallandtake:

𝑣* = �1if𝑛 ∈ 𝑆e1 − 𝜖if𝑛 ∈ 𝑆 − 𝑆e0otherwise

• Thenthegreedyalgorithmselects𝑆e andnoelementsof𝑆 − 𝑆e,soitsvalueis|𝑆e|,but𝑆 achievesatleast 1 − 𝜖 𝑆 > |𝑆e|.∎ 36

WHYSHOULDWECARE?ApproximateSubstitutesandtheIncentiveAuction

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HowWellDoesTheIncentiveAuctionWork?• Can’trunVCGonnational-scaleproblems:

can’tfindanoptimalpacking• RestrictattentiontoallstationswithintwoconstraintsofNewYorkCity• averydenselyconnectedregion• 218stationsmetthiscriterion

• Reverseauctionsimulator(UHFonly)• Simulationassumptions:• 100%participation• 126MHzclearingtarget• valuationsgeneratedbysamplingfromaprominentmodeldueFCCchiefeconomist(beforeherFCCappointment)

• 1mintimeoutgiventoSATFC

PerformanceofIncentiveAuctionAlgorithms

“Greedy”:checkwhetherexistingsolutioncanbedirectlyaugmentedwith

new station

TheSubstitutionIndex• WhydoestheDAalgorithmperformsowell?• Twoconjecturedreasons:• Specialconstraints:theindependentsets𝒞?• Specialvalues:aset𝒪 ⊆ 𝐶 wheretheoptimummaylie?

• “Zeroknowledgecase”:𝒪 = 𝐶.

• Definitions.Giventhegroundset𝒳 andtheconstraints𝒞 andpossibleoptimizers𝒪 thatbothsatisfyfreedisposal,

ℛ∗ 𝒞, 𝒪 ≝ argmaxℛ���jR��T

ℛ⊆𝒞

min�∈𝒪

max��∈ℛ��⊆�

|𝑋′||𝑋|

𝜌 𝒞, 𝒪 ≝ maxℛ���jR��T

ℛ⊆𝒞

min�∈𝒪

max��∈ℛ��⊆�

𝑋e

𝑋 40

ApproximationTheorem

• Giventhegroundset𝒳,any𝒮 ⊆ 𝒫 𝒳 andany𝑣 ∈ ℝN𝒳,definenotationasfollows:

𝑉∗ 𝒮; 𝑣 ≝ max�∈𝒮

; 𝑣*�

*∈�

• Theorem. Thegreedysolutiononℛ∗ approximatestheoptimuminworstcaseasfollows:

minI£8

𝑉∗ ℛ∗; 𝑣𝑉∗ 𝒪; 𝑣 = 𝜌 𝒞, 𝒪 .

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ProofSketch,1• Let

𝑣∗ ∈ argminI£8

𝑉∗ ℛ∗; 𝑣𝑉∗ 𝒪; 𝑣 , 𝜌∗ = 𝑉∗ ℛ∗; 𝑣∗

𝑉∗ 𝒪; 𝑣∗• Amongoptimalsolutions,choose𝑣∗ tobeonewiththesmallestnumberofstrictlypositivecomponents.• Withoutlossofoptimality,werescale𝑣∗ sothatthesmalleststrictlypositivecomponentis1.• Thenextstepwillshowthateverycomponentof𝑣∗ iszeroorone,sothatthevaluesofthetwominimizationproblemsmustexactlycoincide.

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ProofSketch,2• Considerthefamilyofpotentialminimizers𝑣¤(𝛼),where

𝑣¤* 𝛼 ≝ � 𝛼if𝑣*∗ = 1

𝑣*∗otherwise

• Then,𝑣∗ = 𝑣¤(1).Thevalueoftheobjectivefor𝑣¤(𝛼) is

𝜌¤ 𝛼 =𝑉∗ ℛ∗; 𝑣¤ 𝛼𝑉∗ 𝒪; 𝑣¤ 𝛼

=𝛼 �̈� ∩ 𝑋ℛ∗ + ∑ 𝑣*∗�

*∈ 𝒳a�̈ ∩�ℛ∗

𝛼 �̈� ∩ 𝑋𝒪 + ∑ 𝑣*∗�*∈ 𝒳a�̈ ∩�𝒪

where�̈� ≝ 𝑛 𝑣*∗ = 1

𝑋𝒪 ∈ argmax�∈𝒪

; 𝑣*∗�

*∈�

𝑋ℛ∗ ∈ argmax�∈ℛ∗

; 𝑣*∗�

*∈�43

ProofSketch,3

𝜌¤ 𝛼 =𝛼 �̈� ∩ 𝑋ℛ∗ + ∑ 𝑣*∗�

*∈ 𝒳a�̈ ∩�ℛ∗

𝛼 �̈� ∩ 𝑋𝒫 + ∑ 𝑣*∗�*∈ 𝒳a�̈ ∩�𝒪

For𝜌¤ ⋅ toachieveitsminimumof𝜌∗ when𝛼 = 1,itmustbea

constantfunction,whichrequires �̈∩�ℛ∗�̈∩�𝒫

= 𝜌∗.Then,since𝑣∗ istheminimizerwiththefeweststrictlypositiveelements, 𝑛 𝑣*∗ > 1 =∅.∎

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WHATISGOINGONNOW?TheUSIncentiveAuction

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$86.422Billion?Whatdoesitmean?

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TheIncentiveAuction“Stages”:AConceptualIllustration

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Reverse Price

Forward Price

QuantityTraded

LOSS

TVspectrumsupply

Net Revenue > Target

Septem

ber2

015

Desig

ningth

eUSIncentiveAu

ction

ENDThankyou!

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