An

Analysis

of

Alternative

Slot

Auction

Designs

for

Sponsored

Search

S´ebastien

Lahaie

Division

of

Engineering

and

Applied

Sciences

Harvard

University,

Cambridge,

MA

slahaie@eecs.harvard.edu

ABSTRACT

Billions

of

dollars

are

spent

each

year

on

sponsored

search,

a

form

of

advertising

where

merchants

pay

for

placement

alongside

web

search

results.

Slots

for

ad

listings

are

allocated

via

an

auction-style

mechanism

where

the

higher

a

merchant

bids,

the

more

likely

his

ad

is

to

appear

above

other

ads

on

the

page.

In

this

paper

we

analyze

the

incentive,

efficiency,

and

revenue

properties

of

two

slot

auction

designs:

rank

by

bid

(RBB)

and

rank

by

revenue

(RBR),

which

correspond

to

stylized

versions

of

the

mechanisms

currently

used

by

Yahoo!

and

Google,

respectively.

We

also

consider

first-

and

second-price

payment

rules

together

with

each

of

these

allocation

rules,

as

both

have

been

used

historically.

We

consider

both

the

short-run

incomplete

information

setting

and

the

long-run

complete

information

setting.

With

incomplete

information,

neither

RBB

nor

RBR

are

truthful

with

either

first

or

second

pricing.

We

find

that

the

informational

requirements

of

RBB

are

much

weaker

than

those

of

RBR,

but

that

RBR

is

efficient

whereas

RBB

is

not.

We

also

show

that

no

revenue

ranking

of

RBB

and

RBR

is

possible

given

an

arbitrary

distribution

over

bidder

values

and

relevance.

With

complete

information,

we

find

that

no

equilibrium

exists

with

first

pricing

using

either

RBB

or

RBR.

We

show

that

there

typically

exists

a

multitude

of

equilibria

with

second

pricing,

and

we

bound

the

divergence

of

(economic)

value

in

such

equilibria

from

the

value

obtained

assuming

all

merchants

bid

truthfully.

Categories

and

Subject

Descriptors

[Computer

Applications]:

Social

and

Behavioral

Sciences-Economics

General

Terms

Economics,

Theory

INTRODUCTION

Today,

Internet

giants

Google

and

Yahoo!

boast

a

combined

market

capitalization

of

over

billion,

largely

on

the

strength

of

sponsored

search,

the

fastest

growing

component

of

a

resurgent

online

advertising

industry.

PricewaterhouseCoopers

estimates

that

industry-wide

sponsored

search

revenues

were

billion,

or

of

total

Internet

advertising

Industry

watchers

expect

revenues

to

reach

or

exceed

Roughly

of

Google"s

estimated

billion

in

revenue

and

roughly

of

Yahoo!"s

estimated

billion

in

revenue

will

likely

be

attributable

to

sponsored

A

number

of

other

companies-including

LookSmart,

FindWhat,

InterActiveCorp

(Ask

Jeeves),

and

eBay

(Shopping.com)-earn

hundreds

of

millions

of

dollars

of

sponsored

search

revenue

annually.

Sponsored

search

is

a

form

of

advertising

where

merchants

pay

to

appear

alongside

web

search

results.

For

example,

when

a

user

searches

for

used

honda

accord

san

diego

in

a

web

search

engine,

a

variety

of

commercial

entities

(San

Diego

car

dealers,

Honda

Corp,

automobile

information

portals,

classified

ad

aggregators,

eBay,

etc...)

may

bid

to

to

have

their

listings

featured

alongside

the

standard

algorithmic

search

listings.

Advertisers

bid

for

placement

on

the

page

in

an

auction-style

format

where

the

higher

they

bid

the

more

likely

their

listing

will

appear

above

other

ads

on

the

page.

By

convention,

sponsored

search

advertisers

generally

pay

per

click,

meaning

that

they

pay

only

when

a

user

clicks

on

their

ad,

and

do

not

pay

if

their

ad

is

displayed

but

not

clicked.

Though

many

people

claim

to

systematically

ignore

sponsored

search

ads,

Majestic

Research

reports

that

PwC

These

are

rough

back

of

the

envelope

estimates.

Google

and

Yahoo!

revenue

estimates

were

obtained

from

Yahoo!

Finance.

We

assumed

billion

in

industry-wide

sponsored

search

revenues.

We

used

Nielsen/NetRatings

estimates

of

search

engine

market

share

in

the

US,

the

most

monetized

market:

wired-vig.wired.com/news/technology/0,1282,69291,00.html

Using

comScore"s

international

search

engine

market

share

estimates

would

yield

different

estimates:

as

many

as

of

Google

searches

result

in

a

paid

click,

and

that

Google

earns

roughly

nine

cents

on

average

for

every

search

query

they

Usually,

sponsored

search

results

appear

in

a

separate

section

of

the

page

designated

as

sponsored

above

or

to

the

right

of

the

algorithmic

results.

Sponsored

search

results

are

displayed

in

a

format

similar

to

algorithmic

results:

as

a

list

of

items

each

containing

a

title,

a

text

description,

and

a

hyperlink

to

a

corresponding

web

page.

We

call

each

position

in

the

list

a

slot.

Generally,

advertisements

that

appear

in

a

higher

ranked

slot

(higher

on

the

page)

garner

more

attention

and

more

clicks

from

users.

Thus,

all

else

being

equal,

merchants

generally

prefer

higher

ranked

slots

to

lower

ranked

slots.

Merchants

bid

for

placement

next

to

particular

search

queries;

for

example,

Orbitz

and

Travelocity

may

bid

for

las

vegas

hotel

while

Dell

and

HP

bid

for

laptop

computer.

As

mentioned,

bids

are

expressed

as

a

maximum

willingness

to

pay

per

click.

For

example,

a

forty-cent

bid

by

HostRocket

for

web

hosting

means

HostRocket

is

willing

to

pay

up

to

forty

cents

every

time

a

user

clicks

on

their

The

auctioneer

(the

search

evaluates

the

bids

and

allocates

slots

to

advertisers.

In

principle,

the

allocation

decision

can

be

altered

with

each

new

incoming

search

query,

so

in

effect

new

auctions

clear

continuously

over

time

as

search

queries

arrive.

Many

allocation

rules

are

plausible.

In

this

paper,

we

investigate

two

allocation

rules,

roughly

corresponding

to

the

two

allocation

rules

used

by

Yahoo!

and

Google.

The

rank

by

bid

(RBB)

allocation

assigns

slots

in

order

of

bids,

with

higher

ranked

slots

going

to

higher

bidders.

The

rank

by

revenue

(RBR)

allocation

assigns

slots

in

order

of

the

product

of

bid

times

expected

relevance,

where

relevance

is

the

proportion

of

users

that

click

on

the

merchant"s

ad

after

viewing

it.

In

our

model,

we

assume

that

an

ad"s

expected

relevance

is

known

to

the

auctioneer

and

the

advertiser

(but

not

necessarily

to

other

advertisers),

and

that

clickthrough

rate

decays

monotonically

with

lower

ranked

slots.

In

practice,

the

expected

clickthrough

rate

depends

on

a

number

of

factors,

including

the

position

on

the

page,

the

ad

text

(which

in

turn

depends

on

the

identity

of

the

bidder),

the

nature

and

intent

of

the

user,

and

the

context

of

other

ads

and

algorithmic

results

on

the

page,

and

must

be

learned

over

time

by

both

the

auctioneer

and

the

bidder

As

of

this

writing,

to

a

rough

first-order

approximation,

Yahoo!

employs

a

RBB

allocation

and

Google

employs

a

RBR

allocation,

though

numerous

caveats

apply

in

both

cases

when

it

comes

to

the

vagaries

of

real-world

Even

when

examining

a

one-shot

version

of

a

slot

auction,

the

mechanism

differs

from

a

standard

multi-item

Usually

advertisers

also

set

daily

or

monthly

budget

caps;

in

this

paper

we

do

not

model

budget

constraints.

In

the

sponsored

search

industry,

the

auctioneer

and

search

engine

are

not

always

the

same

entity.

For

example

Google

runs

the

sponsored

search

ads

for

AOL

web

search,

with

revenue

being

shared.

Similarly,

Yahoo!

currently

runs

the

sponsored

search

ads

for

MSN

web

search,

though

Microsoft

will

begin

independent

operations

soon.

Here

are

two

among

many

exceptions

to

the

Yahoo!

RBB

and

Google

RBR

assertion:

Yahoo!

excludes

ads

deemed

insufficiently

relevant

either

by

a

human

editor

or

due

to

poor

historical

click

rate;

Google

sets

differing

reserve

prices

depending

on

Google"s

estimate

of

ad

quality.

tion

in

subtle

ways.

First,

a

single

bid

per

merchant

is

used

to

allocate

multiple

non-identical

slots.

Second,

the

bid

is

communicated

not

as

a

direct

preference

over

slots,

but

as

a

preference

for

clicks

that

depend

stochastically

on

slot

allocation.

We

investigate

a

number

of

economic

properties

of

RBB

and

RBR

slot

auctions.

We

consider

the

short-run

incomplete

information

case

in

Section

adapting

and

extending

standard

analyses

of

single-item

auctions.

In

Section

we

turn

to

the

long-run

complete

information

case;

our

characterization

results

here

draw

on

techniques

from

linear

programming.

Throughout,

important

observations

are

highlighted

as

claims

supported

by

examples.

Our

contributions

are

as

follows:

We

show

that

with

multiple

slots,

bidders

do

not

reveal

their

true

values

with

either

RBB

or

RBR,

and

with

either

first-

or

second-pricing.

With

incomplete

information,

we

find

that

the

informational

requirements

of

playing

the

equilibrium

bid

are

much

weaker

for

RBB

than

for

RBR,

because

bidders

need

not

know

any

information

about

each

others"

relevance

(or

even

their

own)

with

RBB.

With

incomplete

information,

we

prove

that

RBR

is

efficient

but

that

RBB

is

not.

We

show

via

a

simple

example

that

no

general

revenue

ranking

of

RBB

and

RBR

is

possible.

We

prove

that

in

a

complete-information

setting,

firstprice

slot

auctions

have

no

pure

strategy

Nash

equilibrium,

but

that

there

always

exists

a

pure-strategy

equilibrium

with

second

pricing.

We

provide

a

constant-factor

bound

on

the

deviation

from

efficiency

that

can

occur

in

the

equilibrium

of

a

second-price

slot

auction.

In

Section

we

specify

our

model

of

bidders

and

the

various

slot

auction

formats.

In

Section

we

study

the

incentive

properties

of

each

format,

asking

in

which

cases

agents

would

bid

truthfully.

There

is

possible

confusion

here

because

the

second-price

design

for

slot

auctions

is

reminiscent

of

the

Vickrey

auction

for

a

single

item;

we

note

that

for

slot

auctions

the

Vickrey

mechanism

is

in

fact

very

different

from

the

second-price

mechanism,

and

so

they

have

different

incentive

In

Section

we

derive

the

Bayes-Nash

equilibrium

bids

for

the

various

auction

formats.

This

is

useful

for

the

efficiency

and

revenue

results

in

later

sections.

It

should

become

clear

in

this

section

that

slot

auctions

in

our

model

are

a

straightforward

generalization

of

single-item

auctions.

Sections

and

address

questions

of

efficiency

and

revenue

under

incomplete

information,

respectively.

In

Section

we

determine

whether

pure-strategy

equilibria

exist

for

the

various

auction

formats,

under

complete

information.

In

Section

we

derive

bounds

on

the

deviation

from

efficiency

in

the

pure-strategy

equilibria

of

secondprice

slot

auctions.

Our

approach

is

positive

rather

than

normative.

We

aim

to

clarify

the

incentive,

efficiency,

and

revenue

properties

of

two

slot

auction

designs

currently

in

use,

under

settings

of

Other

authors

have

also

made

this

observation

incomplete

and

complete

information.

We

do

not

attempt

to

derive

the

optimal

mechanism

for

a

slot

auction.

Related

work.

Feng

et

al.

compare

the

revenue

performance

of

various

ranking

mechanisms

for

slot

auctions

in

a

model

with

incomplete

information,

much

as

we

do

in

Section

but

they

obtain

their

results

via

simulations

whereas

we

perform

an

equilibrium

analysis.

Liu

and

Chen

study

properties

of

slot

auctions

under

incomplete

information.

Their

setting

is

essentially

the

same

as

ours,

except

they

restrict

their

attention

to

a

model

with

a

single

slot

and

a

binary

type

for

bidder

relevance

(high

or

low).

They

find

that

RBR

is

efficient,

but

that

no

general

revenue

ranking

of

RBB

and

RBR

is

possible,

which

agrees

with

our

results.

They

also

take

a

design

approach

and

show

how

the

auctioneer

should

assign

relevance

scores

to

optimize

its

revenue.

Edelman

et

al.

model

the

slot

auction

problem

both

as

a

static

game

of

complete

information

and

a

dynamic

game

of

incomplete

information.

They

study

the

locally

envyfree

equilibria

of

the

static

game

of

complete

information;

this

is

a

solution

concept

motivated

by

certain

bidding

behaviors

that

arise

due

to

the

presence

of

budget

constraints.

They

do

not

view

slot

auctions

as

static

games

of

incomplete

information

as

we

do,

but

do

study

them

as

dynamic

games

of

incomplete

information

and

derive

results

on

the

uniqueness

and

revenue

properties

of

the

resulting

equilibria.

They

also

provide

a

nice

description

of

the

evolution

of

the

market

for

sponsored

search.

Varian

also

studies

slot

auctions

under

a

setting

of

complete

information.

He

focuses

on

symmetric

equilibria,

which

are

a

refinement

of

Nash

equilibria

appropriate

for

slot

auctions.

He

provides

bounds

on

the

revenue

obtained

in

equilibrium.

He

also

gives

bounds

that

can

be

used

to

infer

bidder

values

given

their

bids,

and

performs

some

empirical

analysis

using

these

results.

In

contrast,

we

focus

instead

on

efficiency

and

provide

bounds

on

the

deviation

from

efficiency

in

complete-information

equilibria.

PRELIMINARIES

We

focus

on

a

slot

auction

for

a

single

keyword.

In

a

setting

of

incomplete

information,

a

bidder

knows

only

distributions

over

others"

private

information

(value

per

click

and

relevance).

With

complete

information,

a

bidder

knows

others"

private

information,

and

so

does

not

need

to

rely

on

distributions

to

strategize.

We

first

describe

the

model

for

the

case

with

incomplete

information,

and

drop

the

distributional

information

from

the

model

when

we

come

to

the

complete-information

case

in

Section

The

Model

There

is

a

fixed

number

K

of

slots

to

be

allocated

among

N

bidders.

We

assume

without

loss

of

generality

that

K

N,

since

superfluous

slots

can

remain

blank.

Bidder

i

assigns

a

value

of

Xi

to

each

click

received

on

its

advertisement,

regardless

of

this

advertisement"s

The

probability

that

i"s

advertisement

will

be

clicked

if

viewed

is

Ai

We

refer

to

Ai

as

bidder

i"s

relevance.

We

refer

to

Ri

AiXi

as

bidder

i"s

revenue.

The

Xi,

Ai,

and

Ri

are

random

Indeed

Kitts

et

al.

find

that

in

their

sample

of

actual

click

data,

the

correlation

between

rank

and

conversion

rate

is

not

statistically

significant.

However,

for

the

purposes

of

our

model

it

is

also

important

that

bidders

believe

that

conversion

rate

does

not

vary

with

rank.

variables

and

we

denote

their

realizations

by

xi,

αi,

and

ri

respectively.

The

probability

that

an

advertisement

will

be

viewed

if

placed

in

slot

j

is

γj

We

assume

γ1

γ2

γK.

Hence

bidder

i"s

advertisement

will

have

a

clickthrough

rate

of

γjαi

if

placed

in

slot

j.

Of

course,

an

advertisement

does

not

receive

any

clicks

if

it

is

not

allocated

a

slot.

Each

bidder"s

value

and

relevance

pair

(Xi,

Ai)

is

independently

and

identically

distributed

on

¯x]

according

to

a

continuous

density

function

f

that

has

full

support

on

its

domain.

The

density

f

and

slot

probabilities

γK

are

common

knowledge.

Only

bidder

i

knows

the

realization

xi

of

its

value

per

click

Xi.

Both

bidder

i

and

the

seller

know

the

realization

αi

of

Ai,

but

this

realization

remains

unobservable

to

the

other

bidders.

We

assume

that

bidders

have

quasi-linear

utility

functions.

That

is,

the

expected

utility

to

bidder

i

of

obtaining

the

slot

of

rank

j

at

a

price

of

b

per

click

is

ui(j,

b)

γjαi(xi

b)

If

the

advertising

firms

bidding

in

the

slot

auction

are

riskneutral

and

have

ample

liquidity,

quasi-linearity

is

a

reasonable

assumption.

The

assumptions

of

independence,

symmetry,

and

riskneutrality

made

above

are

all

quite

standard

in

single-item

auction

theory

The

assumption

that

clickthrough

rate

decays

monotonically

with

lower

slots-by

the

same

factors

for

each

agent-is

unique

to

the

slot

auction

problem.

We

view

it

as

a

main

contribution

of

our

work

to

show

that

this

assumption

allows

for

tractable

analysis

of

the

slot

auction

problem

using

standard

tools

from

singleitem

auction

theory.

It

also

allows

for

interesting

results

in

the

complete

information

case.

A

common

model

of

decaying

clickthrough

rate

is

the

exponential

decay

model,

where

γk

δk−1

with

decay

δ

Feng

et

al.

state

that

their

actual

clickthrough

data

is

fitted

extremely

well

by

an

exponential

decay

model

with

δ

Our

model

lacks

budget

constraints,

which

are

an

important

feature

of

real

slot

auctions.

With

budget

constraints

keyword

auctions

cannot

be

considered

independently

of

one

another,

because

the

budget

must

be

allocated

across

multiple

keywords-a

single

advertiser

typically

bids

on

multiple

keywords

relevant

to

his

business.

Introducing

this

element

into

the

model

is

an

important

next

step

for

future

Auction

Formats

In

a

slot

auction

a

bidder

provides

to

the

seller

a

declared

value

per

click

˜xi(xi,

αi)

which

depends

on

his

true

value

and

relevance.

We

often

denote

this

declared

value

(bid)

by

˜xi

for

short.

Since

a

bidder"s

relevance

αi

is

observable

to

the

seller,

the

bidder

cannot

misrepresent

it.

We

denote

the

kth

highest

of

the

N

declared

values

by

˜x(k)

and

the

kth

highest

of

the

N

declared

revenues

by

˜r(k)

where

the

declared

revenue

of

bidder

i

is

˜ri

αi

˜xi.

We

consider

two

types

of

allocation

rules,

rank

by

bid

(RBB)

and

rank

by

revenue

(RBR):

Models

with

budget

constraints

have

begun

to

appear

in

this

research

area.

Abrams

and

Borgs

et

al.

design

multi-unit

auctions

for

budget-constrained

bidders,

which

can

be

interpreted

as

slot

auctions,

with

a

focus

on

revenue

optimization

and

truthfulness.

Mehta

et

al.

address

the

problem

of

matching

user

queries

to

budget-constrained

advertisers

so

as

to

maximize

revenue.

RBB.

Slot

k

goes

to

bidder

i

if

and

only

if

˜xi

˜x(k)

RBR.

Slot

k

goes

to

bidder

i

if

and

only

if

˜ri

˜r(k)

We

will

commonly

represent

an

allocation

by

a

one-to-one

function

σ

[K]

[N],

where

[n]

is

the

set

of

integers

n}.

Hence

slot

k

goes

to

bidder

σ(k).

We

also

consider

two

different

types

of

payment

rules.

Note

that

no

matter

what

the

payment

rule,

a

bidder

that

is

not

allocated

a

slot

will

pay

since

his

listing

cannot

receive

any

clicks.

First-price.

The

bidder

allocated

slot

k,

namely

σ(k),

pays

˜xσ(k)

per

click

under

both

the

RBB

and

RBR

allocation

rules.

Second-price.

If

k

N,

bidder

σ(k)

pays

per

click

under

the

RBB

rule,

and

pays

per

click

under

the

RBR

rule.

If

k

N,

bidder

σ(k)

pays

per

Intuitively,

a

second-price

payment

rule

sets

a

bidder"s

payment

to

the

lowest

bid

it

could

have

declared

while

maintaining

the

same

ranking,

given

the

allocation

rule

used.

Overture

introduced

the

first

slot

auction

design

in

using

a

first-price

RBB

scheme.

Google

then

followed

in

with

a

second-price

RBR

scheme.

In

Overture

(at

this

point

acquired

by

Yahoo!)

then

switched

to

second

pricing

but

still

allocates

using

RBB.

One

possible

reason

for

the

switch

is

given

in

Section

We

assume

that

ties

are

broken

as

follows

in

the

event

that

two

agents

make

the

exact

same

bid

or

declare

the

same

revenue.

There

is

a

permutation

of

the

agents

κ

[N]

[N]

that

is

fixed

beforehand.

If

the

bids

of

agents

i

and

j

are

tied,

then

agent

i

obtains

a

higher

slot

if

and

only

if

κ(i)

κ(j).

This

is

consistent

with

the

practice

in

real

slot

auctions

where

ties

are

broken

by

the

bidders"

order

of

arrival.

INCOMPLETE

INFORMATION

Incentives

It

should

be

clear

that

with

a

first-price

payment

rule,

truthful

bidding

is

neither

a

dominant

strategy

nor

an

ex

post

Nash

equilibrium

using

either

RBB

or

RBR,

because

this

guarantees

a

payoff

of

There

is

always

an

incentive

to

shade

true

values

with

first

pricing.

The

second-price

payment

rule

is

reminiscent

of

the

secondprice

(Vickrey)

auction

used

for

selling

a

single

item,

and

in

a

Vickrey

auction

it

is

a

dominant

strategy

for

a

bidder

to

reveal

his

true

value

for

the

item

However,

using

a

second-price

rule

in

a

slot

auction

together

with

either

allocation

rule

above

does

not

yield

an

incentive-compatible

mechanism,

either

in

dominant

strategies

or

ex

post

Nash

With

a

second-price

rule

there

is

no

incentive

for

a

bidder

to

bid

higher

than

his

true

value

per

click

using

either

RBB

or

RBR:

this

either

leads

to

no

change

We

are

effectively

assuming

a

reserve

price

of

zero,

but

in

practice

search

engines

charge

a

non-zero

reserve

price

per

click.

Unless

of

course

there

is

only

a

single

slot

available,

since

this

is

the

single-item

case.

With

a

single

slot

both

RBB

and

RBR

with

a

second-price

payment

rule

are

dominantstrategy

incentive-compatible.

in

the

outcome,

or

a

situation

in

which

he

will

have

to

pay

more

than

his

value

per

click

for

each

click

received,

resulting

in

a

negative

However,

with

either

allocation

rule

there

may

be

an

incentive

to

shade

true

values

with

second

pricing.

Claim

With

second

pricing

and

K

truthful

bidding

is

not

a

dominant

strategy

nor

an

ex

post

Nash

equilibrium

for

either

RBB

or

RBR.

Example.

There

are

two

agents

and

two

slots.

The

agents

have

relevance

α1

α2

whereas

γ1

and

γ2

Agent

has

a

value

of

per

click,

and

agent

has

a

value

of

per

click.

Let

us

first

consider

the

RBB

rule.

Suppose

agent

bids

truthfully.

If

agent

also

bids

truthfully,

he

wins

the

first

slot

and

obtains

a

payoff

of

However,

if

he

shades

his

bid

down

below

he

obtains

the

second

slot

at

a

cost

of

per

click

yielding

a

payoff

of

Since

the

agents

have

equal

relevance,

the

exact

same

situation

holds

with

the

RBR

rule.

Hence

truthful

bidding

is

not

a

dominant

strategy

in

either

format,

and

neither

is

it

an

ex

post

Nash

equilibrium.

To

find

payments

that

make

RBB

and

RBR

dominantstrategy

incentive-compatible,

we

can

apply

Holmstrom"s

lemma

(see

also

chapter

in

Milgrom

Under

the

restriction

that

a

bidder

with

value

per

click

does

not

pay

anything

(even

if

he

obtains

a

slot,

which

can

occur

if

there

are

as

many

slots

as

bidders),

this

lemma

implies

that

there

is

a

unique

payment

rule

that

achieves

dominant-strategy

incentive

compatibility

for

either

allocation

rule.

For

RBB,

the

bidder

allocated

slot

k

is

charged

per

click

KX

γi)˜x(i)

γK

Note

that

if

K

N,

since

there

is

no

K

bidder.

For

RBR,

the

bidder

allocated

slot

k

is

charged

per

click

ασ(k)

KX

γi)˜r(i)

γK

Using

payment

rule

and

RBR,

the

auctioneer

is

aware

of

the

true

revenues

of

the

bidders

(since

they

reveal

their

values

truthfully),

and

hence

ranks

them

according

to

their

true

revenues.

We

show

in

Section

that

this

allocation

is

in

fact

efficient.

Since

the

VCG

mechanism

is

the

unique

mechanism

that

is

efficient,

truthful,

and

ensures

bidders

with

value

pay

nothing

(by

the

Green-Laffont

theorem

the

RBR

rule

and

payment

scheme

constitute

exactly

the

VCG

mechanism.

In

the

VCG

mechanism

an

agent

pays

the

externality

he

imposes

on

others.

To

understand

payment

in

this

sense,

note

that

the

first

term

is

the

added

utility

(due

to

an

increased

clickthrough

rate)

agents

in

slots

k

to

K

would

receive

if

they

were

all

to

move

up

a

slot;

the

last

term

is

the

utility

that

the

agent

with

the

K

revenue

would

receive

by

obtaining

the

last

slot

as

opposed

to

nothing.

The

leading

coefficient

simply

reduces

the

agent"s

expected

payment

to

a

payment

per

click.

In

a

dynamic

setting

with

second

pricing,

there

may

be

an

incentive

to

bid

higher

than

one"s

true

value

in

order

to

exhaust

competitors"

budgets.

This

phenomenon

is

commonly

called

bid

jamming

or

antisocial

bidding

Equilibrium

Analysis

To

understand

the

efficiency

and

revenue

properties

of

the

various

auction

formats,

we

must

first

understand

which

rankings

of

the

bidders

occur

in

equilibrium

with

different

allocation

and

payment

rule

combinations.

The

following

lemma

essentially

follows

from

the

Monotonic

Selection

Theorem

by

Milgrom

and

Shannon

Lemma

In

a

RBB

(RBR)

auction

with

either

a

firstor

second-price

payment

rule,

the

symmetric

Bayes-Nash

equilibrium

bid

is

strictly

increasing

with

value

revenue).

As

a

consequence

of

this

lemma,

we

find

that

RBB

and

RBR

auctions

allocate

the

slots

greedily

by

the

true

values

and

revenues

of

the

agents,

respectively

(whether

using

firstor

second-price

payment

rules).

This

will

be

relevant

in

Section

below.

For

a

first-price

payment

rule,

we

can

explicitly

derive

the

symmetric

Bayes-Nash

equilibrium

bid

functions

for

RBB

and

RBR

auctions.

The

purpose

of

this

exercise

is

to

lend

qualitative

insights

into

the

parameters

that

influence

an

agent"s

bidding,

and

to

derive

formulae

for

the

expected

revenue

in

RBB

and

RBR

auctions

in

order

to

make

a

revenue

ranking

of

these

two

allocation

rules

(in

Section

Let

G(y)

be

the

expected

resulting

clickthrough

rate,

in

a

symmetric

equilibrium

of

the

RBB

auction

(with

either

payment

rule),

to

a

bidder

with

value

y

and

relevance

α

Let

H(y)

be

the

analogous

quantity

for

a

bidder

with

revenue

y

and

relevance

in

a

RBR

auction.

By

Lemma

a

bidder

with

value

y

will

obtain

slot

k

in

a

RBB

auction

if

y

is

the

kth

highest

of

the

true

realized

values.

The

same

applies

in

a

RBR

auction

when

y

is

the

kth

highest

of

the

true

realized

revenues.

Let

FX

(y)

be

the

distribution

function

for

value,

and

let

FR(y)

be

the

distribution

function

for

revenue.

The

probability

that

y

is

the

kth

highest

out

of

N

values

is

N

k

FX

FX

(y)N−k

whereas

the

probability

that

y

is

the

kth

highest

out

of

N

revenues

is

the

same

formula

with

FR

replacing

FX

Hence

we

have

G(y)

KX

γk

N

k

FX

FX

(y)N−k

The

H

function

is

analogous

to

G

with

FR

replacing

FX

In

the

two

propositions

that

follow,

g

and

h

are

the

derivatives

of

G

and

H

respectively.

We

omit

the

proof

of

the

next

proposition,

because

it

is

almost

identical

to

the

derivation

of

the

equilibrium

bid

in

the

single-item

case

(see

Krishna

Proposition

Proposition

The

symmetric

Bayes-Nash

equilibrium

strategies

in

a

first-price

RBB

auction

are

given

by

˜xB

(x,

α)

G(x)

Z

x

y

g(y)dy

The

first-price

equilibrium

above

closely

parallels

the

firstprice

equilibrium

in

the

single-item

model.

With

a

single

item

g

is

the

density

of

the

second

highest

value

among

all

N

agent

values,

whereas

in

a

slot

auction

it

is

a

weighted

combination

of

the

densities

for

the

second,

third,

etc.

highest

values.

Note

that

the

symmetric

Bayes-Nash

equilibrium

bid

in

a

first-price

RBB

auction

does

not

depend

on

a

bidder"s

relevance

α.

To

see

clearly

why,

note

that

a

bidder

chooses

a

bid

b

so

as

to

maximize

the

objective

(b))(x

b)

and

here

α

is

just

a

leading

constant

factor.

So

dropping

it

does

not

change

the

set

of

optimal

solutions.

Hence

the

equilibrium

bid

depends

only

on

the

value

x

and

function

G,

and

G

in

turn

depends

only

on

the

marginal

cumulative

distribution

of

value

FX

So

really

only

the

latter

needs

to

be

common

knowledge

to

the

bidders.

On

the

other

hand,

we

will

now

see

that

information

about

relevance

is

needed

for

bidders

to

play

the

equilibrium

in

the

first-price

RBR

auction.

So

the

informational

requirements

for

a

first-price

RBB

auction

are

much

weaker

than

for

a

first-price

RBR

auction:

in

the

RBB

auction

a

bidder

need

not

know

his

own

relevance,

and

need

not

know

any

distributional

information

over

others"

relevance

in

order

to

play

the

equilibrium.

Again

we

omit

the

next

proposition"s

proof

since

it

is

so

similar

to

the

one

above.

Proposition

The

symmetric

Bayes-Nash

equilibrium

strategies

in

a

first-price

RBR

auction

are

given

by

˜xR

(x,

α)

αH(αx)

Z

αx

y

h(y)

dy

Here

it

can

be

seen

that

the

equilibrium

bid

is

increasing

with

x,

but

not

necessarily

with

α.

This

should

not

be

much

of

a

concern

to

the

auctioneer,

however,

because

in

any

case

the

declared

revenue

in

equilibrium

is

always

increasing

in

the

true

revenue.

It

would

be

interesting

to

obtain

the

equilibrium

bids

when

using

a

second-price

payment

rule,

but

it

appears

that

the

resulting

differential

equations

for

this

case

do

not

have

a

neat

analytical

solution.

Nonetheless,

the

same

conclusions

about

the

informational

requirements

of

the

RBB

and

RBR

rules

still

hold,

as

can

be

seen

simply

by

inspecting

the

objective

function

associated

with

an

agent"s

bidding

problem

for

the

second-price

case.

Efficiency

A

slot

auction

is

efficient

if

in

equilibrium

the

sum

of

the

bidders"

revenues

from

their

allocated

slots

is

maximized.

Using

symmetry

as

our

equilibrium

selection

criterion,

we

find

that

the

RBB

auction

is

not

efficient

with

either

payment

rule.

Claim

The

RBB

auction

is

not

efficient

with

either

first

or

second

pricing.

Example.

There

are

two

agents

and

one

slot,

with

γ1

Agent

has

a

value

of

per

click

and

relevance

α1

Agent

has

a

value

of

per

click

and

relevance

α2

By

Lemma

agents

are

ranked

greedily

by

value.

Hence

agent

obtains

the

lone

slot,

for

a

total

revenue

of

to

the

agents.

However,

it

is

most

efficient

to

allocate

the

slot

to

agent

for

a

total

revenue

of

Examples

with

more

agents

or

more

slots

are

simple

to

construct

along

the

same

lines.

On

the

other

hand,

under

our

assumptions

on

how

clickthrough

rate

decreases

with

lower

rank,

the

RBR

auction

is

efficient

with

either

payment

rule.

Theorem

The

RBR

auction

is

efficient

with

either

first-

or

second-price

payments

rules.

Proof.

Since

by

Lemma

the

agents"

equilibrium

bids

are

increasing

functions

of

their

revenues

in

the

RBR

auction,

slots

are

allocated

greedily

according

to

true

revenues.

Let

σ

be

a

non-greedy

allocation.

Then

there

are

slots

s,

t

with

s

t

and

rσ(s)

rσ(t).

We

can

switch

the

agents

in

slots

s

and

t

to

obtain

a

new

allocation,

and

the

difference

between

the

total

revenue

in

this

new

allocation

and

the

original

allocation"s

total

revenue

is

γtrσ(s)

γsrσ(t)

γsrσ(s)

γtrσ(t)

(γs

γt)

rσ(t)

rσ(s)

Both

parenthesized

terms

above

are

positive.

Hence

the

switch

has

increased

the

total

revenue

to

the

bidders.

If

we

continue

to

perform

such

switches,

we

will

eventually

reach

a

greedy

allocation

of

greater

revenue

than

the

initial

allocation.

Since

the

initial

allocation

was

arbitrary,

it

follows

that

a

greedy

allocation

is

always

efficient,

and

hence

the

RBR

auction"s

allocation

is

efficient.

Note

that

the

assumption

that

clickthrough

rate

decays

montonically

by

the

same

factors

γK

for

all

agents

is

crucial

to

this

result.

A

greedy

allocation

scheme

does

not

necessarily

find

an

efficient

solution

if

the

clickthrough

rates

are

monotonically

decreasing

in

an

independent

fashion

for

each

agent.

Revenue

To

obtain

possible

revenue

rankings

for

the

different

auction

formats,

we

first

note

that

when

the

allocation

rule

is

fixed

to

RBB,

then

using

either

a

first-price,

second-price,

or

truthful

payment

rule

leads

to

the

same

expected

revenue

in

a

symmetric,

increasing

Bayes-Nash

equilibrium.

Because

a

RBB

auction

ranks

agents

by

their

true

values

in

equilibrium

for

any

of

these

payment

rules

(by

Lemma

it

follows

that

expected

revenue

is

the

same

for

all

these

payment

rules,

following

arguments

that

are

virtually

identical

to

those

used

to

establish

revenue

equivalence

in

the

singleitem

case

(see

e.g.

Proposition

in

Krishna

The

same

holds

for

RBR

auctions;

however,

the

revenue

ranking

of

the

RBB

and

RBR

allocation

rules

is

still

unclear.

Because

of

this

revenue

equivalence

principle,

we

can

choose

whichever

payment

rule

is

most

convenient

for

the

purpose

of

making

revenue

comparisons.

Using

Propositions

and

it

is

a

simple

matter

to

derive

formulae

for

the

expected

revenue

under

both

allocation

rules.

The

payment

of

an

agent

in

a

RBB

auction

is

mB

(x,

α)

αG(x)˜xV

(x,

α)

The

expected

revenue

is

then

N

E

ˆ

mV

(X,

A)

where

the

expectation

is

taken

with

respect

to

the

joint

density

of

value

and

relevance.

The

expected

revenue

formula

for

RBR

auctions

is

entirely

analogous

using

˜xR

(x,

α)

and

the

H

function.

With

these

in

hand

we

can

obtain

revenue

rankings

for

specific

numbers

of

bidders

and

slots,

and

specific

distributions

over

values

and

relevance.

Claim

For

fixed

K,

N,

and

fixed

γK,

no

revenue

ranking

of

RBB

and

RBR

is

possible

for

an

arbitrary

density

f.

Example.

Assume

there

are

bidders,

slots,

and

that

γ1

γ2

Assume

that

value-relevance

pairs

are

uniformly

distributed

over

For

such

a

distribution

with

a

closed-form

formula,

it

is

most

convenient

to

use

the

revenue

formulae

just

derived.

RBB

dominates

RBR

in

terms

of

revenue

for

these

parameters.

The

formula

for

the

expected

revenue

in

a

RBB

auction

yields

whereas

for

RBR

auctions

we

have

Assume

instead

that

with

probability

an

agent"s

valuerelevance

pair

is

and

that

with

probability

it

is

In

this

scenario

it

is

more

convenient

to

appeal

to

formulae

and

In

a

truthful

auction

the

second

agent

will

always

pay

According

to

in

a

truthful

RBB

auction

the

first

agent

makes

an

expected

payment

of

E

ˆ

E

ˆ

E

ˆ

where

we

have

used

the

fact

that

value

and

relevance

are

independently

distributed

for

different

agents.

The

expected

relevance

of

the

agent

with

the

highest

value

is

E

ˆ

The

expected

second

highest

value

is

also

E

ˆ

The

expected

revenue

for

a

RBB

auction

here

is

then

According

to

in

a

truthful

RBR

auction

the

first

agent

makes

an

expected

payment

of

E

ˆ

E

ˆ

In

expectation

the

second

highest

revenue

is

E

ˆ

so

the

expected

revenue

for

a

RBR

auction

is

Hence

in

this

case

the

RBR

auction

yields

higher

expected

This

example

suggests

the

following

conjecture:

when

value

and

relevance

are

either

uncorrelated

or

positively

correlated,

RBB

dominates

RBR

in

terms

of

revenue.

When

value

and

relevance

are

negatively

correlated,

RBR

dominates.

COMPLETE

INFORMATION

In

typical

slot

auctions

such

as

those

run

by

Yahoo!

and

Google,

bidders

can

adjust

their

bids

up

or

down

at

any

time.

As

B¨orgers

et

al.

and

Edelman

et

al.

have

noted,

this

can

be

viewed

as

a

continuous-time

process

in

which

bidders

learn

each

other"s

bids.

If

the

process

stabilizes

the

result

can

then

be

modeled

as

a

Nash

equilibrium

in

pure

strategies

of

the

static

one-shot

game

of

complete

information,

since

each

bidder

will

be

playing

a

best-response

to

the

others"

This

argument

seems

especially

appropriate

for

Yahoo!"s

slot

auction

design

where

all

bids

are

To

be

entirely

rigorous

and

consistent

with

our

initial

assumptions,

we

should

have

constructed

a

continuous

probability

density

with

full

support

over

an

appropriate

domain.

Taking

the

domain

to

be

e.g.

and

a

continuous

density

with

full

support

that

is

sufficiently

concentrated

around

and

with

roughly

equal

mass

around

both,

would

yield

the

same

conclusion.

Claim

should

serve

as

a

word

of

caution,

because

Feng

et

al.

find

through

their

simulations

that

with

a

bivariate

normal

distribution

over

value-relevance

pairs,

and

with

slots,

bidders,

and

δ

RBR

dominates

RBB

in

terms

of

revenue

for

any

level

of

correlation

between

value

and

relevance.

However,

they

assume

that

bidding

behavior

in

a

second-price

slot

auction

can

be

well

approximated

by

truthful

bidding.

We

do

not

claim

that

bidders

will

actually

learn

each

others"

private

information

(value

and

relevance),

just

that

for

a

stable

set

of

bids

there

is

a

corresponding

equilibrium

of

the

complete

information

game.

made

public.

Google

keeps

bids

private,

but

experimentation

can

allow

one

to

discover

other

bids,

especially

since

second

pricing

automatically

reveals

to

an

agent

the

bid

of

the

agent

ranked

directly

below

him.

Equilibrium

Analysis

In

this

section

we

ask

whether

a

pure-strategy

Nash

equilibrium

exists

in

a

RBB

or

RBR

slot

auction,

with

either

first

or

second

pricing.

Before

dealing

with

the

first-price

case

there

is

a

technical

issue

involving

ties.

In

our

model

we

allow

bids

to

be

nonnegative

real

numbers

for

mathematical

convenience,

but

this

can

become

problematic

because

there

is

then

no

bid

that

is

just

higher

than

another.

We

brush

over

such

issues

by

assuming

that

an

agent

can

bid

infinitesimally

higher

than

another.

This

is

imprecise

but

allows

us

to

focus

on

the

intuition

behind

the

result

that

follows.

See

Reny

for

a

full

treatment

of

such

issues.

For

the

remainder

of

the

paper,

we

assume

that

there

are

as

many

slots

as

bidders.

The

following

result

shows

that

there

can

be

no

pure-strategy

Nash

equilibrium

with

first

Note

that

the

argument

holds

for

both

RBB

and

RBR

allocation

rules.

For

RBB,

bids

should

be

interpreted

as

declared

values,

and

for

RBR

as

declared

revenues.

Theorem

There

exists

no

complete

information

Nash

equilibrium

in

pure

strategies

in

the

first-price

slot

auction,

for

any

possible

values

of

the

agents,

whether

using

a

RBB

or

RBR

allocation

rule.

Proof.

Let

σ

[K]

[N]

be

the

allocation

of

slots

to

the

agents

resulting

from

their

bids.

Let

ri

and

bi

be

the

revenue

and

bid

of

the

agent

ranked

ith

respectively.

Note

that

we

cannot

have

bi

or

else

the

agent

in

slot

i

can

make

a

profitable

deviation

by

instead

bidding

bi

for

small

enough

This

does

not

change

its

allocation,

but

increases

its

profit.

Hence

we

must

have

bi

(i.e.

with

one

bidder

bidding

infinitesimally

higher

than

the

other).

Since

this

holds

for

any

two

consecutive

bidders,

it

follows

that

in

a

Nash

equilibrium

all

bidders

must

be

bidding

(since

the

bidder

ranked

last

matches

the

bid

directly

below

him,

which

is

by

default

because

there

is

no

such

bid).

But

this

is

impossible:

consider

the

bidder

ranked

last.

The

identity

of

this

bidder

is

always

clear

given

the

deterministic

tie-breaking

rule.

This

bidder

can

obtain

the

top

spot

and

increase

his

revenue

by

−γK)rK

by

bidding

some

and

for

small

enough

this

is

necessarily

a

profitable

deviation.

Hence

there

is

no

Nash

equilibrium

in

pure

strategies.

On

the

other

hand,

we

find

that

in

a

second-price

slot

auction

there

can

be

a

multitude

of

pure

strategy

Nash

equilibria.

The

next

two

lemmas

give

conditions

that

characterize

the

allocations

that

can

occur

as

a

result

of

an

equilibrium

profile

of

bids,

given

fixed

agent

values

and

revenues.

Then

if

we

can

exhibit

an

allocation

that

satisfies

these

conditions,

there

must

exist

at

least

one

equilibrium.

We

first

consider

the

RBR

case.

B¨orgers

et

al.

have

proven

this

result

in

a

model

with

three

bidders

and

three

slots,

and

we

generalize

their

argument.

Edelman

et

al.

also

point

out

this

non-existence

phenomenon.

They

only

illustrate

the

fact

with

an

example

because

the

result

is

quite

immediate.

Lemma

Given

an

allocation

σ,

there

exists

a

Nash

equilibrium

profile

of

bids

b

leading

to

σ

in

a

second-price

RBR

slot

auction

if

and

only

if

γi

rσ(i)

rσ(j)

for

j

N

and

i

j

Proof.

There

exists

a

desired

vector

b

which

constitutes

a

Nash

equilibrium

if

and

only

if

the

following

set

of

inequalities

can

be

satisfied

(the

variables

are

the

πi

and

bj):

πi

γj(rσ(i)

bj)

∀i,

∀j

i

πi

γj(rσ(i)

∀i,

∀j

i

πi

γi(rσ(i)

∀i

bi

i

N

πi

bi

∀i

Here

rσ(i)

is

the

revenue

of

the

agent

allocated

slot

i,

and

πi

and

bi

may

be

interpreted

as

this

agent"s

surplus

and

declared

revenue,

respectively.

We

first

argue

that

constraints

can

be

removed,

because

the

inequalities

above

can

be

satisfied

if

and

only

if

the

inequalities

without

can

be

satisfied.

The

necessary

direction

is

immediate.

Assume

we

have

a

vector

(π,

b)

which

satisfies

all

inequalities

above

except

Then

there

is

some

i

for

which

bi

Construct

a

new

vector

(π,

b

identical

to

the

original

except

with

bi.

We

now

have

bi

An

agent

in

slot

k

i

sees

the

price

of

slot

i

decrease

from

to

bi,

but

this

does

not

make

i

more

preferred

than

k

to

this

agent

because

we

have

πk

bi)

γi(rσ(k)

bi)

γi(rσ(k)

(i.e.

because

the

agent

in

slot

k

did

not

originally

prefer

slot

i

at

price

bi,

he

will

not

prefer

slot

i

at

price

bi).

A

similar

argument

applies

for

agents

in

slots

k

i

The

agent

in

slot

i

sees

the

price

of

this

slot

go

down,

which

only

makes

it

more

preferred.

Finally,

the

agent

in

slot

i

sees

no

change

in

the

price

of

any

slot,

so

his

slot

remains

most

preferred.

Hence

inequalities

remain

valid

at

(π,

b

We

first

make

this

change

to

the

where

bi

and

index

i

is

smallest.

We

then

recursively

apply

the

change

until

we

eventually

obtain

a

vector

that

satisfies

all

inequalities.

We

safely

ignore

inequalities

from

now

on.

By

the

Farkas

lemma,

the

remaining

inequalities

can

be

satisfied

if

and

only

if

there

is

no

vector

z

such

that

X

i,j

(γj

rσ(i))

zσ(i)j

X

i>j

γjzσ(i)j

X

i<j

∀j

X

j

zσ(i)j

∀i

zσ(i)j

∀i,

∀j

i

zσ(i)i

free

∀i

Note

that

a

variable

of

the

form

zσ(i)i

appears

at

most

once

in

a

constraint

of

type

so

such

a

variable

can

never

be

positive.

Also,

for

all

i

by

constraint

since

such

variables

never

appear

with

another

of

the

form

zσ(i)i.

Now

if

we

wish

to

raise

zσ(i)j

above

by

one

unit

for

j

i,

we

must

lower

zσ(i)i

by

one

unit

because

of

the

constraint

of

type

Because

γjrσ(i)

γirσ(i)

for

i

j,

raising

zσ(i)j

with

i

j

while

adjusting

other

variables

to

maintain

feasibility

cannot

make

the

objective

P

i,j(γjrσ(i))zσ(i)j

positive.

If

this

objective

is

positive,

then

this

is

due

to

some

component

zσ(i)j

with

i

j

being

positive.

Now

for

the

constraints

of

type

if

i

j

then

zσ(i)j

appears

with

(for

j

N).

So

to

raise

the

former

variable

γ−1

j

units

and

maintain

feasibility,

we

must

(I)

lower

zσ(i)i

by

γ−1

j

units,

and

(II)

lower

by

γ−1

units.

Hence

if

the

following

inequalities

hold:

rσ(i)

γi

γj

rσ(i)

for

j

N

and

i

j,

raising

some

zσ(i)j

with

i

j

cannot

make

the

objective

positive,

and

there

is

no

z

that

satisfies

all

inequalities

above.

Conversely,

if

some

inequality

does

not

hold,

the

objective

can

be

made

positive

by

raising

the

corresponding

zσ(i)j

and

adjusting

other

variables

so

that

feasibility

is

just

maintained.

By

a

slight

reindexing,

inequalities

yield

the

statement

of

the

theorem.

The

RBB

case

is

entirely

analogous.

Lemma

Given

an

allocation

σ,

there

exists

a

Nash

equilibrium

profile

of

bids

b

leading

to

σ

in

a

second-price

RBB

slot

auction

if

and

only

if

γi

xσ(i)

xσ(j)

for

j

N

and

i

j

Proof

Sketch.

The

proof

technique

is

the

same

as

in

the

previous

lemma.

The

desired

Nash

equilibrium

exists

if

and

only

if

a

related

set

of

inequalities

can

be

satisfied;

by

the

Farkas

lemma,

this

occurs

if

and

only

if

an

alternate

set

of

inequalities

cannot

be

satisfied.

The

conditions

that

determine

whether

the

latter

holds

are

given

in

the

statement

of

the

lemma.

The

two

lemmas

above

immediately

lead

to

the

following

result.

Theorem

There

always

exists

a

complete

information

Nash

equilibrium

in

pure

strategies

in

the

second-price

RBB

slot

auction.

There

always

exists

an

efficient

complete

information

Nash

equilibrium

in

pure

strategies

in

the

secondprice

RBR

slot

auction.

Proof.

First

consider

RBB.

Suppose

agents

are

ranked

according

to

their

true

values.

Since

xσ(i)

xσ(j)

for

i

j,

the

system

of

inequalities

in

Lemma

is

satisfied,

and

the

allocation

is

the

result

of

some

Nash

equilibrium

bid

profile.

By

the

same

type

of

argument

but

appealing

to

Lemma

for

RBR,

there

exists

a

Nash

equilibrium

bid

profile

such

that

bidders

are

ranked

according

to

their

true

revenues.

By

Theorem

this

latter

allocation

is

efficient.

This

theorem

establishes

existence

but

not

uniqueness.

Indeed

we

expect

that

in

many

cases

there

will

be

multiple

allocations

(and

hence

equilibria)

which

satisfy

the

conditions

of

Lemmas

and

In

particular,

not

all

equilibria

of

a

second-price

RBR

auction

will

be

efficient.

For

instance,

according

to

Lemma

with

two

agents

and

two

slots

any

allocation

can

arise

in

a

RBR

equilibrium

because

no

constraints

apply.

Theorems

and

taken

together

provide

a

possible

explanation

for

Yahoo!"s

switch

from

first

to

second

pricing.

We

saw

in

Section

that

this

does

not

induce

truthfulness

from

bidders.

With

first

pricing,

there

will

always

be

some

bidder

that

feels

compelled

to

adjust

his

bid.

Second

pricing

is

more

convenient

because

an

equilibrium

can

be

reached,

and

this

reduces

the

cost

of

bid

management.

Efficiency

For

a

given

allocation

rule,

we

call

the

allocation

that

would

result

if

the

bidders

reported

their

values

truthfully

the

standard

allocation.

Hence

in

the

standard

RBB

allocation

bidders

are

ranked

by

true

values,

and

in

the

standard

RBR

allocation

they

are

ranked

by

true

revenues.

According

to

Lemmas

and

a

ranking

that

results

from

a

Nash

equilibrium

profile

can

only

deviate

from

the

standard

allocation

by

having

agents

with

relatively

similar

values

or

revenues

switch

places.

That

is,

if

ri

rj

then

with

RBR

agent

j

can

be

ranked

higher

than

i

only

if

the

ratio

rj/ri

is

sufficiently

large;

similarly

for

RBB.

This

suggests

that

the

value

of

an

equilibrium

allocation

cannot

differ

too

much

from

the

value

obtained

in

the

standard

allocation,

and

the

following

theorems

confirms

this.

For

an

allocation

σ

of

slots

to

agents,

we

denote

its

total

value

by

f(σ)

PN

γirσ(i).

We

denote

by

g(σ)

PN

γixσ(i)

allocation

σ"s

value

when

assuming

all

agents

have

identical

relevance,

normalized

to

Let

L

min

min

γi

ff

(where

by

default

Let

ηx

and

ηr

be

the

standard

allocations

when

using

RBB

and

RBR,

respectively.

Theorem

For

an

allocation

σ

that

results

from

a

purestrategy

Nash

equilibrium

of

a

second-price

RBR

slot

auction,

we

have

f(σ)

Lf(ηr).

Proof.

We

number

the

agents

so

that

agent

i

has

the

ith

highest

revenue,

so

rN

Hence

the

standard

allocation

has

value

f(ηr)

PN

γiri.

To

prove

the

theorem,

we

will

make

repeated

use

of

the

fact

thatP

k

akP

k

bk

mink

ak

bk

when

the

ak

and

bk

are

positive.

Note

that

according

to

Lemma

if

agent

i

lies

at

least

two

slots

below

slot

j,

then

rσ(j)

ri

It

may

be

the

case

that

for

some

slot

i,

we

have

σ(i)

i

and

for

slots

k

i

we

have

σ(k)

i.

We

then

say

that

slot

i

is

inverted.

Let

S

be

the

set

of

agents

with

indices

at

least

i

there

are

N

i

of

these.

If

slot

i

is

inverted,

it

is

occupied

by

some

agent

from

S.

Also

all

slots

strictly

lower

than

i

must

be

occupied

by

the

remaining

agents

from

S,

since

σ(k)

i

for

k

i

The

agent

in

slot

i

must

then

have

an

index

σ(i

i

(note

this

means

slot

i

cannot

be

inverted).

Now

there

are

two

cases.

In

the

first

case

we

have

σ(i)

i

Then

γirσ(i)

γiri

γiri

min

γi

γi

ff

γi

In

the

second

case

we

have

σ(i)

Then

since

all

agents

in

S

except

the

one

in

slot

i

lie

strictly

below

slot

i

and

the

agent

in

slot

i

is

not

agent

i

it

must

be

that

agent

is

in

a

slot

strictly

below

slot

This

means

that

it

is

at

least

two

slots

below

the

agent

that

actually

occupies

slot

i,

and

by

Lemma

we

then

have

rσ(i)

Thus,

γirσ(i)

γiri

γirσ(i)

γiri

min

γi

ff

If

slot

i

is

not

inverted,

then

on

one

hand

we

may

have

σ(i)

i,

in

which

case

rσ(i)/ri

On

the

other

hand

we

may

have

σ(i)

i

but

there

is

some

agent

with

index

j

i

that

lies

at

least

two

slots

below

slot

i.

Then

by

Lemma

rσ(i)

rj

ri

We

write

i

I

if

slot

i

is

inverted,

and

i

I

if

neither

i

nor

i

are

inverted.

By

our

arguments

above

two

consecutive

slots

cannot

be

inverted,

so

we

can

write

f(σ)

f(γr)

P

i∈I

γirσ(i)

P

i∈I

γirσ(i)

P

i∈I

(γiri

P

i∈I

γiri

min

min

i∈I

γirσ(i)

γiri

ff

min

i∈I

γirσ(i)

γiri

ffff

L

and

this

completes

the

proof.

Note

that

for

RBR,

the

standard

value

is

also

the

efficient

value

by

Theorem

Also

note

that

for

an

exponential

decay

model,

L

min

˘1

δ

δ

With

δ

(see

Section

the

factor

is

L

so

the

total

value

in

a

pure-strategy

Nash

equilibrium

of

a

second-price

RBR

slot

auction

is

always

within

a

factor

of

of

the

efficient

value

with

such

a

discount.

Again

for

RBB

we

have

an

analogous

result.

Theorem

For

an

allocation

σ

that

results

from

a

purestrategy

Nash

equilibrium

of

a

second-price

RBB

slot

auction,

we

have

g(σ)

Lg(ηx).

Proof

Sketch.

Simply

substitute

bidder

values

for

bidder

revenues

in

the

proof

of

Theorem

and

appeal

to

Lemma

CONCLUSIONS

This

paper

analyzed

stylized

versions

of

the

slot

auction

designs

currently

used

by

Yahoo!

and

Google,

namely

rank

by

bid

(RBB)

and

rank

by

revenue

(RBR),

respectively.

We

also

considered

first

and

second

pricing

rules

together

with

each

of

these

allocation

rules,

since

both

have

been

used

historically.

We

first

studied

the

short-run

setting

with

incomplete

information,

corresponding

to

the

case

where

agents

have

just

approached

the

mechanism.

Our

equilibrium

analysis

revealed

that

RBB

has

much

weaker

informational

requirements

than

RBR,

because

bidders

need

not

know

any

information

about

relevance

(even

their

own)

to

play

the

Bayes-Nash

equilibrium.

However,

RBR

leads

to

an

efficient

allocation

in

equilibrium,

whereas

RBB

does

not.

We

showed

that

for

an

arbitrary

distribution

over

value

and

relevance,

no

revenue

ranking

of

RBB

and

RBR

is

possible.

We

hope

that

the

tools

we

used

to

establish

these

results

(revenue

equivalence,

the

form

of

first-price

equilibria,

the

truthful

payments

rules)

will

help

others

wanting

to

pursue

further

analyses

of

slot

auctions.

We

also

studied

the

long-run

case

where

agents

have

experimented

with

their

bids

and

each

settled

on

one

they

find

optimal.

We

argued

that

a

stable

set

of

bids

in

this

setting

can

be

modeled

as

a

pure-strategy

Nash

equilibrium

of

the

static

game

of

complete

information.

We

showed

that

no

pure-strategy

equilibrium

exists

with

either

RBB

or

RBR

using

first

pricing,

but

that

with

second

pricing

there

always

exists

such

an

equilibrium

(in

the

case

of

RBR,

an

efficient

equilibrium).

In

general

second

pricing

allows

for

multiple

pure-strategy

equilibria,

but

we

showed

that

the

value

of

such

equilibria

diverges

by

only

a

constant

factor

from

the

value

obtained

if

all

agents

bid

truthfully

(which

in

the

case

of

RBR

is

the

efficient

value).

FUTURE

WORK

Introducing

budget

constraints

into

the

model

is

a

natural

next

step

for

future

work.

The

complication

here

lies

in

the

fact

that

budgets

are

often

set

for

entire

campaigns

rather

than

single

keywords.

Assuming

that

the

optimal

choice

of

budget

can

be

made

independent

of

the

choice

of

bid

for

a

specific

keyword,

it

can

be

shown

that

it

is

a

dominant-strategy

to

report

this

optimal

budget

with

one"s

bid.

The

problem

is

then

to

ascertain

that

bids

and

budgets

can

indeed

be

optimized

separately,

or

to

find

a

plausible

model

where

deriving

equilibrium

bids

and

budgets

together

is

tractable.

Identifying

a

condition

on

the

distribution

over

value

and

relevance

that

actually

does

yield

a

revenue

ranking

of

RBB

and

RBR

(such

as

correlation

between

value

and

relevance,

perhaps)

would

yield

a

more

satisfactory

characterization

of

their

relative

revenue

properties.

Placing

bounds

on

the

revenue

obtained

in

a

complete

information

equilibrium

is

also

a

relevant

question.

Because

the

incomplete

information

case

is

such

a

close

generalization

of

the

most

basic

single-item

auction

model,

it

would

be

interesting

to

see

which

standard

results

from

single-item

auction

theory

(e.g.

results

with

risk-averse

bidders,

an

endogenous

number

of

bidders,

asymmetries,

etc...)

automatically

generalize

and

which

do

not,

to

fully

understand

the

structural

differences

between

single-item

and

slot

auctions.

Acknowledgements

David

Pennock

provided

valuable

guidance

throughout

this

project.

I

would

also

like

to

thank

David

Parkes

for

helpful

comments.

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O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O B I O O O O O O B I I I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B I I O O 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O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

alternative slot auction design
ad listing
auction-style mechanism
second-price payment rule
multitude of equilibrium
equilibrium multitude
divergence of value
sponsor search

sponsored search
rank by bid
rank by revenue
incomplete information
second pricing
divergence of economic value
combined market capitalization
resurgent online advertising industry
web search engine
pay per click
search engine
slot allocation
auction theory