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week 121 COS 444 Internet Auctions: Theory and Practice Spring 2009 Ken Steiglitz ken@cs.princeton.edu

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week 122 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).P. Cramton & J.A. Schwartz, “Collusive bidding in the FCC Spectrum Auctions,”

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week 123 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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5 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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week 126 From a first-hand account of more explicit collusion

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week 127 “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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week 128 “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.

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week 129 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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week 1210 Bidder rings (Graham & Marshall 1987)(Graham & Marshall 1987) Stylized facts: 1)They exist and are stable 2)They eliminate competition among ring members; yet ensure ring member with highest value is not undercut 3)Benefits shared by ring members 4)Have open membership 5)Auctioneer responds strategically 6)Try to hide their existence

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week 1211 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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week 1212 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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week 1213 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 PAKT Thus: 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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week 1214

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week 1215 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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week 1216 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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week 1217 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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week 1218 Bilateral trading mechanisms [Myerson & Satterthwaite 83]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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week 1219 Bilateral trading mechanisms The setup: one seller, with private value v 1, distributed with density f 1 > 0 on [a 1, b 1 ] one buyer, with private value v 2, distributed with density f 2 > 0 on [a 2, b 2 ] risk neutral … Notice: not an auction in Riley & Samuelson’s class!

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week 1220 Bilateral trading mechanisms Outline of proof: We use a direct mechanism (p, x), where p (v 1, v 2 ) = prob. of transfer to buyer x (v 1, v 2 ) = expected payment to seller

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week 1221 Bilateral trading mechanisms Similarly and symmetrically:

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week 1222 Bilateral trading mechanisms Incentive-compatible means Individually rational means Ex post efficient means

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week 1223 Bilateral trading mechanisms Incentive-compatible means Individually rational means Ex post efficient means no incentive to lie about v’s participation does not entail expected loss object is sold iff buyer values it more highly

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week 1224 Bilateral trading mechanisms Main result: Main 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.

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week 1225 Proof steps Part 1:Part 1: incentive-compatible and individually rational implies Part 2:Part 2: ex post efficient implies … contradiction! min. E[profit] of seller + min. E[profit] of buyer

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week 1226 Bilateral trading mechanisms Example: Example: Shows f i > 0 is necessary: discrete probs. seller buyer Only profitable transaction is 1 3

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week 1227 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 v 1

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week 1228 Auctions vs. Negotiations [ Bulow & Klemperer 96 ]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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week 1229 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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week 1230 Auctions vs. Negotiations Revenue with optimal reserve, n bidders: Revenue with no reserve, n+1 bidders:

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week 1231 Auctions vs. Negotiations Facts: Why?

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week 1232 Auctions vs. Negotiations Facts: 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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week 1233 Auctions vs. Negotiations Now compare revenue in a no-reserve auction with n+1 bidders, and an optimal auction with n bidders: □

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week 1234 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.Rational frenzies and crashes

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week 1235 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, v max ] Buyer derives surplus (v – p) from a purchase at price p The BK 94 game

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week 1236 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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week 1237 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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week 1238 1)Seller begins offering units at max price v max 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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week 1239 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 Note that this reduces to Vickrey with one item and one buyer

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week 1240 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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week 1241 Why does ω(v) flatten out dramatically?

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week 1242

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week 1243 K=50, L=100 Simulation

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week 1244 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!

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Week 9 1 COS 444 Internet Auctions: Theory and Practice Spring 2008 Ken Steiglitz

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