Robust

Solutions

for

Combinatorial

Auctions

Alan

Holland

Cork

Constraint

Computation

Centre

Department

of

Computer

Science

University

College

Cork,

Ireland

Barry

O"Sullivan

Cork

Constraint

Computation

Centre

Department

of

Computer

Science

University

College

Cork,

Ireland

ABSTRACT

Bids

submitted

in

auctions

are

usually

treated

as

enforceable

commitments

in

most

bidding

and

auction

theory

literature.

In

reality

bidders

often

withdraw

winning

bids

before

the

transaction

when

it

is

in

their

best

interests

to

do

so.

Given

a

bid

withdrawal

in

a

combinatorial

auction,

finding

an

alternative

repair

solution

of

adequate

revenue

without

causing

undue

disturbance

to

the

remaining

winning

bids

in

the

original

solution

may

be

difficult

or

even

impossible.

We

have

called

this

the

Bid-taker"s

Exposure

Problem.

When

faced

with

such

unreliable

bidders,

it

is

preferable

for

the

bid-taker

to

preempt

such

uncertainty

by

having

a

solution

that

is

robust

to

bid

withdrawal

and

provides

a

guarantee

that

possible

withdrawals

may

be

repaired

easily

with

a

bounded

loss

in

revenue.

In

this

paper,

we

propose

an

approach

to

addressing

the

Bidtaker"s

Exposure

Problem.

Firstly,

we

use

the

Weighted

Super

Solutions

framework

from

the

field

of

constraint

programming,

to

solve

the

problem

of

finding

a

robust

solution.

A

weighted

super

solution

guarantees

that

any

subset

of

bids

likely

to

be

withdrawn

can

be

repaired

to

form

a

new

solution

of

at

least

a

given

revenue

by

making

limited

changes.

Secondly,

we

introduce

an

auction

model

that

uses

a

form

of

leveled

commitment

contract

which

we

have

called

mutual

bid

bonds,

to

improve

solution

reparability

by

facilitating

backtracking

on

winning

bids

by

the

bid-taker.

We

then

examine

the

trade-off

between

robustness

and

revenue

in

different

economically

motivated

auction

scenarios

for

different

constraints

on

the

revenue

of

repair

solutions.

We

also

demonstrate

experimentally

that

fewer

winning

bids

partake

in

robust

solutions,

thereby

reducing

any

associated

overhead

in

dealing

with

extra

bidders.

Robust

solutions

can

also

provide

a

means

of

selectively

discriminating

against

distrusted

bidders

in

a

measured

manner.

Categories

and

Subject

Descriptors

[Social

and

Behavioral

Sciences]:

Economics;

[Computers

∗This

work

has

received

support

from

Science

Foundation

Ireland

under

grant

number

The

authors

wish

to

thank

Brahim

Hnich

and

the

anonymous

reviewers

for

their

helpful

comments.

and

Society]:

Electronic

Commerce;

[Artificial

Intelligence]:

Problem

Solving,

Control

Methods,

and

Search.

General

Terms

Algorithms,

Economics,

Reliability.

INTRODUCTION

A

combinatorial

auction

(CA)

provides

an

efficient

means

of

allocating

multiple

distinguishable

items

amongst

bidders

whose

perceived

valuations

for

combinations

of

items

differ.

Such

auctions

are

gaining

in

popularity

and

there

is

a

proliferation

in

their

usage

across

various

industries

such

as

telecoms,

procurement

and

transportation

Revenue

is

the

most

obvious

optimization

criterion

for

such

auctions,

but

another

desirable

attribute

is

solution

robustness.

In

terms

of

combinatorial

auctions,

a

robust

solution

is

one

that

can

withstand

bid

withdrawal

(a

break)

by

making

changes

easily

to

form

a

repair

solution

of

adequate

revenue.

A

brittle

solution

to

a

CA

is

one

in

which

an

unacceptable

loss

in

revenue

is

unavoidable

if

a

winning

bid

is

withdrawn.

In

such

situations

the

bid-taker

may

be

left

with

a

set

of

items

deemed

to

be

of

low

value

by

all

other

bidders.

These

bidders

may

associate

a

higher

value

for

these

items

if

they

were

combined

with

items

already

awarded

to

others,

hence

the

bid-taker

is

left

in

an

undesirable

local

optimum

in

which

a

form

of

backtracking

is

required

to

reallocate

the

items

in

a

manner

that

results

in

sufficient

revenue.

We

have

called

this

the

Bid-taker"s

Exposure

Problem

that

bears

similarities

to

the

Exposure

Problem

faced

by

bidders

seeking

multiple

items

in

separate

single-unit

auctions

but

holding

little

or

no

value

for

a

subset

of

those

items.

However,

reallocating

items

may

be

regarded

as

disruptive

to

a

solution

in

many

real-life

scenarios.

Consider

a

scenario

where

procurement

for

a

business

is

conducted

using

a

CA.

It

would

be

highly

undesirable

to

retract

contracts

from

a

group

of

suppliers

because

of

the

failure

of

a

third

party.

A

robust

solution

that

is

tolerant

of

such

breaks

is

preferable.

Robustness

may

be

regarded

as

a

preventative

measure

protecting

against

future

uncertainty

by

sacrificing

revenue

in

place

of

solution

stability

and

reparability.

We

assume

a

probabilistic

approach

whereby

the

bid-taker

has

knowledge

of

the

reliability

of

bidders

from

which

the

likelihood

of

an

incomplete

transaction

may

be

inferred.

Repair

solutions

are

required

for

bids

that

are

seen

as

brittle

(i.e.

likely

to

break).

Repairs

may

also

be

required

for

sets

of

bids

deemed

brittle.

We

propose

the

use

of

the

Weighted

Super

Solutions

(WSS)

framework

for

constraint

programming,

that

is

ideal

for

establishing

such

robust

solutions.

As

we

shall

see,

this

framework

can

enforce

constraints

on

solutions

so

that

possible

breakages

are

reparable.

This

paper

is

organized

as

follows.

Section

presents

the

Winner

Determination

Problem

(WDP)

for

combinatorial

auctions,

outlines

some

possible

reasons

for

bid

withdrawal

and

shows

how

simply

maximizing

expected

revenue

can

lead

to

intolerable

revenue

losses

for

risk-averse

bid-takers.

This

motivates

the

use

of

robust

solutions

and

Section

introduces

a

constraint

programming

(CP)

framework,

Weighted

Super

Solutions

that

finds

such

solutions.

We

then

propose

an

auction

model

in

Section

that

enhances

reparability

by

introducing

mandatory

mutual

bid

bonds,

that

may

be

seen

as

a

form

of

leveled

commitment

contract

Section

presents

an

extensive

empirical

evaluation

of

the

approach

presented

in

this

paper,

in

the

context

of

a

number

of

well-known

combinatorial

auction

distributions,

with

very

encouraging

results.

Section

discusses

possible

extensions

and

questions

raised

by

our

research

that

deserve

future

work.

Finally,

in

Section

a

number

of

concluding

remarks

are

made.

COMBINATORIAL

AUCTIONS

Before

presenting

the

technical

details

of

our

solution

to

the

Bid-taker"s

Exposure

Problem,

we

shall

present

a

brief

survey

of

combinatorial

auctions

and

existing

techniques

for

handling

bid

withdrawal.

Combinatorial

auctions

involve

a

single

bid-taker

allocating

multiple

distinguishable

items

amongst

a

group

of

bidders.

The

bidtaker

has

a

set

of

m

items

for

sale,

M

m},

and

bidders

submit

a

set

of

bids,

B

Bn}.

A

bid

is

a

tuple

Bj

Sj,

pj

where

Sj

M

is

a

subset

of

the

items

for

sale

and

pj

is

a

price.

The

WDP

for

a

CA

is

to

label

all

bids

as

either

winning

or

losing

so

as

to

maximize

the

revenue

from

winning

bids

without

allocating

any

item

to

more

than

one

bid.

The

following

is

the

integer

programming

formulation

for

the

WDP:

max

n

pjxj

s.t.

j|i∈Sj

xj

∀i

m},

xj

This

problem

is

NP-complete

and

inapproximable

and

is

otherwise

known

as

the

Set

Packing

Problem.

The

above

problem

formulation

assumes

the

notion

of

free

disposal.

This

means

that

the

optimal

solution

need

not

necessarily

sell

all

of

the

items.

If

the

auction

rules

stipulate

that

all

items

must

be

sold,

the

problem

becomes

a

Set

Partition

Problem

The

WDP

has

been

extensively

studied

in

recent

years.

The

fastest

search

algorithms

that

find

optimal

solutions

(e.g.

CABOB

can,

in

practice,

solve

very

large

problems

involving

thousands

of

bids

very

quickly.

The

Problem

of

Bid

Withdrawal

We

assume

an

auction

protocol

with

a

three

stage

process

involving

the

submission

of

bids,

winner

determination,

and

finally

a

transaction

phase.

We

are

interested

in

bid

withdrawals

that

occur

between

the

announcement

of

winning

bids

and

the

end

of

the

transaction

phase.

All

bids

are

valid

until

the

transaction

is

complete,

so

we

anticipate

an

expedient

transaction

In

some

instances

the

transaction

period

may

be

so

lengthy

that

consideration

of

non-winning

bids

as

still

being

valid

may

not

be

fair.

Breaks

that

occur

during

a

lengthy

transaction

phase

are

more

difficult

to

remedy

and

may

require

a

subsequent

auction.

For

example,

if

the

item

is

a

service

contract

for

a

given

period

of

time

and

the

break

occurs

after

partial

fulfilment

of

this

contract,

the

other

An

example

of

a

winning

bid

withdrawal

occurred

in

an

FCC

spectrum

auction

Withdrawals,

or

breaks,

may

occur

for

various

reasons.

Bid

withdrawal

may

be

instigated

by

the

bid-taker

when

Quality

of

Service

agreements

are

broken

or

payment

deadlines

are

not

met.

We

refer

to

bid

withdrawal

by

the

bid-taker

as

item

withdrawal

in

this

paper

to

distinguish

between

the

actions

of

a

bidder

and

the

bid-taker.

Harstad

and

Rothkopf

outlined

several

possibilities

for

breaks

in

single

item

auctions

that

include:

an

erroneous

initial

valuation/bid;

unexpected

events

outside

the

winning

bidder"s

control;

a

desire

to

have

the

second-best

bid

honored;

information

obtained

or

events

that

occurred

after

the

auction

but

before

the

transaction

that

reduces

the

value

of

an

item;

the

revelation

of

competing

bidders"

valuations

infers

reduced

profitability,

a

problem

known

as

the

Winner"s

Curse.

Kastner

et

al.

examined

how

to

handle

perturbations

given

a

solution

whilst

minimizing

necessary

changes

to

that

solution.

These

perturbations

may

include

bid

withdrawals,

change

of

valuation/items

of

a

bid

or

the

submission

of

a

new

bid.

They

looked

at

the

problem

of

finding

incremental

solutions

to

restructure

a

supply

chain

whose

formation

is

determined

using

combinatorial

auctions

Following

a

perturbation

in

the

optimal

solution

they

proceed

to

impose

involuntary

item

withdrawals

from

winning

bidders.

They

formulated

an

incremental

integer

linear

program

(ILP)

that

sought

to

maximize

the

valuation

of

the

repair

solution

whilst

preserving

the

previous

solution

as

much

as

possible.

Being

Proactive

against

Bid

Withdrawal

When

a

bid

is

withdrawn

there

may

be

constraints

on

how

the

solution

can

be

repaired.

If

the

bid-taker

was

freely

able

to

revoke

the

awarding

of

items

to

other

bidders

then

the

solution

could

be

repaired

easily

by

reassigning

all

the

items

to

the

optimal

solution

without

the

withdrawn

bid.

Alternatively,

the

bidder

who

reneged

upon

a

bid

may

have

all

his

other

bids

disqualified

and

the

items

could

be

reassigned

based

on

the

optimum

solution

without

that

bidder

present.

However,

the

bid-taker

is

often

unable

to

freely

reassign

the

items

already

awarded

to

other

bidders.

When

items

cannot

be

withdrawn

from

winning

bidders,

following

the

failure

of

another

bidder

to

honor

his

bid,

repair

solutions

are

restricted

to

the

set

of

bids

whose

items

only

include

those

in

the

bid(s)

that

were

reneged

upon.

We

are

free

to

award

items

to

any

of

the

previously

unsuccessful

bids

when

finding

a

repair

solution.

When

faced

with

uncertainty

over

the

reliability

of

bidders

a

possible

approach

is

to

maximize

expected

revenue.

This

approach

does

not

make

allowances

for

risk-averse

bid-takers

who

may

view

a

small

possibility

of

very

low

revenue

as

unacceptable.

Consider

the

example

in

Table

and

the

optimal

expected

revenue

in

the

situation

where

a

single

bid

may

be

withdrawn.

There

are

three

submitted

bids

for

items

A

and

B,

the

third

being

a

combination

bid

for

the

pair

of

items

at

a

value

of

The

optimal

solution

has

a

value

of

with

the

first

and

second

bids

as

winners.

When

we

consider

the

probabilities

of

failure,

in

the

fourth

column,

the

problem

of

which

solution

to

choose

becomes

more

difficult.

Computing

the

expected

revenue

for

the

solution

with

the

first

and

second

bids

winning

the

items,

denoted

gives:

(200×0.9×0.9)+(2×100×0.9×0.1)+(190×0.1×0.1)

bidders"

valuations

for

the

item

may

have

decreased

in

a

non-linear

fashion.

Table

Example

Combinatorial

Auction.

Items

Bids

A

B

AB

Withdrawal

prob

If

a

single

bid

is

withdrawn

there

is

probability

of

of

a

revenue

of

given

the

fact

that

we

cannot

withdraw

an

item

from

the

other

winning

bidder.

The

expected

revenue

for

is:

We

can

therefore

surmise

that

the

second

solution

is

preferable

to

the

first

based

on

expected

revenue.

Determining

the

maximum

expected

revenue

in

the

presence

of

such

uncertainty

becomes

computationally

infeasible

however,

as

the

number

of

brittle

bids

grows.

A

WDP

needs

to

be

solved

for

all

possible

combinations

of

bids

that

may

fail.

The

possible

loss

in

revenue

for

breaks

is

also

not

tightly

bounded

using

this

approach,

therefore

a

large

loss

may

be

possible

for

a

small

number

of

breaks.

Consider

the

previous

example

where

the

bid

amount

for

becomes

The

expected

revenue

of

becomes

greater

than

that

of

There

are

some

bid-takers

who

may

prefer

the

latter

solution

because

the

revenue

is

never

less

than

but

the

former

solution

returns

revenue

of

only

with

probability

A

risk-averse

bid-taker

may

not

tolerate

such

a

possibility,

preferring

to

sacrifice

revenue

for

reduced

risk.

If

we

modify

our

repair

search

so

that

a

solution

of

at

least

a

given

revenue

is

guaranteed,

the

search

for

a

repair

solution

becomes

a

satisfiability

test

rather

than

an

optimization

problem.

The

approaches

described

above

are

in

contrast

to

that

which

we

propose

in

the

next

section.

Our

approach

can

be

seen

as

preventative

in

that

we

find

an

initial

allocation

of

items

to

bidders

which

is

robust

to

bid

withdrawal.

Possible

losses

in

revenue

are

bounded

by

a

fixed

percentage

of

the

true

optimal

allocation.

Perturbations

to

the

original

solution

are

also

limited

so

as

to

minimize

disruption.

We

regard

this

as

the

ideal

approach

for

real-world

combinatorial

auctions.

DEFINITION

(ROBUST

SOLUTION

FOR

A

CA).

A

robust

solution

for

a

combinatorial

auction

is

one

where

any

subset

of

successful

bids

whose

probability

of

withdrawal

is

greater

than

or

equal

to

α

can

be

repaired

by

reassigning

items

at

a

cost

of

at

most

β

to

other

previously

losing

bids

to

form

a

repair

solution.

Constraints

on

acceptable

revenue,

e.g.

being

a

minimum

percentage

of

the

optimum,

are

defined

in

the

problem

model

and

are

thus

satisfied

by

all

solutions.

The

maximum

cost

of

repair,

β,

may

be

a

fixed

value

that

may

be

thought

of

as

a

fund

for

compensating

winning

bidders

whose

items

are

withdrawn

from

them

when

creating

a

repair

solution.

Alternatively,

β

may

be

a

function

of

the

bids

that

were

withdrawn.

Section

will

give

an

example

of

such

a

mechanism.

In

the

following

section

we

describe

an

ideal

constraint-based

framework

for

the

establishment

of

such

robust

solutions.

FINDING

ROBUST

SOLUTIONS

In

constraint

programming

(CP),

a

constraint

satisfaction

problem

(CSP)

is

modeled

as

a

set

of

n

variables

X

xn},

a

set

of

domains

D

D(xn)},

where

D(xi)

is

the

set

of

finite

possible

values

for

variable

xi

and

a

set

C

Cm}

of

constraints,

each

restricting

the

assignments

of

some

subset

of

the

variables

in

X.

Constraint

satisfaction

involves

finding

values

for

each

of

the

problem

variables

such

that

all

constraints

are

satisfied.

Its

main

advantages

are

its

declarative

nature

and

flexibility

in

tackling

problems

with

arbitrary

side

constraints.

Constraint

optimization

seeks

to

find

a

solution

to

a

CSP

that

optimizes

some

objective

function.

A

common

technique

for

solving

constraint

optimization

problems

is

to

use

branch-and-bound

techniques

that

avoid

exploring

sub-trees

that

are

known

not

to

contain

a

better

solution

than

the

best

found

so

far.

An

initial

bound

can

be

determined

by

finding

a

solution

that

satisfies

all

constraints

in

C

or

by

using

some

heuristic

methods.

A

classical

super

solution

(SS)

is

a

solution

to

a

CSP

in

which,

if

a

small

number

of

variables

lose

their

values,

repair

solutions

are

guaranteed

with

only

a

few

changes,

thus

providing

solution

robustness

It

is

a

generalization

of

both

fault

tolerance

in

CP

and

supermodels

in

propositional

satisfiability

(SAT)

An

(a,b)-super

solution

is

one

in

which

if

at

most

a

variables

lose

their

values,

a

repair

solution

can

be

found

by

changing

at

most

b

other

variables

Super

solutions

for

combinatorial

auctions

minimize

the

number

of

bids

whose

status

needs

to

be

changed

when

forming

a

repair

solution

Only

a

particular

set

of

variables

in

the

solution

may

be

subject

to

change

and

these

are

said

to

be

members

of

the

breakset.

For

each

combination

of

brittle

assignments

in

the

break-set,

a

repair-set

is

required

that

comprises

the

set

of

variables

whose

values

must

change

to

provide

another

solution.

The

cardinality

of

the

repair

set

is

used

to

measure

the

cost

of

repair.

In

reality,

changing

some

variable

assignments

in

a

repair

solution

incurs

a

lower

cost

than

others

thereby

motivating

the

use

of

a

different

metric

for

determining

the

legality

of

repair

sets.

The

Weighted

Super

Solution

(WSS)

framework

considers

the

cost

of

repair

required,

rather

than

simply

the

number

of

assignments

modified,

to

form

an

alternative

solution.

For

CAs

this

may

be

a

measure

of

the

compensation

penalties

paid

to

winning

bidders

to

break

existing

agreements.

Robust

solutions

are

particularly

desirable

for

applications

where

unreliability

is

a

problem

and

potential

breakages

may

incur

severe

penalties.

Weighted

super

solutions

offer

a

means

of

expressing

which

variables

are

easily

re-assigned

and

those

that

incur

a

heavy

cost

Hebrard

et

al.

describe

how

some

variables

may

fail

(such

as

machines

in

a

job-shop

problem)

and

others

may

not.

A

WSS

generalizes

this

approach

so

that

there

is

a

probability

of

failure

associated

with

each

assignment

and

sets

of

variables

whose

assignments

have

probabilities

of

failure

greater

than

or

equal

to

a

threshold

value,

α,

require

repair

solutions.

A

WSS

measures

the

cost

of

repairing,

or

reassigning,

other

variables

using

inertia

as

a

metric.

Inertia

is

a

measure

of

a

variable"s

aversion

to

change

and

depends

on

its

current

assignment,

future

assignment

and

the

breakage

variable(s).

It

may

be

desirable

to

reassign

items

to

different

bidders

in

order

to

find

a

repair

solution

of

satisfactory

revenue.

Compensation

may

have

to

be

paid

to

bidders

who

lose

items

during

the

formation

of

a

repair

solution.

The

inertia

of

a

bid

reflects

the

cost

of

changing

its

state.

For

winning

bids

this

may

reflect

the

necessary

compensation

penalty

for

the

bid-taker

to

break

the

agreement

(if

such

breaches

are

permitted),

whereas

for

previously

losing

bids

this

is

a

free

operation.

The

total

amount

of

compensation

payable

to

bidders

may

depend

upon

other

factors,

such

as

the

cause

of

the

break.

There

is

a

limit

to

how

much

these

overall

repair

costs

should

be,

and

this

is

given

by

the

value

β.

This

value

may

not

be

known

in

advance

and

Algorithm

WSS(int

level,

double

α,

double

β):Boolean

begin

if

level

number

of

variables

then

return

true

choose

unassigned

variable

x

foreach

value

v

in

the

domain

of

x

do

assign

x

v

if

problem

is

consistent

then

foreach

combination

of

brittle

assignments,

A

do

if

¬reparable(A,

β)

then

return

false;

if

then

return

true

unassign

x

return

false

end

may

depend

upon

the

break.

Therefore,

β

may

be

viewed

as

the

fund

used

to

compensate

winning

bidders

for

the

unilateral

withdrawal

of

their

bids

by

the

bid-taker.

In

summary,

an

(α,β)-WSS

allows

any

set

of

variables

whose

probability

of

breaking

is

greater

than

or

equal

to

α

be

repaired

with

changes

to

the

original

robust

solution

with

a

cost

of

at

most

β.

The

depth-first

search

for

a

WSS

(see

pseudo-code

description

in

Algorithm

maintains

arc-consistency

at

each

node

of

the

tree.

As

search

progresses,

the

reparability

of

each

previous

assignment

is

verified

at

each

node

by

extending

a

partial

repair

solution

to

the

same

depth

as

the

current

partial

solution.

This

may

be

thought

of

as

maintaining

concurrent

search

trees

for

repairs.

A

repair

solution

is

provided

for

every

possible

set

of

break

variables,

A.

The

WSS

algorithm

attempts

to

extend

the

current

partial

assignment

by

choosing

a

variable

and

assigning

it

a

value.

Backtracking

may

then

occur

for

one

of

two

reasons:

we

cannot

extend

the

assignment

to

satisfy

the

given

constraints,

or

the

current

partial

assignment

cannot

be

associated

with

a

repair

solution

whose

cost

of

repair

is

less

than

β

should

a

break

occur.

The

procedure

reparable

searches

for

partial

repair

solutions

using

backtracking

and

attempts

to

extend

the

last

repair

found,

just

as

in

solutions

the

differences

being

that

a

repair

is

provided

for

a

set

of

breakage

variables

rather

than

a

single

variable

and

the

cost

of

repair

is

considered.

A

summation

operator

is

used

to

determine

the

overall

cost

of

repair.

If

a

fixed

bound

upon

the

size

of

any

potential

break-set

can

be

formed,

the

WSS

algorithm

is

NPcomplete.

For

a

more

detailed

description

of

the

WSS

search

algorithm,

the

reader

is

referred

to

since

a

complete

description

of

the

algorithm

is

beyond

the

scope

of

this

paper.

EXAMPLE

We

shall

step

through

the

example

given

in

Table

when

searching

for

a

WSS.

Each

bid

is

represented

by

a

single

variable

with

domain

values

of

and

the

former

representing

bid-failure

and

the

latter

bid-success.

The

probability

of

failure

of

the

variables

are

when

they

are

assigned

to

and

otherwise.

The

problem

is

initially

solved

using

an

ILP

solver

such

as

or

CPLEX,

and

the

optimal

revenue

is

found

to

be

A

fixed

percentage

of

this

revenue

can

be

used

as

a

threshold

value

for

a

robust

solution

and

its

repairs.

The

bid-taker

wishes

to

have

a

robust

solution

so

that

if

a

single

winning

bid

is

withdrawn,

a

repair

solution

can

be

formed

without

withdrawing

items

from

any

other

winning

bidder.

This

example

may

be

seen

as

searching

for

a

super

solution,

β

is

because

no

funds

are

available

to

compensate

the

withdrawal

of

items

from

winning

bidders.

The

bid-taker

is

willing

to

compromise

on

revenue,

but

only

by

say,

of

the

optimal

value.

Bids

and

cannot

both

succeed,

since

they

both

require

item

A,

so

a

constraint

is

added

precluding

the

assignment

in

which

both

variables

take

the

value

Similarly,

bids

and

cannot

both

win

so

another

constraint

is

added

between

these

two

variables.

Therefore,

in

this

example

the

set

of

CSP

variables

is

V

whose

domains

are

all

The

constraints

are

and

xi∈V

aixi

where

ai

reflects

the

relevant

bid-amounts

for

the

respective

bid

variables.

In

order

to

find

a

robust

solution

of

optimal

revenue

we

seek

to

maximize

the

sum

of

these

amounts,

max

xi∈V

aixi.

When

all

variables

are

set

to

(see

Figure

branch

this

is

not

a

solution

because

the

minimum

revenue

of

has

not

been

met,

so

we

try

assigning

to

(branch

This

is

a

valid

solution

but

this

variable

is

brittle

because

there

is

a

chance

that

this

bid

may

be

withdrawn

(see

Table

Therefore

we

need

to

determine

if

a

repair

can

be

formed

should

it

break.

The

search

for

a

repair

begins

at

the

first

node,

see

Figure

Notice

that

value

has

been

removed

from

because

this

search

tree

is

simulating

the

withdrawal

of

this

bid.

When

is

set

to

(branch

the

maximum

revenue

solution

in

the

remaining

subtree

has

revenue

of

only

therefore

search

is

discontinued

at

that

node

of

the

tree.

and

are

both

assigned

to

(branches

and

and

the

total

cost

of

both

these

changes

is

still

because

no

compensation

needs

to

be

paid

for

bids

that

change

from

losing

to

winning.

With

now

losing

(branch

this

gives

a

repair

solution

of

Hence

is

reparable

and

therefore

a

WSS.

We

continue

our

search

in

Figure

however,

because

we

are

seeking

a

robust

solution

of

optimal

revenue.

When

is

assigned

to

(branch

we

seek

a

partial

repair

for

this

variable

breaking

(branch

is

not

considered

since

it

offers

insufficient

revenue).

The

repair

search

sets

to

in

a

separate

search

tree,

(not

shown),

and

control

is

returned

to

the

search

for

a

WSS.

is

set

to

(branch

but

this

solution

would

not

produce

sufficient

revenue

so

is

then

set

to

(branch

We

then

attempt

to

extend

the

repair

for

(not

shown).

This

fails

because

the

repair

for

cannot

assign

to

because

the

cost

of

repairing

such

an

assignment

would

be

given

that

the

auction

rules

do

not

permit

the

withdrawal

of

items

from

winning

bids.

A

repair

for

breaking

is

therefore

not

possible

because

items

have

already

been

awarded

to

A

repair

solution

with

assigned

to

does

not

produce

sufficient

revenue

when

is

assigned

to

The

inability

to

withdraw

items

from

winning

bids

implies

that

is

an

irreparable

solution

when

the

minimum

tolerable

revenue

is

greater

than

The

italicized

comments

and

dashed

line

in

Figure

illustrate

the

search

path

for

a

WSS

if

both

of

these

bids

were

deemed

reparable.

Section

introduces

an

alternative

auction

model

that

will

allow

the

bid-taker

to

receive

compensation

for

breakages

and

in

turn

use

this

payment

to

compensate

other

bidders

for

withdrawal

of

items

from

winning

bids.

This

will

enable

the

reallocation

of

items

and

permit

the

establishment

of

as

a

second

WSS

for

this

example.

MUTUAL

BID

BONDS:

A

BACKTRACKING

MECHANISM

Some

auction

solutions

are

inherently

brittle

and

it

may

be

impossible

to

find

a

robust

solution.

If

we

can

alter

the

rules

of

an

auction

so

that

the

bid-taker

can

retract

items

from

winning

bidders,

then

the

reparability

of

solutions

to

such

auctions

may

be

improved.

In

this

section

we

propose

an

auction

model

that

permits

bid

and

item

withdrawal

by

the

bidders

and

bid-taker,

respectively.

We

propose

a

model

that

incorporates

mutual

bid

bonds

to

enable

solution

reparability

for

the

bid-taker,

a

form

of

insurance

against

Insufficient

revenue

Find

repair

solution

for

bid

breakage

Find

partial

repair

for

bid

breakage

Insufficient

revenue

(a)

Extend

partial

repair

for

bid

breakage

(b)

Find

partial

repair

for

bid

breakage

Bid

Bid

Bid

Find

repair

solutions

for

bid

breakages

Insufficient

revenue

(a)

Search

for

WSS.

Insufficient

revenue

Insufficient

revenue

Bid

Bid

Bid

(b)

Search

for

a

repair

for

bid

breakage.

Figure

Search

Tree

for

a

WSS

without

item

withdrawal.

the

winner"s

curse

for

the

bidder

whilst

also

compensating

bidders

in

the

case

of

item

withdrawal

from

winning

bids.

We

propose

that

such

Winner"s

Curse

Bid-taker"s

Exposure

insurance

comprise

a

fixed

percentage,

κ,

of

the

bid

amount

for

all

bids.

Such

mutual

bid

bonds

are

mandatory

for

each

bid

in

our

The

conditions

attached

to

the

bid

bonds

are

that

the

bid-taker

be

allowed

to

annul

winning

bids

(item

withdrawal)

when

repairing

breaks

elsewhere

in

the

solution.

In

the

interests

of

fairness,

compensation

is

paid

to

bidders

from

whom

items

are

withdrawn

and

is

equivalent

to

the

penalty

that

would

have

been

imposed

on

the

bidder

should

he

have

withdrawn

the

bid.

Combinatorial

auctions

impose

a

heavy

computational

burden

on

the

bidder

so

it

is

important

that

the

hedging

of

risk

should

be

a

simple

and

transparent

operation

for

the

bidder

so

as

not

to

further

increase

this

burden

unnecessarily.

We

also

contend

that

it

is

imperative

that

the

bidder

knows

the

potential

penalty

for

withdrawal

in

advance

of

bid

submission.

This

information

is

essential

for

bidders

when

determining

how

aggressive

they

should

be

in

their

bidding

strategy.

Bid

bonds

are

commonplace

in

procurement

for

construction

projects.

Usually

they

are

mandatory

for

all

bids,

are

a

fixed

percentage,

κ,

of

the

bid

amount

and

are

unidirectional

in

that

item

withdrawal

by

the

bid-taker

is

not

permitted.

Mutual

bid

bonds

may

be

seen

as

a

form

of

leveled

commitment

contract

in

which

both

parties

may

break

the

contract

for

the

same

fixed

penalty.

Such

contracts

permit

unilateral

decommitment

for

prespecified

penalties.

Sandholm

et

al.

showed

that

this

can

increase

the

expected

payoffs

of

all

parties

and

enables

deals

that

would

be

impossible

under

full

commitment

In

practice

a

bid

bond

typically

ranges

between

and

of

the

Making

the

insurance

optional

may

be

beneficial

in

some

instances.

If

a

bidder

does

not

agree

to

the

insurance,

it

may

be

inferred

that

he

may

have

accurately

determined

the

valuation

for

the

items

and

therefore

less

likely

to

fall

victim

to

the

winner"s

curse.

The

probability

of

such

a

bid

being

withdrawn

may

be

less,

so

a

repair

solution

may

be

deemed

unnecessary

for

this

bid.

On

the

other

hand

it

decreases

the

reparability

of

solutions.

bid

amount

If

the

decommitment

penalties

are

the

same

for

both

parties

in

all

bids,

κ

does

not

influence

the

reparability

of

a

given

set

of

bids.

It

merely

influences

the

levels

of

penalties

and

compensation

transacted

by

agents.

Low

values

of

κ

incur

low

bid

withdrawal

penalties

and

simulate

a

dictatorial

bid-taker

who

does

not

adequately

compensate

bidders

for

item

withdrawal.

Andersson

and

Sandholm

found

that

myopic

agents

reach

a

higher

social

welfare

quicker

if

they

act

selfishly

rather

than

cooperatively

when

penalties

in

leveled

commitment

contracts

are

low.

Increased

levels

of

bid

withdrawal

are

likely

when

the

penalties

are

low

also.

High

values

of

κ

tend

towards

full-commitment

and

reduce

the

advantages

of

such

Winner"s

Curse

Bid-taker"s

Exposure

insurance.

The

penalties

paid

are

used

to

fund

a

reassignment

of

items

to

form

a

repair

solution

of

sufficient

revenue

by

compensating

previously

successful

bidders

for

withdrawal

of

the

items

from

them.

EXAMPLE

Consider

the

example

given

in

Table

once

more,

where

the

bids

also

comprise

a

mutual

bid

bond

of

of

the

bid

amount.

If

a

bid

is

withdrawn,

the

bidder

forfeits

this

amount

and

the

bid-taker

can

then

compensate

winning

bidders

whose

items

are

withdrawn

when

trying

to

form

a

repair

solution

later.

The

search

for

repair

solutions

for

breaks

to

and

appear

in

Figures

and

When

breaks,

there

is

a

compensation

penalty

paid

to

the

bid-taker

equal

to

that

can

be

used

to

fund

a

reassignment

of

the

items.

We

therefore

set

β

to

and

this

becomes

the

maximum

expenditure

allowed

to

withdraw

items

from

winning

bidders.

β

may

also

be

viewed

as

the

size

of

the

fund

available

to

facilitate

backtracking

by

the

bid-taker.

When

we

extend

the

partial

repair

for

so

that

loses

an

item

(branch

the

overall

cost

of

repair

increases

to

due

to

this

item

withdrawal

by

the

bid-taker,

The

actual

implementation

of

WSS

search

checks

previous

solutions

to

see

if

they

can

repair

breaks

before

searching

for

a

new

repair

solution.

is

a

solution

that

has

already

been

found

so

the

search

for

a

repair

in

this

example

is

not

strictly

necessary

but

is

described

for

pedagogical

reasons.

Bid

Bid

Bid

Insufficient

revenue

(a)

Search

for

a

repair

for

bid

breakage.

Bid

Bid

Bid

Insufficient

revenue

(b)

Search

for

a

repair

for

bid

breakage.

Figure

Repair

Search

Tree

for

breaks

and

κ

and

is

just

within

the

limit

given

by

β.

In

Figure

the

search

path

follows

the

dashed

line

and

sets

to

be

(branch

The

repair

solutions

for

bids

and

can

be

extended

further

by

assigning

to

(branches

and

Therefore,

may

be

considered

a

robust

solution.

Recall,

that

previously

this

was

not

the

case.

Using

mutual

bid

bonds

thus

increases

reparability

and

allows

a

robust

solution

of

revenue

as

opposed

to

as

was

previously

the

case.

EXPERIMENTS

We

have

used

the

Combinatorial

Auction

Test

Suite

(CATS)

to

generate

sample

auction

data.

We

generated

instances

of

problems

in

which

there

are

items

for

sale

and

bids

that

may

be

dominated

in

some

Such

dominated

bids

can

participate

in

repair

solutions

although

they

do

not

feature

in

optimal

solutions.

CATS

uses

economically

motivated

bidding

patterns

to

generate

auction

data

in

various

scenarios.

To

motivate

the

research

presented

in

this

paper

we

use

sensitivity

analysis

to

examine

the

brittleness

of

optimal

solutions

and

hence

determine

the

types

of

auctions

most

likely

to

benefit

from

a

robust

solution.

We

then

establish

robust

solutions

for

CAs

using

the

WSS

framework.

Sensitivity

Analysis

for

the

WDP

We

have

performed

sensitivity

analysis

of

the

following

four

distributions:

airport

take-off/landing

slots

(matching),

electronic

components

(arbitrary),

property/spectrum-rights

(regions)

and

transportation

(paths).

These

distributions

were

chosen

because

they

describe

a

broad

array

of

bidding

patterns

in

different

application

domains.

The

method

used

is

as

follows.

We

first

of

all

determined

the

optimal

solution

using

a

mixed

integer

linear

program

solver

We

then

simulated

a

single

bid

withdrawal

and

re-solved

the

problem

with

the

other

winning

bids

remaining

fixed,

i.e.

there

were

no

involuntary

dropouts.

The

optimal

repair

solution

was

then

determined.

This

process

is

repeated

for

all

winning

bids

in

the

overall

optimal

solution,

thus

assuming

that

all

bids

are

brittle.

Figure

shows

the

average

revenue

of

such

repair

solutions

as

a

percentage

of

the

optimum.

Also

shown

is

the

average

worst-case

scenario

over

auctions.

We

also

implemented

an

auction

rule

that

disallows

bids

from

the

reneging

bidder

participate

in

a

Figure

illustrates

how

the

paths

distribution

is

inherently

the

most

robust

distribution

since

when

any

winning

bid

is

withdrawn

the

solution

can

be

repaired

to

achieve

over

of

the

The

CATS

flags

included

int

prices

with

the

bid

alpha

parameter

set

to

We

assumed

that

all

bids

in

a

given

XOR

bid

with

the

same

dummy

item

were

from

the

same

bidder.

optimal

revenue

on

average

for

auctions

with

more

than

bids.

There

are

some

cases

however

when

such

withdrawals

result

in

solutions

whose

revenue

is

significantly

lower

than

optimum.

Even

in

auctions

with

as

many

as

bids

there

are

occasions

when

a

single

bid

withdrawal

can

result

in

a

drop

in

revenue

of

over

although

the

average

worst-case

drop

in

revenue

is

only

Figure

shows

how

the

matching

distribution

is

more

brittle

on

average

than

paths

and

also

has

an

inferior

worst-case

revenue

on

average.

This

trend

continues

as

the

regions-npv

(Figure

and

arbitrary-npv

(Figure

distributions

are

more

brittle

still.

These

distributions

are

clearly

sensitive

to

bid

withdrawal

when

no

other

winning

bids

in

the

solution

may

be

involuntarily

withdrawn

by

the

bid-taker.

Robust

Solutions

using

WSS

In

this

section

we

focus

upon

both

the

arbitrary-npv

and

regions-npv

distributions

because

the

sensitivity

analysis

indicated

that

these

types

of

auctions

produce

optimal

solutions

that

tend

to

be

most

brittle,

and

therefore

stand

to

benefit

most

from

solution

robustness.

We

ignore

the

auctions

with

bids

because

the

sensitivity

analysis

has

indicated

that

these

auctions

are

inherently

robust

with

a

very

low

average

drop

in

revenue

following

a

bid

withdrawal.

They

would

also

be

very

computationally

expensive,

given

the

extra

complexity

of

finding

robust

solutions.

A

pure

CP

approach

needs

to

be

augmented

with

global

constraints

that

incorporate

operations

research

techniques

to

increase

pruning

sufficiently

so

that

thousands

of

bids

may

be

examined.

Global

constraints

exploit

special-purpose

filtering

algorithms

to

improve

performance

There

are

a

number

of

ways

to

speed

up

the

search

for

a

weighted

super

solution

in

a

CA,

although

this

is

not

the

main

focus

of

our

current

work.

Polynomial

matching

algorithms

may

be

used

in

auctions

whose

bid

length

is

short,

such

as

those

for

airport

landing/take-off

slots

for

example.

The

integer

programming

formulation

of

the

WDP

stipulates

that

a

bid

either

loses

or

wins.

If

we

relax

this

constraint

so

that

bids

can

partially

win,

this

corresponds

to

the

linear

relaxation

of

the

problem

and

is

solvable

in

polynomial

time.

At

each

node

of

the

search

tree

we

can

quickly

solve

the

linear

relaxation

of

the

remaining

problem

in

the

subtree

below

the

current

node

to

establish

an

upper

bound

on

remaining

revenue.

If

this

upper

bound

plus

revenue

in

the

parent

tree

is

less

than

the

current

lower

bound

on

revenue,

search

at

that

node

can

cease.

The

(continuous)

LP

relaxation

thus

provides

a

vital

speed-up

in

the

search

for

weighted

super

solutions,

which

we

have

exploited

in

our

implementation.

The

LP

formulation

is

as

follows:

max

xi∈V

aixi

Revenue(%ofoptimum)

Bids

Average

Repair

Solution

Revenue

Worst-case

Repair

Solution

Revenue

(a)

paths

Revenue(%ofoptimum)

Bids

Average

Repair

Solution

Revenue

Worst-case

Repair

Solution

Revenue

(b)

matching

Revenue(%ofoptimum)

Bids

Average

Repair

Solution

Revenue

Worst-case

Repair

Solution

Revenue

(c)

regions-npv

Revenue(%ofoptimum)

Bids

Average

Repair

Solution

Revenue

Worst-case

Repair

Solution

Revenue

(d)

arbitrary-npv

Figure

Sensitivity

of

bid

distributions

to

single

bid

withdrawal.

s.t.

j|i∈Sj

xj

∀i

m},

xj

xj

R.

Additional

techniques,

that

are

outlined

in

can

aid

the

scalability

of

a

CP

approach

but

our

main

aim

in

these

experiments

is

to

examine

the

robustness

of

various

auction

distributions

and

consider

the

tradeoff

between

robustness

and

revenue.

The

WSS

solver

we

have

developed

is

an

extension

of

the

super

solution

solver

presented

in

This

solver

is,

in

turn,

based

upon

the

EFC

constraint

solver

Combinatorial

auctions

are

easily

modeled

as

a

constraint

optimization

problems.

We

have

chosen

the

branch-on-bids

formulation

because

in

tests

it

worked

faster

than

a

branch-on-items

formulation

for

the

arbitrary-npv

and

regions-npv

distributions.

All

variables

are

binary

and

our

search

mechanism

uses

a

reverse

lexicographic

value

ordering

heuristic.

This

complements

our

dynamic

variable

ordering

heuristic

that

selects

the

most

promising

unassigned

variable

as

the

next

one

in

the

search

tree.

We

use

the

product

of

the

solution

of

the

LP

relaxation

and

the

degree

of

a

variable

to

determine

the

likelihood

of

its

participation

in

a

robust

solution.

High

values

in

the

LP

solution

are

a

strong

indication

of

variables

most

likely

to

form

a

high

revenue

solution

whilst

the

a

variable"s

degree

reflects

the

number

of

other

bids

that

overlap

in

terms

of

desired

items.

Bids

for

large

numbers

of

items

tend

to

be

more

robust,

which

is

why

we

weight

our

robust

solution

search

in

this

manner.

We

found

this

heuristic

to

be

slightly

more

effective

than

the

LP

solution

alone.

As

the

number

of

bids

in

the

auction

increases

however,

there

is

an

increase

in

the

inherent

robustness

of

solutions

so

the

degree

of

a

variable

loses

significance

as

the

auction

size

increases.

Results

Our

experiments

simulate

three

different

constraints

on

repair

solutions.

The

first

is

that

no

winning

bids

are

withdrawn

by

the

bid-taker

and

a

repair

solution

must

return

a

revenue

of

at

least

of

the

optimal

overall

solution.

Secondly,

we

relaxed

the

revenue

constraint

to

of

optimum.

Thirdly,

we

allowed

backtracking

by

the

bid-taker

on

winning

bids

using

mutual

bid

bonds

but

maintaining

the

revenue

constraint

at

of

optimum.

Prior

to

finding

a

robust

solution

we

solved

the

WDP

optimally

using

We

then

set

the

minimum

tolerable

revenue

for

a

solution

to

be

(then

of

the

revenue

of

this

optimal

solution.

We

assumed

that

all

bids

were

brittle,

thus

a

repair

solution

is

required

for

every

bid

in

the

solution.

Initially

we

assume

that

no

backtracking

was

permitted

on

assignments

of

items

to

other

winning

bids

given

a

bid

withdrawal

elsewhere

in

the

solution.

Table

shows

the

percentage

of

optimal

solutions

that

are

robust

for

minimum

revenue

constraints

for

repair

solutions

of

and

of

optimal

revenue.

Relaxing

the

revenue

constraint

on

repair

solutions

to

of

the

optimum

revenue

greatly

increases

the

number

of

optimal

solutions

that

are

robust.

We

also

conducted

experiments

on

the

same

auctions

in

which

backtracking

by

the

bid-taker

is

permitted

using

mutual

bid

bonds.

This

significantly

improves

the

reparability

of

optimal

solutions

whilst

still

maintaining

repair

solutions

of

of

optimum.

An

interesting

feature

of

the

arbitrary-npv

distribution

is

that

optimal

solutions

can

become

more

brittle

as

the

number

of

bids

increases.

The

reason

for

this

is

that

optimal

solutions

for

larger

auctions

have

more

winning

bids.

Some

of

the

optimal

solutions

for

the

smallest

auctions

with

bids

have

only

one

winning

bidder.

If

this

bid

is

withdrawn

it

is

usually

easy

to

find

a

new

repair

solution

within

of

the

previous

optimal

revenue.

Also,

repair

solutions

for

bids

that

contain

a

small

number

of

items

may

be

made

difficult

by

the

fact

that

a

reduced

number

of

bids

cover

only

a

subset

of

those

items.

A

mitigating

factor

is

that

such

bids

form

a

smaller

percentage

of

the

revenue

of

the

optimal

solution

on

average.

We

also

implemented

a

rule

stipulating

that

any

losing

bids

from

Table

Optimal

Solutions

that

are

Inherently

Robust

#Bids

Min

Revenue

arbitrary-npv

repair

repair

MBB

repair

regions-npv

repair

repair

MBB

repair

Table

Occurrence

of

Robust

Solutions

#Bids

Min

Revenue

arbitrary-npv

repair

repair

MBB

repair

regions-npv

repair

repair

MBB

repair

a

withdrawing

bidder

cannot

participate

in

a

repair

solution.

This

acts

as

a

disincentive

for

strategic

withdrawal

and

was

also

used

previously

in

the

sensitivity

analysis.

In

some

auctions,

a

robust

solution

may

not

exist.

Table

shows

the

percentage

of

auctions

that

support

robust

solutions

for

the

arbitrary-npv

and

regions

-npv

distributions.

It

is

clear

that

finding

robust

solutions

for

the

former

distribution

is

particularly

difficult

for

auctions

with

and

bids

when

revenue

constraints

are

of

optimum.

This

difficulty

was

previously

alluded

to

by

the

low

percentage

of

optimal

solutions

that

were

robust

for

these

auctions.

Relaxing

the

revenue

constraint

helps

increase

the

percentage

of

auctions

in

which

robust

solutions

are

achievable

to

and

respectively.

This

improves

the

reparability

of

all

solutions

thereby

increasing

the

average

revenue

of

the

optimal

robust

solution.

It

is

somewhat

counterintuitive

to

expect

a

reduction

in

reparability

of

auction

solutions

as

the

number

of

bids

increases

because

there

tends

to

be

an

increased

number

of

solutions

above

a

revenue

threshold

in

larger

auctions.

The

MBB

auction

model

performs

very

well

however,

and

ensures

that

robust

solutions

are

achievable

for

such

inherently

brittle

auctions

without

sacrificing

over

of

optimal

revenue

to

achieve

repair

solutions.

Figure

shows

the

average

revenue

of

the

optimal

robust

solution

as

a

percentage

of

the

overall

optimum.

Repair

solutions

found

for

a

WSS

provide

a

lower

bound

on

possible

revenue

following

a

bid

withdrawal.

Note

that

in

some

instances

it

is

possible

for

a

repair

solution

to

have

higher

revenue

than

the

original

solution.

When

backtracking

on

winning

bids

by

the

bid-taker

is

disallowed,

this

can

only

happen

when

the

repair

solution

includes

two

or

more

bids

that

were

not

in

the

original.

Otherwise

the

repair

bids

would

participate

in

the

optimal

robust

solution

in

place

of

the

bid

that

was

withdrawn.

A

WSS

guarantees

minimum

levels

of

revenue

for

repair

solutions

but

this

is

not

to

say

that

repair

solutions

cannot

be

improved

upon.

It

is

possible

to

use

an

incremental

algorithm

to

Revenue(%ofoptimum)

Bids

Repair

Revenue:

Min

Optimal

Repair

Revenue:

Min

Optimal

MBB:

Repair

Revenue:

Min

Optimal

(a)

regions-npv

Revenue(%ofoptimum)

Bids

Repair

Revenue:

Min

Optimal

Repair

Revenue:

Min

Optimal

MBB:

Repair

Revenue:

Min

Optimal

(b)

arbitrary-npv

Figure

Revenue

of

optimal

robust

solutions.

determine

an

optimal

repair

solution

following

a

break,

whilst

safe

in

the

knowledge

that

in

advance

of

any

possible

bid

withdrawal

we

can

establish

a

lower

bound

on

the

revenue

of

a

repair.

Kastner

et

al.

have

provided

such

an

incremental

ILP

formulation

Mutual

bid

bonds

facilitate

backtracking

by

the

bid-taker

on

already

assigned

items.

This

improves

the

reparability

of

all

possible

solutions

thus

increasing

the

revenue

of

the

optimal

robust

solution

on

average.

Figure

shows

the

increase

in

revenue

of

robust

solutions

in

such

instances.

The

revenues

of

repair

solutions

are

bounded

by

at

least

of

the

optimum

in

our

experiments

thereby

allowing

a

direct

comparison

with

robust

solutions

already

found

using

the

same

revenue

constraint

but

not

providing

for

backtracking.

It

is

immediately

obvious

that

such

a

mechanism

can

significantly

increase

revenue

whilst

still

maintaining

solution

robustness.

Table

shows

the

number

of

winning

bids

participating

in

optimal

and

optimal

robust

solutions

given

the

three

different

constraints

on

repairing

solutions

listed

at

the

beginning

of

this

section.

As

the

number

of

bids

increases,

more

of

the

optimal

overall

solutions

are

robust.

This

leads

to

a

convergence

in

the

number

of

winning

bids.

The

numbers

in

brackets

are

derived

from

the

sensitivity

analysis

of

optimal

solutions

that

reveals

the

fact

that

almost

all

optimal

solutions

for

auctions

of

bids

are

robust.

We

can

therefore

infer

that

the

average

number

of

winning

bids

in

revenuemaximizing

robust

solutions

converges

towards

that

of

the

optimal

overall

solutions.

A

notable

side-effect

of

robust

solutions

is

that

fewer

bids

participate

in

the

solutions.

It

can

be

clearly

seen

from

Table

that

when

revenue

constraints

on

repair

solutions

are

tight,

there

are

fewer

winning

bids

in

the

optimal

robust

solution

on

average.

This

is

particularly

pronounced

for

smaller

auctions

in

both

distributions.

This

can

win

benefits

for

the

bid-taker

such

as

reduced

overheads

in

dealing

with

fewer

suppliers.

Although

MBBs

aid

solution

Table

Number

of

winning

bids.

#Bids

Solution

arbitrary-npv

Optimal

Repair

Repair

MBB

regions-npv

Optimal

Repair

Repair

MBB

bility,

the

number

of

bids

in

the

solutions

increases

on

average.

This

is

to

be

expected

because

a

greater

fraction

of

these

solutions

are

in

fact

optimal,

as

we

saw

in

Table

DISCUSSION

AND

FUTURE

WORK

Bidding

strategies

can

become

complex

in

non-incentive-compatible

mechanisms

where

winner

determination

is

no

longer

necessarily

optimal.

The

perceived

reparability

of

a

bid

may

influence

the

bid

amount,

with

reparable

bids

reaching

a

lower

equilibrium

point

and

perceived

irreparable

bids

being

more

aggressive.

Penalty

payments

for

bid

withdrawal

also

create

an

incentive

for

more

aggressive

bidding

by

providing

a

form

of

insurance

against

the

winner"s

curse

If

a

winning

bidder"s

revised

valuation

for

a

set

of

items

drops

by

more

than

the

penalty

for

withdrawal

of

the

bid,

then

it

is

in

his

best

interests

to

forfeit

the

item(s)

and

pay

the

penalty.

Should

the

auction

rules

state

that

the

bid-taker

will

refuse

to

sell

the

items

to

any

of

the

remaining

bidders

in

the

event

of

a

withdrawal,

then

insurance

against

potential

losses

will

stimulate

more

aggressive

bidding.

However,

in

our

case

we

are

seeking

to

repair

the

solution

with

the

given

bids.

A

side-effect

of

such

a

policy

is

to

offset

the

increased

aggressiveness

by

incentivizing

reduced

valuations

in

expectation

that

another

bidder"s

successful

bid

is

withdrawn.

Harstad

and

Rothkopf

examined

the

conditions

required

to

ensure

an

equilibrium

position

in

which

bidding

was

at

least

as

aggressive

as

if

no

bid

withdrawal

was

permitted,

given

this

countervailing

incentive

to

under-estimate

a

valuation.

Three

major

results

arose

from

their

study

of

bid

withdrawal

in

a

single

item

auction:

Equilibrium

bidding

is

more

aggressive

with

withdrawal

for

sufficiently

small

probabilities

of

an

award

to

the

second

highest

bidder

in

the

event

of

a

bid

withdrawal;

Equilibrium

bidding

is

more

aggressive

with

withdrawal

if

the

number

of

bidders

is

large

enough;

For

many

distributions

of

costs

and

estimates,

equilibrium

bidding

is

more

aggressive

with

withdrawal

if

the

variability

of

the

estimating

distribution

is

sufficiently

large.

It

is

important

that

mutual

bid

bonds

do

not

result

in

depressed

bidding

in

equilibrium.

An

analysis

of

the

resultant

behavior

of

bidders

must

incorporate

the

possibility

of

a

bidder

winning

an

item

and

having

it

withdrawn

in

order

for

the

bid-taker

to

formulate

a

repair

solution

after

a

break

elsewhere.

Harstad

and

Rothkopf

have

analyzed

bidder

aggressiveness

using

a

strictly

game-theoretic

model

in

which

the

only

reason

for

bid

withdrawal

is

the

winner"s

curse.

They

assumed

all

bidders

were

risk-neutral,

but

surmised

that

it

is

entirely

possible

for

the

bid-taker

to

collect

a

risk

premium

from

risk-averse

bidders

with

the

offer

of

such

insurance.

Combinatorial

auctions

with

mutual

bid

bonds

add

an

extra

incentive

to

bid

aggressively

because

of

the

possibility

of

being

compensated

for

having

a

winning

bid

withdrawn

by

a

bid-taker.

This

is

militated

against

by

the

increased

probability

of

not

having

items

withdrawn

in

a

repair

solution.

We

leave

an

in-depth

analysis

of

the

sufficient

conditions

for

more

aggressive

bidding

for

future

work.

Whilst

the

WSS

framework

provides

ample

flexibility

and

expressiveness,

scalability

becomes

a

problem

for

larger

auctions.

Although

solutions

to

larger

auctions

tend

to

be

naturally

more

robust,

some

bid-takers

in

such

auctions

may

require

robustness.

A

possible

extension

of

our

work

in

this

paper

may

be

to

examine

the

feasibility

of

reformulating

integer

linear

programs

so

that

the

solutions

are

robust.

Hebrard

et

al.

examined

reformulation

of

CSPs

for

finding

super

solutions.

Alternatively,

it

may

be

possible

to

use

a

top-down

approach

by

looking

at

the

k-best

solutions

sequentially,

in

terms

of

revenue,

and

performing

sensitivity

analysis

upon

each

solution

until

a

robust

one

is

found.

In

procurement

settings

the

principle

of

free

disposal

is

often

discounted

and

all

items

must

be

sold.

This

reduces

the

number

of

potential

solutions

and

thereby

reduces

the

reparability

of

each

solution.

The

impact

of

such

a

constraint

on

revenue

of

robust

solutions

is

also

left

for

future

work.

There

is

another

interesting

direction

this

work

may

take,

namely

robust

mechanism

design.

Porter

et

al.

introduced

the

notion

of

fault

tolerant

mechanism

design

in

which

agents

have

private

information

regarding

costs

for

task

completion,

but

also

their

probabilities

of

failure

When

the

bid-taker

has

combinatorial

valuations

for

task

completions

it

may

be

desirable

to

assign

the

same

task

to

multiple

agents

to

ensure

solution

robustness.

It

is

desirable

to

minimize

such

potentially

redundant

task

assignments

but

not

to

the

detriment

of

completed

task

valuations.

This

problem

could

be

modeled

using

the

WSS

framework

in

a

similar

manner

to

that

of

combinatorial

auctions.

In

the

case

where

no

robust

solutions

are

found,

it

is

possible

to

optimize

robustness,

instead

of

revenue,

by

finding

a

solution

of

at

least

a

given

revenue

that

minimizes

the

probability

of

an

irreparable

break.

In

this

manner

the

least

brittle

solution

of

adequate

revenue

may

be

chosen.

CONCLUSION

Fairness

is

often

cited

as

a

reason

for

choosing

the

optimal

solution

in

terms

of

revenue

only

Robust

solutions

militate

against

bids

deemed

brittle,

therefore

bidders

must

earn

a

reputation

for

being

reliable

to

relax

the

reparability

constraint

attached

to

their

bids.

This

may

be

seen

as

being

fair

to

long-standing

business

partners

whose

reliability

is

unquestioned.

Internet-based

auctions

are

often

seen

as

unwelcome

price-gouging

exercises

by

suppliers

in

many

sectors

Traditional

business

partnerships

are

being

severed

by

increased

competition

amongst

suppliers.

Quality

of

Service

can

suffer

because

of

the

increased

focus

on

short-term

profitability

to

the

detriment

of

the

bid-taker

in

the

long-term.

Robust

solutions

can

provide

a

means

of

selectively

discriminating

against

distrusted

bidders

in

a

measured

manner.

As

combinatorial

auction

deployment

moves

from

large

value

auctions

with

a

small

pool

of

trusted

bidders

(e.g.

spectrum-rights

sales)

towards

lower

value

auctions

with

potentially

unknown

bidders

(e.g.

Supply

Chain

Management

solution

robustness

becomes

more

relevant.

As

well

as

being

used

to

ensure

that

the

bid-taker

is

not

left

vulnerable

to

bid

withdrawal,

it

may

also

be

used

to

cement

relationships

with

preferred,

possibly

incumbent,

suppliers.

We

have

shown

that

it

is

possible

to

attain

robust

solutions

for

CAs

with

only

a

small

loss

in

revenue.

We

have

also

illustrated

how

such

solutions

tend

to

have

fewer

winning

bids

than

overall

optimal

solutions,

thereby

reducing

any

overheads

associated

with

dealing

with

more

bidders.

We

have

also

demonstrated

that

introducing

mutual

bid

bonds,

a

form

of

leveled

commitment

contract,

can

significantly

increase

the

revenue

of

optimal

robust

solutions

by

improving

reparability.

We

contend

that

robust

solutions

using

such

a

mechanism

can

allow

a

bid-taker

to

offer

the

possibility

of

bid

withdrawal

to

bidders

whilst

remaining

confident

about

postrepair

revenue

and

also

facilitating

increased

bidder

aggressiveness.

REFERENCES

Martin

Andersson

and

Tuomas

Sandholm.

Leveled

commitment

contracts

with

myopic

and

strategic

agents.

Journal

of

Economic

Dynamics

and

Control,

Special

issue

on

Agent-Based

Computational

Economics.

Fahiem

Bacchus

and

George

Katsirelos.

EFC

solver.

Michael

Berkelaar,

Kjell

Eikland,

and

Peter

Notebaert.

lp

solve

version

Rina

Dechter.

Constraint

Processing.

Morgan

Kaufmann,

Sven

DeVries

and

Rakesh

Vohra.

Combinatorial

auctions:

A

survey.

INFORMS

Journal

on

Computing,

pages

Jim

Ericson.

Reverse

auctions:

Bad

idea.

Line

Sept

Matthew

L.

Ginsberg,

Andrew

J.

Parkes,

and

Amitabha

Roy.

Supermodels

and

Robustness.

In

Proceedings

of

pages

Madison,

WI,

Ronald

M.

Harstad

and

Michael

H.

Rothkopf.

Withdrawable

bids

as

winner"s

curse

insurance.

Operations

Research,

November-December

Emmanuel

Hebrard,

Brahim

Hnich,

and

Toby

Walsh.

Robust

solutions

for

constraint

satisfaction

and

optimization.

In

Proceedings

of

the

European

Conference

on

Artificial

Intelligence,

pages

Emmanuel

Hebrard,

Brahim

Hnich,

and

Toby

Walsh.

Super

solutions

in

constraint

programming.

In

Proceedings

of

CP-AI-OR

pages

Gail

Hohner,

John

Rich,

Ed

Ng,

Grant

Reid,

Andrew

J.

Davenport,

Jayant

R.

Kalagnanam,

Ho

Soo

Lee,

and

Chae

An.

Combinatorial

and

quantity-discount

procurement

auctions

benefit

Mars

Incorporated

and

its

suppliers.

Interfaces,

Alan

Holland

and

Barry

O"Sullivan.

Super

solutions

for

combinatorial

auctions.

In

Ercim-Colognet

Constraints

Workshop

(CSCLP

Springer

LNAI,

Lausanne,

Switzerland,

Alan

Holland

and

Barry

O"Sullivan.

Weighted

super

solutions

for

constraint

programs,

December

Technical

Report:

No.

Selective

Insurance.

Business

insurance.

Ryan

Kastner,

Christina

Hsieh,

Miodrag

Potkonjak,

and

Majid

Sarrafzadeh.

On

the

sensitivity

of

incremental

algorithms

for

combinatorial

auctions.

In

WECWIS,

pages

June

Kevin

Leyton-Brown,

Mark

Pearson,

and

Yoav

Shoham.

Towards

a

universal

test

suite

for

combinatorial

auction

algorithms.

In

ACM

Conference

on

Electronic

Commerce,

pages

Associated

General

Contractors

of

America.

Associated

general

contractors

of

America

white

paper

on

reverse

auctions

for

procurement

of

construction.

ReverseAuctionWhitePaper.pdf,

National

Society

of

Professional

Engineers.

A

basic

guide

to

surety

bonds.

Martin

Pesendorfer

and

Estelle

Cantillon.

Combination

bidding

in

multi-unit

auctions.

Harvard

Business

School

Working

Draft,

Ryan

Porter,

Amir

Ronen,

Yoav

Shoham,

and

Moshe

Tennenholtz.

Mechanism

design

with

execution

uncertainty.

In

Proceedings

of

pages

Jean-Charles

R´egin.

Global

constraints

and

filtering

algorithms.

In

Constraint

and

Integer

ProgrammingTowards

a

Unified

Methodology,

chapter

pages

Kluwer

Academic

Publishers,

Michael

H.

Rothkopf

and

Aleksandar

Peke˘c.

Combinatorial

auction

design.

Management

Science,

November

Michael

H.

Rothkopf,

Aleksandar

Peke˘c,

and

Ronald

M.

Harstad.

Computationally

manageable

combinatorial

auctions.

Management

Science,

Daniel

Sabin

and

Eugene

C.

Freuder.

Contradicting

conventional

wisdom

in

constraint

satisfaction.

In

A.

Cohn,

editor,

Proceedings

of

pages

Tuomas

Sandholm.

Algorithm

for

optimal

winner

determination

in

combinatorial

auctions.

Artificial

Intelligence,

Tuomas

Sandholm

and

Victor

Lesser.

Leveled

Commitment

Contracts

and

Strategic

Breach.

Games

and

Economic

Behavior,

January

Tuomas

Sandholm

and

Victor

Lesser.

Leveled

commitment

contracting:

A

backtracking

instrument

for

multiagent

systems.

AI

Magazine,

Tuomas

Sandholm,

Sandeep

Sikka,

and

Samphel

Norden.

Algorithms

for

optimizing

leveled

commitment

contracts.

In

Proceedings

of

the

pages

Morgan

Kaufmann

Publishers

Inc.,

Tuomas

Sandholm

and

Yunhong

Zhou.

Surplus

equivalence

of

leveled

commitment

contracts.

Artificial

Intelligence,

William

E.

Walsh,

Michael

P.

Wellman,

and

Fredrik

Ygge.

Combinatorial

auctions

for

supply

chain

formation.

In

ACM

Conference

on

Electronic

Commerce,

pages

Rainier

Weigel

and

Christian

Bliek.

On

reformulation

of

constraint

satisfaction

problems.

In

Proceedings

of

pages

Margaret

W.

Wiener.

Access

spectrum

bid

withdrawal.

July

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

enforceable commitment
bid-taker's exposure problem
mutual bid bond
bid withdrawal
mandatory mutual bid bond
constraint program
weight super solution

combinatorial auction
bid
exposure problem
weighted super solution
robustness
constraint programming
set partition problem
winner determination problem