Expressive

Negotiation

over

Donations

to

Charities∗

Vincent

Conitzer

Carnegie

Mellon

University

Forbes

Avenue

Pittsburgh,

PA

USA

conitzer@cs.cmu.edu

Tuomas

Sandholm

Carnegie

Mellon

University

Forbes

Avenue

Pittsburgh,

PA

USA

sandholm@cs.cmu.edu

ABSTRACT

When

donating

money

to

a

(say,

charitable)

cause,

it

is

possible

to

use

the

contemplated

donation

as

negotiating

material

to

induce

other

parties

interested

in

the

charity

to

donate

more.

Such

negotiation

is

usually

done

in

terms

of

matching

offers,

where

one

party

promises

to

pay

a

certain

amount

if

others

pay

a

certain

amount.

However,

in

their

current

form,

matching

offers

allow

for

only

limited

negotiation.

For

one,

it

is

not

immediately

clear

how

multiple

parties

can

make

matching

offers

at

the

same

time

without

creating

circular

dependencies.

Also,

it

is

not

immediately

clear

how

to

make

a

donation

conditional

on

other

donations

to

multiple

charities,

when

the

donator

has

different

levels

of

appreciation

for

the

different

charities.

In

both

these

cases,

the

limited

expressiveness

of

matching

offers

causes

economic

loss:

it

may

happen

that

an

arrangement

that

would

have

made

all

parties

(donators

as

well

as

charities)

better

off

cannot

be

expressed

in

terms

of

matching

offers

and

will

therefore

not

occur.

In

this

paper,

we

introduce

a

bidding

language

for

expressing

very

general

types

of

matching

offers

over

multiple

charities.

We

formulate

the

corresponding

clearing

problem

(deciding

how

much

each

bidder

pays,

and

how

much

each

charity

receives),

and

show

that

it

is

NP-complete

to

approximate

to

any

ratio

even

in

very

restricted

settings.

We

give

a

mixed-integer

program

formulation

of

the

clearing

problem,

and

show

that

for

concave

bids,

the

program

reduces

to

a

linear

program.

We

then

show

that

the

clearing

problem

for

a

subclass

of

concave

bids

is

at

least

as

hard

as

the

decision

variant

of

linear

programming.

Subsequently,

we

show

that

the

clearing

problem

is

much

easier

when

bids

are

quasilinear-for

surplus,

the

problem

decomposes

across

charities,

and

for

payment

maximization,

a

greedy

approach

is

optimal

if

the

bids

are

concave

(although

this

latter

problem

is

weakly

NP-complete

when

the

bids

are

not

concave).

For

the

quasilinear

setting,

we

study

the

mechanism

design

question.

We

show

that

an

ex-post

efficient

mechanism

is

Supported

by

NSF

under

CAREER

Award

Grant

ITR

and

ITR

impossible

even

with

only

one

charity

and

a

very

restricted

class

of

bids.

We

also

show

that

there

may

be

benefits

to

linking

the

charities

from

a

mechanism

design

standpoint.

Categories

and

Subject

Descriptors

[Theory

of

Computation]:

Analysis

of

Algorithms

and

Problem

Complexity;

[Computer

Applications]:

Social

and

Behavioral

Sciences-Economics

General

Terms

Algorithms,

Economics,

Theory

INTRODUCTION

When

money

is

donated

to

a

charitable

(or

other)

cause

(hereafter

referred

to

as

charity),

often

the

donating

party

gives

unconditionally:

a

fixed

amount

is

transferred

from

the

donator

to

the

charity,

and

of

this

transfer

is

contingent

on

other

events-in

particular,

it

is

not

contingent

on

the

amount

given

by

other

parties.

Indeed,

this

is

currently

often

the

only

way

to

make

a

donation,

especially

for

small

donating

parties

such

as

private

individuals.

However,

when

multiple

parties

support

the

same

charity,

each

of

them

would

prefer

to

see

the

others

give

more

rather

than

less

to

this

charity.

In

such

scenarios,

it

is

sensible

for

a

party

to

use

its

contemplated

donation

as

negotiating

material

to

induce

the

others

to

give

more.

This

is

done

by

making

the

donation

conditional

on

the

others"

donations.

The

following

example

will

illustrate

this,

and

show

that

the

donating

parties

as

well

as

the

charitable

cause

may

simultaneously

benefit

from

the

potential

for

such

negotiation.

Suppose

we

have

two

parties,

and

who

are

both

supporters

of

charity

A.

To

either

of

them,

it

would

be

worth

if

A

received

It

follows

neither

of

them

will

be

willing

to

give

unconditionally,

because

However,

if

the

two

parties

draw

up

a

contract

that

says

that

they

will

each

give

both

the

parties

have

an

incentive

to

accept

this

contract

(rather

than

have

no

contract

at

all):

with

the

contract,

the

charity

will

receive

(rather

than

without

a

contract),

which

is

worth

to

each

party,

which

is

greater

than

the

that

that

party

will

have

to

give.

Effectively,

each

party

has

made

its

donation

conditional

on

the

other

party"s

donation,

leading

to

larger

donations

and

greater

happiness

to

all

parties

involved.

One

method

that

is

often

used

to

effect

this

is

to

make

a

matching

offer.

Examples

of

matching

offers

are:

I

will

give

x

dollars

for

every

dollar

donated.,

or

I

will

give

x

dollars

if

the

total

collected

from

other

parties

exceeds

y.

In

our

example

above,

one

of

the

parties

can

make

the

offer

I

will

donate

if

the

other

party

also

donates

at

least

that

much,

and

the

other

party

will

have

an

incentive

to

indeed

donate

so

that

the

total

amount

given

to

the

charity

increases

by

Thus

this

matching

offer

implements

the

contract

suggested

above.

As

a

real-world

example,

the

United

States

government

has

authorized

a

donation

of

up

to

billion

to

the

Global

Fund

to

fight

AIDS,

TB

and

Malaria,

under

the

condition

that

the

American

contribution

does

not

exceed

one

third

of

the

total-to

encourage

other

countries

to

give

more

However,

there

are

several

severe

limitations

to

the

simple

approach

of

matching

offers

as

just

described.

It

is

not

clear

how

two

parties

can

make

matching

offers

where

each

party"s

offer

is

stated

in

terms

of

the

amount

that

the

other

pays.

(For

example,

it

is

not

clear

what

the

outcome

should

be

when

both

parties

offer

to

match

the

other"s

donation.)

Thus,

matching

offers

can

only

be

based

on

payments

made

by

parties

that

are

giving

unconditionally

(not

in

terms

of

a

matching

offer)-or

at

least

there

can

be

no

circular

Given

the

current

infrastructure

for

making

matching

offers,

it

is

impractical

to

make

a

matching

offer

depend

on

the

amounts

given

to

multiple

charities.

For

instance,

a

party

may

wish

to

specify

that

it

will

pay

given

that

charity

A

receives

a

total

of

but

that

it

will

also

count

donations

made

to

charity

B,

at

half

the

rate.

(Thus,

a

total

payment

of

to

charity

A

combined

with

a

total

payment

of

to

charity

B

would

be

just

enough

for

the

party"s

offer

to

take

effect.)

In

contrast,

in

this

paper

we

propose

a

new

approach

where

each

party

can

express

its

relative

preferences

for

different

charities,

and

make

its

offer

conditional

on

its

own

appreciation

for

the

vector

of

donations

made

to

the

different

charities.

Moreover,

the

amount

the

party

offers

to

donate

at

different

levels

of

appreciation

is

allowed

to

vary

arbitrarily

(it

does

need

to

be

a

dollar-for-dollar

(or

n-dollarfor-dollar)

matching

arrangement,

or

an

arrangement

where

the

party

offers

a

fixed

amount

provided

a

given

(strike)

total

has

been

exceeded).

Finally,

there

is

a

clear

interpretation

of

what

it

means

when

multiple

parties

are

making

conditional

offers

that

are

stated

in

terms

of

each

other.

Given

each

combination

of

(conditional)

offers,

there

is

a

(usually)

unique

solution

which

determines

how

much

each

party

pays,

and

how

much

each

charity

is

paid.

However,

as

we

will

show,

finding

this

solution

(the

clearing

problem)

requires

solving

a

potentially

difficult

optimization

problem.

A

large

part

of

this

paper

is

devoted

to

studying

how

difficult

this

problem

is

under

different

assumptions

on

the

structure

of

the

offers,

and

providing

algorithms

for

solving

it.

Typically,

larger

organizations

match

offers

of

private

individuals.

For

example,

the

American

Red

Cross

Liberty

Disaster

Fund

maintains

a

list

of

businesses

that

match

their

customers"

donations

Towards

the

end

of

the

paper,

we

also

study

the

mechanism

design

problem

of

motivating

the

bidders

to

bid

truthfully.

In

short,

expressive

negotiation

over

donations

to

charities

is

a

new

way

in

which

electronic

commerce

can

help

the

world.

A

web-based

implementation

of

the

ideas

described

in

this

paper

can

facilitate

voluntary

reallocation

of

wealth

on

a

global

scale.

Aditionally,

optimally

solving

the

clearing

problem

(and

thereby

generating

the

maximum

economic

welfare)

requires

the

application

of

sophisticated

algorithms.

COMPARISON

TO

COMBINATORIAL

AUCTIONS

AND

EXCHANGES

This

section

discusses

the

relationship

between

expressive

charity

donation

and

combinatorial

auctions

and

exchanges.

It

can

be

skipped,

but

may

be

of

interest

to

the

reader

with

a

background

in

combinatorial

auctions

and

exchanges.

In

a

combinatorial

auction,

there

are

m

items

for

sale,

and

bidders

can

place

bids

on

bundles

of

one

or

more

items.

The

auctioneer

subsequently

labels

each

bid

as

winning

or

losing,

under

the

constraint

that

no

item

can

be

in

more

than

one

winning

bid,

to

maximize

the

sum

of

the

values

of

the

winning

bids.

(This

is

known

as

the

clearing

problem.)

Variants

include

combinatorial

reverse

auctions,

where

the

auctioneer

is

seeking

to

procure

a

set

of

items;

and

combinatorial

exchanges,

where

bidders

can

both

buy

and

and

sell

items

(even

within

the

same

bid).

Other

extensions

include

allowing

for

side

constraints,

as

well

as

the

specification

of

attributes

of

the

items

in

bids.

Combinatorial

auctions

and

exchanges

have

recently

become

a

popular

research

topic

The

problems

of

clearing

expressive

charity

donation

markets

and

clearing

combinatorial

auctions

or

exchanges

are

very

different

in

formulation.

Nevertheless,

there

are

interesting

parallels.

One

of

the

main

reasons

for

the

interest

in

combinatorial

auctions

and

exchanges

is

that

it

allows

for

expressive

bidding.

A

bidder

can

express

exactly

how

much

each

different

allocation

is

worth

to

her,

and

thus

the

globally

optimal

allocation

may

be

chosen

by

the

auctioneer.

Compare

this

to

a

bidder

having

to

bid

on

two

different

items

in

two

different

(one-item)

auctions,

without

any

way

of

expressing

that

(for

instance)

one

item

is

worthless

if

the

other

item

is

not

won.

In

this

scenario,

the

bidder

may

win

the

first

item

but

not

the

second

(because

there

was

another

high

bid

on

the

second

item

that

she

did

not

anticipate),

leading

to

economic

inefficiency.

Expressive

bidding

is

also

one

of

the

main

benefits

of

the

expressive

charity

donation

market.

Here,

bidders

can

express

exactly

how

much

they

are

willing

to

donate

for

every

vector

of

amounts

donated

to

charities.

This

may

allow

bidders

to

negotiate

a

complex

arrangement

of

who

gives

how

much

to

which

charity,

which

is

beneficial

to

all

parties

involved;

whereas

no

such

arrangement

may

have

been

possible

if

the

bidders

had

been

restricted

to

using

simple

matching

offers

on

individual

charities.

Again,

expressive

bidding

is

necessary

to

achieve

economic

efficiency.

Another

parallel

is

the

computational

complexity

of

the

clearing

problem.

In

order

to

achieve

the

full

economic

efficiency

allowed

by

the

market"s

expressiveness

(or

even

come

close

to

it),

hard

computational

problems

must

be

solved

in

combinatorial

auctions

and

exchanges,

as

well

as

in

the

charity

donation

market

(as

we

will

see).

DEFINITIONS

Throughout

this

paper,

we

will

refer

to

the

offers

that

the

donating

parties

make

as

bids,

and

to

the

donating

parties

as

bidders.

In

our

bidding

framework,

a

bid

will

specify,

for

each

vector

of

total

payments

made

to

the

charities,

how

much

that

bidder

is

willing

to

contribute.

(The

contribution

of

this

bidder

is

also

counted

in

the

vector

of

paymentsso,

the

vector

of

total

payments

to

the

charities

represents

the

amount

given

by

all

donating

parties,

not

just

the

ones

other

than

this

bidder.)

The

bidding

language

is

expressive

enough

that

no

bidder

should

have

to

make

more

than

one

bid.

The

following

definition

makes

the

general

form

of

a

bid

in

our

framework

precise.

Definition

In

a

setting

with

m

charities

cm,

a

bid

by

bidder

bj

is

a

function

vj

Rm

R.

The

interpretation

is

that

if

charity

ci

receives

a

total

amount

of

πci

then

bidder

j

is

willing

to

donate

(up

to)

πcm

We

now

define

possible

outcomes

in

our

model,

and

which

outcomes

are

valid

given

the

bids

that

were

made.

Definition

An

outcome

is

a

vector

of

payments

made

by

the

bidders

πbn

and

a

vector

of

payments

received

by

the

charities

πcm

A

valid

outcome

is

an

outcome

where

n

πbj

m

πci

(at

least

as

much

money

is

collected

as

is

given

away);

For

all

j

n,

πbj

πcm

(no

bidder

gives

more

than

she

is

willing

to).

Of

course,

in

the

end,

only

one

of

the

valid

outcomes

can

be

chosen.

We

choose

the

valid

outcome

that

maximizes

the

objective

that

we

have

for

the

donation

process.

Definition

An

objective

is

a

function

from

the

set

of

all

outcomes

to

After

all

bids

have

been

collected,

a

valid

outcome

will

be

chosen

that

maximizes

this

objective.

One

example

of

an

objective

is

surplus,

given

by

n

πbj

m

πci

The

surplus

could

be

the

profits

of

a

company

managing

the

expressive

donation

marketplace;

but,

alternatively,

the

surplus

could

be

returned

to

the

bidders,

or

given

to

the

charities.

Another

objective

is

total

amount

donated,

given

by

m

πci

(Here,

different

weights

could

also

be

placed

on

the

different

charities.)

Finding

the

valid

outcome

that

maximizes

the

objective

is

a

(nontrivial)

computational

problem.

We

will

refer

to

it

as

the

clearing

problem.

The

formal

definition

follows.

Definition

(DONATION-CLEARING).

We

are

given

a

set

of

n

bids

over

charities

cm.

Additionally,

we

are

given

an

objective

function.

We

are

asked

to

find

an

objective-maximizing

valid

outcome.

How

difficult

the

DONATION-CLEARING

problem

is

depends

on

the

types

of

bids

used

and

the

language

in

which

they

are

expressed.

This

is

the

topic

of

the

next

section.

In

general,

the

objective

function

may

also

depend

on

the

bids,

but

the

objective

functions

under

consideration

in

this

paper

do

not

depend

on

the

bids.

The

techniques

presented

in

this

paper

will

typically

generalize

to

objectives

that

take

the

bids

into

account

directly.

A

SIMPLIFIED

BIDDING

LANGUAGE

Specifying

a

general

bid

in

our

framework

(as

defined

above)

requires

being

able

to

specify

an

arbitrary

real-valued

function

over

Rm

Even

if

we

restricted

the

possible

total

payment

made

to

each

charity

to

the

set

s},

this

would

still

require

a

bidder

to

specify

values.

Thus,

we

need

a

bidding

language

that

will

allow

the

bidders

to

at

least

specify

some

bids

more

concisely.

We

will

specify

a

bidding

language

that

only

represents

a

subset

of

all

possible

bids,

which

can

be

described

To

introduce

our

bidding

language,

we

will

first

describe

the

bidding

function

as

a

composition

of

two

functions;

then

we

will

outline

our

assumptions

on

each

of

these

functions.

First,

there

is

a

utility

function

uj

Rm

R,

specifying

how

much

bidder

j

appreciates

a

given

vector

of

total

donations

to

the

charities.

(Note

that

the

way

we

define

a

bidder"s

utility

function,

it

does

not

take

the

payments

the

bidder

makes

into

account.)

Then,

there

is

a

donation

willingness

function

wj

R

R,

which

specifies

how

much

bidder

j

is

willing

to

pay

given

her

utility

for

the

vector

of

donations

to

the

charities.

We

emphasize

that

this

function

does

not

need

to

be

linear,

so

that

utilities

should

not

be

thought

of

as

expressible

in

dollar

amounts.

(Indeed,

when

an

individual

is

donating

to

a

large

charity,

the

reason

that

the

individual

donates

only

a

bounded

amount

is

typically

not

decreasing

marginal

value

of

the

money

given

to

the

charity,

but

rather

that

the

marginal

value

of

a

dollar

to

the

bidder

herself

becomes

larger

as

her

budget

becomes

smaller.)

So,

we

have

πcm

πcm

and

we

let

the

bidder

describe

her

functions

uj

and

wj

separately.

(She

will

submit

these

functions

as

her

bid.)

Our

first

restriction

is

that

the

utility

that

a

bidder

derives

from

money

donated

to

one

charity

is

independent

of

the

amount

donated

to

another

charity.

Thus,

πcm

m

ui

j(πci

(We

observe

that

this

does

not

imply

that

the

bid

function

vj

decomposes

similarly,

because

of

the

nonlinearity

of

wj.)

Furthermore,

each

ui

j

must

be

piecewise

linear.

An

interesting

special

case

which

we

will

study

is

when

each

ui

j

is

a

line:

ui

j(πci

ai

jπci

This

special

case

is

justified

in

settings

where

the

scale

of

the

donations

by

the

bidders

is

small

relative

to

the

amounts

the

charities

receive

from

other

sources,

so

that

the

marginal

use

of

a

dollar

to

the

charity

is

not

affected

by

the

amount

given

by

the

bidders.

The

only

restriction

that

we

place

on

the

payment

willingness

functions

wj

is

that

they

are

piecewise

linear.

One

interesting

special

case

is

a

threshold

bid,

where

wj

is

a

step

function:

the

bidder

will

provide

t

dollars

if

her

utility

exceeds

s,

and

otherwise

Another

interesting

case

is

when

such

a

bid

is

partially

acceptable:

the

bidder

will

provide

t

dollars

if

her

utility

exceeds

s;

but

if

her

utility

is

u

s,

she

is

still

willing

to

provide

ut

s

dollars.

One

might

wonder

why,

if

we

are

given

the

bidders"

utility

functions,

we

do

not

simply

maximize

the

sum

of

the

utilities

rather

than

surplus

or

total

donated.

There

are

several

reasons.

First,

because

affine

transformations

do

not

affect

utility

functions

in

a

fundamental

way,

it

would

be

Of

course,

our

bidding

language

can

be

trivially

extended

to

allow

for

fully

expressive

bids,

by

also

allowing

bids

from

a

fully

expressive

bidding

language,

in

addition

to

the

bids

in

our

bidding

language.

ble

for

a

bidder

to

inflate

her

utility

by

changing

its

units,

thereby

making

her

bid

more

important

for

utility

maximization

purposes.

Second,

a

bidder

could

simply

give

a

payment

willingness

function

that

is

everywhere,

and

have

her

utility

be

taken

into

account

in

deciding

on

the

outcome,

in

spite

of

her

not

contributing

anything.

AVOIDING

INDIRECT

PAYMENTS

In

an

initial

implementation,

the

approach

of

having

donations

made

out

to

a

center,

and

having

a

center

forward

these

payments

to

charities,

may

not

be

desirable.

Rather,

it

may

be

preferable

to

have

a

partially

decentralized

solution,

where

the

donating

parties

write

out

checks

to

the

charities

directly

according

to

a

solution

prescribed

by

the

center.

In

this

scenario,

the

center

merely

has

to

verify

that

parties

are

giving

the

prescribed

amounts.

Advantages

of

this

include

that

the

center

can

keep

its

legal

status

minimal,

as

well

as

that

we

do

not

require

the

donating

parties

to

trust

the

center

to

transfer

their

donations

to

the

charities

(or

require

some

complicated

verification

protocol).

It

is

also

a

step

towards

a

fully

decentralized

solution,

if

this

is

desirable.

To

bring

this

about,

we

can

still

use

the

approach

described

earlier.

After

we

clear

the

market

in

the

manner

described

before,

we

know

the

amount

that

each

donator

is

supposed

to

give,

and

the

amount

that

each

charity

is

supposed

to

receive.

Then,

it

is

straightforward

to

give

some

specification

of

who

should

give

how

much

to

which

charity,

that

is

consistent

with

that

clearing.

Any

greedy

algorithm

that

increases

the

cash

flow

from

any

bidder

who

has

not

yet

paid

enough,

to

any

charity

that

has

not

yet

received

enough,

until

either

the

bidder

has

paid

enough

or

the

charity

has

received

enough,

will

provide

such

a

specification.

(All

of

this

is

assuming

that

bj

πbj

ci

πci

In

the

case

where

there

is

nonzero

surplus,

that

is,

bj

πbj

ci

πci

we

can

distribute

this

surplus

across

the

bidders

by

not

requiring

them

to

pay

the

full

amount,

or

across

the

charities

by

giving

them

more

than

the

solution

specifies.)

Nevertheless,

with

this

approach,

a

bidder

may

have

to

write

out

a

check

to

a

charity

that

she

does

not

care

for

at

all.

(For

example,

an

environmental

activist

who

was

using

the

system

to

increase

donations

to

a

wildlife

preservation

fund

may

be

required

to

write

a

check

to

a

group

supporting

a

right-wing

political

party.)

This

is

likely

to

lead

to

complaints

and

noncompliance

with

the

clearing.

We

can

address

this

issue

by

letting

each

bidder

specify

explicitly

(before

the

clearing)

which

charities

she

would

be

willing

to

make

a

check

out

to.

These

additional

constraints,

of

course,

may

change

the

optimal

solution.

In

general,

checking

whether

a

given

centralized

solution

(with

zero

surplus)

can

be

accomplished

through

decentralized

payments

when

there

are

such

constraints

can

be

modeled

as

a

MAX-FLOW

problem.

In

the

MAX-FLOW

instance,

there

is

an

edge

from

the

source

node

s

to

each

bidder

bj,

with

a

capacity

of

πbj

(as

specified

in

the

centralized

solution);

an

edge

from

each

bidder

bj

to

each

charity

ci

that

the

bidder

is

willing

to

donate

money

to,

with

a

capacity

of

and

an

edge

from

each

charity

ci

to

the

target

node

t

with

capacity

πci

(as

specified

in

the

centralized

solution).

In

the

remainder

of

this

paper,

all

our

hardness

results

apply

even

to

the

setting

where

there

is

no

constraint

on

which

bidders

can

pay

to

which

charity

(that

is,

even

the

problem

as

it

was

specified

before

this

section

is

hard).

We

also

generalize

our

clearing

algorithms

to

the

partially

decentralized

case

with

constraints.

HARDNESS

OF

CLEARING

THE

MARKET

In

this

section,

we

will

show

that

the

clearing

problem

is

completely

inapproximable,

even

when

every

bidder"s

utility

function

is

linear

(with

slope

or

in

each

charity"s

payments),

each

bidder

cares

either

about

at

most

two

charities

or

about

all

charities

equally,

and

each

bidder"s

payment

willingness

function

is

a

step

function.

We

will

reduce

from

(given

a

formula

in

conjunctive

normal

form

(where

each

clause

has

two

literals)

and

a

target

number

of

satisfied

clauses

T,

does

there

exist

an

assignment

of

truth

values

to

the

variables

that

makes

at

least

T

clauses

true?),

which

is

NP-complete

Theorem

There

exists

a

reduction

from

instances

to

DONATION-CLEARING

instances

such

that

If

the

instance

has

no

solution,

then

the

only

valid

outcome

is

the

zero

outcome

(no

bidder

pays

anything

and

no

charity

receives

anything);

Otherwise,

there

exists

a

solution

with

positive

surplus.

Additionally,

the

DONATION-CLEARING

instances

that

we

reduce

to

have

the

following

properties:

Every

ui

j

is

a

line;

that

is,

the

utility

that

each

bidder

derives

from

any

charity

is

linear;

All

the

ui

j

have

slope

either

or

Every

bidder

either

has

at

most

charities

that

affect

her

utility

(with

slope

or

all

charities

affect

her

utility

(with

slope

Every

bid

is

a

threshold

bid;

that

is,

every

bidder"s

payment

willingness

function

wj

is

a

step

function.

Proof.

The

problem

is

in

NP

because

we

can

nondeterministically

choose

the

payments

to

be

made

and

received,

and

check

the

validity

and

objective

value

of

this

outcome.

In

the

following,

we

will

represent

bids

as

follows:

({(ck,

ak)},

s,

t)

indicates

that

uk

j

(πck

akπck

(this

function

is

for

ck

not

mentioned

in

the

bid),

and

wj(uj)

t

for

uj

s,

wj(uj)

otherwise.

To

show

NP-hardness,

we

reduce

an

arbitrary

instance,

given

by

a

set

of

clauses

K

{k}

k,

k)}

over

a

variable

set

V

together

with

a

target

number

of

satisfied

clauses

T,

to

the

following

DONATION-CLEARING

instance.

Let

the

set

of

charities

be

as

follows.

For

every

literal

l

L,

there

is

a

charity

cl.

Then,

let

the

set

of

bids

be

as

follows.

For

every

variable

v,

there

is

a

bid

bv

({(c+v,

(c−v,

For

every

literal

l,

there

is

a

bid

bl

({(cl,

For

every

clause

k

k,

k}

K,

there

is

a

bid

bk

k

k

||K|

Finally,

there

is

a

single

bid

that

values

all

charities

equally:

(cm,

T

||K|

||K|

We

show

the

two

instances

are

equivalent.

First,

suppose

there

exists

a

solution

to

the

instance.

If

in

this

solution,

l

is

true,

then

let

πcl

T

otherwise

πcl

Also,

the

only

bids

that

are

not

accepted

(meaning

the

threshold

is

not

met)

are

the

bl

where

l

is

false,

and

the

bk

such

that

both

of

k,

k

are

false.

First

we

show

that

no

bidder

whose

bid

is

accepted

pays

more

than

she

is

willing

to.

For

each

bv,

either

c+v

or

c−v

receives

at

least

so

this

bidder"s

threshold

has

been

met.

For

each

bl,

either

l

is

false

and

the

bid

is

not

accepted,

or

l

is

true,

cl

receives

at

least

and

the

threshold

has

been

met.

For

each

bk,

either

both

of

k,

k

are

false

and

the

bid

is

not

accepted,

or

at

least

one

of

them

(say

li

k)

is

true

(that

is,

k

is

satisfied)

and

cli

k

receives

at

least

and

the

threshold

has

been

met.

Finally,

because

the

total

amount

received

by

the

charities

is

T

||K|

threshold

has

also

been

met.

The

total

amount

that

can

be

extracted

from

the

accepted

bids

is

at

least

|V

)+|V

|+T

||K|

||K|

T

||K|

||K|

T

||K|

so

there

is

positive

surplus.

So

there

exists

a

solution

with

positive

surplus

to

the

DONATION-CLEARING

instance.

Now

suppose

there

exists

a

nonzero

outcome

in

the

DONATION-CLEARING

instance.

First

we

show

that

it

is

not

possible

(for

any

v

V

that

both

b+v

and

b−v

are

accepted.

For,

this

would

require

that

πc+v

πc−v

The

bids

bv,

b+v,

b−v

cannot

contribute

more

than

so

we

need

another

at

least.

It

is

easily

seen

that

for

any

other

v

accepting

any

subset

of

{bv

b+v

b−v

would

require

that

at

least

as

much

is

given

to

c+v

and

c−v

as

can

be

extracted

from

these

bids,

so

this

cannot

help.

Finally,

all

the

other

bids

combined

can

contribute

at

most

|K|

||K|

||K|

It

follows

that

we

can

interpret

the

outcome

in

the

DONATION-CLEARING

instance

as

a

partial

assignment

of

truth

values

to

variables:

v

is

set

to

true

if

b+v

is

accepted,

and

to

false

if

b−v

is

accepted.

All

that

is

left

to

show

is

that

this

partial

assignment

satisfies

at

least

T

clauses.

First

we

show

that

if

a

clause

bid

bk

is

accepted,

then

either

k

or

k

is

accepted

(and

thus

either

k

or

k

is

set

to

true,

hence

k

is

satisfied).

If

bk

is

accepted,

at

least

one

of

k

and

k

must

be

receiving

at

least

without

loss

of

generality,

say

it

is

k

and

say

k

corresponds

to

variable

k

(that

is,

it

is

k

or

k).

If

k

does

not

receive

at

least

k

is

not

accepted,

and

it

is

easy

to

check

that

the

bids

k

k

k

contribute

(at

least)

less

than

is

paid

to

k

and

k

But

this

is

the

same

situation

that

we

analyzed

before,

and

we

know

it

is

impossible.

All

that

remains

to

show

is

that

at

least

T

clause

bids

are

accepted.

We

now

show

that

is

accepted.

Suppose

it

is

not;

then

one

of

the

bv

must

be

accepted.

(The

solution

is

nonzero

by

assumption;

if

only

some

bk

are

accepted,

the

total

payment

from

these

bids

is

at

most

|K|

||K|

which

is

not

enough

for

any

bid

to

be

accepted;

and

if

one

of

the

bl

is

accepted,

then

the

threshold

for

the

corresponding

bv

is

also

reached.)

For

this

v,

k

k

k

contribute

(at

least)

less

than

the

total

payments

to

c+v

and

c−v.

Again,

the

other

bv

and

bl

cannot

(by

themselves)

help

to

close

this

gap;

and

the

bk

can

contribute

at

most

|K|

||K|

It

follows

that

is

accepted.

Now,

in

order

for

to

be

accepted,

a

total

of

T

||K|

must

be

donated.

Because

is

not

possible

(for

any

v

V

that

both

b+v

and

b−v

are

accepted,

it

follows

that

the

total

payment

by

the

bv

and

the

bl

can

be

at

most

Adding

payment

of

||K|

to

this,

we

still

need

T

||K|

from

the

bk.

But

each

one

of

them

contributes

at

most

||K|

so

at

least

T

of

them

must

be

accepted.

Corollary

Unless

P=NP,

there

is

no

polynomial-time

algorithm

for

approximating

DONATION-CLEARING

(with

either

the

surplus

or

the

total

amount

donated

as

the

objective)

within

any

ratio

f(n),

where

f

is

a

nonzero

function

of

the

size

of

the

instance.

This

holds

even

if

the

DONATIONCLEARING

structures

satisfy

all

the

properties

given

in

Theorem

Proof.

Suppose

we

had

such

a

polynomial

time

algorithm,

and

applied

it

to

the

DONATION-CLEARING

instances

that

were

reduced

from

instances

in

Theorem

It

would

return

a

nonzero

solution

when

the

instance

has

a

solution,

and

a

zero

solution

otherwise.

So

we

can

decide

whether

arbitrary

instances

are

satisfiable

this

way,

and

it

would

follow

that

P=NP.

(Solving

the

problem

to

optimality

is

NP-complete

in

many

other

(noncomparable

or

even

more

restricted)

settings

as

well-we

omit

such

results

because

of

space

constraint.)

This

should

not

be

interpreted

to

mean

that

our

approach

is

infeasible.

First,

as

we

will

show,

there

are

very

expressive

families

of

bids

for

which

the

problem

is

solvable

in

polynomial

time.

Second,

NP-completeness

is

often

overcome

in

practice

(especially

when

the

stakes

are

high).

For

instance,

even

though

the

problem

of

clearing

combinatorial

auctions

is

NP-complete

(even

to

approximate

they

are

typically

solved

to

optimality

in

practice.

MIXED

INTEGER

PROGRAMMING

FORMULATION

In

this

section,

we

give

a

mixed

integer

programming

(MIP)

formulation

for

the

general

problem.

We

also

discuss

in

which

special

cases

this

formulation

reduces

to

a

linear

programming

(LP)

formulation.

In

such

cases,

the

problem

is

solvable

in

polynomial

time,

because

linear

programs

can

be

solved

in

polynomial

time

The

variables

of

the

MIP

defining

the

final

outcome

are

the

payments

made

to

the

charities,

denoted

by

πci

and

the

payments

extracted

from

the

bidders,

πbj

In

the

case

where

we

try

to

avoid

direct

payments

and

let

the

bidders

pay

the

charities

directly,

we

add

variables

πci,bj

indicating

how

much

bj

pays

to

ci,

with

the

constraints

that

for

each

ci,

πci

bj

πci,bj

and

for

each

bj,

πbj

ci

πci,bj

Additionally,

there

is

a

constraint

πci,bj

whenever

bidder

bj

is

unwilling

to

pay

charity

ci.

The

rest

of

the

MIP

can

be

phrased

in

terms

of

the

πci

and

πbj

The

objectives

we

have

discussed

earlier

are

both

linear:

surplus

is

given

by

n

πbj

m

πci

and

total

amount

donated

is

given

by

m

πci

(coefficients

can

be

added

to

represent

different

weights

on

the

different

charities

in

the

objective).

The

constraint

that

the

outcome

should

be

valid

(no

deficit)

is

given

simply

by:

n

πbj

m

πci

For

every

bidder,

for

every

charity,

we

define

an

additional

utility

variable

ui

j

indicating

the

utility

that

this

bidder

derives

from

the

payment

to

this

charity.

The

bidder"s

total

utility

is

given

by

another

variable

uj,

with

the

constraint

that

uj

m

ui

j.

Each

ui

j

is

given

as

a

function

of

πci

by

the

(piecewise

linear)

function

provided

by

the

bidder.

In

order

to

represent

this

function

in

the

MIP

formulation,

we

will

merely

place

upper

bounding

constraints

on

ui

j,

so

that

it

cannot

exceed

the

given

functions.

The

MIP

solver

can

then

push

the

ui

j

variables

all

the

way

up

to

the

constraint,

in

order

to

extract

as

much

payment

from

this

bidder

as

possible.

In

the

case

where

the

ui

j

are

concave,

this

is

easy:

if

(sl,

tl)

and

are

endpoints

of

a

finite

linear

segment

in

the

function,

we

add

the

constraint

that

ui

j

tl

πci

−sl

tl).

If

the

final

(infinite)

segment

starts

at

(sk,

tk)

and

has

slope

d,

we

add

the

constraint

that

ui

j

tk

d(πci

sk).

Using

the

fact

that

the

function

is

concave,

for

each

value

of

πci

the

tightest

upper

bound

on

ui

j

is

the

one

corresponding

to

the

segment

above

that

value

of

πci

and

therefore

these

constraints

are

sufficient

to

force

the

correct

value

of

ui

j.

When

the

function

is

not

concave,

we

require

(for

the

first

time)

some

binary

variables.

First,

we

define

another

point

on

the

function:

(sk

M,

tk

dM),

where

d

is

the

slope

of

the

infinite

segment

and

M

is

any

upper

bound

on

the

πcj

This

has

the

effect

that

we

will

never

be

on

the

infinite

segment

again.

Now,

let

xi,j

l

be

an

indicator

variable

that

should

be

if

πci

is

below

the

lth

segment

of

the

function,

and

otherwise.

To

effect

this,

first

add

a

constraint

k

xi,j

l

Now,

we

aim

to

represent

πci

as

a

weighted

average

of

its

two

neighboring

si,j

l

For

l

k

let

λi,j

l

be

the

weight

on

si,j

l

We

add

the

constraint

λi,j

l

Also,

for

l

k

we

add

the

constraint

λi,j

l

+xl

(where

and

are

defined

to

be

zero),

so

that

indeed

only

the

two

neighboring

si,j

l

have

nonzero

weight.

Now

we

add

the

constraint

πci

si,j

l

λi,j

l

and

now

the

λi,j

l

must

be

set

correctly.

Then,

we

can

set

ui

j

ti,j

l

λi,j

l

(This

is

a

standard

MIP

technique

Finally,

each

πbj

is

bounded

by

a

function

of

uj

by

the

(piecewise

linear)

function

provided

by

the

bidder

(wj).

Representing

this

function

is

entirely

analogous

to

how

we

represented

ui

j

as

a

function

of

πci

(Again

we

will

need

binary

variables

only

if

the

function

is

not

concave.)

Because

we

only

use

binary

variables

when

either

a

utility

function

ui

j

or

a

payment

willingness

function

wj

is

not

concave,

it

follows

that

if

all

of

these

are

concave,

our

MIP

formulation

is

simply

a

linear

program-which

can

be

solved

in

polynomial

time.

Thus:

Theorem

If

all

functions

ui

j

and

wj

are

concave

(and

piecewise

linear),

the

DONATION-CLEARING

problem

can

be

solved

in

polynomial

time

using

linear

programming.

Even

if

some

of

these

functions

are

not

concave,

we

can

simply

replace

each

such

function

by

the

smallest

upper

bounding

concave

function,

and

use

the

linear

programming

formulation

to

obtain

an

upper

bound

on

the

objectivewhich

may

be

useful

in

a

search

formulation

of

the

general

problem.

WHY

ONE

CANNOT

DO

MUCH

BETTER

THAN

LINEAR

PROGRAMMING

One

may

wonder

if,

for

the

special

cases

of

the

DONATIONCLEARING

problem

that

can

be

solved

in

polynomial

time

with

linear

programming,

there

exist

special

purpose

algorithms

that

are

much

faster

than

linear

programming

algorithms.

In

this

section,

we

show

that

this

is

not

the

case.

We

give

a

reduction

from

(the

decision

variant

of)

the

general

linear

programming

problem

to

(the

decision

variant

of)

a

special

case

of

the

DONATION-CLEARING

problem

(which

can

be

solved

in

polynomial

time

using

linear

programming).

(The

decision

variant

of

an

optimization

problem

asks

the

binary

question:

Can

the

objective

value

exceed

o?)

Thus,

any

special-purpose

algorithm

for

solving

the

decision

variant

of

this

special

case

of

the

DONATIONCLEARING

problem

could

be

used

to

solve

a

decision

question

about

an

arbitrary

linear

program

just

as

fast.

(And

thus,

if

we

are

willing

to

call

the

algorithm

a

logarithmic

number

of

times,

we

can

solve

the

optimization

version

of

the

linear

program.)

We

first

observe

that

for

linear

programming,

a

decision

question

about

the

objective

can

simply

be

phrased

as

another

constraint

in

the

LP

(forcing

the

objective

to

exceed

the

given

value);

then,

the

original

decision

question

coincides

with

asking

whether

the

resulting

linear

program

has

a

feasible

solution.

Theorem

The

question

of

whether

an

LP

(given

by

a

set

of

linear

has

a

feasible

solution

can

be

modeled

as

a

DONATION-CLEARING

instance

with

payment

maximization

as

the

objective,

with

charities

and

v

c

bids

(where

v

is

the

number

of

variables

in

the

LP,

and

c

is

the

number

of

constraints).

In

this

model,

each

bid

bj

has

only

linear

ui

j

functions,

and

is

a

partially

acceptable

threshold

bid

(wj(u)

tj

for

u

sj,

otherwise

wj(u)

utj

sj

The

v

bids

corresponding

to

the

variables

mention

only

two

charities

each;

the

c

bids

corresponding

to

the

constraints

mention

only

two

times

the

number

of

variables

in

the

corresponding

constraint.

Proof.

For

every

variable

xi

in

the

LP,

let

there

be

two

charities,

c+xi

and

c−xi

Let

H

be

some

number

such

that

if

there

is

a

feasible

solution

to

the

LP,

there

is

one

in

which

every

variable

has

absolute

value

at

most

H.

In

the

following,

we

will

represent

bids

as

follows:

({(ck,

ak)},

s,

t)

indicates

that

uk

j

(πck

akπck

(this

function

is

for

ck

not

mentioned

in

the

bid),

and

wj(uj)

t

for

uj

s,

wj(uj)

uj

t

s

otherwise.

For

every

variable

xi

in

the

LP,

let

there

be

a

bid

bxi

({(c+xi

(c−xi

c

v

For

every

constraint

i

rj

i

xi

sj

in

the

linear

program,

let

there

be

a

bid

bj

({(c−xi

rj

i

)}i:r

j

i

{(c+xi

−rj

i

)}i:r

j

i

i

|rj

i

|)H

sj,

Let

the

target

total

amount

donated

be

Suppose

there

is

a

feasible

solution

(x∗

x∗

x∗

v)

to

the

LP.

Without

loss

of

generality,

we

can

suppose

that

|x∗

i

H

for

all

i.

Then,

in

the

DONATION-CLEARING

instance,

These

constraints

must

include

bounds

on

the

variables

(including

nonnegativity

bounds),

if

any.

for

every

i,

let

πc+xi

H

x∗

i

and

let

πc−xi

H

x∗

i

(for

a

total

payment

of

to

these

two

charities).

This

allows

us

to

extract

the

maximum

payment

from

the

bids

bxi

-a

total

payment

of

c.

Additionally,

the

utility

of

bidder

bj

is

now

i:r

j

i

rj

i

(H

x∗

i

i:r

j

i

−rj

i

(H

x∗

i

i

|rj

i

|)H

i

rj

i

x∗

i

i

|rj

i

|)H

sj

(where

the

last

inequality

stems

from

the

fact

that

constraint

j

must

be

satisfied

in

the

LP

solution),

so

it

follows

we

can

extract

the

maximum

payment

from

all

the

bidders

bj,

for

a

total

payment

of

c.

It

follows

that

we

can

extract

the

required

payment

from

the

bidders,

and

there

exists

a

solution

to

the

DONATION-CLEARING

instance

with

a

total

amount

donated

of

at

least

Now

suppose

there

is

a

solution

to

the

DONATIONCLEARING

instance

with

a

total

amount

donated

of

at

least

vH.

Then

the

maximum

payment

must

be

extracted

from

each

bidder.

From

the

fact

that

the

maximum

payment

must

be

extracted

from

each

bidder

bxi

it

follows

that

for

each

i,

πc+xi

πc−xi

Because

the

maximum

extractable

total

payment

is

it

follows

that

for

each

i,

πc+xi

πc−xi

Let

x∗

i

πc+xi

H

H

πc−xi

Then,

from

the

fact

that

the

maximum

payment

must

be

extracted

from

each

bidder

bj,

it

follows

that

i

|rj

i

|)H

sj

i:r

j

i

rj

i

πc−xi

i:r

j

i

−rj

i

πc+xi

i:r

j

i

rj

i

(H

x∗

i

i:r

j

i

−rj

i

(H

x∗

i

i

|rj

i

|)H

i

rj

i

x∗

i

Equivalently,

i

rj

i

x∗

i

sj.

It

follows

that

the

x∗

i

constitute

a

feasible

solution

to

the

LP.

QUASILINEAR

BIDS

Another

class

of

bids

of

interest

is

the

class

of

quasilinear

bids.

In

a

quasilinear

bid,

the

bidder"s

payment

willingness

function

is

linear

in

utility:

that

is,

wj

uj.

(Because

the

units

of

utility

are

arbitrary,

we

may

as

well

let

them

correspond

exactly

to

units

of

money-so

we

do

not

need

a

constant

multiplier.)

In

most

cases,

quasilinearity

is

an

unreasonable

assumption:

for

example,

usually

bidders

have

a

limited

budget

for

donations,

so

that

the

payment

willingness

will

stop

increasing

in

utility

after

some

point

(or

at

least

increase

slower

in

the

case

of

a

softer

budget

constraint).

Nevertheless,

quasilinearity

may

be

a

reasonable

assumption

in

the

case

where

the

bidders

are

large

organizations

with

large

budgets,

and

the

charities

are

a

few

small

projects

requiring

relatively

little

money.

In

this

setting,

once

a

certain

small

amount

has

been

donated

to

a

charity,

a

bidder

will

derive

no

more

utility

from

more

money

being

donated

from

that

charity.

Thus,

the

bidders

will

never

reach

a

high

enough

utility

for

their

budget

constraint

(even

when

it

is

soft)

to

take

effect,

and

thus

a

linear

approximation

of

their

payment

willingness

function

is

reasonable.

Another

reason

for

studying

the

quasilinear

setting

is

that

it

is

the

easiest

setting

for

mechanism

design,

which

we

will

discuss

shortly.

In

this

section,

we

will

see

that

the

clearing

problem

is

much

easier

in

the

case

of

quasilinear

bids.

First,

we

address

the

case

where

we

are

trying

to

maximize

surplus

(which

is

the

most

natural

setting

for

mechanism

design).

The

key

observation

here

is

that

when

bids

are

quasilinear,

the

clearing

problem

decomposes

across

charities.

Lemma

Suppose

all

bids

are

quasilinear,

and

surplus

is

the

objective.

Then

we

can

clear

the

market

optimally

by

clearing

the

market

for

each

charity

individually.

That

is,

for

each

bidder

bj,

let

πbj

ci

πbi

j

Then,

for

each

charity

ci,

maximize

bj

πbi

j

πci

under

the

constraint

that

for

every

bidder

bj,

πbi

j

ui

j(πci

Proof.

The

resulting

solution

is

certainly

valid:

first

of

all,

at

least

as

much

money

is

collected

as

is

given

away,

because

bj

πbj

ci

πci

bj

ci

πbi

j

ci

πci

ci

bj

πbi

j

πci

)-and

the

terms

of

this

summation

are

the

objectives

of

the

individual

optimization

problems,

each

of

which

can

be

set

at

least

to

(by

setting

all

the

variables

are

set

to

so

it

follows

that

the

expression

is

nonnegative.

Second,

no

bidder

bj

pays

more

than

she

is

willing

to,

because

uj

−πbj

ci

ui

j(πci

ci

πbi

j

ci

(ui

j(πci

)−πbi

j

)-and

the

terms

of

this

summation

are

nonnegative

by

the

constraints

we

imposed

on

the

individual

optimization

problems.

All

that

remains

to

show

is

that

the

solution

is

optimal.

Because

in

an

optimal

solution,

we

will

extract

as

much

payment

from

the

bidders

as

possible

given

the

πci

all

we

need

to

show

is

that

the

πci

are

set

optimally

by

this

approach.

Let

π∗

ci

be

the

amount

paid

to

charity

πci

in

some

optimal

solution.

If

we

change

this

amount

to

πci

and

leave

everything

else

unchanged,

this

will

only

affect

the

payment

that

we

can

extract

from

the

bidders

because

of

this

particular

charity,

and

the

difference

in

surplus

will

be

bj

ui

j(πci

ui

j(π∗

ci

πci

π∗

ci

This

expression

is,

of

course,

if

πci

π∗

ci

But

now

notice

that

this

expression

is

maximized

as

a

function

of

πci

by

the

decomposed

solution

for

this

charity

(the

terms

without

πci

in

them

do

not

matter,

and

of

course

in

the

decomposed

solution

we

always

set

πbi

j

ui

j(πci

It

follows

that

if

we

change

πci

to

the

decomposed

solution,

the

change

in

surplus

will

be

at

least

(and

the

solution

will

still

be

valid).

Thus,

we

can

change

the

πci

one

by

one

to

the

decomposed

solution

without

ever

losing

any

surplus.

Theorem

When

all

bids

are

quasilinear

and

surplus

is

the

objective,

DONATION-CLEARING

can

be

done

in

linear

time.

Proof.

By

Lemma

we

can

solve

the

problem

separately

for

each

charity.

For

charity

ci,

this

amounts

to

maximizing

bj

ui

j(πci

πci

as

a

function

of

πci

Because

all

its

terms

are

piecewise

linear

functions,

this

whole

function

is

piecewise

linear,

and

must

be

maximized

at

one

of

the

points

where

it

is

nondifferentiable.

It

follows

that

we

need

only

check

all

the

points

at

which

one

of

the

terms

is

nondifferentiable.

Unfortunately,

the

decomposing

lemma

does

not

hold

for

payment

maximization.

Proposition

When

the

objective

is

payment

maximization,

even

when

bids

are

quasilinear,

the

solution

obtained

by

decomposing

the

problem

across

charities

is

in

general

not

optimal

(even

with

concave

bids).

Proof.

Consider

a

single

bidder

placing

the

following

quasilinear

bid

over

two

charities

and

for

πci

πci

−1

otherwise;

πci

The

decomposed

solution

is

for

a

total

donation

of

But

the

solution

is

also

valid,

for

a

total

donation

of

In

fact,

when

payment

maximization

is

the

objective,

DONATION-CLEARING

remains

(weakly)

NP-complete

in

general.

(In

the

remainder

of

the

paper,

proofs

are

omitted

because

of

space

constraint.)

Theorem

DONATION-CLEARING

is

(weakly)

NPcomplete

when

payment

maximization

is

the

objective,

even

when

every

bid

is

concerns

only

one

charity

(and

has

a

stepfunction

utility

function

for

this

charity),

and

is

quasilinear.

However,

when

the

bids

are

also

concave,

a

simple

greedy

clearing

algorithm

is

optimal.

Theorem

Given

a

DONATION-CLEARING

instance

with

payment

maximization

as

the

objective

where

all

bids

are

quasilinear

and

concave,

consider

the

following

algorithm.

Start

with

πci

for

all

charities.

Then,

letting

γci

d

bj

ui

j

(πci

dπci

(at

nondifferentiable

points,

these

derivatives

should

be

taken

from

the

right),

increase

πc∗

i

(where

c∗

i

arg

maxci

γci

until

either

γc∗

i

is

no

longer

the

highest

(in

which

case,

recompute

c∗

i

and

start

increasing

the

corresponding

payment),

or

bj

uj

ci

πci

and

γc∗

i

Finally,

let

πbj

uj.

(A

similar

greedy

algorithm

works

when

the

objective

is

surplus

and

the

bids

are

quasilinear

and

concave,

with

as

only

difference

that

we

stop

increasing

the

payments

as

soon

as

γc∗

i

INCENTIVE

COMPATIBILITY

Up

to

this

point,

we

have

not

discussed

the

bidders"

incentives

for

bidding

any

particular

way.

Specifically,

the

bids

may

not

truthfully

reflect

the

bidders"

preferences

over

charities

because

a

bidder

may

bid

strategically,

misrepresenting

her

preferences

in

order

to

obtain

a

result

that

is

better

to

herself.

This

means

the

mechanism

is

not

strategy-proof.

(We

will

show

some

concrete

examples

of

this

shortly.)

This

is

not

too

surprising,

because

the

mechanism

described

so

far

is,

in

a

sense,

a

first-price

mechanism,

where

the

mechanism

will

extract

as

much

payment

from

a

bidder

as

her

bid

allows.

Such

mechanisms

(for

example,

first-price

auctions,

where

winners

pay

the

value

of

their

bids)

are

typically

not

strategy-proof:

if

a

bidder

reports

her

true

valuation

for

an

outcome,

then

if

this

outcome

occurs,

the

payment

the

bidder

will

have

to

make

will

offset

her

gains

from

the

outcome

completely.

Of

course,

we

could

try

to

change

the

rules

of

the

game-which

outcome

(payment

vector

to

charities)

do

we

select

for

which

bid

vector,

and

which

bidder

pays

how

much-in

order

to

make

bidding

truthfully

beneficial,

and

to

make

the

outcome

better

with

regard

to

the

bidders"

true

preferences.

This

is

the

field

of

mechanism

design.

In

this

section,

we

will

briefly

discuss

the

options

that

mechanism

design

provides

for

the

expressive

charity

donation

problem.

Strategic

bids

under

the

first-price

mechanism

We

first

point

out

some

reasons

for

bidders

to

misreport

their

preferences

under

the

first-price

mechanism

described

in

the

paper

up

to

this

point.

First

of

all,

even

when

there

is

only

one

charity,

it

may

make

sense

to

underbid

one"s

true

valuation

for

the

charity.

For

example,

suppose

a

bidder

would

like

a

charity

to

receive

a

certain

amount

x,

but

does

not

care

if

the

charity

receives

more

than

that.

Additionally,

suppose

that

the

other

bids

guarantee

that

the

charity

will

receive

at

least

x

no

matter

what

bid

the

bidder

submits

(and

the

bidder

knows

this).

Then

the

bidder

is

best

off

not

bidding

at

all

(or

submitting

a

utility

for

the

charity

of

to

avoid

having

to

make

any

payment.

(This

is

known

in

economics

as

the

free

rider

problem

With

multiple

charities,

another

kind

of

manipulation

may

occur,

where

the

bidder

attempts

to

steer

others"

payments

towards

her

preferred

charity.

Suppose

that

there

are

two

charities,

and

three

bidders.

The

first

bidder

bids

if

otherwise;

if

otherwise;

and

if

otherwise.

The

second

bidder

bids

if

otherwise;

(always);

if

otherwise.

Now,

the

third

bidder"s

true

preferences

are

accurately

by

the

bid

if

otherwise;

if

otherwise;

and

if

otherwise.

Now,

it

is

straightforward

to

check

that,

if

the

third

bidder

bids

truthfully,

regardless

of

whether

the

objective

is

surplus

maximization

or

total

donated,

charity

will

receive

at

least

and

charity

will

receive

less

than

The

same

is

true

if

bidder

does

not

place

a

bid

at

all

(as

in

the

previous

type

of

manipulation);

hence

bidder

utility

will

be

in

this

case.

But

now,

if

bidder

reports

everywhere;

if

otherwise

(this

part

of

the

bid

is

truthful);

and

if

otherwise;

then

charity

will

receive

at

least

and

bidder

will

have

to

pay

at

most

Because

up

to

this

amount

of

payment,

one

unit

of

money

corresponds

to

three

units

of

utility

to

bidder

it

follows

his

utility

is

now

at

least

We

observe

that

in

this

case,

the

strategic

bidder

is

not

only

affecting

how

much

the

bidders

pay,

but

also

how

much

the

charities

receive.

Mechanism

design

in

the

quasilinear

setting

There

are

four

reasons

why

the

mechanism

design

approach

is

likely

to

be

most

successful

in

the

setting

of

quasilinear

preferences.

First,

historically,

mechanism

design

has

been

been

most

successful

when

the

quasilinear

assumption

could

be

made.

Second,

because

of

this

success,

some

very

general

mechanisms

have

been

discovered

for

the

quasilinear

setting

(for

instance,

the

VCG

mechanisms

or

the

dAGVA

mechanism

which

we

could

apply

directly

to

the

expressive

charity

donation

problem.

Third,

as

we

saw

in

Section

the

clearing

problem

is

much

easier

in

Formally,

this

means

that

if

the

bidder

is

forced

to

pay

the

full

amount

that

his

bid

allows

for

a

particular

vector

of

payments

to

charities,

the

bidder

is

indifferent

between

this

and

not

participating

in

the

mechanism

at

all.

(Compare

this

to

bidding

truthfully

in

a

first-price

auction.)

this

setting,

and

thus

we

are

less

likely

to

run

into

computational

trouble

for

the

mechanism

design

problem.

Fourth,

as

we

will

show

shortly,

the

quasilinearity

assumption

in

some

cases

allows

for

decomposing

the

mechanism

design

problem

over

the

charities

(as

it

did

for

the

simple

clearing

problem).

Moreover,

in

the

quasilinear

setting

(unlike

in

the

general

setting),

it

makes

sense

to

pursue

social

welfare

(the

sum

of

the

utilities)

as

the

objective,

because

now

units

of

utility

correspond

directly

to

units

of

money,

so

that

we

do

not

have

the

problem

of

the

bidders

arbitrarily

scaling

their

utilities;

and

it

is

no

longer

possible

to

give

a

payment

willingness

function

of

while

still

affecting

the

donations

through

a

utility

function.

Before

presenting

the

decomposition

result,

we

introduce

some

terms

from

game

theory.

A

type

is

a

preference

profile

that

a

bidder

can

have

and

can

report

(thus,

a

type

report

is

a

bid).

Incentive

compatibility

(IC)

means

that

bidders

are

best

off

reporting

their

preferences

truthfully;

either

regardless

of

the

others"

types

(in

dominant

strategies),

or

in

expectation

over

them

(in

Bayes-Nash

equilibrium).

Individual

rationality

(IR)

means

agents

are

at

least

as

well

off

participating

in

the

mechanism

as

not

participating;

either

regardless

of

the

others"

types

(ex-post),

or

in

expectation

over

them

(ex-interim).

A

mechanism

is

budget

balanced

if

there

is

no

flow

of

money

into

or

out

of

the

system-in

general

(ex-post),

or

in

expectation

over

the

type

reports

(ex-ante).

A

mechanism

is

efficient

if

it

(always)

produces

the

efficient

allocation

of

wealth

to

charities.

Theorem

Suppose

all

agents"

preferences

are

quasilinear.

Furthermore,

suppose

that

there

exists

a

single-charity

mechanism

M

that,

for

a

certain

subclass

P

of

(quasilinear)

preferences,

under

a

given

solution

concept

S

(implementation

in

dominant

strategies

or

Bayes-Nash

equilibrium)

and

a

given

notion

of

individual

rationality

R

(ex

post,

ex

interim,

or

none),

satisfies

a

certain

notion

of

budget

balance

(ex

post,

ex

ante,

or

none),

and

is

ex-post

efficient.

Then

there

exists

such

a

mechanism

for

any

number

of

charities.

Two

mechanisms

that

satisfy

efficiency

(and

can

in

fact

be

applied

directly

to

the

multiple-charity

problem

without

use

of

the

previous

theorem)

are

the

VCG

(which

is

incentive

compatible

in

dominant

strategies)

and

dAGVA

(which

is

incentive

compatible

only

in

Bayes-Nash

equilibrium)

mechanisms.

Each

of

them,

however,

has

a

drawback

that

would

probably

make

it

impractical

in

the

setting

of

donations

to

charities.

The

VCG

mechanism

is

not

budget

balanced.

The

dAGVA

mechanism

does

not

satisfy

ex-post

individual

rationality.

In

the

next

subsection,

we

will

investigate

if

we

can

do

better

in

the

setting

of

donations

to

charities.

Impossibility

of

efficiency

In

this

subsection,

we

show

that

even

in

a

very

restricted

setting,

and

with

minimal

requirements

on

IC

and

IR

constraints,

it

is

impossible

to

create

a

mechanism

that

is

efficient.

Theorem

There

is

no

mechanism

which

is

ex-post

budget

balanced,

ex-post

efficient,

and

ex-interim

individually

rational

with

Bayes-Nash

equilibrium

as

the

solution

concept

(even

with

only

one

charity,

only

two

quasilinear

bidders,

with

identical

type

distributions

(uniform

over

two

types,

with

either

both

utility

functions

being

step

functions

or

both

utility

functions

being

concave

piecewise

linear

functions)).

The

case

of

step-functions

in

this

theorem

corresponds

exactly

to

the

case

of

a

single,

fixed-size,

nonexcludable

public

good

(the

public

good

being

that

the

charity

receives

the

desired

amount)-for

which

such

an

impossibility

result

is

already

known

Many

similar

results

are

known,

probably

the

most

famous

of

which

is

the

Myerson-Satterthwaite

impossibility

result,

which

proves

the

impossibility

of

efficient

bilateral

trade

under

the

same

requirements

Theorem

indicates

that

there

is

no

reason

to

decide

on

donations

to

multiple

charities

under

a

single

mechanism

(rather

than

a

separate

one

for

each

charity),

when

an

efficient

mechanism

with

the

desired

properties

exists

for

the

single-charity

case.

However,

because

under

the

requirements

of

Theorem

no

such

mechanism

exists,

there

may

be

a

benefit

to

bringing

the

charities

under

the

same

umbrella.

The

next

proposition

shows

that

this

is

indeed

the

case.

Proposition

There

exist

settings

with

two

charities

where

there

exists

no

ex-post

budget

balanced,

ex-post

efficient,

and

ex-interim

individually

rational

mechanism

with

Bayes-Nash

equilibrium

as

the

solution

concept

for

either

charity

alone;

but

there

exists

an

ex-post

budget

balanced,

ex-post

efficient,

and

ex-post

individually

rational

mechanism

with

dominant

strategies

as

the

solution

concept

for

both

charities

together.

(Even

when

the

conditions

are

the

same

as

in

Theorem

apart

from

the

fact

that

there

are

now

two

charities.)

CONCLUSION

We

introduced

a

bidding

language

for

expressing

very

general

types

of

matching

offers

over

multiple

charities.

We

formulated

the

corresponding

clearing

problem

(deciding

how

much

each

bidder

pays,

and

how

much

each

charity

receives),

and

showed

that

it

is

NP-complete

to

approximate

to

any

ratio

even

in

very

restricted

settings.

We

gave

a

mixed-integer

program

formulation

of

the

clearing

problem,

and

showed

that

for

concave

bids

(where

utility

functions

and

payment

willingness

function

are

concave),

the

program

reduces

to

a

linear

program

and

can

hence

be

solved

in

polynomial

time.

We

then

showed

that

the

clearing

problem

for

a

subclass

of

concave

bids

is

at

least

as

hard

as

the

decision

variant

of

linear

programming,

suggesting

that

we

cannot

do

much

better

than

a

linear

programming

implementation

for

such

bids.

Subsequently,

we

showed

that

the

clearing

problem

is

much

easier

when

bids

are

quasilinear

(where

payment

willingness

functions

are

linear)-for

surplus,

the

problem

decomposes

across

charities,

and

for

payment

maximization,

a

greedy

approach

is

optimal

if

the

bids

are

concave

(although

this

latter

problem

is

weakly

NP-complete

when

the

bids

are

not

concave).

For

the

quasilinear

setting,

we

studied

the

mechanism

design

question

of

making

the

bidders

report

their

preferences

truthfully

rather

than

strategically.

We

showed

that

an

ex-post

efficient

mechanism

is

impossible

even

with

only

one

charity

and

a

very

restricted

class

of

bids.

We

also

showed

that

even

though

the

clearing

problem

decomposes

over

charities

in

the

quasilinear

setting,

there

may

be

benefits

to

linking

the

charities

from

a

mechanism

design

standpoint.

There

are

many

directions

for

future

research.

One

is

to

build

a

web-based

implementation

of

the

(first-price)

mechanism

proposed

in

this

paper.

Another

is

to

study

the

computational

scalability

of

our

MIP/LP

approach.

It

is

also

important

to

identify

other

classes

of

bids

(besides

concave

ones)

for

which

the

clearing

problem

is

tractable.

Much

crucial

work

remains

to

be

done

on

the

mechanism

design

problem.

Finally,

are

there

good

iterative

mechanisms

for

charity

REFERENCES

K.

Arrow.

The

property

rights

doctrine

and

demand

revelation

under

incomplete

information.

In

M.

Boskin,

editor,

Economics

and

human

welfare.

New

York

Academic

Press,

L.

M.

Ausubel

and

P.

Milgrom.

Ascending

auctions

with

package

bidding.

Frontiers

of

Theoretical

Economics,

No.

Article

Y.

Bartal,

R.

Gonen,

and

N.

Nisan.

Incentive

compatible

multi-unit

combinatorial

auctions.

In

Theoretical

Aspects

of

Rationality

and

Knowledge

(TARK

IX),

Bloomington,

Indiana,

USA,

E.

H.

Clarke.

Multipart

pricing

of

public

goods.

Public

Choice,

V.

Conitzer

and

T.

Sandholm.

Complexity

of

mechanism

design.

In

Proceedings

of

the

Annual

Conference

on

Uncertainty

in

Artificial

Intelligence

pages

Edmonton,

Canada,

C.

d"Aspremont

and

L.

A.

G´erard-Varet.

Incentives

and

incomplete

information.

Journal

of

Public

Economics,

M.

R.

Garey,

D.

S.

Johnson,

and

L.

Stockmeyer.

Some

simplified

NP-complete

graph

problems.

Theoretical

Computer

Science,

D.

Goldburg

and

S.

McElligott.

Red

cross

statement

on

official

donation

locations.

Press

release,

pr/

R.

Gonen

and

D.

Lehmann.

Optimal

solutions

for

multi-unit

combinatorial

auctions:

Branch

and

bound

heuristics.

In

Proceedings

of

the

ACM

Conference

on

Electronic

Commerce

(ACM-EC),

pages

Minneapolis,

MN,

Oct.

T.

Groves.

Incentives

in

teams.

Econometrica,

L.

Khachiyan.

A

polynomial

algorithm

in

linear

programming.

Soviet

Math.

Doklady,

R.

Lavi,

A.

Mu"Alem,

and

N.

Nisan.

Towards

a

characterization

of

truthful

combinatorial

auctions.

In

Proceedings

of

the

Annual

Symposium

on

Foundations

of

Computer

Science

(FOCS),

D.

Lehmann,

L.

I.

O"Callaghan,

and

Y.

Shoham.

Truth

revelation

in

rapid,

approximately

efficient

combinatorial

auctions.

Journal

of

the

ACM,

Early

version

appeared

in

Compare,

for

example,

iterative

mechanisms

in

the

combinatorial

auction

setting

A.

Mas-Colell,

M.

Whinston,

and

J.

R.

Green.

Microeconomic

Theory.

Oxford

University

Press,

R.

Myerson

and

M.

Satterthwaite.

Efficient

mechanisms

for

bilateral

trading.

Journal

of

Economic

Theory,

G.

L.

Nemhauser

and

L.

A.

Wolsey.

Integer

and

Combinatorial

Optimization.

John

Wiley

Sons,

Section

page

N.

Nisan.

Bidding

and

allocation

in

combinatorial

auctions.

In

Proceedings

of

the

ACM

Conference

on

Electronic

Commerce

(ACM-EC),

pages

Minneapolis,

MN,

N.

Nisan

and

A.

Ronen.

Computationally

feasible

VCG

mechanisms.

In

Proceedings

of

the

ACM

Conference

on

Electronic

Commerce

(ACM-EC),

pages

Minneapolis,

MN,

D.

C.

Parkes.

iBundle:

An

efficient

ascending

price

bundle

auction.

In

Proceedings

of

the

ACM

Conference

on

Electronic

Commerce

(ACM-EC),

pages

Denver,

CO,

Nov.

M.

H.

Rothkopf,

A.

Pekeˇc,

and

R.

M.

Harstad.

Computationally

manageable

combinatorial

auctions.

Management

Science,

T.

Sandholm.

Algorithm

for

optimal

winner

determination

in

combinatorial

auctions.

Artificial

Intelligence,

Jan.

Conference

version

appeared

at

the

International

Joint

Conference

on

Artificial

Intelligence

(IJCAI),

pp.

Stockholm,

Sweden,

T.

Sandholm,

S.

Suri,

A.

Gilpin,

and

D.

Levine.

CABOB:

A

fast

optimal

algorithm

for

combinatorial

auctions.

In

Proceedings

of

the

Seventeenth

International

Joint

Conference

on

Artificial

Intelligence

(IJCAI),

pages

Seattle,

WA,

J.

Tagliabue.

Global

AIDS

Funds

Is

Given

Attention,

but

Not

Money.

The

New

York

Times,

June

Reprinted

on

NYT

HGAP

fund.html.

W.

Vickrey.

Counterspeculation,

auctions,

and

competitive

sealed

tenders.

Journal

of

Finance,

P.

R.

Wurman

and

M.

P.

Wellman.

AkBA:

A

progressive,

anonymous-price

combinatorial

auction.

In

Proceedings

of

the

ACM

Conference

on

Electronic

Commerce

(ACM-EC),

pages

Minneapolis,

MN,

Oct.

M.

Yokoo.

The

characterization

of

strategy/false-name

proof

combinatorial

auction

protocols:

Price-oriented,

rationing-free

protocol.

In

Proceedings

of

the

Eighteenth

International

Joint

Conference

on

Artificial

Intelligence

(IJCAI),

Acapulco,

Mexico,

Aug.

supporter of charity
charity supporter
donation-clearing
concave bid
donation to charity
market clear

negotiating material
expressive charity donation
combinatorial auction
economic efficiency
bidding framework
bidding language
threshold bid
payment willingness function
linear programming
quasilinearity
incentive compatibility
expressive negotiation
mechanism design