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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Theory Riley Samuelson 81
Quick FP equilibrium with reserve
which gives us immediately
Example
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Theory Riley Samuelson 81

• Revenue at equilibrium
• marginal revenue virtual valuation

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Theory Riley Samuelson 81

• Optimal choice of reserve
• let v0 value to seller
• Total revenue
• Differentiate wrt v and set to zero ?

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Reserves

• The seller chooses reserve b0 to achieve a given
  v .
• In first-price and second-price auctions (but not
  in all the auctions in the Riley-Samuelson class)
  v b0 .
• Proof theres no incentive to bid when our
  value is below b0 , and an incentive to bid when
  our value is above b0 .

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Reserves

• Setting reserve in the second- and first-price
  increases revenue through entirely different
  mechanisms
• In first-price auctions bids are increased.
• In second-price auctions its an equilibrium to
  bid truthfully, but winners are forced to pay
  more.

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All-pay with reserve

• Set E pay from Riley Samuelson 81 b (
  v ) !
• For n2 and uniform vs this gives
• b( v ) v 2/2 v2/2
• Setting E surplus at v 0 gives
• b( v ) v2
• Also, b( v ) b0 (we win only with no
  competition, so bid as low as possible)
• ?Therefore, b0 v2 (not v as before)

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Loser weeps auction, n2

• Winner gets item for free, loser pays his bid!
• Gives us reserve in terms of v
  (evaluate at v )
• b0 v2 / (1-v) using b( v ) b0
• Epay of RS 81 then leads directly to
  equilibrium

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Santa Claus auction, n2

• Winner pays her bid
• Idea give people their expected surplus and try
  to arrange things so bidding truthfully is an
  equilibrium.
• Give people
• Prove truthful bidding is a SBNE

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Santa Claus auction, cont
Suppose 2 bids truthfully. Then
?/?b 0 shows bv
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Matching auction not in Ars

• Bidder 1 may tender an offer on a house,
• b1 b0 reserve
• Bidder 2 currently leases house and has the
  option of matching b1 and buying at that price.
  If bidder 1 doesnt bid, bidder 2 can buy at b0
  if he wants

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Matching auction, cont

• To compare with optimal auctions, choose v ½
• Bidder 2s best strategy Match b1 iff
• v2 b1 else bid ½ iff v2 ½
• Bidder should choose b1 ½ so as to maximize
  expected surplus.
• This turns out to be b1 ½

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Matching auction, cont

• Choose v ½ for comparison
• Bidder 1 tries to max
• (v1-b1 )prob. 2 chooses not to match
• (v1-b1 )b1
• ? b1 0 if v1 lt ½
• ½ if v1 ½

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Matching auction, cont

• Notice
• When ½ lt v2 lt v1 , bibber 2 gets the item, but
  values it less than bidder 1 ? inefficient!
• Erevenue to seller turns out to be 9/24
  (optimal in Ars is 10/24 optimal with no reserve
  is 8/24)
• ? Why is this auction not in Ars ?

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Risk-averse bidders
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Revenue ranking with risk aversion

• Result Suppose bidders utility is concave.
  Then with the assumptions of Ars ,
• RFP RSP
• Proof Let ? be the equilibrium bidding
  function in the risk-averse case, and ß in the
  risk-neutral case.

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Revenue ranking, cont

• In first-price auction,
• Esurplus W (z )u (x - ? (z ) )
• where we bid as if value z , W(z)
• is prob. of winning, etc.

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Constant relative risk aversion (CRRA)

• Defined by utility
• u(t) t ? , ? lt 1
• First-price equilibrium can be found by usual
  methods
• ( u/u t/? helps)
• ?Very similar to risk-neutral form. As if there
  were
• (n-1)/? instead of (n-1) rivals!