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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Some other auctions in Ars
All-pay with reserve

• Consider simple case with n2 and uniform iid
  values on 0,1. We also choose the optimal v
  ½.
• How do we choose b0 to achieve this v ?

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All-pay with reserve

• Standard argument If your value is v you win
  if and only if you have no rival bidder. (This is
  the point of indifference between bidding and not
  bidding, and your expected surplus is 0.)
  Therefore, bid as low as possible! Therefore,
• b(v) b0 . And so b0 v2 ¼ .
• Notice that this is an example where the
  reserve is not equal to the entry value v .

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Loser weeps auction, n2, uniform v

• Winner gets item for free, loser pays his
  bid! Auction is in Ars . The expected payment is
  therefore
• and therefore, choosing v ½ as before
  (optimally),
• To find b0 , set Esurplus 0 at v v ,
  and again argue that b(v) b0 . This gives us

( goes to 8 !)
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Santa Claus auction, n2, uniform v

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

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Santa Claus auction, cont
Suppose 2 bids truthfully and 1 has value v and
bids b. Then
because F(b) prob. winning. For equil.
?
(use reserve b0 v )
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Matching auction not in Ars

• Bidder 1 may tender an offer on a house,
• b1 v
• 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 v
  if he wants to

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Matching auction not in Ars

• To compare with optimal auctions, choose v ½
  uniform iid IPVs on 0,1
• Bidder 2s best strategy If 1 bids, match b1
  iff v2 b1 else bid ½ iff v2 ½
• Bidder 1 should choose b1 ½ so as to maximize
  expected surplus. This turns out to be b1 ½ .
  To see this

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Matching auction not in Ars

• 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 not in Ars

• Notice
• When ½ lt v2 lt v1 , bidder 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)
• BTW, why is this auction not in Ars ?

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Risk aversion

• Intuition Suppose you care more about losses
  than gains. How does that affect your bidding
  strategy in SP auctions? First-price auctions?

recall
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Utility model
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Risk aversion revenue ranking

• 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

• Let the bidder bid as if her value is z, while
  her actual value is x. In a first-price auction,
  her expected surplus is
• where W(z) F(z)n-1 is the prob. of winning.
  As usual, to find an equilibrium, differentiate
  wrt z and set the result to 0
• at z x
• where w(x) W?(x).
• where w(x) W?(x).

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Revenue ranking

• In the risk neutral case this is just
• The utility function is concave

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Revenue ranking

• Using this,
• Now we can see that ??(0) gt ß?(0). If not,
  then there would be an interval near 0 where ?
  ß, and then
• which would be a contradiction.

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Revenue ranking

• Also, its clear that ?(0) ß(0) 0. So ?
  starts out above ß at the origin. To show that it
  stays above ß, consider what would happen should
  it cross, say at x x

A contradiction. ?
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Constant relative risk aversion (CRRA)

• Defined by utility
• In this case we can solve MRS 03
• for
• ?Very similar to risk-neutral form, but bid as if
  there were (n-1)/? instead of (n-1) rivals!