COS 444 Internet Auctions: Theory and Practice

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COS 444 Internet AuctionsTheory and Practice
Spring 2008 Ken Steiglitz
ken_at_cs.princeton.edu
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Multi-unit demand auctions(Ausubel Cramton 98,
Morgan 01)

• Examples FCC spectrum, Treasury debt securities,
  Eurosystem multiple, identical units
• Issues Pay-your-bid (discriminatory) prices v.
  uniform-price efficiency optimality of revenue
• The problem conventional, uniform-price auctions
  provide incentives for demand-reduction

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Multi-unit demand auctions

• Example 1 (Morgan) 2 units supply
• Bidder 1 capacity 2, values 10, 10
• Bidder 2 capacity 1, value 8
• Suppose bidders bid truthfully rank bids
• 10 bidder 1
• 10 bidder 1
• 8 bidder 2 ? first rejected bid
• If buyers pay this, surplus (1) 4
• revenue 16

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Multi-unit demand auctions

• Example 1 But bidder 1 can do better!
• Bidder 1 capacity 2, values 10, 10
• Bidder 2 capacity 1, value 8
• Suppose bidder 1 shades her demand
• 10 bidder 1 for her first unit
• 8 bidder 2 for first unit
• 0 bidder 1 for her 2nd unit ? first rej.
  bid
• If buyers pay this, surplus (1) 10
• surplus (2) 8
  inefficient!
• revenue 0!

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Multi-unit demand auctions

• Thus,
• uniform price ?demand reduction? inefficiency
• The natural generalization of the Vickrey auction
  (winners pay first rejected bid) is not incentive
  compatible and not efficient
• Lots of economists got this wrong!

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Multi-unit demand auctions

• Ausubel Cramton prove, in a simplified model,
  that this example is not pathological
• Proposition There is no efficient equilibrium
  strategy in a uniform-price, multi-unit demand
  auction.
• The appropriate generalization of the Vickrey
  auction is the Vickrey-Clark-Groves (VCG)
  mechanism

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The VCG auction for multi-unit demand

• Return to example 1 2 units supply
• Bidder 1 capacity 2, values 10, 10
• Bidder 2 capacity 1, value 8
• Suppose bidders bid truthfully, and order bids
• 10 bidder 1
• 10 bidder 1
• 8 bidder 2
• Award supply to the highest bidders
• How much does each bidder pay?

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The VCG auction for multi-unit demand

• Define
• social welfare W ( v ) total value received
  by agents, where v is the vector of values
• Then the VCG payment of i is
• W-i ( 0, x-i ) - W-i ( x )
• welfare to others when i bids 0, minus that
  when i bids truthfully
• sum of ki rejected bids (if bidder i gets ki
  items)

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The VCG auction for multi-unit demand

• Example 1 2 units supply
• Bidder 1 capacity 2, values 10, 10
• Bidder 2 capacity 1, value 8
• If bidder 1 bids 0, welfare 8,
• and is 0 when 1 bids truthfully
• ? 1 pays 8 for the 2 items

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The VCG auction for multi-unit demand

• Example 2 3 units supply
• Bidder 1 capacity 2, values 10, 10
• Bidder 2 capacity 1, value 8
• Bidder 3 capacity 1, value 6
• 10 bidder 1
• 10 bidder 1 ? bidder 1 gets 2 items
• 8 bidder 2 ? bidder 2 gets 1 item
• 6 bidder 3
• Welfare when 1 bids 0 14
• Welfare when 1 bids truthfully 8
• ? 1 pays 6 for the 2
  items

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The VCG auction for multi-unit demand

• Example 2, cont
• 3 units supply
• Bidder 1 capacity 2, values 10, 10
• Bidder 2 capacity 1, value 8
• Bidder 3 capacity 1, value 6
• 10 bidder 1 ? bidder 1 gets 2 items
• 8 bidder 2 ? bidder 2 gets 1 item
• 6 bidder 3
• Welfare when 2 bids 0 26
• Welfare when 2 bids truthfully 20
• ? 2 pays 6 for
  the 1 item
• (notice that revenue 12 lt 18 3x6 in
  uniform-price case, so not optimal)

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VCG mechanisms (Krishna 02)

• VCG mechanisms are
• efficient
• incentive-compatible
• (truthful is weakly dominant)
• individually rational
• max-revenue among all such mechanisms
• but not optimal revenue in general,
• and prices are discriminatory, murky

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Bilateral trading mechanisms Myerson
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 1?2
• x (v1 , v2 ) expected payment 1?2

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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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Bilateral trading mechanisms

• Examples
• f i gt 0 is necessary discrete probs.
• Subsidy for efficiency v?1 and v?2 both uniform
  on 0,1

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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 0.5 revenue 1/4 Case
2) Two buyers, no reserve revenue 1/3 gt ¼ ?
One more buyer is worth more than setting reserve
optimally!
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Auctions vs. Negotiations, cont

• 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, cont
Optimal reserve, n bidders No reserve, n1
bidders
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Auctions vs. Negotiations, cont
Facts
QED
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Bidder rings (Graham Marshall 87)

• 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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Bidder rings

• Graham Marshalls 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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Bidder rings

• 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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Bidder rings

• 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 SP among all bids

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Bidder rings

• 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
• 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 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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Term papers due 5pm Tuesday May 13 (Deans Date)

• ? Email me for office hours re term papers