week 6 1

COS 444 Internet Auctions:

Theory and Practice

Spring 2010

Ken Steiglitz ken@cs.princeton.edu

week 6 2

Conditional expectation

Intuitively clear, very useful in auction theory

If x is a random variable with cdf H, and A is an event, define the conditional expectation of x given A:

A

xxdHAprob

AxE )(}{

1]|[

Not defined when prob{ A } = 0.

week 6 3

Two quick examples of conditional expectation

1) Suppose x is uniformly distributed on [0,1]. What is the expected value of x given that it is less than a constant c ≤ 1?

2) Given two independent draws x1 and x2 , what is the expected value of x1, given that x1≤ x2?

week 6 4

Interpretation of FP equil.

Now take a look once more at the equilibrium bidding function for a first-price IPV auction:

1

0

1

)(

)()(

n

v n

fp vF

ydFyvb

I claim this has the form of a conditional expectation. What is the event A?

week 6 5

Interpretation of FP equil.

Claim: event A = you win! = {Y1,(n-1) ≤ v }, where Y1,(n-1) = highest of (n-1) independent draws

To check this•

• Let y = Y1,(n-1) , the “next highest value”. The cdf of y = Y1,(n-1) is F(v)n-1 so the integral in the expected value of y given that you win is

1)(}{ nvFAprob

v n

AydFyydHy

0

1)()(

week 6 6

Interpretation of FP equil.

Therefore,

]wins|[

]|[)(

)1,(1

)1,(1)1,(1

vYE

vYYEvb

n

nnfp

That is, in equilibrium, bid the expected next-highest value conditioned on your winning. …Intuition?

week 6 7

Stronger revenue equivalence

)(

)(}wins{

wins]|[}{)( )1,(1

vP

vbvprob

vYEwinsvprobvP

fp

fp

nsp

Let Psp(v) be the expected payment in equilibrium of a bidder in a SP auction (and similarly for FP).

So SP and FP are revenue equivalent for each v !

week 6 8

Graphical interpretation

• Once again, by parts:

v nn

fpsp dyyFvFPP0

11 )(

week 6 9

Bidder preference revelation

Theorem:Theorem: Suppose there exists a symmetric Bayesian equilibrium in an IPV auction, and assume high bidder wins. Then this equilibrium bidding function is monotonically nondecreasing.

*Thanks to Dilip Abreu for showing me this elegant proof.

week 6 10

Bidder preference revelation

• Proof: Proof: Bid as if your value is z when it’s actually v. Let

w(z) = prob. of winning as fctn. of z

p(z) = exp. payment as fctn. of z

For convenience, let w=w(v), w΄=w(v΄), p=p(v), p΄=p(v΄), for any v, v΄ .

week 6 11

Bidder preference revelation

• From the definition of equilibrium, the expected surplus satisfies:

v·w – p ≥ v·w΄ – p΄ v΄·w΄ – p΄ ≥ v΄·w – p for every v,v΄ . Add: (v – v΄ )·(w – w΄) ≥ 0 .So v > v΄ → w ≥ w΄ → b(v) ≥ b(v΄). □

week 6 12

Example of an IPV auction with no symmetric Bayesian equil.:third-price (see Krishna 02, p. 34)

week 6 13

Riley & Samuelson 1981:Optimal Auctions

• Elegant, landmark paper, constructs the benchmark theory for optimal IPV auctions with reserves

• Paradoxically, gets more powerful results more easily by generalizing

week 6 14

week 6 15

all-pay

3rd-price

war of attr.

losers weepav. others

Santa Claus

last-price

week 6 16

week 6 17

Riley & Samuelson’s class Ars

1. One seller, one indivisible object2. Reserve b0 (open reserve, starting bid)3. n bidders, with valuations vi i=1,…,n4. Values iid according to cdf F, which is strictly

increasing, differentiable, with support [0,1] ( so f > 0 )

5. There is a symmetric equilibrium bidding function b(v) which is strictly increasing (we know by preference revelation it must be nondecreasing)

6. Highest acceptable bid wins7. Rules are anonymous

week 6 18

Abstracting away…

• Bid as if value = z, and denote expected payment of bidder by P(z). Then the expected surplus is

• For an equilibrium, this must be max at z=v, so differentiate and set to 0:

)()( 11 zPzFv n

0)()( 1 xPxFdx

dx n

week 6 19

We need a boundary condition…

• Denote by v* the value at which it becomes profitable to bid positively, called the entry value:

• Now integrate d.e. from v* to our value v1:

1*** )()( nvFvvP

1

*

1*1 )()()(

v

v

nxdFxvPvP

week 6 20

Once more, integrate by parts…

• And use the boundary condition:

• A truly remarkable result! Why?

1

*

11111 )()()(

v

v

nn dxxFvFvvP

week 6 21

Once more, integrate by parts…

• And use the boundary condition:

• A truly remarkable result! Why?

Where is the auction form? FP? SP? Third-price? All-pay? …

1

*

11111 )()()(

v

v

nn dxxFvFvvP

week 6 22

Revenue Equivalence Theorem

Theorem:Theorem: In equilibrium the expected revenue in an (optimal) Riley & Samuelson auction depends only on the entry value v* and not on the form of the auction. □

week 6 23

Marginal revenue, or virtual valuation

• Let’s put some work into this expected revenue:

• And integrate by parts (of course, what else?)…

v

v

n

v

nrs vdFdxxFvFnR

**

)(])([ 11 1

week 6 24

Marginal revenue, or virtual valuation

nvdF

vrs vf

vFvR )(][

1

* )(

)(1

1

*

)()(vrs

nvdFvMRR

)(

)(1)(where

vf

vFvvMR

week 6 25

Interpretation of marginal revenue

Because F(v)n is the cdf of the highest, winning value, we can interpret this as saying:

The expected revenue of an (optimal) Riley & Samuelson auction is the expected marginal revenue of the winner.

week 6 26

Hazard rate

Let the failure time of a device be distributed with pdf f(t) and cdf F(t).

Define the “survival function” = R(t) = prob. of no failure before time t = prob. of survival till time t.

Since F(t) = prob. of failure before t ,

R(t) = 1 – F(t).

week 6 27

Hazard rate

The conditional prob. of failure in the interval (t, t+Δt ] , given survival up to time t , is

The “Hazard Rate” is the limit of this divided by Δt as Δt → 0 :

)(

)()(

tR

ttRtR

F

f

R

RHR

1

week 6 28

Hazard rate

Thus, the marginal revenue, which is key to finding the expected revenue in a Riley-Samuelson auction, is

1/HR is the “Inverse Hazard Rate”

HRv

f

FvMR

11

week 6 29

Why call it “marginal revenue”?

Consider a monopolist seller who makes a take-it-or-leave-it offer to a single seller at a price p. The buyer has value distribution F, so the prob. of her accepting the offer is 1–F(p). Think of this as the buyer’s demand curve. She buys, on the average, quantity q = 1–F(p) at price p. Or, what is the same thing, the seller offers price

p(q) = F-1(1– q) to sell quantity q.

after Krishna 02, BR 89

week 6 30

Why call it “marginal revenue”?

The revenue function of the seller is therefore q·p(q) = q F-1(1-q) , the revenue derived from selling quantity q. The derivative of this wrt q is by definition the marginal revenue of the monopolist :

F-1(1-q) = p, so this is

□

))1(()1(

11

qFF

qqF

)(

)(1)(

pf

pFppMR