Docstoc
EXCLUSIVE OFFER FOR DOCSTOC USERS
Try the all-new QuickBooks Online for FREE.  No credit card required.

9

Document Sample
9 Powered By Docstoc
					COS 444
Internet Auctions:
Theory and Practice


         Spring 2008

         Ken Steiglitz
         ken@cs.princeton.edu

week 9                          1
Theory: Riley & Samuelson 81
  Quick FP equilibrium with reserve:
                         v
P(v)  vF(v)   n 1
                        F ( x)      n 1
                                             dx  b(v) F (v)   n 1
                        v*

  which gives us immediately:
                             v
                       
                                           n 1
                                  F ( x)          dx
            b(v)  v 
                             v*

                                  F (v) n1
         Example…
week 9                                                          2
Theory: Riley & Samuelson 81

  Revenue at equilibrium:
                1
         Rrs   MR(v)dF (v)        n
                v*

                       1  F (v )
          MR (v)  v 
                          f (v )
  = “marginal revenue” = “virtual valuation”
week 9                                         3
Theory: Riley & Samuelson 81
Optimal choice of reserve
 let v0 = value to seller
 Total revenue =
                        1
         v0 F (v* )   MR(v) dF (v)
                  n                           n
                        v*

Differentiate wrt v* and set to zero 
                           1  F (v* )
           MR (v* )  v*               v0
                              f (v* )
week 9                                            4
    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: there’s no incentive to bid when our
  value is below b0 , and an incentive to bid
  when our value is above b0 .

    week 9                                  5
     Reserves
   Setting reserve in the second- and first-
    price increases revenue through entirely
    different mechanisms:

o   In first-price auctions bids are increased.

o   In second-price auctions it’s an equilibrium
    to bid truthfully, but winners are forced to
    pay more.

     week 9                                       6
     All-pay with reserve
 Set E[ pay ] from Riley & Samuelson 81
   =b(v) !
 • For n=2 and uniform v’s this gives
            b( v ) = v 2/2 + v*2/2
 •       Setting E[ surplus at v* ] = 0 gives
            b( v* ) = v*2
 •       Also, b( v* ) = b0 (we win only with no
  competition, so bid as low as possible)
 Therefore, b0 = v*2 (not v* as before)
week 9                                        8
Loser weeps auction, n=2

Winner gets item for free, loser pays his bid!
E[value]  E[pay]  vF(v)  b(v)(1  F (v))
Gives us reserve in terms of v*
  (evaluate at v* ):
     b0 = v*2 / (1-v*) … using b( v* ) = b0

E[pay] of R&S 81 then leads directly to equilibrium

week 9                                           10
Santa Claus auction, n=2
    Winner pays her bid
    Idea: give people their expected surplus and
     try to arrange things so bidding truthfully is
     an equilibrium.
    Give people
                      b
                     v*
                           F ( y)dy
    Prove: truthful bidding is a SBNE …

week 9                                            12
Santa Claus auction, con’t

Suppose 2 bids truthfully. Then
              b
E[surplus]  F ( y)dy  (v  b) F (b)
              v*

 ∂∕∂b = 0 shows b=v



week 9                               13
    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 doesn’t bid, bidder 2
      can buy at b0 if he wants

    week 9                                     14
    Matching auction, con’t
•   To compare with optimal auctions, choose
    v* = ½
•   Bidder 2’s 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 …

    week 9                                15
     Matching auction, con’t

•   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 < ½
           = ½ if v1 ≥ ½

    week 9                                    16
Matching auction, con’t
 Notice:
 When ½ < v2 < v1 , bibber 2 gets the
 item, but values it less than bidder 1 
 inefficient!
 E[revenue 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 ?
week 9                                  17
Risk-averse bidders
Revenue ranking with risk aversion

    Result: Suppose bidders’ utility is
    concave. Then with the assumptions
    of Ars ,
             RFP ≥ RSP

    Proof: Let γ be the equilibrium
    bidding function in the risk-averse
    case, and β in the risk-neutral case.
  week 9                                    19
Revenue ranking, con’t

In first-price auction,

E[surplus] = W (z )·u (x − γ (z ) )

where we bid as if value = z , W(z)
is prob. of winning, … etc.


week 9                                20
Constant relative risk aversion
(CRRA)
Defined by utility
         u(t) = t ρ , ρ < 1
First-price equilibrium can be found by usual methods
 ( u/u’ = t/ρ helps):
                           v
                       
                                        ( n 1) / 
                               F (t )                 dt
          b (v )  v  0
                                        ( n 1) / 
                               F (v )
Very similar to risk-neutral form. As if there were
(n-1)/ρ instead of (n-1) rivals!
week 9                                                     21

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:7
posted:9/11/2012
language:Unknown
pages:21