COS 444 Internet Auctions: Theory and Practice

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COS 444 Internet AuctionsTheory and Practice
Spring 2009 Ken Steiglitz
ken_at_cs.princeton.edu
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Bidder collusion

• Of course, in general, bidders want to reduce
  competition, the seller wants to increase
  competition
• Weve 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 account of 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 Harveys De Motu Cordis, the earliest
  printed book on the circulation of the blood,
  which was coming up for auction at Hodgsons. 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 hed 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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See Cassady 1967 for lots of details about real
ring operations

• 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 sellers revenue.
• Organizing a ring is often a very complex
  operation.
• Mentions that in sale of timber rights by U.S.
  govt. 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.

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Bidder rings (Graham Marshall 1987)

• Stylized facts
• They exist and are stable
• They eliminate competition among ring members
  yet ensure ring member with highest value is not
  undercut
• Benefits shared by ring members
• Have open membership
• Auctioneer responds strategically
• Try to hide their existence

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• Graham Marshalls 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)
• Appoint ring center, who pays P to each ring
  member, P to be determined below
• Each ring member submits a sealed bid to the ring
  center
• Winner is advised to submit her winning bid at
  main auction other ring members submit only
  meaningless bids
• 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 d max P - P , 0 , where P
  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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• The quantity d is the amount stolen from the
  main auctioneer, the booty
• The ring center receives and distributes
• Ed sub-winner wins main auction
• ? so his budget is balanced, in expectation
• Each ring member receives
• P Ed sub-winner wins main auction/K

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Bidder rings

• Graham and Marshall prove
• Truthful bidding in the PAKT, and following the
  recommendation of the ring center is a SBNE
  weakly dominant strategy (incentive compatible)
• Voluntary participation is advantageous
  (individually rational)
• 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
• Optimal main reserve is an increasing function of
  ring size K
• Expected surplus of ring member is a decreasing
  function of reserve prices
• Expected surplus of ring member is an increasing
  function of ring size K
• so
  best to be secretive

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Bilateral trading mechanismsMyerson
Satterthwaite 83

• An impossibility result
• The following desirable characteristics of
  bilateral trade (not an auction)
• efficient
• incentive-compatible
• individually rational
• cannot all be achieved simultaneously!

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Bilateral trading mechanisms

• The setup
• one seller, with private value v?1 , distributed
  with density f1 gt 0 on a1 , b1
• one buyer, with private value v?2 , distributed
  with density f2 gt 0 on a2 , b2
• risk neutral
• Notice not an auction in Riley
  Samuelsons 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

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Bilateral trading mechanisms

• Similarly and symmetrically

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Bilateral trading mechanisms

• Incentive-compatible means
• Individually rational means
• Ex post efficient means

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Bilateral trading mechanisms

• Incentive-compatible means
• Individually rational means
• Ex post efficient means

no incentive to lie about vs
participation does not entail expected loss
object is sold iff buyer values it more highly
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Bilateral trading mechanisms

• 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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Proof steps

• Part 1 incentive-compatible and individually
  rational implies
• Part 2 ex post efficient implies
• 
  contradiction!

min. Eprofit of seller min. Eprofit of buyer
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Bilateral trading mechanisms

• Example Shows
• f i gt 0 is necessary
• discrete probs.

buyer
seller
Only profitable transaction is 1?3
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Bilateral trading mechanisms

• 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 Eprofit gt0
• Efficient trade occurs only when v1ltv2

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Auctions vs. Negotiations Bulow Klemperer 96

Simple example IPV, uniform Case 1) Optimal
auction optimal mechanism with one buyer.
Optimal entry value v 1/2 revenue 1/4 Case
2) Two buyers, no reserve revenue 1/3 gt 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
Revenue with optimal reserve, n bidders
Revenue with no reserve, n1 bidders
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Auctions vs. Negotiations
Facts
Why?
Why?
Why?
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Auctions vs. Negotiations
Facts
distribution fctn. of max. of n draws, integrate
only where M 0
distribution fctn. of max. of n1 draws
expected revenue with only one buyer!
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Auctions vs. Negotiations
Now compare revenue in a no-reserve auction with
n1 bidders, and an optimal auction with n
bidders
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Rational frenzies and crashes, J. Bulow P.
Klemperer, J. Political Economy, 102, pp. 1-23,
1994.

• 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

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The BK 94 game

• K identical units for sale, one seller, KL
  risk-neutral potential buyers, each wanting to
  buy a single unit
• IPVs, drawn from F(v) on 0, vmax
• Buyer derives surplus (v p) from a purchase at
  price p

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The simple motivating idea WTP

• Suppose youre in a simple single-item Vickrey
  auction with IPVs 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 youre in a simple single-item Vickrey
  auction with IPVs 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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Dynamics of BK 94 game

• Seller begins offering units at max price vmax
  and lowers it until a purchase occurs, at price p
• (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 kl bidders offering to buy the
  remaining k units, go to 1) and restart the game
  with these kl bidders competing for the
  remaining k units. All previous sales remain
  valid.

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Solution to game

• With k units and (kl) bidders remaining a
  symmetric equilibrium strategy is offer to buy
  at price p if and only if p ?(v), where

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 ?(v) (see BK 94).
• The interesting dynamics are a consequence of the
  shape of ?(v). When k goes down, ?(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 ?(v) flatten out dramatically?
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Simulation
K50 , L100
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Term papers due 5pm Tuesday May 12 (Deans Date)

• ? Email me for office hours re term papers

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