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```					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

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
week 9                                                     21

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