week 7 1

COS 444 Internet Auctions:

Theory and Practice

Spring 2009

Ken Steiglitz ken@cs.princeton.edu

week 7 2

Field experiment

“A Test of the Revenue Equivalence Theorem using Field Experiments on eBay”

T. Hossain, J. Morgan, 2004

We have just seen that Riley & Samuelson 1981 predicts that the revenue for a wide class of auctions depends only on the entry value (v*), also called the “effective reserve”. This paper uses eBay to field-test this prediction.

week 7 3

The experiment

• 80 auctions in all, 40 for CDs, 40 for Xbox games. Four copies each of 10 CDs, four copies each of 10 Xboxes.

• Private values is a good assumption.

• Auctions were held for v* = $4 (low effective reserve) and v* = $8 (high effective reserve). For each of these cases, the opening bid was varied and the shipping charges adjusted to achieve the desired v*.

week 7 4

The experiment

Treat. A: Opening bid = $4.00 ship = $0.00

Treat. B: Opening bid = $0.01 ship = $3.99

Treat. C: Opening bid = $6.00 ship = $2.00

Treat. D: Opening bid = $2.00 ship = $6.00

All other experimental variables held as close to fixed as possible, order randomized

V* = $4

V* = $8

week 7 5

Revenue results

A: low v* , high opening bid

B: low v* , low opening bid

C: high v* , high opening bid

D: high v* , low opening bid

Explain…?

Higher revenue in B

rev. eq. for CDs,

Higher revenue in D for Xbox games

week 7 6

Revenue results

A: low v* , high opening bid

B: low v* , low opening bid

C: high v* , high opening bid

D: high v* , low opening bid

Explain…?

Higher revenue in B

rev. eq. for CDs,V* > 50% retail,people notice!

Higher revenue in D for Xbox games

week 7 7

Explanations of revenue results

• Mental accounting (Kahneman & Tversky 84; Thaler 85). Modeled in Hossain & Morgan.

• Salience• Bidders suspicious of free shipping• Love of winning• Costly search (usual searches ignore shipping)• Sequential auctions

week 7 8

Hypothesis testing

• We often want to test the statistical significance of observations (as in

Hossein-Morgan 04)

• Many common tests use normal distributions and their derivatives

• The one-tailed binomial test is the simplest

• Such tests can easily be abused, and are often blindly applied

week 7 9

Using the one-sided binomial test in Hossein & Morgan 04

Consider Treatment A (v* = $4, high opening bid)

vs. Treatment B (v* = $4, low opening bid)

Null hypothesis: A and B are rev. equiv.

One-sided alternative: rev. in B > A

Data: B>A 9/10 for CDs, 7/10 for Xboxes

16/20

week 7 10

One-sided binomial test• Bernoulli trials: n independent coin flips, say in

this case with a coin that comes up heads with prob. p

• So we ask what the probability is that we get 16 or more heads out of 20 flips if the coin is fair (one-sided test of null hypothesis)

• Add these for k = 16, … , 20

knpkpk

nheadskprob

)1(}{

week 7 11

table of cumulative binomial distribution

Weight of tail up and including k=4, for n=20 = 0.0059

Hossein & Morgan 04, p. 11: “The p-value of the one-sided binomialtest is 0.005, which implies that we can reject the null hypothesisimplied by the revenue equivalence theorem at the 99.5% level.

week 7 12

Warning: the Normal approximation

• When n is “reasonably” large, the binomial distribution is well approximated by the normal distribution… usually that means n > ~ 30. If you use normal tables for this problem you get a one-sided p value of 0.00368 --- not very close to the true value.

week 7 13

Warning: inference and priors

This test tells us Prob (DATA|NULL).We might worry more about Prob (NULL|DATA)Bayes’ Rule tells us

But do we know the priors: P(NULL)? P(DATA)?

)(

)()|()|(

DATAP

NULLPNULLDATAPDATANULLP

week 7 14

Back to optimal IPV auctions

Introduce v0 = value of the item to the seller, which we’ve taken to be 0 till now, and

which we will often do in the future. Then the total expected revenue is

The first term is due to the possibility that all

values are below v* and the seller retains the item.

1

*0*

)()()(v

nntotalrs vdFvMRvFvR

week 7 15

Optimal reserve b0

We now ask the the question: how should the seller choose the reserve (opening bid) b0 optimally---that is, to max exp. rev.? b0 determines v*, so we differentiate wrt v* :

0)()()()()( *1

***1

*0 vfvnFvMRvfvnFv nn

0*)( vvMR

0*

** )(

)(1v

vf

vFv

week 7 16

Optimal reserve b0

Notice: v* does does not depend on the number of bidders, nor on the particular form of the auction!

In the uniform case with v0=0, eg, F(x) = x, and v* = ½ , for any auction in Ars.

0*

** )(

)(1v

vf

vFv

week 7 17

Optimal reserve b0

Lemma:Lemma: In a first- or second-price auction in Ars, v* = b0 .

Proof:Proof: In either FP or SP there is no incentive to bid if your value ≤ b0 . Therefore v* ≥ b0 . On the other hand, as soon as our value reaches b0+ ε we can realize a positive expected surplus. The point at which we are indifferent to bidding is therefore v* = b0 . □

week 7 18

week 7 19

Optimal reserve b0

Notice that in FP and SP auctions in the class Ars the seller’s optimal reserve is

… above the seller’s value! Intuition?

0*

*0*0 )(

)(1v

vf

vFvvb

week 7 20

FP equilibrium with reserve b0

The next question: What is the equilibrium when there is a positive reserve? A slick way to do this is to recall the E[pay] from the beginning of Riley & Samuelson 81:

We got this when we abstracted the payment away from the particular type of auction. But in FP:

1

*

11111 )()()(

v

v

nn dxxFvFvvP

1111 )()()( nvFvbvP

week 7 21

FP equilibrium with reserve b0

Therefore,

Simple example: v* = b0 = ½, F(v) = v, n = 2. Then

else0

if)(

)(

)( *1

1

* vvyF

dyyFvvb n

v

v

n

v

vvb

8

1

2)(

week 7 22

week 7 23

FP equilibrium with reserve b0

Checking revenue… use (with v0 = 0 )

v* = 0 : Revenue = 4/12

v* = ½ : Revenue = 5/12 > 4/12 Notice that the revenue increase is a won

tradeoff for seller: he rejects bids below ½ , but forces increased bidding in equilibrium when bidder values are above ½ .

1

*

)()(v

ntotalrs vdFvMRR

week 7 24

SP equilibrium with reserve b0

Not a problem: Vickrey’s argument works again: just bid truthfully, there can never be an advantage to deviating from truthful bidding.

But the mechanism for increasing revenue with a reserve is completely different from that in FP. Now the increase in payments results from bids above b0 being reduced to b0 rather than the second-highest bid when it’s below b0 .

Notice that this requires much less in the way of strategic thinking on the part of the bidders.

week 7 25

Reserves: testing the benchmark theory

“Field Experiments on the Effects of Reserve Prices in Auctions: More Magic on the Internet” Lucking-Reiley, 2000

Pre-eBay, first-price sealed-bid auctions with control over open reserve

The unique window in the history of civilization when auction experiments like this were possible (recall also LR 1999)

week 7 26

Reserves: testing the benchmark theory

Design 1 (within cards): Binary variable: no-reserve vs. reserve

The familiar setup with pairs of matched Magic cards:

• Treatment 1: 86 cards, no reserve• Treatment 2: same cards, one week later, reserve• Treatments 3,4: same experiments, different cards,

reverse time order

week 7 27

Reserves: testing the benchmark theory

Design 2 (between cards): Continuous variable: reserve level = varying percentage of Cloister price (“catalog”)

• Auctions 1 & 2: 99 cards, 9 at 10%; 9 at 20%; …, 9 at 110% catalog

• Auctions 3 & 4: equal numbers of cards at 10%; 20%; 30%; 40%; 50%; 100%; 110%; 120%; 130%; 140%; and 150% catalog

week 7 28

Reserves: testing the benchmark theory

R&S 81 IPV Theory predicts that higher (open) reserves b0 :

• reduces # of bidders OK

• decreases prob. of sale OK

• increases price conditional on sale OK

• increases total revenue NO!

• Bidders respond strategically to increased reserve OK

week 7 29open reserve

week 7 30

“Optimal” reserves

Lucking-Riley 2000, p. 22:

“After spending months observing this market environment and after running auctions myself, it is hard for me to imagine how an auctioneer in a real-world environment could ever have enough information to choose precisely the optimal reserve price.”