week 12 1

COS 444 Internet Auctions:

Theory and Practice

Spring 2009

Ken Steiglitz ken@cs.princeton.edu

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Bidder collusion

• Of course, in general, bidders want to reduce competition, the seller wants to increase competition

• We’ve seen examples on eBay of hypothetical implicit bidder collusion (Rasmusen 2006), and likely seller shill bidding

• Collusive bidding may be easier in multi-item auctions• P. Cramton

& J.A. Schwartz, “Collusive bidding in the FCC Spectrum Auctions,” J. Regulatory Economics, 1999, describe (highly) probable collusion in simultaneous ascending price auctions for FCC licenses (analogous to English for multiple items).

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• Code bids: use the trailing digits of the bid (often multi-millions of dollars) “to tell other bidders on which licenses to bid or not bid. … Oftentimes, a bidder (the sender) would use these code bids as retaliation against another bidder (the receiver) who was bidding on a license desired by the sender. The sender would raise the price on some market the receiver wanted, and use the trailing digits to tell the receiver on which license to cease bidding. ”

Cramton & Schwartz 1999

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• Here is their disclaimer:

“Disclaimer: For this analysis, we show that several bidders apparently used signaling to coordinate on

license allocations. This apparent signaling may be coincidental. The claims we make concerning a

bidders signaling are based on circumstantial evidence, and though we may attach some meaning to help

explain certain patterns of bidding, this meaning should be taken as our hypothesis only. We make no

claims concerning the actual intent of the bidders.”

Cramton & Schwartz 1999

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From

a first-hand accountof more explicit collusion

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“Marks & Co. were kings of the book ring. They were one of the five leading firms of antiquarian booksellers who never bid against each other in the auction rooms. One member of the ring would be allowed to buy a book for a nominal sum, say £100. As soon as the auction was over* the five conspirators would hurry to their nearest safe-house – usually a Lyons tea shop – and conduct a private auction. If one of them bought the book for £500, the £400 profit would be divided in cash amongst the other four†. This process was called a ‘knock-out’, and Frank Doel once blew an entire operation.

Between Silk and Cyanide, Leo Marks, Harper Collins, London, 1998.

†nota bene* a post-auction knock-out

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“A famous heart specialist names Evan Bedford instructed him to bid up to £300 for an edition of Harvey’s De Motu Cordis, the earliest printed book on the circulation of the blood, which was coming up for auction at Hodgson’s. Too busy with his own Hartley Street salesroom to attend the auction himself, he telephoned Frank at home late at night demanding to know why the book had been sold to another dealer for £200 when he’d authorized Frank to bid three. Frank confided that it had been sold in the knock-out for £600. The irate physician immediately undertook to have the whole question of the book ring raised in the House of Commons, which caused cardiac arrest amongst its five participants.

Between Silk and Cyanide, Leo Marks, Harper Collins, London, 1998.

week 12 9

• He describes post-auction knockouts, “…the ring holds a private sale to liquidate [the goods] and divide them among ring members.” (p. 180). Notice that the knock-out is an example where the utility of a buyer includes some utility of the seller---since each prospective buyer has a stake in the seller’s revenue.

• Organizing a ring is often a very complex operation.• Mentions that in sale of timber rights by U.S. gov’t.

collusion is common; these sales have strong common-value features, similar to spectrum auctions.

• Mentions Australian wool trade---the most complex buyer collusion known to Cassady. (p. 187) One buyer belonged to thirteen two-member and thirteen three-member rings.

See Cassady 1967 for lots of details about real ring operations

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Bidder rings (Graham & Marshall 1987)

Stylized facts:1) They exist and are stable2) They eliminate competition among ring

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

3) Benefits shared by ring members4) Have open membership5) Auctioneer responds strategically6) Try to hide their existence

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Graham & Marshall’s theoretical model: Second-price pre- auction knockout (PAKT)

• IPV, risk neutral

• Value distributions F, common knowledge

• Identity of winner & price paid common knowledge

• Membership of ring known only to ring members

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Pre-auction knock-out (PAKT):

1) Appoint ring center, who pays P to each ring member, P to be determined below

2) Each ring member submits a sealed bid to the ring center

3) Winner is advised to submit her winning bid at main auction; other ring members submit only meaningless bids

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

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If sub-winner wins main auction, she pays:• Main auctioneer P* = SP at main auction• Ring center δ = max{ P̃ − P* , 0 }, where P̃

= SP in PAKTThus: If the sub-winner wins main auction, she

pays in total the SP among all bids (which would have happened without the ring)

and… the profit is passed along to the ring center

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The quantity δ is the amount “stolen” from the main auctioneer, the “booty”

The ring center receives and distributes

E[δ | sub-winner wins main auction]

so his budget is balanced, in expectation

Each ring member receives

P = E[δ | sub-winner wins main auction]/K

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Bidder rings

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

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

2) Voluntary participation is advantageous (individually rational)

3) Efficient (buyer with highest value gets item)

In fact, the whole thing is equivalent to a Vickrey

auction

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Bidder rings

Main auctioneer responds strategically by increasing reserves or shill-bidding

Graham& Marshall also prove that

1) Optimal main reserve is an increasing function of ring size K

2) Expected surplus of ring member is a decreasing function of reserve prices

3) Expected surplus of ring member is an increasing function of ring size K

… so best to be secretive

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Bilateral trading mechanisms[Myerson & Satterthwaite 83]

An impossibility result:

The following desirable characteristics of bilateral trade (not an auction):

1) efficient

2) incentive-compatible

3) individually rational

cannot all be achieved simultaneously!

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Bilateral trading mechanisms

The setup:• one seller, with private value v1 ,

distributed with density f1 > 0 on [a1 , b1 ]• one buyer, with private value v2 ,

distributed with density f2 > 0 on [a2 , b2 ]• risk neutral

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

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Bilateral trading mechanisms

• Outline of proof: We use a direct mechanism (p, x), where

p (v1 , v2 ) = prob. of transfer to buyer

x (v1 , v2 ) = expected payment to seller

seller]topayment[E)(),()( 2222111

2

2

dttftvxvxb

a

buyertosellingprob.)(),()( 2222111

2

2

dttftvpvpb

a

seller]ofE[profit)()()( 1111111 vpvvxvU

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Bilateral trading mechanisms

• Similarly and symmetrically:

]buyer frompayment[E)(),()( 1112122

1

1

dttfvtxvxb

a

buyertotransferprob.)(),()( 1112122

1

1

dttfvtpvpb

a

buyer]ofE[profit)()()( 2222222 vxvpvvU

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Bilateral trading mechanisms

• Incentive-compatible means

• Individually rational means

• Ex post efficient means

],[ˆ,)ˆ()ˆ()( 11111111111 bavvvpvvxvU ],[ˆ,)ˆ()ˆ()( 2222222222 bavvvxvpvvU

],[0)( 11111 bavvU ],[0)( 22222 bavvU

else0

if1),( 21

21

vvvvp

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Bilateral trading mechanisms

• Incentive-compatible means

• Individually rational means

• Ex post efficient means

],[ˆ,)ˆ()ˆ()( 11111111111 bavvvpvvxvU ],[ˆ,)ˆ()ˆ()( 2222222222 bavvvxvpvvU

],[0)( 11111 bavvU ],[0)( 22222 bavvU

else0

if1),( 21

21

vvvvp

no incentiveto lie about v’s

participation does not entail expected loss

object is sold iff buyer values it more highly

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Bilateral trading mechanisms

Main resultMain result:: If

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

Furthermore,

is the smallest lump-sum subsidy to achieve efficiency.

],[],[ 2211 baba

dttFtFb

a

)()](1[1

2

12

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Proof steps• Part 1:Part 1: incentive-compatible and individually

rational implies

• Part 2:Part 2: ex post efficient implies

… contradiction!

0)()( 2211 aUbU

min. E[profit] of seller + min. E[profit] of buyer

1

2

0)()](1[)()( 122211

b

adttFtFaUbU

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Bilateral trading mechanisms

Example:Example: Shows

f i > 0 is necessary:

discrete probs.

4/14/14

4/14/11

30

seller

buyer

Only profitable transaction is 13

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Bilateral trading mechanisms

Claim:Claim: “sell at price 2 if both are willing, else no trade” is incentive compatible, individually rational, and efficient.

• Incentive compatible: truthful reporting is an equilibrium (check)

• Individually rational: E[profit] >0

• Efficient: trade occurs only when v1<v2

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Auctions vs. Negotiations [ Bulow & Klemperer 96 ]

Simple example: IPV, uniform

Case 1)Case 1) Optimal auction = optimal mechanism with one buyer. Optimal entry value v* = 1/2; revenue = 1/4

Case 2)Case 2) Two buyers, no reserve; revenue = 1/3 > 1/4

One more buyer is worth more than setting reserve optimally!

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Auctions vs. Negotiations

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 (IPV) auction (and hence optimal mechanism) with n bidders

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Auctions vs. Negotiations

n

vvdFvM )()(

1

*

Revenue with optimal reserve, n bidders:

1

0

1)()( nvdFvM

Revenue with no reserve, n+1 bidders:

1

2/14/12 )1( dxx

3/12 2

0

1

)1( dxx

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Auctions vs. Negotiations

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

0 nn vM,...,vMvdFvM

Facts:

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

1

*nv

n vMvMvdFvM

0)](E[ vM

Why?

Why?

Why?

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Auctions vs. Negotiations

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

0 nn vM,...,vMvdFvM

Facts:

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

1

*nv

n vMvMvdFvM

0)](E[ vM

distribution fctn. of max. of n+1 draws

distribution fctn. of max. of n draws, integrate only where M ≥ 0

expected revenue with only one buyer!

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Auctions vs. Negotiations

Now compare revenue in a no-reserve auction with n+1 bidders, and an optimal auction with n bidders:

0

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

)}](),(),...,([max{

],E[optimal]1,reserveE[no

1

11

1

0

11

*

n

nn

v

nn

vMvME

vMvMvME

dFMdFM

nn

□

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• Asset markets are volatile! Common wisdom attributes to irrational behavior, market imperfections, market failure

• This paper offers a model of a simple situation in which completely rational behavior leads to “frenzies” and “crashes”

• Uses IPV auction theory and the RET in a dynamic setting

• An elegant economic idealization makes the point

“Rational frenzies and crashes,” J. Bulow & P. Klemperer, J. Political Economy, 102, pp. 1-23, 1994.

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• K identical units for sale, one seller, K+L risk-neutral potential buyers, each wanting to buy a single unit

• IPV’s, drawn from F(v) on [0, vmax]

• Buyer derives surplus (v – p) from a purchase at price p

The BK 94 game

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The simple motivating idea… WTP

• Suppose you’re in a simple single-item Vickrey auction with IPV’s that are uniform on [0,1], and you have value v.

• You are made a take-it-or-leave-it offer at price p. Should you accept it?

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The simple motivating idea… WTP

• Suppose you’re in a simple single-item Vickrey auction with IPV’s that are uniform on [0,1], and you have value v.

• You are made a take-it-or-leave-it offer at price p. Should you accept it?

• Well, if and only if p ≤ E(second-price | v wins)

= “Willingness To Pay”

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1) Seller begins offering units at max price vmax and lowers it until a purchase occurs, at price p

2) (NEW SALE) When a purchase occurs, every buyer gets an invitation to purchase 1 unit at price p.

Either: (a) (FRENZY) all goods are sold at p game ends (b) (FRENZY) not all goods are sold at p, no one is left

willing to buy at that price then go to 1) and continue lowering price until another NEW SALE takes place

(c) (EXCESS DEMAND) More buyers want to buy at price p than there are units remaining. Then if there are k+l bidders offering to buy the remaining k units, go to 1) and restart the game with these k+l bidders competing for the remaining k units. All previous sales remain valid.

Dynamics of BK 94 game

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Solution to game

• With k units and (k+l) bidders remaining: a symmetric equilibrium strategy is: offer to buy at price p if and only if p ≤ ω(v), where

pay" towillingess"

])1(|values)(ofouthighest)1[(E

]win|auctionpricerejected-firstainprice[E)(

vstklkstk

v

Note that this reduces to Vickrey with one item and one buyer

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• This follows from a straightforward generalization of the RET: any mechanism selling K identical items to the bidders with the K highest values in a unit-demand auction has in equilibrium the same expected payment conditional on winning, namely ω(v) (see B&K 94).

• The interesting dynamics are a consequence of the shape of ω(v). When k goes down, ω(v) goes up, and this changes the next threshold drastically and a bunch of buyers may jump in all at once!

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Why does ω(v) flatten out dramatically?

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K=50 , L=100

Simulation

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Term papers due 5pm Tuesday May 12 (Dean’s Date)

Email me for office hours re term papers

Neshmet Bark of Osiris, on a bronze drachm of M. Aurelius, Alexandria, Egypt. E. 2160,174/5 AD.

It’s been fun!