On

Cheating

in

Sealed-Bid

Auctions

Ryan

Porter

rwporter@stanford.edu

Yoav

Shoham

shoham@stanford.edu

Computer

Science

Department

Stanford

University

Stanford,

CA

ABSTRACT

Motivated

by

the

rise

of

online

auctions

and

their

relative

lack

of

security,

this

paper

analyzes

two

forms

of

cheating

in

sealed-bid

auctions.

The

first

type

of

cheating

we

consider

occurs

when

the

seller

spies

on

the

bids

of

a

second-price

auction

and

then

inserts

a

fake

bid

in

order

to

increase

the

payment

of

the

winning

bidder.

In

the

second

type,

a

bidder

cheats

in

a

first-price

auction

by

examining

the

competing

bids

before

deciding

on

his

own

bid.

In

both

cases,

we

derive

equilibrium

strategies

when

bidders

are

aware

of

the

possibility

of

cheating.

These

results

provide

insights

into

sealedbid

auctions

even

in

the

absence

of

cheating,

including

some

counterintuitive

results

on

the

effects

of

overbidding

in

a

first-price

auction.

Categories

and

Subject

Descriptors

[Computer

Applications]:

Social

and

Behavioral

Sciences-Economics

General

Terms

Economics,

Security

INTRODUCTION

Among

the

types

of

auctions

commonly

used

in

practice,

sealed-bid

auctions

are

a

good

practical

choice

because

they

require

little

communication

and

can

be

completed

almost

instantly.

Each

bidder

simply

submits

a

bid,

and

the

winner

is

immediately

determined.

However,

sealed-bid

auctions

do

require

that

the

bids

be

kept

private

until

the

auction

clears.

The

increasing

popularity

of

online

auctions

only

makes

this

disadvantage

more

troublesome.

At

an

auction

house,

with

all

participants

present,

it

is

difficult

to

examine

a

bid

that

another

bidder

gave

directly

to

the

auctioneer.

However,

in

an

online

auction

the

auctioneer

is

often

little

more

than

a

server

with

questionable

security;

and,

since

all

participants

are

in

different

locations,

one

can

anonymously

attempt

to

break

into

the

server.

In

this

paper,

we

present

a

game

theoretic

analysis

of

how

bidders

should

behave

when

they

are

aware

of

the

possibility

of

cheating

that

is

based

on

knowledge

of

the

bids.

We

investigate

this

type

of

cheating

along

two

dimensions:

whether

it

is

the

auctioneer

or

a

bidder

who

cheats,

and

which

variant

(either

first

or

second-price)

of

the

sealed-bid

auction

is

used.

Note

that

two

of

these

cases

are

trivial.

In

our

setting,

there

is

no

incentive

for

the

seller

to

submit

a

shill

bid

in

a

first

price

auction,

because

doing

so

would

either

cancel

the

auction

or

not

affect

the

payment

of

the

winning

bidder.

In

a

second-price

auction,

knowing

the

competing

bids

does

not

help

a

bidder

because

it

is

dominant

strategy

to

bid

truthfully.

This

leaves

us

with

two

cases

that

we

examine

in

detail.

A

seller

can

profitably

cheat

in

a

second-price

auction

by

looking

at

the

bids

before

the

auction

clears

and

submitting

an

extra

bid.

This

possibility

was

pointed

out

as

early

as

the

seminal

paper

that

introduced

this

type

of

auction.

For

example,

if

the

bidders

in

an

eBay

auction

each

use

a

proxy

bidder

(essentially

creating

a

second-price

auction),

then

the

seller

may

be

able

to

break

into

eBay"s

server,

observe

the

maximum

price

that

a

bidder

is

willing

to

pay,

and

then

extract

this

price

by

submitting

a

shill

bid

just

below

it

using

a

false

identity.

We

assume

that

there

is

no

chance

that

the

seller

will

be

caught

when

it

cheats.

However,

not

all

sellers

are

willing

to

use

this

power

(or,

not

all

sellers

can

successfully

cheat).

We

assume

that

each

bidder

knows

the

probability

with

which

the

seller

will

cheat.

Possible

motivation

for

this

knowledge

could

be

a

recently

published

expos´e

on

seller

cheating

in

eBay

auctions.

In

this

setting,

we

derive

an

equilibrium

bidding

strategy

for

the

case

in

which

each

bidder"s

value

for

the

good

is

independently

drawn

from

a

common

distribution

(with

no

further

assumptions

except

for

continuity

and

differentiability).

This

result

shows

how

first

and

second-price

auctions

can

be

viewed

as

the

endpoints

of

a

spectrum

of

auctions.

But

why

should

the

seller

have

all

the

fun?

In

a

first-price

auction,

a

bidder

must

bid

below

his

value

for

the

good

(also

called

shaving

his

bid)

in

order

to

have

positive

utility

if

he

wins.

To

decide

how

much

to

shave

his

bid,

he

must

trade

off

the

probability

of

winning

the

auction

against

how

much

he

will

pay

if

he

does

win.

Of

course,

if

he

could

simply

examine

the

other

bids

before

submitting

his

own,

then

his

problem

is

solved:

bid

the

minimum

necessary

to

win

the

auction.

In

this

setting,

our

goal

is

to

derive

an

equilibrium

bidding

strategy

for

a

non-cheating

bidder

who

is

aware

of

the

possibility

that

he

is

competing

against

cheating

bidders.

When

bidder

values

are

drawn

from

the

commonly-analyzed

uniform

distribution,

we

show

the

counterintuitive

result

that

the

possibility

of

other

bidders

cheating

has

no

effect

on

the

equilibrium

strategy

of

an

honest

bidder.

This

result

is

then

extended

to

show

the

robustness

of

the

equilibrium

of

a

first-price

auction

without

the

possibility

of

cheating.

We

conclude

this

section

by

exploring

other

distributions,

including

some

in

which

the

presence

of

cheating

bidders

actually

induces

an

honest

bidder

to

lower

its

bid.

The

rest

of

the

paper

is

structured

as

follows.

In

Section

we

formalize

the

setting

and

present

our

results

for

the

case

of

a

seller

cheating

in

a

second

price

auction.

Section

covers

the

case

of

bidders

cheating

in

a

first-price

auction.

In

Section

we

quantify

the

effects

that

the

possibility

of

cheating

has

on

an

honest

seller

in

the

two

settings.

We

discuss

related

work,

including

other

forms

of

cheating

in

auctions,

in

Section

before

concluding

with

Section

All

proofs

and

derivations

are

found

in

the

appendix.

SECOND-PRICE

AUCTION,

CHEATING

SELLER

In

this

section,

we

consider

a

second-price

auction

in

which

the

seller

may

cheat

by

inserting

a

shill

bid

after

observing

all

of

the

bids.

The

formulation

for

this

section

will

be

largely

reused

in

the

following

section

on

bidders

cheating

in

a

first-price

auction.

While

no

prior

knowledge

of

game

theory

or

auction

theory

is

assumed,

good

introductions

can

be

found

in

and

respectively.

Formulation

The

setting

consists

of

N

bidders,

or

agents,

(indexed

by

i

n)

and

a

seller.

Each

agent

has

a

type

θi

drawn

from

a

continuous

range,

which

represents

the

agent"s

value

for

the

good

being

Each

agent"s

type

is

independently

drawn

from

a

cumulative

distribution

function

(cdf

F

over

where

and

We

assume

that

F(·)

is

strictly

increasing

and

differentiable

over

the

interval

Call

the

probability

density

function

(pdf

f(θi)

F

(θi),

which

is

the

derivative

of

the

cdf.

Each

agent

knows

its

own

type

θi,

but

only

the

distribution

over

the

possible

types

of

the

other

agents.

A

bidding

strategy

for

an

agent

bi

maps

its

type

to

its

Let

θ

θn)

be

the

vector

of

types

for

all

agents,

and

θ−i

θn)

be

the

vector

of

all

types

except

for

that

of

agent

i.

We

can

then

combine

the

vectors

so

that

θ

(θi,

θ−i).

We

also

define

the

vector

of

bids

as

b(θ)

bn(θn)),

and

this

vector

without

We

can

restrict

the

types

to

the

range

without

loss

of

generality

because

any

distribution

over

a

different

range

can

be

normalized

to

this

range.

We

thus

limit

agents

to

deterministic

bidding

strategies,

but,

because

of

our

continuity

assumption,

there

always

exists

a

pure

strategy

equilibrium.

the

bid

of

agent

i

as

b−i(θ−i).

Let

be

the

value

of

the

highest

bid

of

the

vector

b(θ),

with

a

corresponding

definition

for

An

agent

obviously

wins

the

auction

if

its

bid

is

greater

than

all

other

bids,

but

ties

complicate

the

formulation.

Fortunately,

we

can

ignore

the

case

of

ties

in

this

paper

because

our

continuity

assumption

will

make

them

a

zero

probability

event

in

equilibrium.

We

assume

that

the

seller

does

not

set

a

reserve

If

the

seller

does

not

cheat,

then

the

winning

agent

pays

the

highest

bid

by

another

agent.

On

the

other

hand,

if

the

seller

does

cheat,

then

the

winning

agent

will

pay

its

bid,

since

we

assume

that

a

cheating

seller

would

take

full

advantage

of

its

power.

Let

the

indicator

variable

µc

be

if

the

seller

cheats,

and

otherwise.

The

probability

that

the

seller

cheats,

Pc

is

known

by

all

We

can

then

write

the

payment

of

the

winning

agent

as

follows.

pi(b(θ),

µc

µc

bi(θi)

µc

Let

µ(·)

be

an

indicator

function

that

takes

an

inequality

as

an

argument

and

returns

is

if

it

holds,

and

otherwise.

The

utility

for

agent

i

is

zero

if

it

does

not

win

the

auction,

and

the

difference

between

its

valuation

and

its

price

if

it

does.

ui(b(θ),

µc

θi)

µ

bi(θi)

θi

pi(b(θ),

µc

We

will

be

concerned

with

the

expected

utility

of

an

agent,

with

the

expectation

taken

over

the

types

of

the

other

agents

and

over

whether

or

not

the

seller

cheats.

By

pushing

the

expectation

inward

so

that

it

is

only

over

the

price

(conditioned

on

the

agent

winning

the

auction),

we

can

write

the

expected

utility

as:

Eθ−i,µc

[ui(b(θ),

µc

θi)]

Prob

bi(θi)

θi

Eθ−i,µc

pi(b(θ),

µc

bi(θi)

We

assume

that

all

agents

are

rational,

expected

utility

maximizers.

Because

of

the

uncertainty

over

the

types

of

the

other

agents,

we

will

be

looking

for

a

Bayes-Nash

equilibrium.

A

vector

of

bidding

strategies

b∗

is

a

Bayes-Nash

equilibrium

if

for

each

agent

i

and

each

possible

type

θi,

agent

i

cannot

increase

its

expected

utility

by

using

an

alternate

bidding

strategy

bi,

holding

the

bidding

strategies

for

all

other

agents

fixed.

Formally,

b∗

is

a

Bayes-Nash

equilibrium

if:

∀i,

θi,

bi

Eθ−i,µc

ui

b∗

i

(θi),

b∗

−i(θ−i)

µc

θi

Eθ−i,µc

ui

bi(θi),

b∗

−i(θ−i)

µc

θi

Equilibrium

We

first

present

the

Bayes-Nash

equilibrium

for

an

arbitrary

distribution

F(·).

This

simplifies

the

analysis,

but

all

of

our

results

can

be

applied

to

the

case

in

which

the

seller

announces

a

reserve

price

before

the

auction

begins.

Note

that

common

knowledge

is

not

necessary

for

the

existence

of

an

equilibrium.

Theorem

In

a

second-price

auction

in

which

the

seller

cheats

with

probability

Pc

it

is

a

Bayes-Nash

equilibrium

for

each

agent

to

bid

according

to

the

following

strategy:

bi(θi)

θi

θi

F(

P

c

(x)dx

F(

P

c

(θi)

It

is

useful

to

consider

the

extreme

points

of

Pc

Setting

Pc

yields

the

correct

result

for

a

first-price

auction

(see,

e.g.,

In

the

case

of

Pc

this

solution

is

not

defined.

However,

in

the

limit,

bi(θi)

approaches

θi

as

Pc

approaches

which

is

what

we

expect

as

the

auction

approaches

a

standard

second-price

auction.

The

position

of

Pc

is

perhaps

surprising.

For

example,

the

linear

combination

bi(θi)

θi

Pc

θi

F

(x)dx

F

of

the

equilibrium

bidding

strategies

of

first

and

second-price

auctions

would

have

also

given

us

the

correct

bidding

strategies

for

the

cases

of

Pc

and

Pc

Continuum

of

Auctions

An

alternative

perspective

on

the

setting

is

as

a

continuum

between

first

and

second-price

auctions.

Consider

a

probabilistic

sealed-bid

auction

in

which

the

seller

is

honest,

but

the

price

paid

by

the

winning

agent

is

determined

by

a

weighted

coin

flip:

with

probability

Pc

it

is

his

bid,

and

with

probability

Pc

it

is

the

second-highest

bid.

By

adjusting

Pc

we

can

smoothly

move

between

a

first

and

second-price

auction.

Furthermore,

the

fact

that

this

probabilistic

auction

satisfies

the

properties

required

for

the

Revenue

Equivalence

Theorem

(see,

e.g.,

provides

a

way

to

verify

that

the

bidding

strategy

in

Equation

is

the

symmetric

equilibrium

of

this

auction

(see

the

alternative

proof

of

Theorem

in

the

appendix).

Special

Case:

Uniform

Distribution

Another

way

to

try

to

gain

insight

into

Equation

is

by

instantiating

the

distribution

of

types.

We

now

consider

the

often-studied

uniform

distribution:

F(θi)

θi.

Corollary

In

a

second-price

auction

in

which

the

seller

cheats

with

probability

Pc

and

F(θi)

θi,

it

is

a

Bayes-Nash

equilibrium

for

each

agent

to

bid

according

to

the

following

strategy:

bi(θi)

N

N

Pc

θi

This

equilibrium

bidding

strategy,

parameterized

by

Pc

can

be

viewed

as

an

interpolation

between

two

well-known

results.

When

Pc

the

bidding

strategy

is

now

welldefined

(each

agent

bids

its

true

type),

while

when

Pc

we

get

the

correct

result

for

a

first-price

auction:

each

agent

bids

according

to

the

strategy

bi(θi)

N

θi.

FIRST-PRICE

AUCTION,

CHEATING

AGENTS

We

now

consider

the

case

in

which

the

seller

is

honest,

but

there

is

a

chance

that

agents

will

cheat

and

examine

the

other

bids

before

submitting

their

own

(or,

alternatively,

they

will

revise

their

bid

before

the

auction

clears).

Since

this

type

of

cheating

is

pointless

in

a

second-price

auction,

we

only

analyze

the

case

of

a

first-price

auction.

After

revising

the

formulation

from

the

previous

section,

we

present

a

fixed

point

equation

for

the

equilibrium

strategy

for

an

arbitrary

distribution

F(·).

This

equation

will

be

useful

for

the

analysis

the

uniform

distribution,

in

which

we

show

that

the

possibility

of

cheating

agents

does

not

change

the

equilibrium

strategy

of

honest

agents.

This

result

has

implications

for

the

robustness

of

the

symmetric

equilibrium

to

overbidding

in

a

standard

first-price

auction.

Furthermore,

we

find

that

for

other

distributions

overbidding

actually

induces

a

competing

agent

to

shave

more

off

of

its

bid.

Formulation

It

is

clear

that

if

a

single

agent

is

cheating,

he

will

bid

(up

to

his

valuation)

the

minimum

amount

necessary

to

win

the

auction.

It

is

less

obvious,

though,

what

will

happen

if

multiple

agents

cheat.

One

could

imagine

a

scenario

similar

to

an

English

auction,

in

which

all

cheating

agents

keep

revising

their

bids

until

all

but

one

cheater

wants

the

good

at

the

current

winning

bid.

However,

we

are

only

concerned

with

how

an

honest

agent

should

bid

given

that

it

is

aware

of

the

possibility

of

cheating.

Thus,

it

suffices

for

an

honest

agent

to

know

that

it

will

win

the

auction

if

and

only

if

its

bid

exceeds

every

other

honest

agent"s

bid

and

every

cheating

agent"s

type.

This

intuition

can

be

formalized

as

the

following

discriminatory

auction.

In

the

first

stage,

each

agent"s

payment

rule

is

determined.

With

probability

Pa

the

agent

will

pay

the

second

highest

bid

if

it

wins

the

auction

(essentially,

he

is

a

cheater),

and

otherwise

it

will

have

to

pay

its

bid.

These

selections

are

recorded

by

a

vector

of

indicator

variables

µa

µan

where

µai

denotes

that

agent

i

pays

the

second

highest

bid.

Each

agent

knows

the

probability

Pa

but

does

not

know

the

payment

rule

for

all

other

agents.

Otherwise,

this

auction

is

a

standard,

sealed-bid

auction.

It

is

thus

a

dominant

strategy

for

a

cheater

to

bid

its

true

type,

making

this

formulation

strategically

equivalent

to

the

setting

outlined

in

the

previous

paragraph.

The

expression

for

the

utility

of

an

honest

agent

in

this

discriminatory

auction

is

as

follows.

ui(b(θ),

µa

θi)

θi

bi(θ)

j=i

µaj

µ

bi(θi)

θj

µaj

µ

bi(θi)

bj(θj)

Equilibrium

Our

goal

is

to

find

the

equilibrium

in

which

all

cheating

agents

use

their

dominant

strategy

of

bidding

truthfully

and

honest

agents

bid

according

to

a

symmetric

bidding

strategy.

Since

we

have

left

F(·)

unspecified,

we

cannot

present

a

closed

form

solution

for

the

honest

agent"s

bidding

strategy,

and

instead

give

a

fixed

point

equation

for

it.

Theorem

In

a

first-price

auction

in

which

each

agent

cheats

with

probability

Pa

it

is

a

Bayes-Nash

equilibrium

for

each

non-cheating

agent

i

to

bid

according

to

the

strategy

that

is

a

fixed

point

of

the

following

equation:

bi(θi)

θi

θi

Pa

F(bi(x))

Pa)

F(x)

dx

Pa

F(bi(θi))

Pa)

F(θi)

Special

Case:

Uniform

Distribution

Since

we

could

not

solve

Equation

in

the

general

case,

we

can

only

see

how

the

possibility

of

cheating

affects

the

equilibrium

bidding

strategy

for

particular

instances

of

F(·).

A

natural

place

to

start

is

uniform

distribution:

F(θi)

θi.

Recall

the

logic

behind

the

symmetric

equilibrium

strategy

in

a

first-price

auction

without

cheating:

bi(θi)

N

θi

is

the

optimal

tradeoff

between

increasing

the

probability

of

winning

and

decreasing

the

price

paid

upon

winning,

given

that

the

other

agents

are

bidding

according

to

the

same

strategy.

Since

in

the

current

setting

the

cheating

agents

do

not

shave

their

bid

at

all

and

thus

decrease

an

honest

agent"s

probability

of

winning

(while

obviously

not

affecting

the

price

that

an

honest

agent

pays

if

he

wins),

it

is

natural

to

expect

that

an

honest

agent

should

compensate

by

increasing

his

bid.

The

idea

is

that

sacrificing

some

potential

profit

in

order

to

regain

some

of

the

lost

probability

of

winning

would

bring

the

two

sides

of

the

tradeoff

back

into

balance.

However,

it

turns

out

that

the

equilibrium

bidding

strategy

is

unchanged.

Corollary

In

a

first-price

auction

in

which

each

agent

cheats

with

probability

Pa

and

F(θi)

θi,

it

is

a

BayesNash

equilibrium

for

each

non-cheating

agent

to

bid

according

to

the

strategy

bi(θi)

N

θi.

This

result

suggests

that

the

equilibrium

of

a

first-price

auction

is

particularly

robust

when

types

are

drawn

from

the

uniform

distribution,

since

the

best

response

is

unaffected

by

deviations

of

the

other

agents

to

the

strategy

of

always

bidding

their

type.

In

fact,

as

long

as

all

other

agents

shave

their

bid

by

a

fraction

(which

can

differ

across

the

agents)

no

greater

than

N

it

is

still

a

best

response

for

the

remaining

agent

to

bid

according

to

the

equilibrium

strategy.

Note

that

this

result

holds

even

if

other

agents

are

shaving

their

bid

by

a

negative

fraction,

and

are

thus

irrationally

bidding

above

their

type.

Theorem

In

a

first-price

auction

where

F(θi)

θi,

if

each

agent

j

i

bids

according

a

strategy

bj(θj)

N

θj,

where

αj

then

it

is

a

best

response

for

the

remaining

agent

i

to

bid

according

to

the

strategy

bi(θi)

N

θi.

Obviously,

these

strategy

profiles

are

not

equilibria

(unless

each

αj

because

each

agent

j

has

an

incentive

to

set

αj

The

point

of

this

theorem

is

that

a

wide

range

of

possible

beliefs

that

an

agent

can

hold

about

the

strategies

of

the

other

agents

will

all

lead

him

to

play

the

equilibrium

strategy.

This

is

important

because

a

common

(and

valid)

criticism

of

equilibrium

concepts

such

as

Nash

and

BayesNash

is

that

they

are

silent

on

how

the

agents

converge

on

a

strategy

profile

from

which

no

one

wants

to

deviate.

However,

if

the

equilibrium

strategy

is

a

best

response

to

a

large

set

of

strategy

profiles

that

are

out

of

equilibrium,

then

it

seems

much

more

plausible

that

the

agents

will

indeed

converge

on

this

equilibrium.

It

is

important

to

note,

though,

that

while

this

equilibrium

is

robust

against

arbitrary

deviations

to

strategies

that

shave

less,

it

is

not

robust

to

even

a

single

agent

shaving

more

off

of

its

bid.

In

fact,

if

we

take

any

strategy

profile

consistent

with

the

conditions

of

Theorem

and

change

a

single

agent

j"s

strategy

so

that

its

corresponding

αj

is

negative,

then

agent

i"s

best

response

is

to

shave

more

than

N

off

of

its

bid.

Effects

of

Overbidding

for

Other

Distributions

A

natural

question

is

whether

the

best

response

bidding

strategy

is

similarly

robust

to

overbidding

by

competing

agents

for

other

distributions.

It

turns

out

that

Theorem

holds

for

all

distributions

of

the

form

F(θi)

(θi)k

where

k

is

some

positive

integer.

However,

taking

a

simple

linear

combination

of

two

such

distributions

to

produce

F(θi)

θ2

i

+θi

yields

a

distribution

in

which

an

agent

should

actually

shave

its

bid

more

when

other

agents

shave

their

bids

less.

In

the

example

we

present

for

this

distribution

(with

the

details

in

the

appendix),

there

are

only

two

players

and

the

deviation

by

one

agent

is

to

bid

his

type.

However,

it

can

be

generalized

to

a

higher

number

of

agents

and

to

other

deviations.

Example

In

a

first-price

auction

where

F(θi)

θ2

i

+θi

and

N

if

agent

always

bids

its

type

then,

for

all

θ1

agent

best

response

bidding

strategy

is

strictly

less

than

the

bidding

strategy

of

the

symmetric

equilibrium.

We

also

note

that

the

same

result

holds

for

the

normalized

exponential

distribution

(F(θi)

eθi

−1

It

is

certainly

the

case

that

distributions

can

be

found

that

support

the

intuition

given

above

that

agents

should

shave

their

bid

less

when

other

agents

are

doing

likewise.

Examples

include

F(θi)

−1

θ2

i

θi

(the

solution

to

the

system

of

equations:

F

(θi)

and

and

F(θi)

It

would

be

useful

to

relate

the

direction

of

the

change

in

the

best

response

bidding

strategy

to

a

general

condition

on

F(·).

Unfortunately,

we

were

not

able

to

find

such

a

condition,

in

part

because

the

integral

in

the

symmetric

bidding

strategy

of

a

first-price

auction

cannot

be

solved

without

knowing

F(·)

(or

at

least

some

restrictions

on

it).

We

do

note,

however,

that

the

sign

of

the

second

derivative

of

F(θi)/f(θi)

is

an

accurate

predictor

for

all

of

the

distributions

that

we

considered.

REVENUE

LOSS

FOR

AN

HONEST

SELLER

In

both

of

the

settings

we

covered,

an

honest

seller

suffers

a

loss

in

expected

revenue

due

to

the

possibility

of

cheating.

The

equilibrium

bidding

strategies

that

we

derived

allow

us

to

quantify

this

loss.

Although

this

is

as

far

as

we

will

take

the

analysis,

it

could

be

applied

to

more

general

settings,

in

which

the

seller

could,

for

example,

choose

the

market

in

which

he

sells

his

good

or

pay

a

trusted

third

party

to

oversee

the

auction.

In

a

second-price

auction

in

which

the

seller

may

cheat,

an

honest

seller

suffers

due

the

fact

that

the

agents

will

shave

their

bids.

For

the

case

in

which

agent

types

are

drawn

from

the

uniform

distribution,

every

agent

will

shave

its

bid

by

P

c

c

which

is

thus

also

the

fraction

by

which

an

honest

seller"s

revenue

decreases

due

to

the

possibility

of

cheating.

Analysis

of

the

case

of

a

first-price

auction

in

which

agents

may

cheat

is

not

so

straightforward.

If

Pa

(each

agent

cheats

with

certainty),

then

we

simply

have

a

second-price

auction,

and

the

seller"s

expected

revenue

will

be

unchanged.

Again

considering

the

uniform

distribution

for

agent

types,

it

is

not

surprising

that

Pa

causes

the

seller

to

lose

the

most

revenue.

However,

even

in

this

worst

case,

the

percentage

of

expected

revenue

lost

is

significantly

less

than

it

is

for

the

second-price

auction

in

which

Pc

as

shown

in

Table

It

turns

out

that

setting

Pc

would

make

the

expected

loss

of

these

two

settings

comparable.

While

this

comparison

between

the

settings

is

unlikely

to

be

useful

for

a

seller,

it

is

interesting

to

note

that

agent

suspicions

of

possible

cheating

by

the

seller

are

in

some

sense

worse

than

agents

actually

cheating

themselves.

Percentage

of

Revenue

lost

for

an

Honest

Seller

Agents

Second-Price

Auction

First-Price

Auction

(Pc

(Pa

Table

The

percentage

of

expected

revenue

lost

by

an

honest

seller

due

to

the

possibility

of

cheating

in

the

two

settings

considered

in

this

paper.

Agent

valuations

are

drawn

from

the

uniform

distribution.

RELATED

WORK

Existing

work

covers

another

dimension

along

which

we

could

analyze

cheating:

altering

the

perceived

value

of

N.

In

this

paper,

we

have

assumed

that

N

is

known

by

all

of

the

bidders.

However,

in

an

online

setting

this

assumption

is

rather

tenuous.

For

example,

a

bidder"s

only

source

of

information

about

N

could

be

a

counter

that

the

seller

places

on

the

auction

webpage,

or

a

statement

by

the

seller

about

the

number

of

potential

bidders

who

have

indicated

that

they

will

participate.

In

these

cases,

the

seller

could

arbitrarily

manipulate

the

perceived

N.

In

a

first-price

auction,

the

seller

obviously

has

an

incentive

to

increase

the

perceived

value

of

N

in

order

to

induce

agents

to

bid

closer

to

their

true

valuation.

However,

if

agents

are

aware

that

the

seller

has

this

power,

then

any

communication

about

N

to

the

agents

is

cheap

talk,

and

furthermore

is

not

credible.

Thus,

in

equilibrium

the

agents

would

ignore

the

declared

value

of

N,

and

bid

according

to

their

own

prior

beliefs

about

the

number

of

agents.

If

we

make

the

natural

assumption

of

a

common

prior,

then

the

setting

reduces

to

the

one

tackled

by

which

derived

the

equilibrium

bidding

strategies

of

a

first-price

auction

when

the

number

of

bidders

is

drawn

from

a

known

distribution

but

not

revealed

to

any

of

the

bidders.

Of

course,

instead

of

assuming

that

the

seller

can

always

exploit

this

power,

we

could

assume

that

it

can

only

do

so

with

some

probability

that

is

known

by

the

agents.

The

analysis

would

then

proceed

in

a

similar

manner

as

that

of

our

cheating

seller

model.

The

other

interesting

case

of

this

form

of

cheating

is

by

bidders

in

a

first-price

auction.

Bidders

would

obviously

want

to

decrease

the

perceived

number

of

agents

in

order

to

induce

their

competition

to

lower

their

bids.

While

it

is

Note

that

we

have

not

considered

the

costs

of

the

seller.

Thus,

the

expected

loss

in

profit

could

be

much

greater

than

the

numbers

that

appear

here.

unreasonable

for

bidders

to

be

able

to

alter

the

perceived

N

arbitrarily,

collusion

provides

an

opportunity

to

decrease

the

perceived

N

by

having

only

one

of

a

group

of

colluding

agents

participate

in

the

auction.

While

the

non-colluding

agents

would

account

for

this

possibility,

as

long

as

they

are

not

certain

of

the

collusion

they

will

still

be

induced

to

shave

more

off

of

their

bids

than

they

would

if

the

collusion

did

not

take

place.

This

issue

is

tackled

in

Other

types

of

collusion

are

of

course

related

to

the

general

topic

of

cheating

in

auctions.

Results

on

collusion

in

first

and

second-price

auctions

can

be

found

in

and

respectively.

The

work

most

closely

related

to

our

first

setting

is

which

also

presents

a

model

in

which

the

seller

may

cheat

in

a

second-price

auction.

In

their

setting,

the

seller

is

a

participant

in

the

Bayesian

game

who

decides

between

running

a

first-price

auction

(where

profitable

cheating

is

never

possible)

or

second-price

auction.

The

seller

makes

this

choice

after

observing

his

type,

which

is

his

probability

of

having

the

opportunity

and

willingness

to

cheat

in

a

second-price

auction.

The

bidders,

who

know

the

distribution

from

which

the

seller"s

type

is

drawn,

then

place

their

bid.

It

is

shown

that,

in

equilibrium,

only

a

seller

with

the

maximum

probability

of

cheating

would

ever

choose

to

run

a

second-price

auction.

Our

work

differs

in

that

we

focus

on

the

agents"

strategies

in

a

second-price

auction

for

a

given

probability

of

cheating

by

the

seller.

An

explicit

derivation

of

the

equilibrium

strategies

then

allows

us

relate

first

and

second-price

auctions.

An

area

of

related

work

that

can

be

seen

as

complementary

to

ours

is

that

of

secure

auctions,

which

takes

the

point

of

view

of

an

auction

designer.

The

goals

often

extend

well

beyond

simply

preventing

cheating,

including

properties

such

as

anonymity

of

the

bidders

and

nonrepudiation

of

bids.

Cryptographic

methods

are

the

standard

weapon

of

choice

here

(see

CONCLUSION

In

this

paper

we

presented

the

equilibria

of

sealed-bid

auctions

in

which

cheating

is

possible.

In

addition

to

providing

strategy

profiles

that

are

stable

against

deviations,

these

results

give

us

with

insights

into

both

first

and

second-price

auctions.

The

results

for

the

case

of

a

cheating

seller

in

a

second-price

auction

allow

us

to

relate

the

two

auctions

as

endpoints

along

a

continuum.

The

case

of

agents

cheating

in

a

first-price

auction

showed

the

robustness

of

the

first-price

auction

equilibrium

when

agent

types

are

drawn

from

the

uniform

distribution.

We

also

explored

the

effect

of

overbidding

on

the

best

response

bidding

strategy

for

other

distributions,

and

showed

that

even

for

relatively

simple

distributions

it

can

be

positive,

negative,

or

neutral.

Finally,

results

from

both

of

our

settings

allowed

us

to

quantify

the

expected

loss

in

revenue

for

a

seller

due

to

the

possibility

of

cheating.

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Franklin

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M.

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The

Design

and

Implementation

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Secure

Auction

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In

Proc.

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Symp.

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D.

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Game

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D.

Graham

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R.

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M.

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J.

D.

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R.

Harstad,

J.

Kagel,

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P.

Klemperer.

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A

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the

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K.

Leyton-Brown,

Y.

Shoham,

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M.

Tennenholtz.

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In

R.

McAfee

and

J.

McMillan.

Bidding

rings.

The

American

Economic

Review,

M.

Naor,

B.

Pinkas,

and

R.

Sumner.

Privacy

preserving

auctions

and

mechanism

design.

In

J.

Riley

and

W.

Samuelson.

Optimal

auctions.

American

Economic

Review,

M.

Rothkopf

and

R.

Harstad.

Two

models

of

bid-taker

cheating

in

vickrey

auctions.

The

Journal

of

Business,

W.

Vickrey.

Counterspeculations,

auctions,

and

competitive

sealed

tenders.

Journal

of

Finance,

APPENDIX

Theorem

In

a

second-price

auction

in

which

the

seller

cheats

with

probability

Pc

it

is

a

Bayes-Nash

equilibrium

for

each

agent

to

bid

according

to

the

following

strategy:

bi(θi)

θi

θi

F(

P

c

(x)dx

F(

P

c

(θi)

Proof.

To

find

an

equilibrium,

we

start

by

guessing

that

there

exists

an

equilibrium

in

which

all

agents

bid

according

to

the

same

function

bi(θi),

because

the

game

is

symmetric.

Further,

we

guess

that

bi(θi)

is

strictly

increasing

and

differentiable

over

the

range

We

can

also

assume

that

because

negative

bids

are

not

allowed

and

a

positive

bid

is

not

rational

when

the

agent"s

valuation

is

Note

that

these

are

not

assumptions

on

the

setting-

they

are

merely

limitations

that

we

impose

on

our

search.

Let

Φi

be

the

inverse

function

of

bi(θi).

That

is,

it

takes

a

bid

for

agent

i

as

input

and

returns

the

type

θi

that

induced

this

bid.

Recall

Equation

Eθ−i,µc

ui(b(θ),

µc

θi)

Prob

bi(θi)

θi

Eθ−i,µc

pi(b(θ),

µc

bi(θi)

The

probability

that

a

single

other

bid

is

below

that

of

agent

i

is

equal

to

the

cdf

at

the

type

that

would

induce

a

bid

equal

to

that

of

agent

i,

which

is

formally

written

as

F(Φi(bi(θi))).

Since

all

agents

are

independent,

the

probability

that

all

other

bids

are

below

agent

i"s

is

simply

this

term

raised

the

(N

power.

Thus,

we

can

re-write

the

expected

utility

as:

Eθ−i,µc

ui(b(θ),

µc

θi)

(Φi(bi(θi)))·

θi

Eθ−i,µc

pi(b(θ),

µc

bi(θi)

We

now

solve

for

the

expected

payment.

Plugging

Equation

(which

gives

the

price

for

the

winning

agent)

into

the

term

for

the

expected

price

in

Equation

and

then

simplifying

the

expectation

yields:

Eθ−i,µc

pi(b(θ),

µc

bi(θi)

Eθ−i,µc

µc

bi(θi)

µc

bi(θi)

Pc

bi(θi)

Pc

Eθ−i

bi(θi)

Pc

bi(θi)

Pc

bi(θi)

pdf

bi(θi)

Note

that

the

integral

on

the

last

line

is

taken

up

to

bi(θi)

because

we

are

conditioning

on

the

fact

that

bi(θi)

To

derive

the

pdf

of

given

this

condition,

we

start

with

the

cdf.

For

a

given

value

the

probability

that

any

one

agent"s

bid

is

less

than

this

value

is

equal

to

We

then

condition

on

the

agent"s

bid

being

below

bi(θi)

by

dividing

by

F(Φi(bi(θi))).

The

cdf

for

the

N

agents

is

then

this

value

raised

to

the

(N

power.

cdf

bi(θi)

The

pdf

is

then

the

derivative

of

the

cdf

with

respect

to

pdf

bi(θi)

N

Substituting

the

pdf

into

Equation

and

pulling

terms

out

of

the

integral

that

do

not

depend

on

yields:

Eθ−i,µc

pi(b(θ),

µc

bi(θi)

Pc

bi(θi)+

Pc

(N

bi(θi)

Plugging

the

expected

price

back

into

the

expected

utility

equation

and

distributing

(Φi(bi(θi))),

yields:

Eθ−i,µc

ui(b(θ),

µc

θi)

(Φi(bi(θi)))

θi−

(Φi(bi(θi)))

Pc

bi(θi)−

Pc

(N

bi(θi)

We

are

now

ready

to

optimize

the

expected

utility

by

taking

the

derivative

with

respect

to

bi(θi)

and

setting

it

to

Note

that

we

do

not

need

to

solve

the

integral,

because

it

will

disappear

when

the

derivative

is

taken

(by

application

of

the

Fundamental

Theorem

of

Calculus).

(Φi(bi(θi)))·f(Φi(bi(θi)))·Φi(bi(θi))·θi−

(Φi(bi(θi)))

Pc

Pc

(Φi(bi(θi)))·f(Φi(bi(θi)))·Φi(bi(θi))·bi(θi)−

(Φi(bi(θi)))·f(Φi(bi(θi)))·Φi(bi(θi))

Dividing

through

by

(Φi(bi(θi)))

and

combining

like

terms

yields:

θi

Pc

bi(θi)

Pc

bi(θi)

(N

f(Φi(bi(θi)))

Φi(bi(θi))

Pc

F(Φi(bi(θi)))

Simplifying

the

expression

and

rearranging

terms

produces:

bi(θi)

θi

Pc

F(Φi(bi(θi)))

(N

f(Φi(bi(θi)))

Φi(bi(θi))

To

further

simplify,

we

use

the

formula

f

(x)

g

(f(x))

where

g(x)

is

the

inverse

function

of

f(x).

Plugging

in

function

from

our

setting

gives

us:

Φi(bi(θi))

bi(θi)

Applying

both

this

equation

and

Φi(bi(θi))

θi

yields:

bi(θi)

θi

Pc

F(θi)

bi(θi)

(N

f(θi)

Attempts

at

a

derivation

of

the

solution

from

this

point

proved

fruitless,

but

we

are

at

a

point

now

where

a

guessed

solution

can

be

quickly

verified.

We

used

the

solution

for

the

first-price

auction

(see,

e.g.,

as

our

starting

point

to

find

the

answer:

bi(θi)

θi

θi

F(

P

c

(x)dx

F(

P

c

(θi)

To

verify

the

solution,

we

first

take

its

derivative:

bi(θi)

1−

P

c

(θi)

P

c

F(

P

c

(θi)

f(θi)

θi

F(

P

c

(x)dx

P

c

(θi)

This

simplifies

to:

bi(θi)

P

c

f(θi)

θi

F(

P

c

(x)dx

F(

P

c

(θi)

We

then

plug

this

derivative

into

the

equation

we

derived

bi(θi)

θi

Pc

F(θi)

P

c

f(θi)

θi

F(

P

c

(x)dx

(N

f(θi)

F(

P

c

(θi)

Cancelling

terms

yields

Equation

verifying

that

our

guessed

solution

is

correct.

Alternative

Proof

of

Theorem

The

following

proof

uses

the

Revenue

Equivalence

Theorem

(RET)

and

the

probabilistic

auction

given

as

an

interpretation

of

our

cheating

seller

setting.

In

a

first-price

auction

without

the

possibility

of

cheating,

the

expected

payment

for

an

agent

with

type

θi

is

simply

the

product

of

its

bid

and

the

probability

that

this

bid

is

the

highest.

For

the

symmetric

equilibrium,

this

is

equal

to:

(θi)

θi

θi

(x)dx

For

our

probabilistic

auction,

the

expected

payment

of

the

winning

agent

is

a

weighted

average

of

its

bid

and

the

second

highest

bid.

For

the

bi(·)

we

found

in

the

original

interpretation

of

the

setting,

it

can

be

written

as

follows.

(θi)

Pc

θi

θi

F(

P

c

(x)dx

F(

P

c

(θi)

Pc

θi

x−

x

F(

P

c

(y)dy

F(

P

c

(x)

·(N

(x)·f(x)dx

By

the

RET,

the

expected

payments

will

be

the

same

in

the

two

auctions.

Thus,

we

can

verify

our

equilibrium

bidding

strategy

by

showing

that

the

expected

payment

in

the

two

auctions

is

equal.

Since

the

expected

payment

is

zero

at

θi

for

both

functions,

it

suffices

to

verify

that

the

derivatives

of

the

expected

payment

functions

with

respect

to

θi

are

equal,

for

an

arbitrary

value

θi.

Thus,

we

need

to

verify

the

following

equation:

(θi)

(N

(θi)

f(θi)

θi

(θi)

Pc

(θi)

1−

P

c

F(

P

c

(θi)

f(θi)

θi

F(

P

c

(x)dx

P

c

(θi)

(N

(θi)

f(θi)

θi

θi

F(

P

c

(x)dx

F(

P

c

(θi)

θi−

θi

F(

P

c

(y)dy

F(

P

c

(θi)

(θi)·f(θi)

This

simplifies

to:

Pc

P

c

(θi)

f(θi)

θi

F(

P

c

(x)dx

F(

P

c

(θi)

(N

(θi)

f(θi)

θi

F(

P

c

(x)dx

F(

P

c

(θi)

θi

F(

P

c

(y)dy

F(

P

c

(θi)

(θi)·f(θi)

After

distributing

Pc

the

right-hand

side

of

this

equation

cancels

out,

and

we

have

verified

our

equilibrium

bidding

strategy.

Corollary

In

a

second-price

auction

in

which

the

seller

cheats

with

probability

Pc

and

F(θi)

θi,

it

is

a

Bayes-Nash

equilibrium

for

each

agent

to

bid

according

to

the

following

strategy:

bi(θi)

N

N

Pc

θi

Proof.

Plugging

F(θi)

θi

into

Equation

(repeated

as

we

get:

bi(θi)

θi

θi

x(

P

c

dx

θi

P

c

θi

P

c

c

θi

c

P

c

θi

P

c

θi

Pc

N

Pc

θi

N

N

Pc

θi

Theorem

In

a

first-price

auction

in

which

each

agent

cheats

with

probability

Pa

it

is

a

Bayes-Nash

equilibrium

for

each

non-cheating

agent

i

to

bid

according

to

the

strategy

that

is

a

fixed

point

of

the

following

equation:

bi(θi)

θi

θi

Pa

F(bi(x))

Pa)

F(x)

dx

Pa

F(bi(θi))

Pa)

F(θi)

Proof.

We

make

the

same

guesses

about

the

equilibrium

strategy

to

aid

our

search

as

we

did

in

the

proof

of

Theorem

When

simplifying

the

expectation

of

this

setting"s

utility

equation

we

use

the

fact

that

the

probability

that

agent

i

will

have

a

higher

bid

than

another

honest

agent

is

still

F(Φi(bi(θi))),

while

the

probability

is

F(bi(θi))

if

the

other

agent

cheats.

The

probability

that

agent

i

beats

a

single

other

agent

is

then

a

weighted

average

of

these

two

probabilities.

Thus,

we

can

write

agent

i"s

expected

utility

as:

Eθ−i,µa

ui(b(θ),

µa

θi)

θi

bi(θi)

Pa

F(bi(θi))

Pa

F(Φi(bi(θi)))

As

before,

to

find

the

equilibrium

bi(θi),

we

take

the

derivative

and

set

it

to

zero:

θi

bi(θi)

(N

Pa

F(bi(θi))

Pa

F(Φi(bi(θi)))

Pa

f(bi(θi))

Pa

f(Φi(bi(θi)))

Φi(bi(θi))

Pa

F(bi(θi))

Pa

F(Φi(bi(θi)))

Applying

the

equations

Φi(bi(θi))

bi(θi)

and

Φi(bi(θi))

θi,

and

dividing

through,

produces:

θi

bi(θi)

(N

Pa

f(bi(θi))

Pa

f(θi)

bi(θi)

Pa

F(bi(θi))

Pa

F(θi)

Rearranging

terms

yields:

bi(θi)

θi

Pa

F(bi(θi))

Pa)

F(θi)

bi(θi)

(N

Pa

f(bi(θi))

bi(θi)

Pa)

f(θi)

In

this

setting,

because

we

leave

F(·)

unspecified,

we

cannot

present

a

closed

form

solution.

However,

we

can

simplify

the

expression

by

removing

its

dependence

on

bi(θi).

bi(θi)

θi

θi

Pa

F(bi(x))

Pa)

F(x)

dx

Pa

F(bi(θi))

Pa)

F(θi)

To

verify

Equation

first

take

its

derivative:

bi(θi)

1−

(N

Pa

F(bi(θi))

Pa

F(θi)

Pa

f(bi(θi))

bi(θi))

Pa

f(θi)

θi

Pa

F(bi(x))

Pa

F(x)

dx

Pa

F(bi(θi))

Pa)

F(θi)

This

equation

simplifies

to:

bi(θi)

(N

−1)·

Pa

)·f(θi)

θi

Pa

F(bi(x))

Pa

F(x)

dx

Pa

F(bi(θi))

Pa)

F(θi)

N

Plugging

this

equation

into

the

bi(θi)

in

the

numerator

of

Equation

yields

Equation

verifying

the

solution.

Corollary

In

a

first-price

auction

in

which

each

agent

cheats

with

probability

Pa

and

F(θi)

θi,

it

is

a

BayesNash

equilibrium

for

each

non-cheating

agent

to

bid

according

to

the

strategy

bi(θi)

N

θi.

Proof.

Instantiating

the

fixed

point

equation

and

repeated

as

with

F(θi)

θi

yields:

bi(θi)

θi

θi

Pa

bi(x)

Pa

x

dx

Pa

bi(θi)

Pa)

θi

We

can

plug

the

strategy

bi(θi)

N

θi

into

this

equation

in

order

to

verify

that

it

is

a

fixed

point.

bi(θi)

θi

θi

Pa

N

x

Pa

x

dx

Pa

N

θi

Pa)

θi

θi

θi

dx

θ

i

θi

N

θN

i

θ

i

N

N

θi

Theorem

In

a

first-price

auction

where

F(θi)

θi,

if

each

agent

j

i

bids

according

a

strategy

bj(θj)

N

θj,

where

αj

then

it

is

a

best

response

for

the

remaining

agent

i

to

bid

according

to

the

strategy

bi(θi)

N

θi.

Proof.

We

again

use

Φj

as

the

inverse

of

bj(θj).

For

all

j

i

in

this

setting,

Φj(x)

N

x.

The

probability

that

agent

i

has

a

higher

bid

than

a

single

agent

j

is

F(Φj(bi(θi)))

N

bi(θi).

Note,

however,

that

since

Φj(·)

is

only

defined

over

the

range

it

must

be

the

case

that

which

is

why

αj

is

necessary,

in

addition

to

being

sufficient.

Assuming

that

bi(θi)

N

θi,

then

indeed

Φj(bi(θi))

is

always

welldefined.

We

will

now

show

that

this

assumption

is

correct.

The

expected

utility

for

agent

i

can

then

be

written

as:

Eθ−i

ui(b(θ),

θi)

Πj=i

N

N

αj

bi(θi)

θi

−bi(θ)

Πj=i

N

N

αj

θi·

bi(θi)

bi(θi)

N

Taking

the

derivative

with

respect

to

bi(θi),

setting

it

to

zero,

and

dividing

out

Πj=i

N

yields:

θi

(N

bi(θi)

N

bi(θi)

This

simplifies

to

the

solution:

bi(θi)

N

θi.

Full

Version

of

Example

In

a

first-price

auction

where

F(θi)

θ2

i

+θi

and

N

if

agent

always

bids

its

type

then,

for

all

θ1

agent

best

response

bidding

strategy

is

strictly

less

than

the

bidding

strategy

of

the

symmetric

equilibrium.

After

calculating

the

symmetric

equilibrium

in

which

both

agents

shave

their

bid

by

the

same

amount,

we

find

the

best

response

to

an

agent

who

instead

does

not

shave

its

bid.

We

then

show

that

this

best

response

is

strictly

less

than

the

equilibrium

strategy.

To

find

the

symmetric

equilibrium

bidding

strategy,

we

instantiate

N

in

the

general

formula

the

equation

found

in

plug

in

F(θi)

θ2

i

+θi

and

simplify:

bi(θi)

θi

θi

F(x)dx

F(θi)

θi−

θi

x)dx

i

θi)

θi−

θ3

i

θ2

i

θ2

i

θi

θ2

i

θi

θi

We

now

derive

the

best

response

for

agent

to

the

strategy

denoting

the

best

response

strategy

b∗

to

distinguish

it

from

the

symmetric

case.

The

probability

of

agent

winning

is

F(b∗

which

is

the

probability

that

agent

type

is

less

than

agent

bid.

Thus,

agent

expected

utility

is:

ui((b∗

F(b∗

θ1

b∗

(b∗

b∗

θ1

b∗

Taking

the

derivative

with

respect

to

b∗

setting

it

to

zero,

and

then

rearranging

terms

gives

us:

b∗

θ1

(b∗

θ1

b∗

(b∗

b∗

θ1

Of

the

two

solutions

of

this

equation,

one

always

produces

a

negative

bid.

The

other

is:

b∗

θ1

θ2

θ1

We

now

need

to

show

that

b∗

holds

for

all

θi

Substituting

in

for

both

terms,

and

then

simplifying

the

inequality

gives

us:

θ2

θ1

θ1

θ1

θ2

θ1

θ2

θ1

θ2

θ1

Since

θ1

we

can

square

both

sides

of

the

inequality,

which

then

allows

us

to

verify

the

inequality

for

all

θ1

θ4

θ2

θ4

θ2

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

sealed-bid
bidsecond-price auction
first-price auction
sealed-bid auction

cheating
auction
agent
case
seller
payment
possibility
game theory
cheat