week 8 1

COS 444 Internet Auctions:

Theory and Practice

Spring 2009

Ken Steiglitz ken@cs.princeton.edu

week 8 2

Some other auctions in Ars

Consider simple case with n=2 and uniform iid values on [0,1]. We also choose the optimal v* = ½.

How do we choose b0 to achieve this v* ?

All-pay with reserve

*

2*

211 ,)()(][)(

22*

vvvv

dyyFvvFpayEvbv

v

nn

*

2*

22 ,)(][

22vv

vvvbvsurplusE

week 8 3

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 = v*2 = ¼ .

Notice that this is an example where the reserve is not equal to the entry value v* .

All-pay with reserve

week 8 4

week 8 5

Loser weeps auction, n=2, 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 E[surplus] = 0 at v = v* , and again argue that b(v*) = b0 . This gives us

][)1)((*

2 payEvvbdxxvv

v

2/1*

2*

0 1

v

vb

2/14/1,)1(2

)(2

v

v

vvb ( goes to ∞ !)

week 8 6

week 8 7

Santa Claus auction, n=2, 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

b

vdyyF

*

)(

week 8 8

Santa Claus auction, con’t

)()()(E[surplus]*

bFbvdyyFb

v

Suppose 2 bids truthfully and 1 has value v and bids b. Then

because F(b) = prob. winning. For equil.:

0)()()()(/ bFbfbbfvbFb

vb □ (use reserve b0 = v* )

week 8 9

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 doesn’t bid, bidder 2 can buy at v* if he wants to

week 8 10

• To compare with optimal auctions, choose v* = ½; uniform iid IPV’s on [0,1]

• Bidder 2’s 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

Matching auction: not in Ars

week 8 11

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

Matching auction: not in Ars

□

week 8 12

Notice:

When ½ < v2 < v1 , bidder 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)

BTW, …why is this auction not in Ars ?

Matching auction: not in Ars

week 8 13

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

Risk aversion

recall

week 8 14

Utility model

week 8 15

Risk aversion & revenue ranking

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

RFP ≥ RSP

Proof:Proof: Let γ be the equilibrium bidding function in the risk-averse case, and β in the risk-neutral case.

week 8 16

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).

))(()( zxuzW

))((

))((

)(

)()(

xxu

xxu

xW

xwx

week 8 17

Revenue ranking

In the risk neutral case this is just:

The utility function is concave:

))(()(

)()( xx

xW

xwx

xxuxu )(/)(

week 8 18

Revenue ranking

Using this,

Now we can see that γ΄(0) > β΄(0). If not, then there would be an interval near 0 where γ ≤ β, and then

which would be a contradiction.

)())(()(

)()( xxx

xW

xwx

))(()(

)()( xx

xW

xwx

week 8 19

Revenue ranking Also, it’s clear that γ(0) = β(0) = 0. So γ starts out above β at the

origin. To show that it stays above β, consider what would happen should it cross, say at x = x* :

)(

))(()(

)(

))(()(

)()(

*

****

*

****

**

x

xxxW

xw

xxxW

xwx

A contradiction. □

week 8 20

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)/ρ instead of (n−1) rivals!

/)1(

0

/)1(

)(

)()(

n

v n

vF

dyyFvv

concavefor1,)( ttu

))((

))((

)(

)()(

xxu

xxu

xW

xwx