Generalized

Value

Decomposition

and

Structured

Multiattribute

Auctions

Yagil

Engel

and

Michael

P.

Wellman

University

of

Michigan,

Computer

Science

Engineering

Hayward

St,

Ann

Arbor,

MI

USA

{yagil,wellman}@umich.edu

ABSTRACT

Multiattribute

auction

mechanisms

generally

either

remain

agnostic

about

traders"

preferences,

or

presume

highly

restrictive

forms,

such

as

full

additivity.

Real

preferences

often

exhibit

dependencies

among

attributes,

yet

may

possess

some

structure

that

can

be

usefully

exploited

to

streamline

communication

and

simplify

operation

of

a

multiattribute

auction.

We

develop

such

a

structure

using

the

theory

of

measurable

value

functions,

a

cardinal

utility

representation

based

on

an

underlying

order

over

preference

differences.

A

set

of

local

conditional

independence

relations

over

such

differences

supports

a

generalized

additive

preference

representation,

which

decomposes

utility

across

overlapping

clusters

of

related

attributes.

We

introduce

an

iterative

auction

mechanism

that

maintains

prices

on

local

clusters

of

attributes

rather

than

the

full

space

of

joint

configurations.

When

traders"

preferences

are

consistent

with

the

auction"s

generalized

additive

structure,

the

mechanism

produces

approximately

optimal

allocations,

at

approximate

VCG

prices.

Categories

and

Subject

Descriptors:

[Computer

Applications]:

Social

and

Behavioral

Sciences-Economics

General

Terms:

Algorithms,

Economics

INTRODUCTION

Multiattribute

trading

mechanisms

extend

traditional,

price-only

mechanisms

by

facilitating

the

negotiation

over

a

set

of

predefined

attributes

representing

various

non-price

aspects

of

the

deal.

Rather

than

negotiating

over

a

fully

defined

good

or

service,

a

multiattribute

mechanism

delays

commitment

to

specific

configurations

until

the

most

promising

candidates

are

identified.

For

example,

a

procurement

department

of

a

company

may

use

a

multiattribute

auction

to

select

a

supplier

of

hard

drives.

Supplier

offers

may

be

evaluated

not

only

over

the

price

they

offer,

but

also

over

various

qualitative

attributes

such

as

volume,

RPM,

access

time,

latency,

transfer

rate,

and

so

on.

In

addition,

suppliers

may

offer

different

contract

conditions

such

as

warranty,

delivery

time,

and

service.

In

order

to

account

for

traders"

preferences,

the

auction

mechanism

must

extract

evaluative

information

over

a

complex

domain

of

multidimensional

configurations.

Constructing

and

communicating

a

complete

preference

specification

can

be

a

severe

burden

for

even

a

moderate

number

of

attributes,

therefore

practical

multiattribute

auctions

must

either

accommodate

partial

specifications,

or

support

compact

expression

of

preferences

assuming

some

simplified

form.

By

far

the

most

popular

multiattribute

form

to

adopt

is

the

simplest:

an

additive

representation

where

overall

value

is

a

linear

combination

of

values

associated

with

each

attribute.

For

example,

several

recent

proposals

for

iterative

multiattribute

auctions

require

additive

preference

representations.

Such

additivity

reduces

the

complexity

of

preference

specification

exponentially

(compared

to

the

general

discrete

case),

but

precludes

expression

of

any

interdependencies

among

the

attributes.

In

practice,

however,

interdependencies

among

natural

attributes

are

quite

common.

For

example,

the

buyer

may

exhibit

complementary

preferences

for

size

and

access

time

(since

the

performance

effect

is

more

salient

if

much

data

is

involved),

or

may

view

a

strong

warranty

as

a

good

substitute

for

high

reliability

ratings.

Similarly,

the

seller"s

production

characteristics

(such

as

increasing

access

time

is

harder

for

larger

hard

drives)

can

easily

violate

additivity.

In

such

cases

an

additive

value

function

may

not

be

able

to

provide

even

a

reasonable

approximation

of

real

preferences.

On

the

other

hand,

fully

general

models

are

intractable,

and

it

is

reasonable

to

expect

multiattribute

preferences

to

exhibit

some

structure.

Our

goal,

therefore,

is

to

identify

the

subtler

yet

more

widely

applicable

structured

representations,

and

exploit

these

properties

of

preferences

in

trading

mechanisms.

We

propose

an

iterative

auction

mechanism

based

on

just

such

a

flexible

preference

structure.

Our

approach

is

inspired

by

the

design

of

an

iterative

multiattribute

procurement

auction

for

additive

preferences,

due

to

Parkes

and

Kalagnanam

(PK)

PK

propose

two

types

of

iterative

auctions:

the

first

(NLD)

makes

no

assumptions

about

traders"

preferences,

and

lets

sellers

bid

on

the

full

multidimensional

attribute

space.

Because

NLD

maintains

an

exponential

price

structure,

it

is

suitable

only

for

small

domains.

The

other

auction

(AD)

assumes

additive

buyer

valuation

and

seller

cost

functions.

It

collects

sell

bids

per

attribute

level

and

for

a

single

discount

term.

The

price

of

a

configuration

is

defined

as

the

sum

of

the

prices

of

the

chosen

attribute

levels

minus

the

discount.

The

auction

we

propose

also

supports

compact

price

spaces,

albeit

for

levels

of

clusters

of

attributes

rather

than

singletons.

We

employ

a

preference

decomposition

based

on

generalized

additive

independence

(GAI),

a

model

flexible

enough

to

accommodate

interdependencies

to

the

exact

degree

of

accuracy

desired,

yet

providing

a

compact

functional

form

to

the

extent

that

interdependence

can

be

limited.

Given

its

roots

in

multiattribute

utility

theory

the

GAI

condition

is

defined

with

respect

to

the

expected

utility

function.

To

apply

it

for

modeling

values

for

certain

outcomes,

therefore,

requires

a

reinterpretation

for

preference

under

certainty.

To

this

end,

we

exploit

the

fact

that

auction

outcomes

are

associated

with

continuous

prices,

which

provide

a

natural

scale

for

assessing

magnitude

of

preference.

We

first

lay

out

a

representation

framework

for

preferences

that

captures,

in

addition

to

simple

orderings

among

attribute

configuration

values,

the

difference

in

the

willingness

to

pay

(wtp)

for

each.

That

is,

we

should

be

able

not

only

to

compare

outcomes

but

also

decide

whether

the

difference

in

quality

is

worth

a

given

difference

in

price.

Next,

we

build

a

direct,

formally

justified

link

from

preference

statements

over

priced

outcomes

to

a

generalized

additive

decomposition

of

the

wtp

function.

After

laying

out

this

infrastructure,

we

employ

this

representation

tool

for

the

development

of

a

multiattribute

iterative

auction

mechanism

that

allows

traders

to

express

their

complex

preferences

in

GAI

format.

We

then

study

the

auction"s

allocational,

computational,

and

practical

properties.

In

Section

we

present

essential

background

on

our

representation

framework,

the

measurable

value

function

(MVF).

Section

develops

new

multiattribute

structures

for

MVF,

supporting

generalized

additive

decompositions.

Next,

we

show

the

applicability

of

the

theoretical

framework

to

preferences

in

trading.

The

rest

of

the

paper

is

devoted

to

the

proposed

auction

mechanism.

MULTIATTRIBUTE

PREFERENCES

As

mentioned,

most

tools

facilitating

expression

of

multiattribute

value

for

trading

applications

assume

that

agents"

preferences

can

be

represented

in

an

additive

form.

By

way

of

background,

we

start

by

introducing

the

formal

prerequisites

justifying

the

additive

representation,

as

provided

by

multiattribute

utility

theory.

We

then

present

the

generalized

additive

form,

and

develop

the

formal

underpinnings

for

measurable

value

needed

to

extend

this

model

to

the

case

of

choice

under

certainty.

Preferential

Independence

Let

Θ

denote

the

space

of

possible

outcomes,

with

a

preference

relation

(weak

total

order)

over

Θ.

Let

A

am}

be

a

set

of

attributes

describing

Θ.

Capital

letters

denote

subsets

of

variables,

small

letters

(with

or

without

numeric

subscripts)

denote

specific

variables,

and

¯X

denotes

the

complement

of

X

with

respect

to

A.

We

indicate

specific

variable

assignments

with

prime

signs

or

superscripts.

To

represent

an

instantiation

of

subsets

X,

Y

at

the

same

time

we

use

a

sequence

of

instantiation

symbols,

as

in

X

Y

DEFINITION

A

set

of

attributes

Y

A

is

preferentially

independent

(PI)

of

its

complement

Z

A

Y

if

the

conditional

preference

order

over

Y

given

a

fixed

level

of

Z

is

the

same

regardless

of

the

choice

of

In

other

words,

the

preference

order

over

the

projection

of

A

on

the

attributes

in

Y

is

the

same

for

any

instantiation

of

the

attributes

in

Z.

DEFINITION

A

am}

is

mutually

preferentially

independent

(MPI)

if

any

subset

of

A

is

preferentially

independent

of

its

complement.

The

preference

relation

when

no

uncertainty

is

modeled

is

usually

represented

by

a

value

function

v

The

following

fundamental

result

greatly

simplifies

the

value

function

representation.

THEOREM

A

preference

order

over

set

of

attributes

A

has

an

additive

value

function

representation

am)

mX

vi(ai)

iff

A

is

mutually

preferential

independent.

Essentially,

the

additive

forms

used

in

trading

mechanisms

assume

mutual

preferential

independence

over

the

full

set

of

attributes,

including

the

money

attribute.

Intuitively

that

means

that

willingness

to

pay

for

value

of

an

attribute

or

attributes

cannot

be

affected

by

the

value

of

other

attributes.

A

cardinal

value

function

representing

an

ordering

over

certain

outcomes

need

not

in

general

coincide

with

the

cardinal

utility

function

that

represents

preference

over

lotteries

or

expected

utility

(EU).

Nevertheless,

EU

functions

may

possess

structural

properties

analogous

to

that

for

value

functions,

such

as

additive

decomposition.

Since

the

present

work

does

not

involve

decisions

under

uncertainty,

we

do

not

provide

a

full

exposition

of

the

EU

concept.

However

we

do

make

frequent

reference

to

the

following

additive

independence

relations.

DEFINITION

Let

X,

Y,

Z

be

a

partition

of

the

set

of

attributes

A.

X

and

Y

are

conditionally

additive

independent

given

Z,

denoted

as

CAI(X,

Y

Z),

if

preferences

over

lotteries

on

A

depend

only

on

their

marginal

conditional

probability

distributions

over

X

and

Y

DEFINITION

Let

Ig

A

such

that

Sg

Ii

A.

Ig

are

called

generalized

additive

independent

(GAI)

if

preferences

over

lotteries

on

A

depend

only

on

their

marginal

distributions

over

Ig.

An

(expected)

utility

function

u(·)

can

be

decomposed

additively

according

to

its

(possibly

overlapping)

GAI

sub-configurations.

THEOREM

Let

Ig

be

GAI.

Then

there

exist

functions

fg

such

that

am)

g

X

fr(Ir).

What

is

now

known

as

the

GAI

condition

was

originally

introduced

by

Fishburn

for

EU,

and

was

named

GAI

and

brought

to

the

attention

of

AI

researchers

by

Bacchus

and

Grove

Graphical

models

and

elicitation

procedures

for

GAI

decomposable

utility

were

developed

for

EU

for

a

cardinal

representation

of

the

ordinal

value

function

and

for

an

ordinal

preference

relations

corresponding

to

a

TCP-net

structure

by

Brafman

et

al.

Apart

from

the

work

on

GAI

in

the

context

of

preference

handling

that

were

discussed

above,

GAI

have

been

recently

used

in

the

context

of

mechanism

design

by

Hyafil

and

Boutilier

as

an

aid

in

direct

revelation

mechanisms.

As

shown

by

Bacchus

and

Grove

GAI

structure

can

be

identified

based

on

a

set

of

CAI

conditions,

which

are

much

easier

to

detect

and

verify.

In

general,

utility

functions

may

exhibit

GAI

structure

not

based

on

CAI.

However,

to

date

all

proposals

for

reasoning

and

eliciting

utility

in

GAI

form

take

advantage

of

the

GAI

structure

primarily

to

the

extent

that

it

represents

a

collection

of

CAI

conditions.

For

example,

GAI

trees

employ

triangulation

of

the

CAI

map,

and

Braziunas

and

Boutilier"s

conditional

set

Cj

of

a

set

Ij

corresponds

to

the

CAI

separating

set

of

Ij.

Since

the

CAI

condition

is

also

defined

based

on

preferences

over

lotteries,

we

cannot

apply

Bacchus

and

Grove"s

result

without

first

establishing

an

alternative

framework

based

on

priced

outcomes.

We

develop

such

a

framework

using

the

theory

of

measurable

value

functions,

ultimately

producing

a

GAI

decomposition

(Eq.

of

the

wtp

function.

Readers

interested

primarily

in

the

multiattribute

auction

and

willing

to

grant

the

well-foundedness

of

the

preference

structure

may

skip

down

to

Section

Measurable

Value

Functions

Trading

decisions

represent

a

special

case

of

decisions

under

certainty,

where

choices

involve

multiattribute

outcomes

and

corresponding

monetary

payments.

In

such

problems,

the

key

decision

often

hinges

on

relative

valuations

of

price

differences

compared

to

differences

in

alternative

configurations

of

goods

and

services.

Theoretically,

price

can

be

treated

as

just

another

attribute,

however,

such

an

approach

fails

to

exploit

the

special

character

of

the

money

dimension,

and

can

significantly

add

to

complexity

due

to

the

inherent

continuity

and

typical

wide

range

of

possible

monetary

outcome

values.

We

build

on

the

fundamental

work

of

Dyer

and

Sarin

on

measurable

value

functions

(MVFs).

As

we

show

below,

wtp

functions

in

a

quasi-linear

setting

can

be

interpreted

as

MVFs.

However

we

first

present

the

MVF

framework

in

a

more

generic

way,

where

the

measurement

is

not

necessarily

monetary.

We

present

the

essential

definitions

and

refer

to

Dyer

and

Sarin

for

more

detailed

background

and

axiomatic

treatment.

The

key

concept

is

that

of

preference

difference.

Let

θ1

θ2

ϑ1

ϑ2

Θ

such

that

θ1

θ2

and

ϑ1

ϑ2

θ1

denotes

the

preference

difference

between

θ2

and

θ1

interpreted

as

the

strength,

or

degree,

to

which

θ2

is

preferred

over

θ1

Let

denote

a

preference

order

over

Θ

Θ.

We

interpret

the

statement

θ1

ϑ1

as

the

preference

of

ϑ2

over

ϑ1

is

at

least

as

strong

as

the

preference

of

θ2

over

θ1

We

use

the

symbol

to

represent

equality

of

preference

differences.

DEFINITION

u

D

is

a

measurable

value

function

(MVF)

wrt

if

for

any

θ1

θ2

ϑ1

ϑ2

D,

θ1

ϑ1

Note

that

an

MVF

can

also

be

used

as

a

value

function

representing

since

[θ

θ]

[θ

θ]

iff

θ

θ

DEFINITION

Attribute

set

X

A

is

called

difference

independent

of

¯X

if

for

any

two

assignments

¯X

¯X

¯X

¯X

¯X

¯X

for

any

assignment

¯X

Or,

in

words,

the

preference

differences

on

assignments

to

X

given

a

fixed

level

of

¯X

do

not

depend

on

the

particular

level

chosen

for

¯X.

As

with

additive

independence

for

EU,

this

condition

is

stronger

than

preferential

independence

of

X.

Also

analogously

to

EU,

mutual

preferential

independence

combined

with

other

conditions

leads

to

additive

decomposition

of

the

MVF.

Moreover,

Dyer

and

Sarin

have

defined

analogs

of

utility

independence

for

MVF,

and

worked

out

a

parallel

set

of

decomposition

results.

ADVANCED

MVF

STRUCTURES

Conditional

Difference

Independence

Our

first

step

is

to

generalize

Definition

to

a

conditional

version.

DEFINITION

Let

X,

Y,

Z

be

a

partition

of

the

set

of

attributes

A.

X

is

conditionally

difference

independent

of

Y

given

Z,

denoted

as

CDI(X,

Y

Z),

if

instantiations

ˆZ,

Y

Y

Y

ˆZ,

Y

ˆZ]

Y

ˆZ,

Y

ˆZ].

Since

the

conditional

set

is

always

the

complement,

we

sometimes

leave

it

implicit,

using

the

abbreviated

notation

CDI(X,

Y

CDI

leads

to

a

decomposition

similar

to

that

obtained

from

CAI

LEMMA

Let

u(A)

be

an

MVF

representing

preference

differences.

Then

CDI(X,

Y

Z)

iff

u(A)

Y,

Z)

u(X,

Y

Z)

Y

Z).

To

complete

the

analogy

with

CAI,

we

generalize

Lemma

as

follows.

PROPOSITION

CDI(X,

Y

Z)

iff

there

exist

functions

Z)

and

Z),

such

that

u(X,

Y,

Z)

Z)

Z).

An

immediate

result

of

Proposition

is

that

CDI

is

a

symmetric

relation.

The

conditional

independence

condition

is

much

more

applicable

than

the

unconditional

one.

For

example,

if

attributes

a

X

and

b

X

are

complements

or

substitutes,

X

cannot

be

difference

independent

of

¯X.

However,

X

{a}

may

still

be

CDI

of

¯X

given

a.

GAI

Structure

for

MVF

A

single

CDI

condition

decomposes

the

value

function

into

two

parts.

We

seek

a

finer-grain

global

decomposition

of

the

utility

function,

similar

to

that

obtained

from

mutual

preferential

independence.

For

this

purpose

we

are

now

ready

to

employ

the

results

of

Bacchus

and

Grove

who

establish

that

the

CAI

condition

has

a

perfect

map

that

is,

there

exists

a

graph

whose

nodes

correspond

to

the

set

A,

and

its

node

separation

reflects

exactly

the

complete

set

of

CAI

conditions

on

A.

Moreover,

they

show

that

the

utility

function

decomposes

over

the

set

of

maximal

cliques

of

the

perfect

map.

Their

proofs

can

be

easily

adapted

to

CDI,

since

they

only

rely

on

the

decomposition

property

of

CAI

that

is

also

implied

by

CDI

according

to

Proposition

THEOREM

Let

G

(A,

E)

be

a

perfect

map

for

the

CDI

conditions

on

A.

Then

u(A)

g

X

fr(Ir),

where

Ig

are

(overlapping)

subsets

of

A,

each

corresponding

to

a

maximal

clique

of

G.

Given

Theorem

we

can

now

identify

an

MVF

GAI

structure

from

a

collection

of

CDI

conditions.

The

CDI

conditions,

in

turn,

are

particularly

intuitive

to

detect

when

the

preference

differences

carry

a

direct

interpretation,

as

in

the

case

with

monetary

differences

discussed

below.

Moreover,

the

assumption

or

detection

of

CDI

conditions

can

be

performed

incrementally,

until

the

MVF

is

decomposed

to

a

reasonable

dimension.

This

is

in

contrast

with

the

fully

additive

decomposition

of

MVF

that

requires

mutual

preferential

independence

Theorem

defines

a

decomposition

structure,

but

to

represent

the

actual

MVF

we

need

to

specify

the

functions

over

the

cliques.

The

next

theorem

establishes

that

the

functional

constituents

of

MVF

are

the

same

as

those

for

GAI

decompositions

as

defined

by

Fishburn

for

EU.

We

adopt

the

following

conventional

notation.

Let

m)

be

a

predefined

vector

called

the

reference

outcome.

For

any

I

A,

the

function

u([I])

stands

for

the

projection

of

u(A)

to

I

where

the

rest

of

the

attributes

are

fixed

at

their

reference

levels.

THEOREM

Let

G

(A,

E)

be

a

perfect

map

for

the

CDI

condition

on

A,

and

Ig}

a

set

of

maximal

cliques

as

defined

in

Theorem

Then

the

functional

decomposition

from

that

theorem

can

be

defined

as

and

for

r

g

fr

u([Ir])

X

u([

k\

Iis

Ir])

The

proof

directly

shows

that

if

graph

G

(A,

E)

is

a

perfect

map

of

CDI,

u(A)

decomposes

to

a

sum

over

the

functions

defined

in

Thus

this

proof

does

not

rely

on

the

decomposition

result

of

Theorem

only

on

the

existence

of

the

perfect

map.

To

summarize,

the

results

of

this

section

generalize

additive

MVF

theory.

In

particular

it

justifies

the

application

of

methods

recently

developed

under

the

EU

framework

to

representation

of

value

under

certainty.

WILLINGNESS-TO-PAY

AS

AN

MVF

Construction

In

this

section

we

apply

measurable

value

to

represent

differences

of

willingness

to

pay

for

outcomes.

We

assume

that

the

agent

has

a

preference

order

over

outcome

space,

represented

by

a

set

of

attributes

A,

and

an

attribute

p

representing

monetary

consequence.

Note

that

in

evaluating

a

purchase

decision,

p

would

correspond

to

the

agent"s

money

holdings

net

of

the

transaction

(i.e.,

wealth

after

purchase),

not

the

purchase

price.

An

outcome

in

this

space

is

represented

for

example

by

(θ

p

where

θ

is

an

instantiation

of

A

and

p

is

a

value

of

p.

We

further

assume

that

preferences

are

quasi-linear

in

p,

that

is

there

exists

a

value

function

of

the

form

v(A,

p)

u(A)

L(p),

where

L

is

a

positive

linear

The

quasi-linear

form

immediately

qualifies

money

as

a

measure

of

preference

differences,

and

establishes

a

monetary

scale

for

u(A).

DEFINITION

Let

v(A,

p)

u(A)+L(p)

represent

where

p

is

the

attribute

representing

money.

We

call

u(A)

a

willingnessto-pay

(wtp)

function.

Note

that

wtp

may

also

refer

to

the

seller"s

willingness

to

accept

function.

The

wtp

u(A)

is

a

cardinal

function,

unique

up

to

a

positive

linear

transformation.

Since

p

p

L(p

p

(where

θ1

Θ,

the

domain

of

A)

the

wtp

function

can

be

used

to

choose

among

priced

outcomes.

This

proof

and

most

other

proofs

in

this

paper

are

omitted

for

space

consideration,

and

are

available

in

an

online

appendix.

In

many

procurement

applications,

the

deals

in

question

are

small

relative

to

the

enterprises

involved,

so

the

quasi-linearity

assumption

is

warranted.

This

assumption

can

be

relaxed

to

a

condition

called

corresponding

tradeoffs

which

does

not

require

the

value

over

money

to

be

linear.

To

simplify

the

presentation,

however,

we

maintain

the

stronger

assumption.

Naturally,

elicitation

of

wtp

function

is

most

intuitive

when

using

direct

monetary

values.

In

other

words,

we

elicit

a

function

in

which

L(p)

p,

so

v(A,

p)

u(A)

p.

We

define

a

reference

outcome

and

assuming

continuity

of

p,

for

any

assignment

ˆθ

there

exists

a

ˆp

such

that

(ˆθ,

ˆp)

As

v

is

normalized

such

that

ˆp

is

interpreted

as

the

wtp

for

ˆθ,

or

the

reserve

price

of

ˆθ.

PROPOSITION

The

wtp

function

is

an

MVF

over

differences

in

the

reserve

prices.

We

note

that

the

wtp

function

is

used

extensively

in

economics,

and

that

all

the

development

in

Section

could

be

performed

directly

in

terms

of

wtp,

relying

on

quasi-linearity

for

preference

measurement,

and

without

formalization

using

MVFs.

This

formalization

however

aligns

this

work

with

the

fundamental

difference

independence

theory

by

Dyer

and

Sarin.

In

addition

to

facilitating

the

detection

of

GAI

structure,

the

CDI

condition

supports

elicitation

using

local

queries,

similar

to

how

CAI

is

used

by

Braziunas

and

Boutilier

We

adopt

their

definition

of

conditional

set

of

Ir,

noted

here

Sr,

as

the

set

of

neighbors

of

attributes

in

Ir

not

including

the

attributes

of

Ir.

Clearly,

Sr

is

the

separating

set

of

Ir

in

the

CDI

map,

hence

CDI(Ir,

Vr),

where

Vr

A

(Ir

Sr).

From

the

definition

of

CDI,

for

any

V

r

V

r

we

have:

r

r

V

r

r

r

V

r

r

r

V

r

r

r

V

r

Eliciting

the

wtp

function

therefore

amounts

to

eliciting

the

utility

(wtp)

of

one

full

outcome

(the

reference

outcome

θ0

and

then

obtaining

the

function

over

each

maximal

clique

using

monetary

differences

between

its

possible

assignments

(technique

known

as

pricing

out

keeping

the

variables

in

the

conditional

set

fixed.

These

ceteris

paribus

elicitation

queries

are

local

in

the

sense

that

the

agent

does

not

need

to

consider

the

values

of

the

rest

of

the

attributes.

Furthermore,

in

eliciting

MVFs

we

can

avoid

the

global

scaling

step

that

is

required

for

EU

functions.

Since

the

preference

differences

are

extracted

with

respect

to

specific

amounts

of

the

attribute

p,

the

utility

is

already

scaled

according

to

that

external

measure.

Hence,

once

the

conditional

utility

functions

u([Ij])

are

obtained,

we

can

calculate

u(A)

according

to

This

last

step

may

require

(in

the

worst

case)

computation

of

a

number

of

terms

that

is

exponential

in

the

number

of

max

cliques.

In

practice

however

we

do

not

expect

the

intersection

of

the

cliques

to

go

that

deep;

intersection

of

more

than

just

a

few

max

cliques

would

normally

be

empty.

To

take

advantage

of

that

we

can

use

the

search

algorithm

suggested

by

Braziunas

and

Boutilier

which

efficiently

finds

all

the

nonempty

intersections

for

each

clique.

Optimization

As

shown,

the

wtp

function

can

be

used

directly

for

pairwise

comparisons

of

priced

outcomes.

Another

preference

query

often

treated

in

the

literature

is

optimization,

or

choice

of

best

outcome,

possibly

under

constraints.

Typical

decisions

about

exchange

of

a

good

or

service

exhibit

what

we

call

first-order

preferential

independence

(FOPI),

under

which

most

or

all

single

attributes

have

a

natural

ordering

of

quality,

independent

of

the

values

of

the

For

example,

when

choosing

a

PC

we

always

prefer

more

memory,

faster

CPU,

longer

warranty,

and

so

on.

Under

FOPI,

the

unconstrained

optimization

of

This

should

not

be

mistaken

with

the

highly

demanding

condition

of

mutual

preferential

independence,

that

requires

all

tradeoffs

between

attributes

to

be

independent.

unpriced

outcomes

is

trivial,

hence

we

consider

choice

among

attribute

points

with

prices.

Since

any

outcome

can

be

best

given

enough

monetary

compensation,

this

problem

is

not

well-defined

unless

the

combinations

are

constrained

somehow.

A

particularly

interesting

optimization

problem

arises

in

the

context

of

negotiation,

where

we

consider

the

utility

of

both

buyers

and

sellers.

The

multiattribute

matching

problem

(MMP)

is

concerned

with

finding

an

attribute

point

that

maximizes

the

surplus

of

a

trade,

or

the

difference

between

the

utilities

of

the

buyer

and

the

seller,

ub(A)

us(A).

GAI,

as

an

additive

decomposition,

has

the

property

that

if

ub

and

us

are

in

GAI

form

then

ub(A)−us(A)

is

in

GAI

form

as

well.

We

can

therefore

use

combinatorial

optimization

procedures

for

GAI

decomposition,

based

on

the

well

studied

variable

elimination

schemes

(e.g.,

to

find

the

best

trading

point.

Similarly,

this

optimization

can

be

done

to

maximize

surplus

between

a

trader"s

utility

function

and

a

pricing

system

that

assigns

a

price

to

each

level

of

each

GAI

element,

and

this

way

guide

traders

to

their

optimal

bidding

points.

In

the

rest

of

the

paper

we

develop

a

multiattribute

procurement

auction

that

builds

on

this

idea.

GAI

IN

MULTIATTRIBUTE

AUCTIONS

The

Multiattribute

Procurement

Problem

In

the

procurement

setting

a

single

buyer

wishes

to

procure

a

single

good,

in

some

configuration

θ

Θ

from

one

of

the

candidate

sellers

sn.

The

buyer

has

some

private

valuation

function

(wtp)

ub

Θ

R,

and

similarly

each

seller

si

has

a

private

valuation

function

(willingness-to-accept).

For

compliance

with

the

procurement

literature

we

refer

to

seller

si"s

valuation

as

a

cost

function,

denoted

by

ci.

The

multiattribute

allocation

problem

(MAP)

is

the

welfare

optimization

problem

in

procurement

over

a

discrete

domain,

and

it

is

defined

as:

i∗

θ∗

arg

max

i,θ

(ub(θ)

ci(θ)).

To

illustrate

the

need

for

a

GAI

price

space

we

consider

the

case

of

traders

with

non-additive

preferences

bidding

in

an

additive

price

space

such

as

in

PK"s

auction

AD.

If

the

buyer"s

preferences

are

not

additive,

choosing

preferred

levels

per

attribute

(as

in

auction

AD)

admits

undesired

combinations

and

fails

to

guide

the

sellers

to

the

efficient

configurations.

Non-additive

sellers

face

an

exposure

problem,

somewhat

analogous

to

traders

with

complementary

preferences

that

participate

in

simultaneous

auctions.

A

value

for

attribute

a

may

be

optimal

given

that

the

value

of

another

attribute

b

is

and

arbitrarily

suboptimal

given

other

values

of

b.

Therefore

bidding

and

may

result

in

a

poor

allocation

if

the

seller

is

outbid

on

but

left

holding

Instead

of

assuming

full

additivity,

the

auction

designer

can

come

up

with

a

GAI

preference

structure

that

captures

the

set

of

common

interdependencies

between

attributes.

If

traders

could

bid

on

clusters

of

interdependent

attributes,

it

would

solve

the

problems

discussed

above.

For

example,

if

a

and

b

are

interdependent

(meaning

CDI(a,

b)

does

not

hold),

we

should

be

able

to

bid

on

the

cluster

ab.

If

b

in

turn

depends

on

c,

we

need

another

cluster

bc.

This

is

still

better

than

a

general

pricing

structure

that

solicits

bids

for

the

cluster

abc.

We

stress

that

each

trader

may

have

a

different

set

of

interdependencies,

and

therefore

to

be

completely

general

the

If

only

the

sellers

are

non-additive,

the

auction

design

could

potentially

alleviate

this

problem

by

collecting

a

new

set

of

bids

each

round

and

forgetting

bids

from

previous

rounds,

and

also

guiding

non-additive

sellers

to

bid

on

only

one

level

per

attribute

in

order

to

avoid

undesired

combinations.

ya

yb

yc

Z

Z

Z

(i)

©a,

b

©b,

c

(ii)

Figure

(i)

CDI

map

for

{a,

b,

c},

reflecting

the

single

condition

CDI(a,

c).

(ii)

The

corresponding

GAI

network.

GAI

structure

needs

to

account

for

However,

in

practice

many

domains

have

natural

dependencies

that

are

mutual

to

traders.

GAI

Trees

Assume

that

preferences

of

all

traders

are

reflected

in

a

GAI

structure

Ig.

We

call

each

Ir

a

GAI

element,

and

any

assignment

to

Ir

a

sub-configuration.

We

use

θr

to

denote

the

subconfiguration

formed

by

projecting

configuration

θ

to

element

Ir.

DEFINITION

Let

α

be

an

assignment

to

Ir

and

β

an

assignment

to

Ir

The

sub-configurations

α

and

β

are

consistent

if

for

any

attribute

aj

Ir

Ir

α

and

β

agree

on

the

value

of

aj.

A

collection

ν

of

sub-configurations

is

consistent

if

all

pairs

α,

β

ν

are

consistent.

The

collection

is

called

a

cover

if

it

contains

exactly

one

sub-configuration

αr

corresponding

to

each

element

Ir,

r

g}.

Note

that

a

consistent

cover

αg}

represents

a

full

configuration,

which

we

denote

by

αg).

A

GAI

network

is

a

graph

G

whose

nodes

correspond

to

the

GAI

elements

Ig,

with

an

edge

between

Ir,

Ir

iff

Ir

Ir

Equivalently,

a

GAI

network

is

the

clique

graph

of

a

CDI-map.

In

order

to

justify

the

compact

pricing

structure

we

require

that

for

any

set

of

optimal

configurations

(wrt

a

given

utility

function),

with

a

corresponding

collection

of

sub-configurations

γ,

all

consistent

covers

in

γ

must

be

optimal

configurations

as

well.

To

ensure

this

(see

Lemmas

and

we

assume

a

GAI

decomposition

in

the

form

of

a

tree

or

a

forest

(the

GAI

tree).

A

tree

structure

can

be

achieved

for

any

set

of

CDI

conditions

by

triangulation

of

the

CDI-map

prior

to

construction

of

the

clique

graph

(GAI

networks

and

GAI

trees

are

defined

by

Gonzales

and

Perny

who

also

provide

a

triangulation

algorithm).

Under

GAI,

the

buyer"s

value

function

ub

and

sellers"

cost

functions

ci

can

be

decomposed

as

in

We

use

fb,r

and

fi,r

to

denote

the

local

functions

of

buyer

and

sellers

(respectively),

according

to

For

example,

consider

the

procurement

of

a

good

with

three

attributes,

a,

b,

c.

Each

attribute"s

domain

has

two

values

(e.g.,

is

the

domain

of

A).

Let

the

GAI

structure

be

{a,

b},

{b,

c}.

Figure

shows

the

simple

CDI

map

and

the

corresponding

GAI

network,

which

is

a

GAI

tree.

Here,

subconfigurations

are

assignments

of

the

form

and

so

on.

The

set

of

sub-configurations

is

a

consistent

cover,

corresponding

to

the

configuration

In

contrast,

the

set

is

inconsistent.

The

GAI

Auction

We

define

an

iterative

multiattribute

auction

that

maintains

a

GAI

pricing

structure:

that

is,

a

price

pt

corresponding

to

each

subconfiguration

of

each

GAI-tree

element.

The

price

of

a

configuration

θ

at

time

t

is

defined

as

pt

(θ)

g

X

pt

(θr)

Δ.

We

relax

this

requirement

in

Section

Bidders

submit

sub-bids

on

sub-configurations

and

on

an

additional

global

discount

term

Sub-bids

are

always

submitted

for

current

prices,

and

need

to

be

resubmitted

at

each

round,

therefore

they

do

not

need

to

explicitly

carry

the

price.

The

set

of

full

bids

of

a

seller

contains

all

consistent

covers

that

can

be

generated

from

that

seller"s

current

set

of

sub-bids.

The

existence

of

a

full

bid

over

a

configuration

θ

represents

the

seller"s

willingness

to

accept

the

price

pt

(θ)

for

supplying

θ.

At

the

start

of

the

auction,

the

buyer

reports

(to

the

auction,

not

to

sellers)

her

complete

valuation

in

GAI

form.

The

initial

prices

of

sub-configurations

are

set

at

some

level

above

the

buyer"s

valuations,

that

is,

(θr)

fb,r(θr)

for

all

θr.

The

discount

Δ

is

initialized

to

zero.

The

auction

has

the

dynamics

of

a

descending

clock

auction:

at

each

round

t,

bids

are

collected

for

current

prices

and

then

prices

are

reduced

according

to

price

rules.

A

seller

is

considered

active

in

a

round

if

she

submits

at

least

one

full

bid.

In

round

t

only

sellers

who

where

active

in

round

t

are

allowed

to

participate,

and

the

auction

terminates

when

no

more

than

a

single

seller

is

active.

We

denote

the

set

of

sub-bids

submitted

by

si

by

Bt

i

and

the

corresponding

set

of

full

bids

is

Bt

i

{θ

θg)

Θ

∀r.θr

Bt

i

In

our

example,

a

seller

could

submit

sub-bids

on

a

set

of

subconfigurations

such

as

and

and

that

combines

to

a

full

bid

on

The

auction

proceeds

in

two

phases.

In

the

first

phase

(A),

at

each

round

t

the

auction

computes

a

set

of

preferred

sub-configurations

Mt

Section

shows

how

to

define

Mt

to

ensure

convergence,

and

Section

shows

how

to

efficiently

compute

it.

In

phase

A,

the

auction

adjusts

prices

after

each

round,

reducing

the

price

of

every

sub-configuration

that

has

received

a

bid

but

is

not

in

the

preferred

set.

Let

be

the

prespecified

price

increment

parameter.

Specifically,

the

phase

A

price

change

rule

is

applied

to

all

θr

Sn

Bt

i

Mt

(θr)

max(pt

(θr)

g

fb,r(θr)).

[A]

The

RHS

maximum

ensures

that

prices

do

not

get

reduced

below

the

buyer"s

valuation

in

phase

A.

Let

Mt

denote

the

set

of

configurations

that

are

consistent

covers

in

Mt

Mt

{θ

θg)

Θ

∀r.θr

Mt

The

auction

switches

to

phase

B

when

all

active

sellers

have

at

least

one

full

bid

in

the

buyer"s

preferred

set:

∀i.

Bt

i

Bt

i

Mt

[SWITCH]

Let

T

be

the

round

at

which

[SWITCH]

becomes

true.

At

this

point,

the

auction

selects

the

buyer-optimal

full

bid

ηi

for

each

seller

si.

ηi

arg

max

θ∈BT

i

(ub(θ)

pT

(θ)).

In

phase

B,

si

may

bid

only

on

ηi.

The

prices

of

sub-configurations

are

fixed

at

pT

during

this

phase.

The

only

adjustment

in

phase

B

is

to

Δ,

which

is

increased

in

every

round

by

The

auction

terminates

when

at

most

one

seller

(if

exactly

one,

designate

it

sˆi)

is

active.

There

are

four

distinct

cases:

All

sellers

drop

out

in

phase

A

(i.e.,

before

rule

[SWITCH]

holds).

The

auction

returns

with

no

allocation.

The

discount

term

could

be

replaced

with

a

uniform

price

reduction

across

all

sub-configurations.

All

active

sellers

drop

out

in

the

same

round

in

phase

B.

The

auction

selects

the

best

seller

(sˆi)

from

the

preceding

round,

and

applies

the

applicable

case

below.

The

auction

terminates

in

phase

B

with

a

final

price

above

buyer"s

valuation,

pT

(ηˆi)

Δ

ub(ηˆi).

The

auction

offers

the

winner

sˆi

an

opportunity

to

supply

ηˆi

at

price

ub(ηˆi).

The

auction

terminates

in

phase

B

with

a

final

price

pT

(ηˆi)−

Δ

ub(ηˆi).

This

is

the

ideal

situation,

where

the

auction

allocates

the

chosen

configuration

and

seller

at

this

resulting

price.

The

overall

auction

is

described

by

high-level

pseudocode

in

Algorithm

As

explained

in

Section

the

role

of

phase

A

is

to

guide

the

traders

to

their

efficient

configurations.

Phase

B

is

a

one-dimensional

competition

over

the

surplus

that

remaining

seller

candidates

can

provide

to

the

buyer.

In

Section

we

discuss

the

computational

tasks

associated

with

the

auction,

and

Section

provides

a

detailed

example.

Algorithm

GAI-based

multiattribute

auction

collect

a

reported

valuation,

ˆv

from

the

buyer

set

high

initial

prices,

(θr)

on

each

level

θr,

and

set

Δ

while

not

[SWITCH]

do

collect

sub-bids

from

sellers

compute

Mt

apply

price

change

by

[A]

end

while

compute

ηi

while

more

than

one

active

seller

do

increase

Δ

by

collect

bids

on

(ηi,

Δ)

from

sellers

end

while

implement

allocation

and

payment

to

winning

seller

Economic

Analysis

When

the

optimal

solution

to

MAP

provides

negative

welfare

and

sellers

do

not

bid

below

their

cost,

the

auction

terminates

in

phase

A,

no

trade

occurs

and

the

auction

is

trivially

efficient.

We

therefore

assume

throughout

the

analysis

that

the

optimal

(seller,configuration)

pair

provides

non-negative

welfare.

The

buyer

profit

from

a

configuration

θ

is

defined

πb(θ)

ub(θ)

p(θ)

and

similarly

πi(θ)

p(θ)

ci(θ)

is

the

profit

of

si.

In

addition,

for

μ

g}

we

denote

the

corresponding

set

of

subconfigurations

by

θμ,

and

define

the

profit

from

a

configuration

θ

over

the

subset

μ

as

πb(θμ)

X

r∈μ

(fb,r(θr)

p(θr)).

πi(θμ)

is

defined

similarly

for

si.

Crucially,

for

any

μ

and

its

complement

¯μ

and

for

any

trader

τ,

πτ

(θ)

πτ

(θμ)

πτ

(θ¯μ).

The

function

σi

Θ

R

represents

the

welfare,

or

surplus

function

ub(·)

ci(·).

For

any

price

system

p,

σi(θ)

πb(θ)

πi(θ).

We

drop

the

t

superscript

in

generic

statements

involving

price

and

profit

functions,

understanding

that

all

usage

is

with

respect

to

the

(currently)

applicable

prices.

Since

we

do

not

assume

anything

about

the

buyer"s

strategy,

the

analysis

refers

to

profit

and

surplus

with

respect

to

the

face

value

of

the

buyer"s

report.

The

functions

πi

and

σi

refer

to

the

true

cost

functions

of

si.

DEFINITION

A

seller

is

called

Straightforward

Bidder

(SB)

if

at

each

round

t

she

bids

on

Bt

i

as

follows:

if

maxθ∈Θ

πt

i

(θ)

then

Bt

i

Otherwise

let

Ωt

i

arg

max

θ∈Θ

πt

i

(θ)

Bt

i

{θr

θ

Ωt

i,

r

g}}.

Intuitively,

an

SB

seller

follows

a

myopic

best

response

strategy

(MBR),

meaning

they

bid

myopically

rather

than

strategically

by

optimizing

their

profit

with

respect

to

current

prices.

To

calculate

Bt

i

sellers

need

to

optimize

their

current

profit

function,

as

discussed

in

Section

The

following

lemma

bridges

the

apparent

gap

between

the

compact

pricing

and

bid

structure

and

the

global

optimization

performed

by

the

traders.

LEMMA

Let

Ψ

be

a

set

of

configurations,

all

maximizing

profit

for

a

trader

τ

(seller

or

buyer)

at

the

relevant

prices.

Let

Φ

{θr

θ

Ψ,

r

g}.

Then

any

consistent

cover

in

Φ

is

also

a

profit-maximizing

configuration

for

τ.

Proof

sketch

(full

proof

in

the

online

appendix):

A

source

of

an

element

θr

is

a

configuration

˜θ

Ψ

from

which

it

originated

(meaning,

˜θr

θr).

Starting

from

the

supposedly

suboptimal

cover

θ1

we

build

a

series

of

covers

θ1

θL

At

each

θj

we

flip

the

value

of

a

set

of

sub-configurations

μj

corresponding

to

a

subtree,

with

the

sub-configurations

of

the

configuration

ˆθj

Ψ

which

is

the

source

of

the

parent

γj

of

μj

That

ensures

that

all

elements

in

μj

{γj}

have

a

mutual

source

ˆθj

We

show

that

all

θj

are

consistent

and

that

they

must

all

be

suboptimal

as

well,

and

since

all

elements

of

θL

have

a

mutual

source,

meaning

θL

ˆθL

Ψ,

it

contradicts

optimality

of

Ψ.

COROLLARY

For

SB

seller

si,

∀t,

∀θ

Bt

i

πt

i

(θ

max

θ∈Θ

πt

i

(θ).

Next

we

consider

combinations

of

configurations

that

are

only

within

some

δ

of

optimality.

LEMMA

Let

Ψ

be

a

set

of

configurations,

all

are

within

δ

of

maximizing

profit

for

a

trader

τ

at

the

prices,

and

Φ

defined

as

in

Lemma

Then

any

consistent

cover

in

Φ

is

within

δg

of

maximizing

utility

for

τ.

This

bound

is

tight,

that

is

for

any

GAI

tree

and

a

non-trivial

domain

we

can

construct

a

set

Ψ

as

above

in

which

there

exists

a

consistent

cover

whose

utility

is

exactly

δg

below

the

maximal.

Next

we

formally

define

Mt

For

connected

GAI

trees,

Mt

is

the

set

of

sub-configurations

that

are

part

of

a

configuration

within

of

optimal.

When

the

GAI

tree

is

in

fact

a

forest,

we

apportion

the

error

proportionally

across

the

disconnected

trees.

Let

G

be

comprised

of

trees

Gh.

We

use

θj

to

denote

the

projection

of

a

configuration

θ

on

the

tree

Gj

and

gj

denotes

the

number

of

GAI

elements

in

Gj

Mt

j

{θr

πt

b(θj)

max

θj

∈Θj

πt

b(θj

gj

g

r

Gj

Then

define

Mt

Sh

Mt

j.

Let

ej

gj

−1

denote

the

number

of

edges

in

Gj

We

define

the

connectivity

parameter,

e

ej

As

shown

below,

this

connectivity

parameter

is

an

important

factor

in

the

performance

of

the

auction.

COROLLARY

∀θ

Mt

πt

b(θ

max

θ∈Θ

πt

b(θ)

(e

In

the

fully

additive

case

this

loss

of

efficiency

reduces

to

On

the

other

extreme,

if

the

GAI

network

is

connected

then

g.

We

also

note

that

without

assuming

any

preference

structure,

meaning

that

the

CDI

map

is

fully

connected,

g

and

the

efficiency

loss

is

again

Lemmas

through

show

that

through

the

price

system,

the

choice

of

buyer

preferred

configurations,

and

price

change

rules,

Phase

A

leads

the

buyer

and

each

of

the

sellers

to

their

mutually

efficient

configuration.

LEMMA

maxθ∈Θ

πt

b(θ)

does

not

change

in

any

round

t

of

phase

A.

PROOF.

We

prove

the

lemma

per

each

tree

Gj.

The

optimal

values

for

disconnected

components

are

independent

of

each

other

hence

if

the

maximal

profit

for

each

component

does

not

change

the

combined

maximal

profit

does

not

change

as

well.

If

the

price

of

θj

was

reduced

during

phase

A,

that

is

(θj)

pt

(θj

δ,

it

must

be

the

case

that

some

w

gj

sub-configurations

of

θj

are

not

in

Mt

j,

and

δ

w

g

The

definition

of

Mt

j

ensures

πt

b(θj

max

θ∈Θ

πt

b(θj)

gj

g

Therefore,

b

(θ

πt

(θ

δ

πt

(θ

w

g

max

θ∈Θ

πt

b(θj).

This

is

true

for

any

configuration

whose

profit

improves,

therefore

the

maximal

buyer

profit

does

not

change

during

phase

A.

LEMMA

The

price

of

at

least

one

sub-configuration

must

be

reduced

at

every

round

in

phase

A.

PROOF.

In

each

round

t

T

of

phase

A

there

exists

an

active

seller

i

for

whom

Bt

i

Mt

However

to

be

active

in

round

t,

Bt

i

Let

ˆθ

Bt

i

If

∀r.ˆθr

Mt

then

ˆθ

Mt

by

definition

of

Mt

Therefore

there

must

be

ˆθr

Mt

We

need

to

prove

that

for

at

least

one

of

these

sub-configurations,

πt

b(ˆθr)

to

ensure

activation

of

rule

[A].

Assume

for

contradiction

that

for

any

ˆθr

¯Mt

πt

b(ˆθr)

For

simplicity

we

assume

that

for

any

θr,

π1

b

(θr)

is

some

product

of

g

(that

can

be

easily

done),

and

that

ensures

that

πt

b(ˆθr)

because

once

profit

hits

it

cannot

increase

by

rule

[A].

If

ˆθr

¯Mt

∀r

g

then

πt

b(ˆθ)

This

contradicts

Lemma

since

we

set

high

initial

prices.

Therefore

some

of

the

sub-configurations

of

ˆθ

are

in

Mt

and

WLOG

we

assume

it

is

ˆθ1,

ˆθk.

To

be

in

Mt

these

k

sub-configurations

must

have

been

in

some

preferred

full

configuration,

meaning

there

exists

θ

Mt

such

that

θ

ˆθk,

θg)

Since

ˆθ

Mt

It

must

be

that

case

that

πt

b(ˆθ)

πt

b(θ

Therefore

πt

θg)

πt

ˆθg)

Hence

for

at

least

one

r

{k

g},

πt

b(θr)

contradicting

rule

[A].

LEMMA

When

the

solution

to

MAP

provides

positive

surplus,

and

at

least

the

best

seller

is

SB,

the

auction

must

reach

phase

B.

PROOF.

By

Lemma

prices

must

go

down

in

every

round

of

phase

A.

Rule

[A]

sets

a

lower

bound

on

all

prices

therefore

the

auction

either

terminates

in

phase

A

or

must

reach

condition

[SWITCH].

We

set

the

initial

prices

are

high

such

that

maxθ∈Θ

π1

b

(θ)

and

by

Lemma

maxθ∈Θ

πt

b(θ)

during

phase

A.

We

assume

that

the

efficient

allocation

(θ∗

i∗

provides

positive

welfare,

that

is

σi∗

(θ∗

πt

b(θ∗

πt

i∗

(θ∗

si∗

is

SB

therefore

she

will

leave

the

auction

only

when

πt

i∗

(θ∗

This

can

happen

only

when

πt

b(θ∗

therefore

si∗

does

not

drop

in

phase

A

hence

the

auction

cannot

terminate

before

reaching

condition

[SWITCH].

LEMMA

For

SB

seller

si,

ηi

is

(e

-efficient.

PROOF.

ηi

is

chosen

to

maximize

the

buyer"s

surplus

out

of

Bt

i

at

the

end

of

phase

A.

Since

Bt

i

Mt

clearly

ηi

Mt

From

Corollary

and

Corollary

for

any

˜θ,

πT

b

(ηi)

πT

b

(˜θ)

(e

πT

i

(ηi)

πT

i

(˜θ)

σi(ηi)

σi(˜θ)

(e

This

establishes

the

approximate

bilateral

efficiency

of

the

results

of

Phase

A

(at

this

point

under

the

assumption

of

SB).

Based

on

Phase

B"s

simple

role

as

a

single-dimensional

bidding

competition

over

the

discount,

we

next

assert

that

the

overall

result

is

efficient

under

SB,

which

in

turn

proves

to

be

an

approximately

ex-post

equilibrium

strategy

in

the

two

phases.

LEMMA

If

sellers

si

and

sj

are

SB,

and

si

is

active

at

least

as

long

as

sj

is

active

in

phase

B,

then

σi(ηi)

max

θ∈Θ

σj(θ)

(e

THEOREM

Given

a

truthful

buyer

and

SB

sellers,

the

auction

is

-efficient:

the

surplus

of

the

final

allocation

is

within

(e

of

the

maximal

surplus.

Following

PK,

we

rely

on

an

equivalence

to

the

one-sided

VCG

auction

to

establish

incentive

properties

for

the

sellers.

In

the

onesided

multiattribute

VCG

auction,

buyer

and

sellers

report

valuation

and

cost

functions

ˆub,

ˆci,

and

the

buyer

pays

the

sell-side

VCG

payment

to

the

winning

seller.

DEFINITION

Let

(θ∗

i∗

be

the

optimal

solution

to

MAP.

Let

(˜θ,˜i)

be

the

best

solution

to

MAP

when

i∗

does

not

participate.

The

sell-side

VCG

payment

is

V

CG(ˆub,

ˆci)

ˆub(θ∗

(ˆub(˜θ)

ˆc˜i(˜θ)).

It

is

well-known

that

truthful

bidding

is

a

dominant

strategy

for

sellers

in

the

one-sided

VCG

auction.

It

is

also

shown

by

PK

that

the

maximal

regret

for

buyers

from

bidding

truthfully

in

this

mechanism

is

ub(θ∗

ci∗

(θ∗

(ub(˜θ)

ˆc˜i(˜θ)),

that

is,

the

marginal

product

of

the

efficient

seller.

Usually

in

iterative

auctions

the

VCG

outcome

is

only

nearly

achieved,

but

the

deviation

is

bounded

by

the

minimal

price

change.

We

show

a

similar

result,

and

therefore

define

δ-VCG

payments.

DEFINITION

Sell-side

δ-VCG

payment

for

MAP

is

a

payment

p

such

that

V

CG(ˆub,

ˆci)

δ

p

V

CG(ˆub,

ˆci)

δ.

When

payment

is

guaranteed

to

be

δ-VCG

sellers

can

only

affect

their

payment

within

that

range,

therefore

their

gain

by

falsely

reporting

their

cost

is

bounded

by

LEMMA

When

sellers

are

SB,

the

payment

in

the

end

of

GAI

auction

is

sell-side

(e

-VCG.

THEOREM

SB

is

an

ex-post

Nash

Equilibrium

for

sellers

in

GAI

auction.

That

is,

sellers

cannot

gain

more

than

by

deviating.

In

practice,

however,

sellers

are

unlikely

to

have

the

information

that

would

let

them

exploit

that

potential

gain.

They

are

much

more

likely

to

lose

from

bidding

on

their

less

attractive

configurations.

Computation

and

Complexity

The

size

of

the

price

space

maintained

in

the

auction

is

equal

to

the

total

number

of

sub-configurations,

meaning

it

is

exponential

in

maxr

|Ir|.

This

is

also

equivalent

to

the

tree-width

(plus

one)

of

the

original

CDI-map.

For

the

purpose

of

the

computational

analysis

let

dj

denote

the

domain

of

attribute

aj,

and

I

Sg

Q

j∈Ir

dj,

the

collection

of

all

sub-configurations.

The

first

purpose

of

this

sub-section

is

to

show

that

the

complexity

of

all

the

computations

required

for

the

auction

depends

only

on

|I|,

i.e.,

no

computation

depends

on

the

size

of

the

full

exponential

domain.

We

are

first

concerned

with

the

computation

of

Mt

Since

Mt

grows

monotonically

with

t,

a

naive

application

of

optimization

algorithm

to

generate

the

best

outcomes

sequentially

might

end

up

enumerating

significant

portions

of

the

fully

exponential

domain.

However

as

shown

below

this

plain

enumeration

can

be

avoided.

PROPOSITION

The

computation

of

Mt

can

be

done

in

time

Moreover,

the

total

time

spent

on

this

task

throughout

the

auction

is

O(|I|(|I|

T)).

The

bounds

are

in

practice

significantly

lower,

based

on

results

on

similar

problems

from

the

probabilistic

reasoning

literature

One

of

the

benefits

of

the

compact

pricing

structure

is

the

compact

representation

it

lends

for

bids:

sellers

submit

only

sub-bids,

and

therefore

the

number

of

them

submitted

and

stored

per

seller

is

bounded

by

|I|.

Since

the

computation

tasks:

Bt

i

rule

[SWITCH]

and

choice

of

ηi

are

all

involving

the

set

Bt

i

it

is

important

to

note

that

their

performance

only

depend

on

the

size

of

the

set

Bt

i

since

they

are

all

subsumed

by

the

combinatorial

optimization

task

over

Bt

i

or

Bt

i

Mt

Next,

we

analyze

the

number

of

rounds

it

takes

for

the

auction

to

terminate.

Phase

B

requires

πT

i

Since

this

is

equivalent

to

price-only

auctions,

the

concern

is

only

with

the

time

complexity

of

phase

A.

Since

prices

cannot

go

below

fb,r(θr),

an

upper

bound

on

the

number

of

rounds

required

is

T

X

θr∈I

(θr)

fb,r(θr))

g

However

phase

A

may

converge

faster.

Let

the

initial

negative

profit

chosen

by

the

auctioneer

be

m

maxθ∈Θ

π1

b

(θ).

In

the

worst

case

phase

A

needs

to

run

until

∀θ

Θ.πb(θ)

m.

This

happens

for

example

when

∀θr

I.pt

(θr)

fb,r(θr)

m

g

In

general,

the

closer

the

initial

prices

reflect

buyer

valuation,

the

faster

phase

A

converges.

One

extreme

is

to

choose

(θr)

fb

Table

GAI

utility

functions

for

the

example

domain.

fb

represents

the

buyer"s

valuation,

and

and

costs

of

the

sellers

and

fb,r(θr)

m

g

That

would

make

phase

A

redundant,

at

the

cost

of

full

initial

revelation

of

buyer"s

valuation

as

done

in

other

mechanisms

discussed

below.

Between

this

option

and

the

other

extreme,

which

is

∀α,

ˆα

I,

(α)

(ˆα)

the

auctioneer

has

a

range

of

choices

to

determine

the

right

tradeoff

between

convergence

time

and

information

revelation.

In

the

example

below

the

choice

of

a

lower

initial

price

for

the

domain

of

provides

some

speedup

by

revealing

a

harmless

amount

of

information.

Another

potential

concern

is

the

communication

cost

associated

with

the

Japanese

auction

style.

The

sellers

need

to

send

their

bids

over

and

over

again

at

each

round.

A

simple

change

can

be

made

to

avoid

much

of

the

redundant

communication:

the

auction

can

retain

sub-bids

from

previous

rounds

on

sub-configurations

whose

price

did

not

change.

Since

combinations

of

sub-bids

from

different

rounds

can

yield

sub-optimal

configurations,

each

sub-bid

should

be

tagged

with

the

number

of

the

latest

round

in

which

it

was

submitted,

and

only

consistent

combinations

from

the

same

round

are

considered

to

be

full

bids.

With

this

implementation

sellers

need

not

resubmit

their

bid

until

a

price

of

at

least

one

sub-configuration

has

changed.

Example

We

use

the

example

settings

introduced

in

Section

Recall

that

the

GAI

structure

is

{a,

b},

{b,

c}

(note

that

e

Table

shows

the

GAI

utilities

for

the

buyer

and

the

two

sellers

The

efficient

allocation

is

with

a

surplus

of

The

maximal

surplus

of

the

second

best

seller,

is

achieved

by

and

We

set

all

initial

prices

over

to

and

all

initial

prices

over

to

We

set

meaning

that

price

reduction

for

sub-configurations

is

Though

with

these

numbers

it

is

not

guaranteed

by

Theorem

we

expect

to

win

on

either

the

efficient

allocation

or

on

which

provides

a

surplus

of

The

reason

is

that

these

are

the

only

two

configurations

which

are

within

(e

of

being

efficient

for

(therefore

one

of

them

must

be

chosen

by

Phase

A),

and

both

provide

more

than

surplus

over

most

efficient

configuration

(and

this

is

sufficient

in

order

to

win

in

Phase

B).

Table

shows

the

progress

of

phase

A.

Initially

all

configuration

have

the

same

cost

so

sellers

bid

on

their

lowest

cost

configuration

which

is

for

both

(with

profit

to

and

to

and

that

translates

to

sub-bids

on

and

contains

the

sub-configurations

and

of

the

highest

value

configuration

Price

is

therefore

decreased

on

and

After

the

price

change,

has

higher

profit

on

and

she

therefore

bids

on

and

Now

(round

their

prices

go

down,

reducing

the

profit

on

to

and

therefore

in

round

prefers

(profit

After

the

next

price

change

the

configurations

and

both

become

optimal

(profit

and

the

subbids

and

capture

the

two.

These

configurations

stay

optimal

for

another

round

with

profit

At

this

point

has

a

full

bid

(in

fact

two

full

bids:

and

in

and

t

Table

Auction

progression

in

phase

A.

Sell

bids

and

designation

of

Mt

(using

are

shown

below

the

price

of

each

subconfiguration.

therefore

she

no

longer

changes

her

bids

since

the

price

of

her

optimal

configurations

does

not

decrease.

sticks

to

during

the

first

four

rounds,

switching

to

in

round

It

takes

four

more

rounds

for

and

Mt

to

converge

After

round

the

auction

sets

η1

(which

yields

more

buyer

profit

than

and

η2

For

the

next

round

Δ

increased

by

for

each

subsequent

round.

Note

that

and

therefore

πT

In

round

Δ

meaning

and

that

causes

to

drop

out,

setting

the

final

allocation

to

and

That

leaves

the

buyer

with

a

profit

of

and

with

a

profit

of

less

than

below

the

VCG

profit

The

welfare

achieved

in

this

case

is

optimal.

To

illustrate

how

some

efficiency

loss

could

occur

consider

the

case

that

In

that

case,

in

round

the

configuration

provides

the

same

profit

as

and

bids

on

both.

While

is

no

longer

optimal

after

the

price

change,

remains

optimal

on

subsequent

rounds

because

Mt

and

the

price

change

of

affects

both

and

the

efficient

configuration

When

phase

A

ends

so

the

auction

terminates

with

the

slightly

suboptimal

configuration

and

surplus

DISCUSSION

Preferential

Assumptions

A

key

aspect

in

implementing

GAI

based

auctions

is

the

choice

of

the

preference

structure,

that

is,

the

elements

Ig}.

In

some

domains

the

structure

can

be

more

or

less

robust

over

time

and

over

different

decision

makers.

When

this

is

not

the

case,

extracting

reliable

structure

from

sellers

(in

the

form

of

CDI

conditions)

is

a

serious

challenge.

This

could

have

been

a

deal

breaker

for

such

domains,

but

in

fact

it

can

be

overcome.

It

turns

out

that

we

can

run

this

auction

without

any

assumptions

on

sellers"

preference

structure.

The

only

place

where

this

assumption

is

used

in

our

analysis

is

for

Lemma

If

sellers

whose

preference

structure

does

not

agree

with

the

one

used

by

the

auction

are

guided

to

submit

only

one

full

bid

at

each

round,

or

a

set

of

bids

that

does

not

yield

undesired

consistent

combinations,

all

the

properties

of

the

auction

still

hold.

Locally,

the

sellers

can

optimize

their

profit

functions

using

the

union

of

their

GAI

structure

with

the

auction"s

structure.

It

is

therefore

essential

only

that

the

buyer"s

preference

structure

is

accurately

modeled.

Of

course,

capturing

sellers"

structures

as

well

is

still

preferred

since

it

can

speed

up

the

execution

and

let

sellers

take

advantage

of

the

compact

bid

representation.

In

both

cases

the

choice

of

clusters

may

significantly

affect

the

complexity

of

the

price

structure

and

the

runtime

of

the

auction.

It

is

sometimes

better

to

ignore

some

weaker

interdependencies

in

order

to

reduce

dimensionality.

The

complexity

of

the

structure

also

affects

the

efficiency

of

the

auction

through

the

value

of

e.

Information

Revelation

Properties

In

considering

information

properties

of

this

mechanism

we

compare

to

the

standard

approach

for

iterative

multiattribute

auctions,

which

is

based

on

the

theoretical

foundations

of

Che

In

most

of

these

mechanisms

the

buyer

reveals

a

scoring

function

and

then

the

mechanism

solicits

bids

from

the

sellers

(the

mechanisms

suggested

by

Beil

and

Wein

is

different

since

buyers

can

modify

their

scoring

function

each

round,

but

the

goal

there

is

to

maximize

the

buyer"s

profit).

Whereas

these

iterative

procurement

mechanisms

tend

to

relieve

the

burden

of

information

revelation

from

the

sellers,

a

major

drawback

is

that

the

buyer"s

utility

function

must

be

revealed

to

the

sellers

before

receiving

any

commitment.

In

the

mechanisms

suggested

by

PK

and

in

our

GAI

auction

above,

buyer

information

is

revealed

only

in

exchange

for

sell

commitments.

In

particular,

sellers

learn

nothing

(beyond

the

initial

price

upper

bound,

which

can

be

arbitrarily

loose)

about

the

utility

of

configurations

for

which

no

bid

was

submitted.

When

bids

are

submitted

for

a

configuration

θ,

sellers

would

be

able

to

infer

its

utility

relative

to

the

current

preferred

configurations

only

after

the

price

of

θ

is

driven

down

sufficiently

to

make

it

a

preferred

configuration

as

well.

Conclusions

We

propose

a

novel

exploitation

of

preference

structure

in

multiattribute

auctions.

Rather

than

assuming

full

additivity,

or

no

structure

at

all,

we

model

preferences

using

the

GAI

decomposition.

We

developed

an

iterative

auction

mechanism

directly

relying

on

the

decomposition,

and

also

provided

direct

means

of

constructing

the

representation

from

relatively

simple

statements

of

willingnessto-pay.

Our

auction

mechanism

generalizes

PK"s

preference

modeling,

while

in

essence

retaining

their

information

revelation

properties.

It

allows

for

a

range

of

tradeoffs

between

accuracy

of

preference

representation

and

both

the

complexity

of

the

pricing

structure

and

efficiency

of

the

auction,

as

well

as

tradeoffs

between

buyer"s

information

revelation

and

the

time

required

for

convergence.

ACKNOWLEDGMENTS

This

work

was

supported

in

part

by

NSF

grants

and

and

the

STIET

program

under

NSF

IGERT

grant

We

are

grateful

to

comments

from

anonymous

reviewers.

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