Auction Theory Stephane

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    Auction TheoryAuction Theory

    an introductionan introduction

    DAI HardsDAI Hards

    October 16October 16thth

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    IntroductionIntroduction

    Auctions are the most widely-studied economicmechanism.

    Auctions refer to arbitrary resource allocation problemswith self-motivated participants: Auctioneer and bidders

    Auction (selling item(s)): one buyer, multiple bidders)e.g. selling a cd on eBay

    Reverse Auction (buying item(s)): one buyer, multiple

    sellerse.g. procurement

    Well discuss auction, though the same theory holds forreverse auction

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    Historical noteHistorical note

    Reports that auctions was used in BabylonReports that auctions was used in Babylon500 B.C.500 B.C.

    193 A.D. After having killed Emperor193 A.D. After having killed EmperorPertinax, Prtorian Guard sold the RomanPertinax, Prtorian Guard sold the RomanEmpire by means of an AuctionEmpire by means of an Auction

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    Where auctions are usedWhere auctions are used

    nowadays?nowadays? Treasury auctions (bill, notes, Treasury bonds,Treasury auctions (bill, notes, Treasury bonds,

    securities)securities) Has been used to transfer assets from public to privateHas been used to transfer assets from public to private

    sectorsector Right to drill oil, offRight to drill oil, off--shore oil leaseshore oil lease Use the EM spectrumUse the EM spectrum Government and private corporations solicit deliveryGovernment and private corporations solicit delivery

    price offers of productsprice offers of products

    Private firms sell products (flowers, fish, tobacco,Private firms sell products (flowers, fish, tobacco,livestock, diamonds)livestock, diamonds) Internet auctionsInternet auctions

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    QuestionsQuestions

    Information problem: the seller has usuallyInformation problem: the seller has usuallyincomplete information about buyers valuationsincomplete information about buyers valuations

    (else, he just need to set the price as the(else, he just need to set the price as themaximum valuation of the buyer)maximum valuation of the buyer) what pricingwhat pricingscheme performs well even in incompletescheme performs well even in incompleteinformation setting (is auction better suited for ainformation setting (is auction better suited for a

    given problem? Does a type of auction yieldgiven problem? Does a type of auction yieldgreater revenue?)greater revenue?)

    For the buyer, what are good bidding strategies?For the buyer, what are good bidding strategies?

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    TerminologyTerminology

    Criterion of comparison:Criterion of comparison: Revenue:Revenue: expected selling priceexpected selling price Efficiency:Efficiency: the object ends up in the hands of the person who values itthe object ends up in the hands of the person who values it

    the mostthe most(resale does not yield to efficiency)(resale does not yield to efficiency)

    Private ValuePrivate Value:: no bidder knows with certainty the valuation of the otherno bidder knows with certainty the valuation of the otherbidders, and knowledge of the other bidders valuation would not affect thebidders, and knowledge of the other bidders valuation would not affect thevalue of the particular biddervalue of the particular bidder

    Pure common valuePure common value: the actual value is the same for ever bidders: the actual value is the same for ever biddersbut bidders have different private information about the what that valuebut bidders have different private information about the what that valueactually isactually is (e.g. auction of an oil field and the amount of oil is unknown, different(e.g. auction of an oil field and the amount of oil is unknown, different

    bidders have different geological signals, learning another signal would change thebidders have different geological signals, learning another signal would change thevaluation of a bidder).valuation of a bidder).

    Correlated value:Correlated value: agents value of an item depends partly on its ownagents value of an item depends partly on its ownpreferences and partly on others values for itpreferences and partly on others values for it

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    Agents care about utility, notvaluation

    Auctions are really lotteries, so you must compareexpected utility rather than utility.

    Risk attitude speak about the shape of the utilityfunction: linear utility function refers to risk-neutrality

    optimize her/his expected payoff

    Concave utility function refers to risk-aversion (u>0 and u0 and u>0)

    The types of utility functions, and the associated riskattitudes of agents, are among the most importantconcepts in Bayesian games, and in particular inauctions. Most theoretical results about auction aresensitive to the risk attitude of the bidders.

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    OutlineOutline

    SingleSingle--item Auctionsitem Auctions Common auctions formsCommon auctions forms

    Equivalence between auctionsEquivalence between auctions

    Revenue equivalenceRevenue equivalence

    MultiMulti--unit Auctionunit Auction

    MultiMulti--item Auctionitem Auction

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    Single Item AuctionSingle Item Auction

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    EnglishEnglish

    (first(first--price openprice open--cry = ascending)cry = ascending) Protocol:Protocol: Each bidder is free to raise his bid. When no bidder isEach bidder is free to raise his bid. When no bidder is

    willing to raise, the auction ends, and the highest bidder wins thewilling to raise, the auction ends, and the highest bidder wins theitem at the price of his biditem at the price of his bid

    Strategy:Strategy: Series of bids as a function of agents private value, hisSeries of bids as a function of agents private value, hisprior estimates of others valuations, and past bidsprior estimates of others valuations, and past bids

    Best strategy:Best strategy: In private value auctions, bidders dominantIn private value auctions, bidders dominantstrategy is to always bid a small amount more than current higheststrategy is to always bid a small amount more than current highestbid, and stop when his private value price is reachedbid, and stop when his private value price is reached

    Variations:Variations: In correlated value auctions, auctioneer often increases price atIn correlated value auctions, auctioneer often increases price at

    a constant rate or as he thinks is appropriate (japonese auction)a constant rate or as he thinks is appropriate (japonese auction) OpenOpen--exit: Bidder has to openly declare exit without reexit: Bidder has to openly declare exit without re--enteringentering

    possibility => More info to other bidders about the agentspossibility => More info to other bidders about the agentsvaluationvaluation

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    FirstFirst--price sealedprice sealed--bidbid

    Protocol:Protocol: Each bidder submits one bid without knowingEach bidder submits one bid without knowingothers bids. The highest bidder wins the item at theothers bids. The highest bidder wins the item at theprice of his bidprice of his bid

    Single round of biddingSingle round of bidding Strategy:Strategy: Bid as a function of agents private value andBid as a function of agents private value andhis prior estimates of others valuationshis prior estimates of others valuations

    Best strategy:Best strategy: No dominant strategy in generalNo dominant strategy in general Strategic underbidding & counterspeculationStrategic underbidding & counterspeculation

    Can determine Nash equilibrium strategies via commonCan determine Nash equilibrium strategies via commonknowledge assumptions about the probabilityknowledge assumptions about the probabilitydistributions from which valuations are drawndistributions from which valuations are drawn

    VariantVariant: k: kthth priceprice

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    ExampleExample

    Values are uniformly distributed on [0,1]Values are uniformly distributed on [0,1]The equilibrium bid is (NThe equilibrium bid is (N--1)*x/N1)*x/N

    WhereWhere

    x is the valuation of the bidderx is the valuation of the bidder

    N is the number of biddersN is the number of bidders

    (proof)(proof)

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    Dutch (descending)Dutch (descending)

    ProtocolProtocol: Auctioneer continuously lowers the price until a bidder: Auctioneer continuously lowers the price until a biddertakes the item at the current pricetakes the item at the current price

    Strategically equivalent to firstStrategically equivalent to first--price sealedprice sealed--bid protocol in allbid protocol in allauction settingsauction settings

    StrategyStrategy: Bid as a function of agents private value and his prior: Bid as a function of agents private value and his priorestimates of others valuationsestimates of others valuations

    Best strategyBest strategy: No dominant strategy in general: No dominant strategy in general Lying (downLying (down--biasing bids) & counterspeculationbiasing bids) & counterspeculation Possible to determine Nash equilibrium strategies via commonPossible to determine Nash equilibrium strategies via common

    knowledge assumptions regarding the probability distributions of othersknowledge assumptions regarding the probability distributions of others

    valuesvalues Requires multiple rounds of posting current priceRequires multiple rounds of posting current price

    Dutch flower market, Ontario tobacco auction, Filenes basement,Dutch flower market, Ontario tobacco auction, Filenes basement,WaldenbooksWaldenbooks

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    Vickrey (= secondVickrey (= second--price sealed bid)price sealed bid)

    ProtocolProtocol: Each bidder submits one bid without knowing (!) others: Each bidder submits one bid without knowing (!) othersbids. Highest bidder wins item at 2nd highest pricebids. Highest bidder wins item at 2nd highest price

    StrategyStrategy: Bid as a function of agents private value & his prior: Bid as a function of agents private value & his priorestimates of others valuationsestimates of others valuations

    Best strategyBest strategy: In a private value auction with risk neutral bidders,: In a private value auction with risk neutral bidders,Vickrey is strategically equivalent to English. In such settings,Vickrey is strategically equivalent to English. In such settings,dominant strategy is to bid ones true valuationdominant strategy is to bid ones true valuation No counterspeculationNo counterspeculation Independent of others bidding plans, operating environments,Independent of others bidding plans, operating environments,

    capabilities...capabilities...

    Single round of biddingSingle round of bidding Widely advocated for computational multiagent systemsWidely advocated for computational multiagent systems Old [Vickrey 1961], but not widely used among humansOld [Vickrey 1961], but not widely used among humans Revelation principleRevelation principle ------ proxy bidder agents on www.ebay.com,proxy bidder agents on www.ebay.com,

    www.webauction.com, www.onsale.comwww.webauction.com, www.onsale.com

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    All PayAll Pay (e.g.lobbying activity)(e.g.lobbying activity)

    ProtocolProtocol: Each bidder is free to raise his bid. When no: Each bidder is free to raise his bid. When nobidder is willing to raise, the auction ends, and thebidder is willing to raise, the auction ends, and thehighest bidder wins the item. All bidders have to payhighest bidder wins the item. All bidders have to pay

    their last bidtheir last bid StrategyStrategy: Series of bids as a function of agents private: Series of bids as a function of agents privatevalue, his prior estimates of others valuations, and pastvalue, his prior estimates of others valuations, and pastbidsbids

    Best strategyBest strategy: ?: ? In private value settings it can be computed (low bids)In private value settings it can be computed (low bids)

    Potentially long bidding processPotentially long bidding process VariationsVariations

    Each agent pays only part of his highest bidEach agent pays only part of his highest bid Each agents payment is a function of the highest bid of allEach agents payment is a function of the highest bid of all

    agentsagents

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    In a NutshellIn a Nutshell

    English AuctionEnglish AuctionSecond-Price Sealed Bid

    i.e Vickrey

    Second-Price Sealed Bid

    i.e Vickrey

    First-Price Sealed BidFirst-Price Sealed Bid Dutch Descending PriceDutch Descending Price

    Strong

    Weak

    Private Value

    Sealed Bid Format Open Format

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    Setting for Private Value AuctionsSetting for Private Value Auctions

    N potential bidders. BidderN potential bidders. Bidder ii is assigned a value ofis assigned a value of XXii to theto theobjectobject EachEach XXii is i.i.d. on some interval [0,is i.i.d. on some interval [0,] according to the] according to the

    cumulative distribution Fcumulative distribution F

    Bidders I knows her/hisBidders I knows her/his xxii and also that other bidders values areand also that other bidders values arei.i.d. according to Fi.i.d. according to F

    Bidders are risk neutral (seek to maximize their expectedBidders are risk neutral (seek to maximize their expectedpayoffs)payoffs)

    The number of bidders and the distribution F are commonThe number of bidders and the distribution F are common

    knowledge.knowledge. SymmetrySymmetryThe distribution of values is the same for all bidders. WE canThe distribution of values is the same for all bidders. WE can

    consider that all bidders are alike, hence an optimal biddingconsider that all bidders are alike, hence an optimal biddingstrategy for one should also be an optimal strategy for thestrategy for one should also be an optimal strategy for theothersothers symmetric equilibriumsymmetric equilibrium

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    Results for private value auctionsResults for private value auctions

    Dutch strategically equivalent to firstDutch strategically equivalent to first--price sealedprice sealed--bidbid Risk neutral agents => Vickrey strategically equivalentRisk neutral agents => Vickrey strategically equivalent

    to Englishto English

    All four protocols allocate item efficiently (assuming noAll four protocols allocate item efficiently (assuming noreservation price for the auctioneer)reservation price for the auctioneer)

    English & Vickrey have dominant strategiesEnglish & Vickrey have dominant strategies no effortno effortwasted in counterspeculationwasted in counterspeculation

    Which of the four auction mechanisms gives highestWhich of the four auction mechanisms gives highestexpected revenue to the seller?expected revenue to the seller?Assuming valuations are drawn independently & agents are riskAssuming valuations are drawn independently & agents are risk

    neutral: The four mechanisms haveneutral: The four mechanisms have equalequalexpected revenue!expected revenue!

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    Reserve Price in Private ValuesReserve Price in Private Values

    A seller can reserve the right to not sell the object if the price isA seller can reserve the right to not sell the object if the price isbelow abelow a reserved price rreserved price r

    Bidders with valueBidders with value xx

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    Revenue Equivalence TheoremRevenue Equivalence Theorem

    In all auctions for k units with the following properties Buyers are risk neutral

    Private Value, with values independently and identically

    distributed over [a,b] (technicality distribution must beatomless)

    Each bidder demands at most 1 unit

    Auction allocates the units to the k highest bids (efficiency)

    The bidder with the lowest valuation has a surplus of 0 (i.e. abidder with a value of 0 has an expected payment of 0)

    a buyer with a given valuation will make the sameexpected payment, and therefore all such auctions havethe same expected revenue

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    Application of the RevenueApplication of the Revenue

    Equivalence TheoremEquivalence TheoremHelps to find some equilibrium strategyHelps to find some equilibrium strategy

    Ex: compute the equilibrium bid in an allEx: compute the equilibrium bid in an allpay auction or in a third price auctionpay auction or in a third price auction

    In the case where the number of biddersIn the case where the number of biddersis uncertain, we can compute theis uncertain, we can compute the

    equilibrium bid strategy for a first priceequilibrium bid strategy for a first priceauctionauction

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    Revenue equivalence ceases toRevenue equivalence ceases to

    hold if agents are not riskhold if agents are not risk--neutralneutral Risk averse AgentsRisk averse Agents

    for bidders:for bidders:Dutch, firstDutch, first--price sealedprice sealed--bid Vickrey, Englishbid Vickrey, English

    Compared to a risk neutral bidder, a risk averse bidder willCompared to a risk neutral bidder, a risk averse bidder willbid higher (buy insurance against the possibility ofbid higher (buy insurance against the possibility ofloosing)loosing)

    (utility of winning with a lower bid

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    Revenue equivalence ceases toRevenue equivalence ceases to

    hold if it is not Private Valuehold if it is not Private ValueResults for nonResults for non--private value auctionsprivate value auctions Dutch strategically equivalent to firstDutch strategically equivalent to first--price sealedprice sealed--bidbid Vickrey not strategically equivalent to EnglishVickrey not strategically equivalent to English

    All four protocols allocate item efficientlyAll four protocols allocate item efficiently Winners curse: each bidder must recognize that she/he wins theWinners curse: each bidder must recognize that she/he wins theobjects only if she/he has the highest signal, failure to take intoobjects only if she/he has the highest signal, failure to take intoaccount the bad news about others signal can lead the bidder to payaccount the bad news about others signal can lead the bidder to paymore than the prize it is worth.more than the prize it is worth. Common value auctions:Common value auctions:

    Agent should lie (bid low) even in Vickrey & English Revelation to Agent should lie (bid low) even in Vickrey & English Revelation toproxy bidders?proxy bidders?

    Thrm (revenue nonThrm (revenue non--equivalence ). With more than 2 bidders, theequivalence ). With more than 2 bidders, theexpected revenues are not the same:expected revenues are not the same:English Vickrey Dutch = firstEnglish Vickrey Dutch = first--price sealed bidprice sealed bid

    v 1 ! E [ v | v 1 , b ( v 2 ) b ( v1 ), ..., b ( vN ) b ( v 1 )]

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    Results for nonResults for non--private valueprivate value

    auctionsauctions Common knowledge that auctioneer hasCommon knowledge that auctioneer has

    private infoprivate info

    Q: What info should the auctioneer release ?Q: What info should the auctioneer release ?

    A: auctioneer is best off releasing all of itA: auctioneer is best off releasing all of it

    No news is worst newsNo news is worst news

    Mitigates the winners curseMitigates the winners curse

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    The revelation principle

    (mechanism Design)

    In a revelation mechanism agents are asked to reporttheir types (e.g.valuations for the good), and an action(e.g. decision on the winner and his/her payment) will

    be based the agents announcement. In general, agents may cheat about their types, but:

    Any mechanism that implements certain behavior (e.g. agood is allocated to the agent with the highestvaluation,v, and he pays (1-1/n)v) can be replaced byanother mechanism that implements the same behaviorand where truth-revealing is in equilibrium.

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    MultiMulti--unit Auctionunit Auction

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    Auctions with multipleAuctions with multiple

    indistinguishable units for saleindistinguishable units for sale ExamplesExamples

    IBM stocksIBM stocks

    Barrels of oilBarrels of oil

    Pork belliesPork bellies

    TransTrans--Atlantic backbone bandwidth from NYCAtlantic backbone bandwidth from NYC

    to Paristo Paris

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    Setting for sealed bid auctionsSetting for sealed bid auctions

    Each bidder sends a bid vector indicatingEach bidder sends a bid vector indicatinghow much she/he is willing to pay forhow much she/he is willing to pay for

    each additional uniteach additional unit Can be understood as a demand functionCan be understood as a demand function

    Number of units

    Value of the bid

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    Pricing rulesPricing rules

    Auctioning multiple indistinguishable unitsAuctioning multiple indistinguishable unitsof an itemof an item

    The discriminatory (or pay your bid)The discriminatory (or pay your bid)auctionauction

    The uniform price auctionThe uniform price auction

    The Vickrey auctionThe Vickrey auction

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    Discriminatory auctionDiscriminatory auction

    Each bidder pays an amount equal to theEach bidder pays an amount equal to thesum of his bids that are among the Ksum of his bids that are among the K

    highest of the N*K bids submitted.highest of the N*K bids submitted.

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    UniformUniform--price Auctionprice Auction

    Any price between the highest loosing bidAny price between the highest loosing bidand the lowest winning bid is possibleand the lowest winning bid is possible

    can choose the highest losing bidcan choose the highest losing bid

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    Vickrey AuctionVickrey Auction

    Basic principle is the same as the VickreyBasic principle is the same as the Vickrey--ClarkeClarke--Groves mechanism (see MechanismGroves mechanism (see Mechanism

    Design)Design)A bidder who wins k units pays the kA bidder who wins k units pays the k

    highest losing bids of the other biddershighest losing bids of the other bidders

    For bidder i to win the kFor bidder i to win the kthth unit, is kunit, is kththhighest bid must defeat the khighest bid must defeat the kthth lowestlowestcompeting bidcompeting bid

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    Some Open AuctionsSome Open Auctions

    Dutch AuctionsDutch Auctions English AuctionsEnglish AuctionsAusubel AuctionsAusubel Auctions

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    MultiMulti--item auctionsitem auctions

    multiple distinguishable items formultiple distinguishable items for

    salesale

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    Bundle bidding scenarioBundle bidding scenario

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    Bundle bidding scenarioBundle bidding scenario

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    Bundle bidding scenarioBundle bidding scenario

    (console, television, cd player $1000)

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    Bundle bidding scenarioBundle bidding scenario

    (television, music system, computer, $1600)

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    Bundle bidding scenarioBundle bidding scenario

    (cd player, console, music system $400)

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    Bundle bidding scenarioBundle bidding scenario

    ((console, television, cd player $1000),

    (television, music system, computer, $1600),

    (cd player, console, music system $ 400))

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    Bundle bidding scenarioBundle bidding scenario

    ((Computer, television, cd player $1000),

    (television, music system, console, $600),

    (cd player, console, music system $400))

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    Bundle bidding scenarioBundle bidding scenario

    ((Computer, television, cd player $1000),

    (television, music system, console, $600),

    (cd player, console, music system $400))

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    Bundle bidding scenarioBundle bidding scenario

    ((Computer, television, cd player $1000),

    (television, music system, console, $600),

    (cd player, console, music system $400))

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    MultipleMultiple--item auctionsitem auctions

    Auction of multiple, distinguishable itemsAuction of multiple, distinguishable items Bidders have preferences over itemBidders have preferences over item

    combinationscombinations

    Combinatorial auctionsCombinatorial auctions Bids can be submitted over item bundlesBids can be submitted over item bundles

    Winner selection: combinatorial optimizationWinner selection: combinatorial optimization NPNP--completecomplete

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    SourceSource

    Vijay Krishna: Auction Theory (Academic Press)Vijay Krishna: Auction Theory (Academic Press) Paul Klemperer: Auction Theory: A guide to thePaul Klemperer: Auction Theory: A guide to the

    literature (Journal of Economics Survey)literature (Journal of Economics Survey) Elmar Wolfstetter: Auctions An IntroductionElmar Wolfstetter: Auctions An Introduction Tuomas Sandholm COURSE: CS 15Tuomas Sandholm COURSE: CS 15--892892

    Foundations of Electronic Marketplaces (CMU)Foundations of Electronic Marketplaces (CMU)