Auction TheoryAuction Theoryan introductionan introduction

DAI HardsDAI Hards

October 16October 16thth

IntroductionIntroduction

• Auctions are the most widely-studied economic mechanism.

• Auctions refer to arbitrary resource allocation problems with 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 sellers e.g. procurement

• We’ll discuss auction, though the same theory holds for reverse auction

Historical noteHistorical note

• Reports that auctions was used in Reports that auctions was used in Babylon 500 B.C.Babylon 500 B.C.

• 193 A.D. After having killed Emperor 193 A.D. After having killed Emperor Pertinax, Prætorian Guard sold the Pertinax, Prætorian Guard sold the Roman Empire by means of an Roman Empire by means of an AuctionAuction

Where auctions are used Where 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 Has been used to transfer assets from public to

private sectorprivate sector• Right to drill oil, off-shore oil leaseRight to drill oil, off-shore oil lease• Use the EM spectrumUse the EM spectrum• Government and private corporations solicit Government and private corporations solicit

delivery price offers of products delivery price offers of products • Private firms sell products (flowers, fish, tobacco, Private firms sell products (flowers, fish, tobacco,

livestock, diamonds)livestock, diamonds)• Internet auctionsInternet auctions

QuestionsQuestions

• Information problem: the seller has usually Information problem: the seller has usually incomplete information about buyers’ incomplete information about buyers’ valuations (else, he just need to set the price valuations (else, he just need to set the price as the maximum valuation of the buyer) as the maximum valuation of the buyer) what pricing scheme performs well even in what pricing scheme performs well even in incomplete information setting (is auction incomplete information setting (is auction better suited for a given problem? Does a type better suited for a given problem? Does a type of auction yield greater revenue?)of auction yield greater revenue?)

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

TerminologyTerminology

• Criterion of comparison:Criterion of comparison:– Revenue:Revenue: expected selling price expected selling price– Efficiency:Efficiency: the object ends up in the hands of the person who the object ends up in the hands of the person who

values it the most values it the most (resale does not yield to efficiency)(resale does not yield to efficiency)

• Private ValuePrivate Value:: no bidder knows with certainty the valuation of no bidder knows with certainty the valuation of the other bidders, and knowledge of the other bidders’ valuation the other bidders, and knowledge of the other bidders’ valuation would not affect the value of the particular bidder would not affect the value of the particular bidder

• Pure common valuePure common value: the actual value is the same for ever : the actual value is the same for ever bidders but bidders have different private information about the bidders but bidders have different private information about the what that value actually is what that value actually is (e.g. auction of an oil field and the amount of (e.g. auction of an oil field and the amount of oil is unknown, different bidders have different geological signals, learning oil is unknown, different bidders have different geological signals, learning another signal would change the valuation of a bidder). another signal would change the valuation of a bidder).

• Correlated value:Correlated value: agent’s value of an item depends partly on its agent’s value of an item depends partly on its own preferences and partly on others’ values for itown preferences and partly on others’ values for it

Agents care about utility, not valuation

• Auctions are really lotteries, so you must compare expected utility rather than utility.

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

optimize her/his expected payoff– Concave utility function refers to risk-aversion (u’>0 and

u’’<0)– convex utility function refers to risk-seeking (u’>0 and

u’’>0)

• The types of utility functions, and the associated risk attitudes of agents, are among the most important concepts in Bayesian games, and in particular in auctions. Most theoretical results about auction are sensitive to the risk attitude of the bidders.

OutlineOutline

• Single-item AuctionsSingle-item Auctions– Common auctions formsCommon auctions forms– Equivalence between auctionsEquivalence between auctions– Revenue equivalenceRevenue equivalence

• Multi-unit AuctionMulti-unit Auction

• Multi-item AuctionMulti-item Auction

Single Item AuctionSingle Item Auction

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

bidder is willing to raise, the auction ends, and the highest bidder is willing to raise, the auction ends, and the highest bidder wins the item at the price of his bidbidder wins the item at the price of his bid

• Strategy:Strategy: Series of bids as a function of agent’s private Series of bids as a function of agent’s private value, his prior estimates of others’ valuations, and past bidsvalue, his prior estimates of others’ valuations, and past bids

• Best strategy:Best strategy: In private value auctions, bidder’s dominant In private value auctions, bidder’s dominant strategy is to always bid a small amount more than current strategy is to always bid a small amount more than current highest bid, and stop when his private value price is reachedhighest bid, and stop when his private value price is reached

• Variations:Variations:– In correlated value auctions, auctioneer often increases In correlated value auctions, auctioneer often increases

price at a constant rate or as he thinks is appropriate price at a constant rate or as he thinks is appropriate (japonese auction)(japonese auction)

– Open-exit: Bidder has to openly declare exit without re-Open-exit: Bidder has to openly declare exit without re-entering possibility => More info to other bidders about entering possibility => More info to other bidders about the agent’s valuationthe agent’s valuation

First-price sealed-bidFirst-price sealed-bid

• Protocol:Protocol: Each bidder submits one bid without Each bidder submits one bid without knowing others’ bids. The highest bidder wins the knowing others’ bids. The highest bidder wins the item at the price of his biditem at the price of his bid

• Single round of biddingSingle round of bidding• Strategy:Strategy: Bid as a function of agent’s private Bid as a function of agent’s private

value and his prior estimates of others’ valuationsvalue and his prior estimates of others’ valuations• Best strategy:Best strategy: No dominant strategy in general No dominant strategy in general• Strategic underbidding & counterspeculationStrategic underbidding & counterspeculation• Can determine Nash equilibrium strategies via Can determine Nash equilibrium strategies via

common knowledge assumptions about the common knowledge assumptions about the probability distributions from which valuations are probability distributions from which valuations are drawndrawn

• VariantVariant: k: kthth price price

ExampleExample

• Values are uniformly distributed on Values are uniformly distributed on [0,1][0,1]

The equilibrium bid is (N-1)*x/NThe equilibrium bid is (N-1)*x/N

Where Where – 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)

Dutch (descending)Dutch (descending)

• ProtocolProtocol: Auctioneer continuously lowers the price until a : Auctioneer continuously lowers the price until a bidder takes the item at the current pricebidder takes the item at the current price

• Strategically equivalent to first-price sealed-bid protocol in Strategically equivalent to first-price sealed-bid protocol in all auction settingsall auction settings

• StrategyStrategy: Bid as a function of agent’s private value and his : Bid as a function of agent’s private value and his prior estimates of others’ valuationsprior estimates of others’ valuations

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

knowledge assumptions regarding the probability distributions knowledge assumptions regarding the probability distributions of others’ valuesof others’ values

– Requires multiple rounds of posting current priceRequires multiple rounds of posting current price• Dutch flower market, Ontario tobacco auction, Filene’s Dutch flower market, Ontario tobacco auction, Filene’s

basement, Waldenbooksbasement, Waldenbooks

Vickrey (= second-price sealed Vickrey (= second-price sealed bid)bid)• ProtocolProtocol: Each bidder submits one bid without knowing (!) : Each bidder submits one bid without knowing (!)

others’ bids. Highest bidder wins item at 2nd highest priceothers’ bids. Highest bidder wins item at 2nd highest price• StrategyStrategy: Bid as a function of agent’s private value & his : Bid as a function of agent’s private value & his

prior estimates of others’ valuationsprior estimates of others’ valuations• Best strategyBest strategy: In a private value auction with risk neutral : In a private value auction with risk neutral

bidders, Vickrey is strategically equivalent to English. In bidders, Vickrey is strategically equivalent to English. In such settings, dominant strategy is to bid one’s true such settings, dominant strategy is to bid one’s true valuationvaluation– 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 principle --- proxy bidder agents on Revelation principle --- proxy bidder agents on

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

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

• ProtocolProtocol: Each bidder is free to raise his bid. : Each bidder is free to raise his bid. When no bidder is willing to raise, the auction When no bidder is willing to raise, the auction ends, and the highest bidder wins the item. All ends, and the highest bidder wins the item. All bidders have to pay their last bidbidders have to pay their last bid

• StrategyStrategy: Series of bids as a function of agent’s : Series of bids as a function of agent’s private value, his prior estimates of others’ private value, his prior estimates of others’ valuations, and past bidsvaluations, and past bids

• 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 agent’s payment is a function of the highest bid of Each agent’s payment is a function of the highest bid of

all agentsall agents

In a NutshellIn a Nutshell

English AuctionEnglish Auction

Second-Price Sealed Bid

i.e Vickrey

Second-Price Sealed Bid

i.e Vickrey

First-Price Sealed Bid

First-Price Sealed Bid

Dutch Descending Price

Dutch Descending Price

Strong

Weak

Private Value

Sealed Bid Format Open Format

Setting for Private Value Setting for Private Value AuctionsAuctionsN potential bidders. Bidder N potential bidders. Bidder ii is assigned a value of is assigned a value of XXii

to the objectto the object– Each Each XXii is i.i.d. on some interval [0, is i.i.d. on some interval [0,ωω] according to the ] according to the

cumulative distribution Fcumulative distribution F– Bidders I knows her/his Bidders I knows her/his xxii and also that other bidders and also that other bidders

values are i.i.d. according to Fvalues are i.i.d. according to F– Bidders are risk neutral (seek to maximize their Bidders are risk neutral (seek to maximize their

expected payoffs)expected payoffs)– The number of bidders and the distribution F are The number of bidders and the distribution F are

common knowledge.common knowledge.

• SymmetrySymmetryThe distribution of values is the same for all bidders. WE The distribution of values is the same for all bidders. WE

can consider that all bidders are alike, hence an optimal can consider that all bidders are alike, hence an optimal bidding strategy for one should also be an optimal bidding strategy for one should also be an optimal strategy for the others strategy for the others symmetric equilibrium symmetric equilibrium

Results for private value Results for private value auctionsauctions• Dutch strategically equivalent to first-price sealed-bidDutch strategically equivalent to first-price sealed-bid• Risk neutral agents => Vickrey strategically Risk neutral agents => Vickrey strategically

equivalent to Englishequivalent to English• All four protocols allocate item efficiently (assuming All four protocols allocate item efficiently (assuming

no reservation price for the auctioneer)no reservation price for the auctioneer)• English & Vickrey have dominant strategies English & Vickrey have dominant strategies no no

effort wasted in counterspeculationeffort wasted in counterspeculation• Which of the four auction mechanisms gives highest Which of the four auction mechanisms gives highest

expected revenue to the seller?expected revenue to the seller?Assuming valuations are drawn independently & agents are Assuming valuations are drawn independently & agents are

risk neutral: The four mechanisms haverisk neutral: The four mechanisms have equalequal expected expected revenue!revenue!

Reserve Price in Private Reserve Price in Private ValuesValues• A seller can reserve the right to not sell the object if the A seller can reserve the right to not sell the object if the

price is below a price is below a reserved price rreserved price r Bidders with value Bidders with value xx<<r r are excluded from the auctionare excluded from the auction Bidders change their strategy (can be computed)Bidders change their strategy (can be computed)• A revenue maximizing seller should always set a reserve A revenue maximizing seller should always set a reserve

price price r r that exceeds her or his valuation that exceeds her or his valuation xx00Proof: compute the expected payoff of the seller, differentiate it with respect to the Proof: compute the expected payoff of the seller, differentiate it with respect to the

reserve price and observe it the derivative is positive at reserve price and observe it the derivative is positive at xx00

• Entry Fees: the auctioneer can use an entry fee: a Entry Fees: the auctioneer can use an entry fee: a nonrefundable amount that the bidder has to pay to nonrefundable amount that the bidder has to pay to participate to the auction.participate to the auction.

Note : there is a way to fix the entry fee such it is equivalent to using a Note : there is a way to fix the entry fee such it is equivalent to using a reserved price as far as the agents that are excluded are concerned.reserved price as far as the agents that are excluded are concerned.

• Trade-off: may improve revenue at the expense of Trade-off: may improve revenue at the expense of efficiency efficiency (if seller set a reservation price which is too high)(if seller set a reservation price which is too high)

Revenue Equivalence Revenue Equivalence TheoremTheorem• 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 be atomless)

– 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.

a bidder with a value of 0 has an expected payment of 0)

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

Application of the Revenue Application of the Revenue Equivalence TheoremEquivalence Theorem

Helps to find some equilibrium strategyHelps to find some equilibrium strategy

• Ex: compute the equilibrium bid in an Ex: compute the equilibrium bid in an all pay auction or in a third price all pay auction or in a third price auctionauction

• In the case where the number of In the case where the number of bidders is uncertain, we can compute bidders is uncertain, we can compute the equilibrium bid strategy for a first the equilibrium bid strategy for a first price auctionprice auction

Revenue equivalence ceases to Revenue equivalence ceases to hold if agents are not risk-hold if agents are not risk-neutralneutral

• Risk averse AgentsRisk averse Agents– for bidders: for bidders: Dutch, first-price sealed-bid ≥ Vickrey, EnglishDutch, first-price sealed-bid ≥ Vickrey, EnglishCompared to a risk neutral bidder, a risk averse Compared to a risk neutral bidder, a risk averse

bidder will bid higher (“buy” insurance against the bidder will bid higher (“buy” insurance against the possibility of loosing)possibility of loosing)

(utility of winning with a lower bid <(utility of winning with a lower bid <utility consequenceutility consequence

loosing the object) loosing the object)

– For auctioneer auctioneer: For auctioneer auctioneer: Dutch, first-price sealed-bid ≤ Vickrey, EnglishDutch, first-price sealed-bid ≤ Vickrey, English

• Risk-Seeking Agents– The expected revenue in third-price is greater than

the expected revenue in second-price (English)– Under constant risk-attitude: (k+1)-price is

preferable to k-price

Revenue equivalence ceases to Revenue equivalence ceases to hold if it is not Private Valuehold if it is not Private Value

Results for non-private value auctionsResults for non-private value auctions• Dutch strategically equivalent to first-price sealed-bidDutch strategically equivalent to first-price sealed-bid• Vickrey not strategically equivalent to EnglishVickrey not strategically equivalent to English• All four protocols allocate item efficientlyAll four protocols allocate item efficiently• Winner’s curse: each bidder must recognize that she/he wins Winner’s curse: each bidder must recognize that she/he wins

the objects only if she/he has the highest signal, failure to the objects only if she/he has the highest signal, failure to take into account the bad news about others’ signal can lead take into account the bad news about others’ signal can lead the bidder to pay more than the prize it is worth.the bidder to pay more than the prize it is worth.– Common value auctions: Common value auctions:

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

• Thrm (revenue non-equivalence ). With more than 2 bidders, Thrm (revenue non-equivalence ). With more than 2 bidders, the expected revenues are not the same: the expected revenues are not the same: English ≥ Vickrey ≥ Dutch = first-price sealed bidEnglish ≥ Vickrey ≥ Dutch = first-price sealed bid

˜ v 1

E [ v | ˆ v 1

, b ( ˆ v 2

) b ( ˆ v 1

), ..., b ( ˆ v N

) b ( ˆ v 1

)]

Results for non-private value Results for non-private value auctionsauctions

• Common knowledge that auctioneer Common knowledge that auctioneer has private info has private info

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

A: auctioneer is best off releasing all of A: auctioneer is best off releasing all of itit

• ““No news is worst news”No news is worst news”

• Mitigates the winner’s curseMitigates the winner’s curse

The revelation principle(mechanism Design)

• In a revelation mechanism agents are asked to report their 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. a good is allocated to the agent with the highest valuation,v, and he pays (1-1/n)v) can be replaced by another mechanism that implements the same behavior and where truth-revealing is in equilibrium.

Multi-unit AuctionMulti-unit Auction

Auctions with multiple Auctions with multiple indistinguishable units for saleindistinguishable units for sale

• ExamplesExamples– IBM stocksIBM stocks– Barrels of oilBarrels of oil– Pork belliesPork bellies– Trans-Atlantic backbone bandwidth from Trans-Atlantic backbone bandwidth from

NYC to ParisNYC to Paris– ……

Setting for sealed bid Setting for sealed bid auctionsauctions

• Each bidder sends a “bid vector” Each bidder sends a “bid vector” indicating how much she/he is willing indicating how much she/he is willing to pay for each additional unitto pay for each additional unit

Can be understood as a demand Can be understood as a demand functionfunction

Number of units

Value of the bid

Pricing rulesPricing rules

Auctioning multiple indistinguishable Auctioning multiple indistinguishable units of an itemunits of an item

• The discriminatory (or “pay your The discriminatory (or “pay your bid”) auctionbid”) auction

• The uniform price auctionThe uniform price auction

• The Vickrey auctionThe Vickrey auction

Discriminatory auctionDiscriminatory auction

• Each bidder pays an amount equal to Each bidder pays an amount equal to the sum of his bids that are among the sum of his bids that are among the K highest of the N*K bids the K highest of the N*K bids submitted.submitted.

Uniform-price AuctionUniform-price Auction

• Any price between the highest Any price between the highest loosing bid and the lowest winning loosing bid and the lowest winning bid is possiblebid is possible

can choose the highest losing bidcan choose the highest losing bid

Vickrey AuctionVickrey Auction

• Basic principle is the same as the Basic principle is the same as the Vickrey-Clarke-Groves mechanism Vickrey-Clarke-Groves mechanism (see Mechanism Design)(see Mechanism Design)

• A bidder who wins k units pays the k A bidder who wins k units pays the k highest losing bids of the other highest losing bids of the other biddersbidders

• For bidder i to win the kFor bidder i to win the kthth unit, i’s k unit, i’s kth th

highest bid must defeat the khighest bid must defeat the kth th lowest lowest competing bidcompeting bid

Some Open AuctionsSome Open Auctions

• Dutch AuctionsDutch Auctions

• English AuctionsEnglish Auctions

• Ausubel AuctionsAusubel Auctions

Multi-item auctionsMulti-item auctions

multiple distinguishable multiple distinguishable items for saleitems for sale

Bundle bidding scenarioBundle bidding scenario

Bundle bidding scenarioBundle bidding scenario

Bundle bidding scenarioBundle bidding scenario

(console, television, cd player $1000)

Bundle bidding scenarioBundle bidding scenario

(television, music system, computer, $1600)

Bundle bidding scenarioBundle bidding scenario

(cd player, console, music system $400)

Bundle bidding scenarioBundle bidding scenario

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

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

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

Bundle bidding scenarioBundle bidding scenario

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

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

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

Bundle bidding scenarioBundle bidding scenario

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

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

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

Bundle bidding scenarioBundle bidding scenario

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

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

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

Multiple-item auctionsMultiple-item auctions

• Auction of multiple, distinguishable itemsAuction of multiple, distinguishable items

• Bidders have preferences over item Bidders have preferences over item combinations combinations

• Combinatorial auctionsCombinatorial auctions– Bids can be submitted over item bundles Bids can be submitted over item bundles – Winner selection: combinatorial optimizationWinner selection: combinatorial optimization

•NP-completeNP-complete

SourceSource

• Vijay Krishna: Auction Theory (Academic Vijay Krishna: Auction Theory (Academic Press)Press)

• Paul Klemperer: Auction Theory: A guide to Paul Klemperer: Auction Theory: A guide to the literature (Journal of Economics Survey)the literature (Journal of Economics Survey)

• Elmar Wolfstetter: Auctions An IntroductionElmar Wolfstetter: Auctions An Introduction

• Tuomas Sandholm COURSE: CS 15-892 Tuomas Sandholm COURSE: CS 15-892 Foundations of Electronic Marketplaces Foundations of Electronic Marketplaces (CMU)(CMU)