COS 444 Internet Auctions: Theory and Practice

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COS 444 COS 444 Internet Auctions:Internet Auctions:Theory and PracticeTheory and Practice

Spring 2008

Ken Steiglitz [email protected]

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Multi-unit demand Multi-unit demand auctionsauctions(Ausubel & Cramton 98, Morgan 01)(Ausubel & Cramton 98, Morgan 01)• Examples: FCC spectrum, Treasury Examples: FCC spectrum, Treasury

debt securities, Eurosystem: debt securities, Eurosystem: multiple, multiple, identical unitsidentical units

• Issues: Pay-your-bid (discriminatory) Issues: Pay-your-bid (discriminatory) prices v. uniform-price; efficiency; prices v. uniform-price; efficiency; optimality of revenueoptimality of revenue

• The problem: conventional, The problem: conventional, uniform-uniform-priceprice auctions provide incentives for auctions provide incentives for demand-reductiondemand-reduction

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Multi-unit demand Multi-unit demand auctionsauctionsExample 1: (Morgan) Example 1: (Morgan) 2 units supply2 units supply Bidder 1:Bidder 1: capacity 2, values $10, $10 capacity 2, values $10, $10 Bidder 2:Bidder 2: capacity 1, value $8 capacity 1, value $8

Suppose bidders bid truthfully; rank bids:Suppose bidders bid truthfully; rank bids: $10 bidder 1$10 bidder 1 10 bidder 110 bidder 1 8 bidder 2 8 bidder 2 first rejected bid first rejected bidIf buyers pay this, surplus (1) = $4If buyers pay this, surplus (1) = $4 revenue = $16revenue = $16

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Multi-unit demand Multi-unit demand auctionsauctionsExample 1: But bidder 1 can do better!Example 1: But bidder 1 can do better! Bidder 1:Bidder 1: capacity 2, values $10, $10 capacity 2, values $10, $10 Bidder 2:Bidder 2: capacity 1, value $8 capacity 1, value $8

Suppose bidder 1 shades her demand:Suppose bidder 1 shades her demand: $10 bidder 1 for her first unit$10 bidder 1 for her first unit 8 bidder 2 for first unit8 bidder 2 for first unit 0 bidder 1 for her 20 bidder 1 for her 2ndnd unit unit first rej. bid first rej. bidIf buyers pay this, surplus (1) = $10 If buyers pay this, surplus (1) = $10 surplus (2) = $8 surplus (2) = $8 inefficient!inefficient! revenue = $0!revenue = $0!

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Multi-unit demand Multi-unit demand auctionsauctionsThus, Thus,

uniform price uniform price demand reductiondemand reduction inefficiencyinefficiency

The natural generalization of the Vickrey The natural generalization of the Vickrey auction (winners pay first rejected bid) is auction (winners pay first rejected bid) is not incentive compatible and not efficientnot incentive compatible and not efficient

Lots of economists got this wrong!Lots of economists got this wrong!

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Multi-unit demand Multi-unit demand auctionsauctionsAusubel & Cramton prove, in a simplified Ausubel & Cramton prove, in a simplified

model, that this example is not model, that this example is not pathological:pathological:

Proposition:Proposition: There is no efficient equilibrium There is no efficient equilibrium strategy in a uniform-price, multi-unit strategy in a uniform-price, multi-unit demand auction. demand auction.

The appropriate generalization of the Vickrey The appropriate generalization of the Vickrey auction is the Vickrey-Clark-Groves (VCG) auction is the Vickrey-Clark-Groves (VCG) mechanism…mechanism…

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The VCG auction for multi-unit The VCG auction for multi-unit demanddemand

Return to example 1: 2 units supplyReturn to example 1: 2 units supply Bidder 1:Bidder 1: capacity 2, values $10, $10 capacity 2, values $10, $10 Bidder 2:Bidder 2: capacity 1, value $8 capacity 1, value $8

Suppose bidders bid truthfully, and order Suppose bidders bid truthfully, and order bids:bids:

$10 bidder 1$10 bidder 1 10 bidder 1 10 bidder 1 8 bidder 28 bidder 2 Award supply to the highest biddersAward supply to the highest bidders … … How much does each bidder pay?How much does each bidder pay?

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The VCG auction for multi-unit The VCG auction for multi-unit demanddemandDefine:Define: social welfaresocial welfare = = W W ( ( v v ) = total value received ) = total value received

by agents, where by agents, where vv is the vector of values is the vector of values

Then the Then the VCG paymentVCG payment of i is of i is WW-i-i ( 0, ( 0, xx-i-i ) − ) − WW-i-i ( ( x x ))= welfare to = welfare to othersothers when when ii bids 0, bids 0, minusminus that that

when when i i bids truthfullybids truthfully= sum of = sum of kkii rejected bids (if bidder rejected bids (if bidder ii gets gets kkii

items)items)

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The VCG auction for multi-unit The VCG auction for multi-unit demanddemandExample 1: Example 1: 2 units supply2 units supply Bidder 1:Bidder 1: capacity 2, values $10, $10 capacity 2, values $10, $10 Bidder 2:Bidder 2: capacity 1, value $8 capacity 1, value $8

If bidder 1 bids 0, If bidder 1 bids 0, welfarewelfare = $8, = $8,and is $0 when 1 bids truthfully…and is $0 when 1 bids truthfully… 1 pays $8 for the 2 items1 pays $8 for the 2 items

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The VCG auction for multi-unit The VCG auction for multi-unit demanddemandExample 2: Example 2: 3 units supply3 units supply Bidder 1:Bidder 1: capacity 2, values $10, $10 capacity 2, values $10, $10 Bidder 2:Bidder 2: capacity 1, value $8 capacity 1, value $8 Bidder 3:Bidder 3: capacity 1, value $6 capacity 1, value $6

$10 bidder 1$10 bidder 1 10 bidder 1 10 bidder 1 bidder 1 gets 2 items bidder 1 gets 2 items 8 bidder 2 8 bidder 2 bidder 2 gets 1 item bidder 2 gets 1 item 6 bidder 36 bidder 3

Welfare when 1 bids 0 = $14 Welfare when 1 bids 0 = $14 Welfare when 1 bids truthfully = $8 Welfare when 1 bids truthfully = $8 1 pays $6 for the 2 items1 pays $6 for the 2 items

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The VCG auction for multi-unit The VCG auction for multi-unit demanddemand

Example 2, con’t Example 2, con’t 3 units supply3 units supply Bidder 1:Bidder 1: capacity 2, values $10, $10 capacity 2, values $10, $10 Bidder 2:Bidder 2: capacity 1, value $8 capacity 1, value $8 Bidder 3:Bidder 3: capacity 1, value $6 capacity 1, value $6

$10 bidder 1 $10 bidder 1 bidder 1 gets 2 items bidder 1 gets 2 items 8 bidder 2 8 bidder 2 bidder 2 gets 1 item bidder 2 gets 1 item 6 bidder 36 bidder 3

Welfare when 2 bids 0 = $26 Welfare when 2 bids 0 = $26 Welfare when 2 bids truthfully = $20 Welfare when 2 bids truthfully = $20 2 pays $6 for the 1 item2 pays $6 for the 1 item(notice that revenue = $12 < $18 =3x$6 in uniform-(notice that revenue = $12 < $18 =3x$6 in uniform-

price case, so not optimal)price case, so not optimal)

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VCG mechanismsVCG mechanisms (Krishna 02)(Krishna 02)

VCG mechanisms areVCG mechanisms are• efficientefficient• incentive-compatible incentive-compatible (truthful is weakly dominant)(truthful is weakly dominant)• individually rationalindividually rational• max-revenue among all such max-revenue among all such

mechanismsmechanisms

… … but not optimal revenue in general,but not optimal revenue in general,and prices are discriminatory, “murky”and prices are discriminatory, “murky”

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Bilateral trading mechanismsBilateral trading mechanisms [Myerson & Satterthwaite 83][Myerson & Satterthwaite 83]

An impossibility result:An impossibility result:The following desirable characteristics The following desirable characteristics

of bilateral trade (not an auction):of bilateral trade (not an auction):1)1) efficientefficient2)2) incentive-compatibleincentive-compatible3)3) individually rationalindividually rationalCannot all be achieved simultaneously!Cannot all be achieved simultaneously!

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Bilateral trading mechanismsBilateral trading mechanisms

The setup:The setup:• one seller, with private value one seller, with private value vv11 , ,

distributed with density distributed with density ff1 1 > 0 on [> 0 on [aa11 , , bb1 1 ]]• one buyer, with private value one buyer, with private value vv22 , ,

distributed with density distributed with density ff22 > 0 on [ > 0 on [aa22 , , bb2 2 ]]• risk neutralrisk neutral

… … Notice: not an auction in Riley & Notice: not an auction in Riley & Samuelson’s class!Samuelson’s class!

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Bilateral trading mechanismsBilateral trading mechanismsOutline of proof: We use a direct Outline of proof: We use a direct

mechanism (mechanism (p, x p, x ): ): where where p p ((vv1 1 , v, v2 2 ) = prob. of transfer ) = prob. of transfer

1122 x x ((vv1 1 , v, v2 2 ) = expected payment ) = expected payment

1122

]1sellerrev[E)(),()( 22221112

2

dttftvxvxb

a

2to1sellingprob.)(),()( 22221112

2

dttftvpvpb

a

1]E[profit)()()( 1111111 vpvvxvU

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Bilateral trading mechanismsBilateral trading mechanismsMain resultMain result: If: If

then no incentive-compatible individually rational then no incentive-compatible individually rational trading mechanism can be (ex post) efficient.trading mechanism can be (ex post) efficient.

Furthermore,Furthermore,

is the smallest lump-sum subsidy to achieve is the smallest lump-sum subsidy to achieve efficiency.efficiency.

],[],[ 2211 baba

dttFtFb

a

)()](1[1

2

12

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Bilateral trading mechanismsBilateral trading mechanismsExamplesExamples1.1. ff i i > 0 is necessary: discrete > 0 is necessary: discrete

probs.probs.2.2. Subsidy for efficiency: Subsidy for efficiency: vv1 1 and and vv22

both uniform on [0,1]both uniform on [0,1]

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Auctions vs. NegotiationsAuctions vs. Negotiations (Bulow & Klemperer 96)(Bulow & Klemperer 96)Simple example: IPV, uniform

Case 1) Optimal auction = optimal mechanism with one buyer. Optimal entry value v* = 0.5; revenue = 1/4

Case 2) Case 2) Two buyers, no reserve; revenue Two buyers, no reserve; revenue = 1/3 > ¼= 1/3 > ¼

One more buyer is worth more than One more buyer is worth more than setting reserve optimally!setting reserve optimally!

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Auctions vs. Negotiations, con’tAuctions vs. Negotiations, con’t

Bulow & Klemperer 96 generalize to any F,any number of bidders…

A no-reserve auction with n +1 bidders is more profitable than an optimal auction(and hence optimal mechanism) with n bidders

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Auctions vs. Negotiations, con’tAuctions vs. Negotiations, con’t

n

vvdFvM )()(

1

*

Optimal reserve, n bidders:

No reserve, n+1 bidders

1

0

1)()( nvdFvM

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Auctions vs. Negotiations, con’tAuctions vs. Negotiations, con’t

)}]()([max{E)()( 1111

0 nn v,...,MvMvdFvM

Facts:

… QED

}]0),(),...,([max{E )()( 1

1

*nv

n vMvMvdFvM

0)]([E vM

0)](E[ vM

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Bidder rings Bidder rings (Graham & Marshall (Graham & Marshall 87)87)Stylized factsStylized facts1)1) They exist and are stableThey exist and are stable2)2) They eliminate competition among They eliminate competition among

ring members; yet ensure ring ring members; yet ensure ring member with highest value is not member with highest value is not undercutundercut

3)3) Benefits shared by ring membersBenefits shared by ring members4)4) Have open membershipHave open membership5)5) Auctioneer responds strategicallyAuctioneer responds strategically6)6) Try to hide their existenceTry to hide their existence

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Bidder ringsBidder ringsGraham & Marshall’s model: Second-Graham & Marshall’s model: Second-

price pre-auction knockout (PAKT)price pre-auction knockout (PAKT)i.i. IPV, risk neutralIPV, risk neutralii.ii. Value distributions F, common Value distributions F, common

knowledgeknowledgeiii.iii. Identity of winner & price paid Identity of winner & price paid

common knowledgecommon knowledgeiv.iv. Membership of ring known only to Membership of ring known only to

ring membersring members

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Bidder ringsBidder ringsPre-auction knock-out (PAKT):Pre-auction knock-out (PAKT):1)1) Appoint Appoint ring centerring center, who pays , who pays P P to each to each

ring member, ring member, P P to be determined below to be determined below2)2) Each ring member submits a sealed bid Each ring member submits a sealed bid

to the ring centerto the ring center3)3) Winner is advised to submit her winning Winner is advised to submit her winning

bid at main auction; other ring members bid at main auction; other ring members submit only meaningless bidssubmit only meaningless bids

4)4) If the winner at the sub-auction (If the winner at the sub-auction (sub-sub-winnerwinner) also wins main auction, she pays:) also wins main auction, she pays:

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Bidder ringsBidder ringsIf sub-winner wins main auction, she If sub-winner wins main auction, she

pays:pays:o Main auctioneer Main auctioneer P*P* = SP at main = SP at main

auctionauctiono Ring center Ring center δδ = = max{ max{ PP ̃ − ̃ − P*P* , ,

0 }, where 0 }, where PP ̃ = SP in PAKT̃ = SP in PAKTThus: If the sub-winner wins main Thus: If the sub-winner wins main

auction, she pays auction, she pays SP among all SP among all bidsbids

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Bidder ringsBidder ringsThe quantity The quantity δδ is the amount “stolen” is the amount “stolen”

from the main auctioneer, the “booty”from the main auctioneer, the “booty”

The ring center receives and distributes The ring center receives and distributes E[E[δδ | sub-winner wins main auction] | sub-winner wins main auction] so his budget is balancedso his budget is balanced

Each ring member receives Each ring member receives P = E[P = E[δδ | sub-winner wins main auction]/ | sub-winner wins main auction]/KK

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Bidder ringsBidder ringsGraham and Marshall prove:Graham and Marshall prove:1)1) Truthful bidding in the PAKT, and following Truthful bidding in the PAKT, and following

the recommendation of the ring center is the recommendation of the ring center is SBNE & weakly dominant strategy SBNE & weakly dominant strategy (incentive compatible)(incentive compatible)

2)2) Voluntary participation is advantageous Voluntary participation is advantageous (individually rational)(individually rational)

3)3) Efficient Efficient (buyer with highest value gets (buyer with highest value gets item)item)

In fact, the whole thing is equivalent to a Vickrey auctionIn fact, the whole thing is equivalent to a Vickrey auction

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Bidder ringsBidder ringsMain auctioneer responds strategically by Main auctioneer responds strategically by

increasing reserves or shill-biddingincreasing reserves or shill-biddingGraham& Marshall also prove thatGraham& Marshall also prove that1.1. Optimal main reserve is an increasing Optimal main reserve is an increasing

function of ring size function of ring size KK2.2. Expected surplus of ring member is a Expected surplus of ring member is a

decreasing function of reserve pricesdecreasing function of reserve prices3.3. Expected surplus of ring member is an Expected surplus of ring member is an

increasing function of ring size increasing function of ring size KKSo best to be secretiveSo best to be secretive

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Term papers due 5pm Term papers due 5pm Tuesday May 13 (Dean’s Tuesday May 13 (Dean’s Date)Date) Email me for office hours re term Email me for office hours re term

paperspapers