Weak

Monotonicity

Suffices

for

Truthfulness

on

Convex

Domains

Michael

Saks

Dept.

of

Mathematics

Rutgers

University

Frelinghuysen

Road

Piscataway,

NJ,

saks@math.rutgers.edu

Lan

Yu

Dept.

of

Computer

Science

Rutgers

University

Frelinghuysen

Road

Piscataway,

NJ,

lanyu@paul.rutgers.edu

ABSTRACT

Weak

monotonicity

is

a

simple

necessary

condition

for

a

social

choice

function

to

be

implementable

by

a

truthful

mechanism.

Roberts

showed

that

it

is

sufficient

for

all

social

choice

functions

whose

domain

is

unrestricted.

Lavi,

Mu"alem

and

Nisan

proved

the

sufficiency

of

weak

monotonicity

for

functions

over

order-based

domains

and

Gui,

Muller

and

Vohra

proved

sufficiency

for

order-based

domains

with

range

constraints

and

for

domains

defined

by

other

special

types

of

linear

inequality

constraints.

Here

we

show

the

more

general

result,

conjectured

by

Lavi,

Mu"alem

and

Nisan

that

weak

monotonicity

is

sufficient

for

functions

defined

on

any

convex

domain.

Categories

and

Subject

Descriptors

[Social

and

Behavioral

Sciences]:

Economics;

[Computers

and

Society]:

Electronic

Commerce-payment

schemes

General

Terms

Theory,

Economics

INTRODUCTION

Social

choice

theory

centers

around

the

general

problem

of

selecting

a

single

outcome

out

of

a

set

A

of

alternative

outcomes

based

on

the

individual

preferences

of

a

set

P

of

players.

A

method

for

aggregating

player

preferences

to

select

one

outcome

is

called

a

social

choice

function.

In

this

paper

we

assume

that

the

range

A

is

finite

and

that

each

player"s

preference

is

expressed

by

a

valuation

function

which

assigns

to

each

possible

outcome

a

real

number

representing

the

benefit

the

player

derives

from

that

outcome.

The

ensemble

of

player

valuation

functions

is

viewed

as

a

valuation

matrix

with

rows

indexed

by

players

and

columns

by

outcomes.

A

major

difficulty

connected

with

social

choice

functions

is

that

players

can

not

be

required

to

tell

the

truth

about

their

preferences.

Since

each

player

seeks

to

maximize

his

own

benefit,

he

may

find

it

in

his

interest

to

misrepresent

his

valuation

function.

An

important

approach

for

dealing

with

this

problem

is

to

augment

a

given

social

choice

function

with

a

payment

function,

which

assigns

to

each

player

a

(positive

or

negative)

payment

as

a

function

of

all

of

the

individual

preferences.

By

carefully

choosing

the

payment

function,

one

can

hope

to

entice

each

player

to

tell

the

truth.

A

social

choice

function

augmented

with

a

payment

function

is

called

a

mechanism

and

the

mechanism

is

said

to

implement

the

social

choice

function.

A

mechanism

is

truthful

(or

to

be

strategyproof

or

to

have

a

dominant

strategy)

if

each

player"s

best

strategy,

knowing

the

preferences

of

the

others,

is

always

to

declare

his

own

true

preferences.

A

social

choice

function

is

truthfully

implementable,

or

truthful

if

it

has

a

truthful

implementation.

(The

property

of

truthful

implementability

is

sometimes

called

dominant

strategy

incentive

compatibility).

This

framework

leads

naturally

to

the

question:

which

social

choice

functions

are

truthful?

This

question

is

of

the

following

general

type:

given

a

class

of

functions

(here,

social

choice

functions)

and

a

property

that

holds

for

some

of

them

(here,

truthfulness),

characterize

the

property.

The

definition

of

the

property

itself

provides

a

characterization,

so

what

more

is

needed?

Here

are

some

useful

notions

of

characterization:

Recognition

algorithm.

Give

an

algorithm

which,

given

an

appropriate

representation

of

a

function

in

the

class,

determines

whether

the

function

has

the

property.

Parametric

representation.

Give

an

explicit

parametrized

family

of

functions

and

show

that

each

function

in

the

The

usual

definition

of

mechanism

is

more

general

than

this

(see

Chapter

or

the

mechanisms

we

consider

here

are

usually

called

direct

revelation

mechanisms.

family

has

the

property,

and

that

every

function

with

the

property

is

in

the

family.

A

third

notion

applies

in

the

case

of

hereditary

properties

of

functions.

A

function

g

is

a

subfunction

of

function

f,

or

f

contains

g,

if

g

is

obtained

by

restricting

the

domain

of

f.

A

property

P

of

functions

is

hereditary

if

it

is

preserved

under

taking

subfunctions.

Truthfulness

is

easily

seen

to

be

hereditary.

Sets

of

obstructions.

For

a

hereditary

property

P,

a

function

g

that

does

not

have

the

property

is

an

obstruction

to

the

property

in

the

sense

that

any

function

containing

g

doesn"t

have

the

property.

An

obstruction

is

minimal

if

every

proper

subfunction

has

the

property.

A

set

of

obstructions

is

complete

if

every

function

that

does

not

have

the

property

contains

one

of

them

as

a

subfunction.

The

set

of

all

functions

that

don"t

satisfy

P

is

a

complete

(but

trivial

and

uninteresting)

set

of

obstructions;

one

seeks

a

set

of

small

(ideally,

minimal)

obstructions.

We

are

not

aware

of

any

work

on

recognition

algorithms

for

the

property

of

truthfulness,

but

there

are

significant

results

concerning

parametric

representations

and

obstruction

characterizations

of

truthfulness.

It

turns

out

that

the

domain

of

the

function,

i.e.,

the

set

of

allowed

valuation

matrices,

is

crucial.

For

functions

with

unrestricted

domain,

i.e.,

whose

domain

is

the

set

of

all

real

matrices,

there

are

very

good

characterizations

of

truthfulness.

For

general

domains,

however,

the

picture

is

far

from

complete.

Typically,

the

domains

of

social

choice

functions

are

specified

by

a

system

of

constraints.

For

example,

an

order

constraint

requires

that

one

specified

entry

in

some

row

be

larger

than

another

in

the

same

row,

a

range

constraint

places

an

upper

or

lower

bound

on

an

entry,

and

a

zero

constraint

forces

an

entry

to

be

These

are

all

examples

of

linear

inequality

constraints

on

the

matrix

entries.

Building

on

work

of

Roberts

Lavi,

Mu"alem

and

Nisan

defined

a

condition

called

weak

monotonicity

(WMON).

(Independently,

in

the

context

of

multi-unit

auctions,

Bikhchandani,

Chatterji

and

Sen

identified

the

same

condition

and

called

it

nondecreasing

in

marginal

utilities

(NDMU).)

The

definition

of

W-MON

can

be

formulated

in

terms

of

obstructions:

for

some

specified

simple

set

F

of

functions

each

having

domains

of

size

a

function

satisfies

W-MON

if

it

contains

no

function

from

F.

The

functions

in

F

are

not

truthful,

and

therefore

W-MON

is

a

necessary

condition

for

truthfulness.

Lavi,

Mu"alem

and

Nisan

showed

that

W-MON

is

also

sufficient

for

truthfulness

for

social

choice

functions

whose

domain

is

order-based,

i.e.,

defined

by

order

constraints

and

zero

constraints,

and

Gui,

Muller

and

Vohra

extended

this

to

other

domains.

The

domain

constraints

considered

in

both

papers

are

special

cases

of

linear

inequality

constraints,

and

it

is

natural

to

ask

whether

W-MON

is

sufficient

for

any

domain

defined

by

such

constraints.

Lavi,

Mu"alem

and

Nisan

conjectured

that

W-MON

suffices

for

convex

domains.

The

main

result

of

this

paper

is

an

affirmative

answer

to

this

conjecture:

Theorem

For

any

social

choice

function

having

convex

domain

and

finite

range,

weak

monotonicity

is

necessary

and

sufficient

for

truthfulness.

Using

the

interpretation

of

weak

monotonicity

in

terms

of

obstructions

each

having

domain

size

this

provides

a

complete

set

of

minimal

obstructions

for

truthfulness

within

the

class

of

social

choice

functions

with

convex

domains.

The

two

hypotheses

on

the

social

choice

function,

that

the

domain

is

convex

and

that

the

range

is

finite,

can

not

be

omitted

as

is

shown

by

the

examples

given

in

section

Related

Work

There

is

a

simple

and

natural

parametrized

set

of

truthful

social

choice

functions

called

affine

maximizers.

Roberts

showed

that

for

functions

with

unrestricted

domain,

every

truthful

function

is

an

affine

maximizer,

thus

providing

a

parametrized

representation

for

truthful

functions

with

unrestricted

domain.

There

are

many

known

examples

of

truthful

functions

over

restricted

domains

that

are

not

affine

maximizers

(see

and

Each

of

these

examples

has

a

special

structure

and

it

seems

plausible

that

there

might

be

some

mild

restrictions

on

the

class

of

all

social

choice

functions

such

that

all

truthful

functions

obeying

these

restrictions

are

affine

maximizers.

Lavi,

Mu"alem

and

Nisan

obtained

a

result

in

this

direction

by

showing

that

for

order-based

domains,

under

certain

technical

assumptions,

every

truthful

social

choice

function

is

almost

an

affine

maximizer.

There

are

a

number

of

results

about

truthfulness

that

can

be

viewed

as

providing

obstruction

characterizations,

although

the

notion

of

obstruction

is

not

explicitly

discussed.

For

a

player

i,

a

set

of

valuation

matrices

is

said

to

be

i-local

if

all

of

the

matrices

in

the

set

are

identical

except

for

row

i.

Call

a

social

choice

function

i-local

if

its

domain

is

ilocal

and

call

it

local

if

it

is

i-local

for

some

i.

The

following

easily

proved

fact

is

used

extensively

in

the

literature:

Proposition

The

social

choice

function

f

is

truthful

if

and

only

if

every

local

subfunction

of

f

is

truthful.

This

implies

that

the

set

of

all

local

non-truthful

functions

comprises

a

complete

set

of

obstructions

for

truthfulness.

This

set

is

much

smaller

than

the

set

of

all

non-truthful

functions,

but

is

still

far

from

a

minimal

set

of

obstructions.

Rochet

Rozenshtrom

and

Gui,

Muller

and

Vohra

identified

a

necessary

and

sufficient

condition

for

truthfulness

(see

lemma

below)

called

the

nonnegative

cycle

property.

This

condition

can

be

viewed

as

providing

a

minimal

complete

set

of

non-truthful

functions.

As

is

required

by

proposition

each

function

in

the

set

is

local.

Furthermore

it

is

one-to-one.

In

particular

its

domain

has

size

at

most

the

number

of

possible

outcomes

|A|.

As

this

complete

set

of

obstructions

consists

of

minimal

non-truthful

functions,

this

provides

the

optimal

obstruction

characterization

of

non-truthful

functions

within

the

class

of

all

social

choice

functions.

But

by

restricting

attention

to

interesting

subclasses

of

social

choice

functions,

one

may

hope

to

get

simpler

sets

of

obstructions

for

truthfulness

within

that

class.

The

condition

of

weak

monotonicity

mentioned

earlier

can

be

defined

by

a

set

of

obstructions,

each

of

which

is

a

local

function

of

domain

size

exactly

Thus

the

results

of

Lavi,

Mu"alem

and

Nisan

and

of

Gui,

Muller

and

Vohra

give

a

very

simple

set

of

obstructions

for

truthfulness

within

certain

subclasses

of

social

choice

functions.

Theorem

extends

these

results

to

a

much

larger

subclass

of

functions.

Weak

Monotonicity

and

the

Nonnegative

Cycle

Property

By

proposition

a

function

is

truthful

if

and

only

if

each

of

its

local

subfunctions

is

truthful.

Therefore,

to

get

a

set

of

obstructions

for

truthfulness,

it

suffices

to

obtain

such

a

set

for

local

functions.

The

domain

of

an

i-local

function

consists

of

matrices

that

are

fixed

on

all

rows

but

row

i.

Fix

such

a

function

f

and

let

D

RA

be

the

set

of

allowed

choices

for

row

i.

Since

f

depends

only

on

row

i

and

row

i

is

chosen

from

D,

we

can

view

f

as

a

function

from

D

to

A.

Therefore,

f

is

a

social

choice

function

having

one

player;

we

refer

to

such

a

function

as

a

single

player

function.

Associated

to

any

single

player

function

f

with

domain

D

we

define

an

edge-weighted

directed

graph

Hf

whose

vertex

set

is

the

image

of

f.

For

convenience,

we

assume

that

f

is

surjective

and

so

this

image

is

A.

For

each

a,

b

A,

x

(a)

there

is

an

edge

ex(a,

b)

from

a

to

b

with

weight

x(a)

x(b).

The

weight

of

a

set

of

edges

is

just

the

sum

of

the

weights

of

the

edges.

We

say

that

f

satisfies:

the

nonnegative

cycle

property

if

every

directed

cycle

has

nonnegative

weight.

the

nonnegative

two-cycle

property

if

every

directed

cycle

between

two

vertices

has

nonnegative

weight.

We

say

a

local

function

g

satisfies

nonnegative

cycle

property/nonnegative

two-cycle

property

if

its

associated

single

player

function

f

does.

The

graph

Hf

has

a

possibly

infinite

number

of

edges

between

any

two

vertices.

We

define

Gf

to

be

the

edgeweighted

directed

graph

with

exactly

one

edge

from

a

to

b,

whose

weight

δab

is

the

infimum

(possibly

of

all

of

the

edge

weights

ex(a,

b)

for

x

(a).

It

is

easy

to

see

that

Hf

has

the

nonnegative

cycle

property/nonnegative

two-cycle

property

if

and

only

if

Gf

does.

Gf

is

called

the

outcome

graph

of

f.

The

weak

monotonicity

property

mentioned

earlier

can

be

defined

for

arbitrary

social

choice

functions

by

the

condition

that

every

local

subfunction

satisfies

the

nonnegative

two-cycle

property.

The

following

result

was

obtained

by

Rochet

in

a

slightly

different

form

and

rediscovered

by

Rozenshtrom

and

Gui,

Muller

and

Vohra

Lemma

A

local

social

choice

function

is

truthful

if

and

only

if

it

has

the

nonnegative

cycle

property.

Thus

a

social

choice

function

is

truthful

if

and

only

if

every

local

subfunction

satisfies

the

nonnegative

cycle

property.

In

light

of

this,

theorem

follows

from:

Theorem

For

any

surjective

single

player

function

f

D

A

where

D

is

a

convex

subset

of

RA

and

A

is

finite,

the

nonnegative

two-cycle

property

implies

the

nonnegative

cycle

property.

This

is

the

result

we

will

prove.

Overview

of

the

Proof

of

Theorem

Let

D

RA

be

convex

and

let

f

D

A

be

a

single

player

function

such

that

Gf

has

no

negative

two-cycles.

We

want

to

conclude

that

Gf

has

no

negative

cycles.

For

two

vertices

a,

b,

let

δ∗

ab

denote

the

minimum

weight

of

any

path

from

a

to

b.

Clearly

δ∗

ab

δab.

Our

proof

shows

that

the

δ∗

-weight

of

every

cycle

is

exactly

from

which

theorem

follows.

There

seems

to

be

no

direct

way

to

compute

δ∗

and

so

we

proceed

indirectly.

Based

on

geometric

considerations,

we

identify

a

subset

of

paths

in

Gf

called

admissible

paths

and

a

subset

of

admissible

paths

called

straight

paths.

We

prove

that

for

any

two

outcomes

a,

b,

there

is

a

straight

path

from

a

to

b

(lemma

and

corollary

and

all

straight

paths

from

a

to

b

have

the

same

weight,

which

we

denote

ρab

(theorem

We

show

that

ρab

δab

(lemma

and

that

the

ρ-weight

of

every

cycle

is

The

key

step

to

this

proof

is

showing

that

the

ρ-weight

of

every

directed

triangle

is

(lemma

It

turns

out

that

ρ

is

equal

to

δ∗

(corollary

although

this

equality

is

not

needed

in

the

proof

of

theorem

To

expand

on

the

above

summary,

we

give

the

definitions

of

an

admissible

path

and

a

straight

path.

These

are

somewhat

technical

and

rely

on

the

geometry

of

f.

We

first

observe

that,

without

loss

of

generality,

we

can

assume

that

D

is

(topologically)

closed

(section

In

section

for

each

a

A,

we

enlarge

the

set

(a)

to

a

closed

convex

set

Da

D

in

such

a

way

that

for

a,

b

A

with

a

b,

Da

and

Db

have

disjoint

interiors.

We

define

an

admissible

path

to

be

a

sequence

of

outcomes

ak)

such

that

each

of

the

sets

Ij

Daj

is

nonempty

(section

An

admissible

path

is

straight

if

there

is

a

straight

line

that

meets

one

point

from

each

of

the

sets

in

order

(section

Finally,

we

mention

how

the

hypotheses

of

convex

domain

and

finite

range

are

used

in

the

proof.

Both

hypotheses

are

needed

to

show:

the

existence

of

a

straight

path

from

a

to

b

for

all

a,

b

(lemma

that

the

ρ-weight

of

a

directed

triangle

is

(lemma

The

convex

domain

hypothesis

is

also

needed

for

the

convexity

of

the

sets

Da

(section

The

finite

range

hypothesis

is

also

needed

to

reduce

theorem

to

the

case

that

D

is

closed

(section

and

to

prove

that

every

straight

path

from

a

to

b

has

the

same

δ-weight

(theorem

REDUCTION

TO

CLOSED

DOMAIN

We

first

reduce

the

theorem

to

the

case

that

D

is

closed.

Write

DC

for

the

closure

of

D.

Since

A

is

finite,

DC

(a))C

Thus

for

each

v

DC

D,

there

is

an

a

a(v)

A

such

that

v

(a))C

Extend

f

to

the

function

g

on

DC

by

defining

g(v)

a(v)

for

v

DC

D

and

g(v)

f(v)

for

v

D.

It

is

easy

to

check

that

δab(g)

δab(f)

for

all

a,

b

A

and

therefore

it

suffices

to

show

that

the

nonnegative

two-cycle

property

for

g

implies

the

nonnegative

cycle

property

for

g.

Henceforth

we

assume

D

is

convex

and

closed.

A

DISSECTION

OF

THE

DOMAIN

In

this

section,

we

construct

a

family

of

closed

convex

sets

{Da

a

A}

with

disjoint

interiors

whose

union

is

D

and

satisfying

(a)

Da

for

each

a

A.

Let

Ra

{v

∀b

A,

v(a)

v(b)

δab}.

Ra

is

a

closed

polyhedron

containing

(a).

The

next

proposition

implies

that

any

two

of

these

polyhedra

intersect

only

on

their

boundary.

Proposition

Let

a,

b

A.

If

v

Ra

∩Rb

then

v(a)−

v(b)

δab

−δba.

Da

Db

Dc

Dd

De

v

w

x

y

z

u

p

Figure

A

domain

with

outcomes.

Proof.

v

Ra

implies

v(a)

v(b)

δab

and

v

Rb

implies

v(b)−v(a)

δba

which,

by

the

nonnegative

two-cycle

property,

implies

v(a)

v(b)

δab.

Thus

v(a)

v(b)

δab

and

by

symmetry

v(b)

v(a)

δba.

Finally,

we

restrict

the

collection

of

sets

{Ra

a

A}

to

the

domain

D

by

defining

Da

Ra

D

for

each

a

A.

Clearly,

Da

is

closed

and

convex,

and

contains

(a).

Therefore

S

a∈A

Da

D.

Also,

by

proposition

any

point

v

in

Da

Db

satisfies

v(a)

v(b)

δab

−δba.

PATHS

AND

D-SEQUENCES

A

path

of

size

k

is

a

sequence

−→a

ak)

with

each

ai

A

(possibly

with

repetition).

We

call

−→a

an

ak)path.

For

a

path

−→a

we

write

|−→a

for

the

size

of

−→a

−→a

is

simple

if

the

ai"s

are

distinct.

For

b,

c

A

we

write

Pbc

for

the

set

of

(b,

c)-paths

and

SPbc

for

the

set

of

simple

(b,

c)-paths.

The

δ-weight

of

path

−→a

is

defined

by

δ(−→a

A

D-sequence

of

order

k

is

a

sequence

−→u

uk)

with

each

ui

D

(possibly

with

repetition).

We

call

−→u

a

uk)-sequence.

For

a

D-sequence

−→u

we

write

ord(u)

for

the

order

of

−→u

For

v,

w

D

we

write

Svw

for

the

set

of

(v,

w)-sequences.

A

compatible

pair

is

a

pair

(−→a

−→u

where

−→a

is

a

path

and

−→u

is

a

D-sequence

satisfying

ord(−→u

|−→a

and

for

each

i

[k],

both

and

ui

belong

to

Dai

We

write

C(−→a

for

the

set

of

D-sequences

−→u

that

are

compatible

with

−→a

We

say

that

−→a

is

admissible

if

C(−→a

is

nonempty.

For

−→u

C(−→a

we

define

∆−→a

(−→u

|−→a

X

(ui(ai)

For

v,

w

D

and

b,

c

A,

we

define

Cvw

bc

to

be

the

set

of

compatible

pairs

(−→a

−→u

such

that

−→a

Pbc

and

−→u

Svw

To

illustrate

these

definitions,

figure

gives

the

dissection

of

a

domain,

a

plane,

into

five

regions

Da,

Db,

Dc,

Dd,

De.

D-sequence

(v,

w,

x,

y,

z)

is

compatible

with

both

path

(a,

b,

c,

e)

and

path

(a,

b,

d,

e);

D-sequence

(v,

w,

u,

y,

z)

is

compatible

with

a

unique

path

(a,

b,

d,

e).

D-sequence

(x,

w,

p,

y,

z)

is

compatible

with

a

unique

path

(b,

a,

d,

e).

Hence

(a,

b,

c,

e),

(a,

b,

d,

e)

and

(b,

a,

d,

e)

are

admissible

paths.

However,

path

(a,

c,

d)

or

path

(b,

e)

is

not

admissible.

Proposition

For

any

compatible

pair

(−→a

−→u

∆−→a

(−→u

δ(−→a

Proof.

Let

k

ord(−→u

|−→a

By

the

definition

of

a

compatible

pair,

ui

Dai

for

i

[k

ui(ai)

from

proposition

Therefore,

∆−→a

(−→u

(ui(ai)

δ(−→a

Lemma

Let

b,

c

A

and

let

−→a

−→a

Pbc.

If

C(−→a

C(−→a

then

δ(−→a

δ(−→a

Proof.

Let

−→u

be

a

D-sequence

in

C(−→a

C(−→a

By

proposition

δ(−→a

∆−→a

(−→u

and

δ(−→a

∆−→a

(−→u

it

suffices

to

show

∆−→a

(−→u

∆−→a

(−→u

Let

k

ord(−→u

|−→a

|−→a

Since

∆−→a

(−→u

(ui(ai)

(ui(ai)

(ui(ai)

∆−→a

(−→u

∆−→a

(−→u

((ui(ai)

(ui(ai)

((ui(ai)

ui(ai))

Noticing

both

and

ui

belong

to

Dai

Dai

we

have

by

proposition

δaiai

ui(ai)

ui(ai).

Hence

∆−→a

(−→u

∆−→a

(−→u

LINEAR

D-SEQUENCES

AND

STRAIGHT

PATHS

For

v,

w

D

we

write

vw

for

the

(closed)

line

segment

joining

v

and

w.

A

D-sequence

−→u

of

order

k

is

linear

provided

that

there

is

a

sequence

of

real

numbers

λ0

λ1

λk

such

that

ui

λiuk.

In

particular,

each

ui

belongs

to

For

v,

w

D

we

write

Lvw

for

the

set

of

linear

(v,

w)-sequences.

For

b,

c

A

and

v,

w

D

we

write

LCvw

bc

for

the

set

of

compatible

pairs

(−→a

−→u

such

that

−→a

Pbc

and

−→u

Lvw

For

a

path

−→a

we

write

L(−→a

for

the

set

of

linear

sequences

compatible

with

−→a

We

say

that

−→a

is

straight

if

L(−→a

For

example,

in

figure

D-sequence

(v,

w,

x,

y,

z)

is

linear

while

(v,

w,

u,

y,

z),

(x,

w,

p,

y,

z),

and

(x,

v,

w,

y,

z)

are

not.

Hence

path

(a,

b,

c,

e)

and

(a,

b,

d,

e)

are

both

straight.

However,

path

(b,

a,

d,

e)

is

not

straight.

Lemma

Let

b,

c

A

and

v

Db,

w

Dc.

There

is

a

simple

path

−→a

and

D-sequence

−→u

such

that

(−→a

−→u

LCvw

bc

Furthermore,

for

any

such

path

−→a

δ(−→a

v(b)

v(c).

Proof.

By

the

convexity

of

D,

any

sequence

of

points

on

vw

is

a

D-sequence.

If

b

c,

singleton

path

−→a

(b)

and

D-sequence

−→u

(v,

w)

are

obviously

compatible.

δ(−→a

v(b)

v(c).

So

assume

b

c.

If

Db

∩Dc

∩vw

we

pick

an

arbitrary

x

from

this

set

and

let

−→a

(b,

c)

SPbc,

−→u

(v,

x,

w)

Lvw

Again

it

is

easy

to

check

the

compatibility

of

(−→a

−→u

Since

v

Db,

v(b)

v(c)

δbc

δ(−→a

For

the

remaining

case

b

c

and

Db

∩Dc

∩vw

notice

v

w

otherwise

v

w

Db

Dc

vw.

So

we

can

define

λx

for

every

point

x

on

vw

to

be

the

unique

number

in

such

that

x

λx)v

λxw.

For

convenience,

we

write

x

y

for

λx

λy.

Let

Ia

Da

vw

for

each

a

A.

Since

D

∪a∈ADa,

we

have

vw

∪a∈AIa.

Moreover,

by

the

convexity

of

Da

and

vw,

Ia

is

a

(possibly

trivial)

closed

interval.

We

begin

by

considering

the

case

that

Ib

and

Ic

are

each

a

single

point,

that

is,

Ib

{v}

and

Ic

{w}.

Let

S

be

a

minimal

subset

of

A

satisfying

∪s∈SIs

vw.

For

each

s

S,

Is

is

maximal,

i.e.,

not

contained

in

any

other

It,

for

t

S.

In

particular,

the

intervals

{Is

s

S}

have

all

left

endpoints

distinct

and

all

right

endpoints

distinct

and

the

order

of

the

left

endpoints

is

the

same

as

that

of

the

right

endpoints.

Let

k

|S|

and

index

S

as

in

the

order

defined

by

the

right

endpoints.

Denote

the

interval

Iai

by

[li,

ri].

Thus

and

the

fact

that

these

intervals

cover

vw

implies

v,

w

and

for

all

i

k

ri

which

further

implies

li

ri.

Now

we

define

the

path

−→a

ak)

with

b,

ak

c

and

as

above.

Define

the

linear

D-sequence

−→u

uk)

by

v,

uk

w

and

for

i

ui

ri.

It

follows

immediately

that

(−→a

−→u

LCvw

bc

Neither

b

nor

c

is

in

S

since

lb

rb

and

lc

rc.

Thus

−→a

is

simple.

Finally

to

show

δ(−→a

v(b)

v(c),

we

note

v(b)

v(c)

v(ak)

(v(ai)

and

δ(−→a

∆−→a

(−→u

(ui(ai)

(ri(ai)

For

two

outcomes

d,

e

A,

let

us

define

fde(z)

z(d)−z(e)

for

all

z

D.

It

suffices

to

show

(ri)

(v)

for

i

k

Fact

For

d,

e

A,

fde(z)

is

a

linear

function

of

z.

Furthermore,

if

x

Dd

and

y

De

with

x

y,

then

fde(x)

x(d)

x(e)

δde

−δed

−(y(e)

y(d))

fde(y).

Therefore

fde(z)

is

monotonically

nonincreasing

along

the

line

xy

as

z

moves

in

the

direction

from

x

to

y.

Applying

this

fact

with

d

ai,

e

x

li

and

y

ri

gives

the

desired

conclusion.

This

completes

the

proof

for

the

case

that

Ib

{v}

and

Ic

{w}.

For

general

Ib,

Ic,

rb

lc

otherwise

Db

Dc

vw

Ib

Ic

Let

v

rb

and

w

lc.

Clearly

we

can

apply

the

above

conclusion

to

v

Db,

w

Dc

and

get

a

compatible

pair

(−→a

−→u

LCv

w

bc

with

−→a

simple

and

δ(−→a

v

(b)

v

(c).

Define

the

linear

D-sequence

−→u

by

v,

uk

w

and

ui

ui

for

i

[k

(−→a

−→u

LCvw

bc

is

evident.

Moreover,

applying

the

above

fact

with

d

b,

e

c,

x

v

and

y

w,

we

get

v(b)

v(c)

v

(b)

v

(c)

δ(−→a

Corollary

For

any

b,

c

A

there

is

a

straight

(b,

c)path.

The

main

result

of

this

section

(theorem

says

that

for

any

b,

c

A,

every

straight

(b,

c)-path

has

the

same

δ-weight.

To

prove

this,

we

first

fix

v

Db

and

w

Dc

and

show

(lemma

that

every

straight

(b,

c)-path

compatible

with

some

linear

(v,

w)-sequence

has

the

same

δ-weight

ρbc(v,

w).

We

then

show

in

theorem

that

ρbc(v,

w)

is

the

same

for

all

choices

of

v

Db

and

w

Dc.

Lemma

For

b,

c

A,

there

is

a

function

ρbc

Db

Dc

R

satisfying

that

for

any

(−→a

−→u

LCvw

bc

δ(−→a

ρbc(v,

w).

Proof.

Let

(−→a

−→u

(−→a

−→u

LCvw

bc

It

suffices

to

show

δ(−→a

δ(−→a

To

do

this

we

construct

a

linear

(v,

w)-sequence

−→u

and

paths

−→a

−→a

Pbc,

both

compatible

with

−→u

satisfying

δ(−→a

δ(−→a

and

δ(−→a

δ(−→a

Lemma

implies

δ(−→a

δ(−→a

which

will

complete

the

proof.

Let

|−→a

ord(−→u

k

and

|−→a

ord(−→u

l.

We

select

−→u

to

be

any

linear

(v,

w)-sequence

ut)

such

that

−→u

and

−→u

are

both

subsequences

of

−→u

i.e.,

there

are

indices

ik

t

and

jl

t

such

that

−→u

uik

and

−→u

ujl

We

now

construct

a

(b,

c)-path

−→a

compatible

with

−→u

such

that

δ(−→a

δ(−→a

(An

analogous

construction

gives

−→a

compatible

with

−→u

such

that

δ(−→a

δ(−→a

This

will

complete

the

proof.

−→a

is

defined

as

follows:

for

j

t,

a∗

j

ar

where

r

is

the

unique

index

satisfying

j

ir.

Since

both

and

uir

ur

belong

to

Dar

uj

Dar

for

j

ir

by

the

convexity

of

Dar

The

compatibility

of

(−→a

−→u

follows

immediately.

Clearly,

a∗

b

and

a∗

t

ak

c,

so

−→a

Pbc.

Furthermore,

as

δa∗

j

a∗

δarar

for

each

r

[k],

j

ir,

δ(−→a

δa∗

ir

a∗

δ(−→a

We

are

now

ready

for

the

main

theorem

of

the

section:

Theorem

ρbc

is

a

constant

map

on

Db

Dc.

Thus

for

any

b,

c

A,

every

straight

(b,

c)-path

has

the

same

δweight.

Proof.

For

a

path

−→a

(v,

w)

is

compatible

with

−→a

if

there

is

a

linear

(v,

w)-sequence

compatible

with

−→a

We

write

CP(−→a

for

the

set

of

pairs

(v,

w)

compatible

with

−→a

ρbc

is

constant

on

CP(−→a

because

for

each

(v,

w)

CP(−→a

ρbc(v,

w)

δ(−→a

By

lemma

we

also

haveS

−→a

∈SPbc

CP(−→a

Db

×Dc.

Since

A

is

finite,

SPbc,

the

set

of

simple

paths

from

b

to

c,

is

finite

as

well.

Next

we

prove

that

for

any

path

−→a

CP(−→a

is

closed.

Let

((vn

wn

n

N)

be

a

convergent

sequence

in

CP(−→a

and

let

(v,

w)

be

the

limit.

We

want

to

show

that

(v,

w)

CP(−→a

For

each

n

N,

since

(vn

wn

CP(−→a

there

is

a

linear

(vn

wn

)-sequence

un

compatible

with

−→a

i.e.,

there

are

λn

λn

λn

k

(k

|−→a

such

that

un

j

λn

j

)vn

λn

j

wn

(j

k).

Since

for

each

n,

λn

(λn

λn

λn

k

belongs

to

the

closed

bounded

set

we

can

choose

an

infinite

subset

I

N

such

that

the

sequence

(λn

n

I)

converges.

Let

λ

λk)

be

the

limit.

Clearly

λ0

λ1

λk

Define

the

linear

(v,

w)-sequence

−→u

by

uj

λj

)v

λj

w

(j

k).

Then

for

each

j

k},

uj

is

the

limit

of

the

sequence

(un

j

n

I).

For

j

each

un

j

belongs

to

the

closed

set

Daj

so

uj

Daj

Similarly,

for

j

k

each

un

j

belongs

to

the

closed

set

so

uj

Hence

(−→a

−→u

is

compatible,

implying

(v,

w)

CP(−→a

Now

we

have

Db

Dc

covered

by

finitely

many

closed

subsets

on

each

of

them

ρbc

is

a

constant.

Suppose

for

contradiction

that

there

are

(v,

w),

(v

w

Db

Dc

such

that

ρbc(v,

w)

ρbc(v

w

L

λ)v

λv

λ)w

λw

λ

is

a

line

segment

in

Db

×Dc

by

the

convexity

of

Db,

Dc.

Let

{(x,

y)

L

ρbc(x,

y)

ρbc(v,

w)}

and

L

Clearly

(v,

w)

(v

w

Let

P

{−→a

SPbc

δ(−→a

ρbc(v,

w)}.

`S

−→a

∈P

CP(−→a

L,

S

−→a

∈SPbc−P

CP(−→a

L

are

closed

by

the

finiteness

of

P.

This

is

a

contradiction,

since

it

is

well

known

(and

easy

to

prove)

that

a

line

segment

can

not

be

expressed

as

the

disjoint

union

of

two

nonempty

closed

sets.

Summarizing

corollary

lemma

and

theorem

we

have

Corollary

For

any

b,

c

A,

there

is

a

real

number

ρbc

with

the

property

that

There

is

at

least

one

straight

(b,

c)-path

of

δ-weight

ρbc

and

Every

straight

(b,

c)-path

has

δ-weight

ρbc.

PROOF

OF

THEOREM

Lemma

ρbc

δbc

for

all

b,

c

A.

Proof.

For

contradiction,

suppose

ρbc

δbc

By

the

definition

of

δbc,

there

exists

v

(b)

Db

with

v(b)

v(c)

δbc

ρbc.

Pick

an

arbitrary

w

Dc.

By

lemma

there

is

a

compatible

pair

(−→a

−→u

LCvw

bc

with

δ(−→a

v(b)

v(c).

Since

−→a

is

a

straight

(b,

c)-path,

ρbc

δ(−→a

v(b)

v(c),

leading

to

a

contradiction.

Define

another

edge-weighted

complete

directed

graph

Gf

on

vertex

set

A

where

the

weight

of

arc

(a,

b)

is

ρab.

Immediately

from

lemma

the

weight

of

every

directed

cycle

in

Gf

is

bounded

below

by

its

weight

in

Gf

To

prove

theorem

it

suffices

to

show

the

zero

cycle

property

of

Gf

i.e.,

every

directed

cycle

has

weight

zero.

We

begin

by

considering

two-cycles.

Lemma

ρbc

ρcb

for

all

b,

c

A.

Proof.

Let

−→a

be

a

straight

(b,

c)-path

compatible

with

linear

sequence

−→u

let

−→a

be

the

reverse

of

−→a

and

−→u

the

reverse

of

−→u

Obviously,

(−→a

−→u

is

compatible

as

well

and

thus

−→a

is

a

straight

(c,

b)-path.

Therefore,

ρbc

ρcb

δ(−→a

δ(−→a

where

the

final

equality

uses

proposition

Next,

for

three

cycles,

we

first

consider

those

compatible

with

linear

triples.

Lemma

If

there

are

collinear

points

u

Da,

v

Db,

w

Dc

(a,

b,

c

A),

ρab

ρbc

ρca

Proof.

First,

we

prove

for

the

case

where

v

is

between

u

and

w.

From

lemma

there

are

compatible

pairs

(−→a

−→u

LCuv

ab

(−→a

−→u

LCvw

bc

Let

|−→a

ord(−→u

k

and

|−→a

ord(−→u

l.

We

paste

−→a

and

−→a

together

as

−→a

(a

ak,

al

c),

−→u

and

−→u

as

−→u

(u

uk

v

ul

w).

Clearly

(−→a

−→u

LCuw

ac

and

δ(−→a

δak

δai

δ(−→a

δbb

δ(−→a

δ(−→a

δ(−→a

Therefore,

ρac

δ(−→a

δ(−→a

δ(−→a

ρab

ρbc.

Moreover,

ρac

−ρca

by

lemma

so

we

get

ρab

ρbc

ρca

Now

suppose

w

is

between

u

and

v.

By

the

above

argument,

we

have

ρac

ρcb

ρba

and

by

lemma

ρab

ρbc

ρca

−ρba

ρcb

ρac

The

case

that

u

is

between

v

and

w

is

similar.

Now

we

are

ready

for

the

zero

three-cycle

property:

Lemma

ρab

ρbc

ρca

for

all

a,

b,

c

A.

Proof.

Let

S

{(a,

b,

c)

ρab

ρbc

ρca

and

for

contradiction,

suppose

S

S

is

finite.

For

each

a

A,

choose

va

Da

arbitrarily

and

let

T

be

the

convex

hull

of

{va

a

A}.

For

each

(a,

b,

c)

S,

let

Rabc

Da

Db

Dc

Clearly,

each

Rabc

is

nonempty

and

compact.

Moreover,

by

lemma

no

(u,

v,

w)

Rabc

is

collinear.

Define

f

R

by

f(u,

v,

w)

|v−u|+|w−v|+|u−w|.

For

(a,

b,

c)

S,

the

restriction

of

f

to

the

compact

set

Rabc

attains

a

minimum

m(a,

b,

c)

at

some

point

(u,

v,

w)

Rabc

by

the

continuity

of

f,

i.e.,

there

exists

a

triangle

∆uvw

of

minimum

perimeter

within

T

with

u

Da,

v

Db,

w

Dc.

Choose

(a∗

b∗

c∗

S

so

that

m(a∗

b∗

c∗

is

minimum

and

let

(u∗

v∗

w∗

Ra∗b∗c∗

be

a

triple

achieving

it.

Pick

an

arbitrary

point

p

in

the

interior

of

∆u∗

v∗

w∗

By

the

convexity

of

domain

D,

there

is

d

A

such

that

p

Dd.

Consider

triangles

∆u∗

pw∗

∆w∗

pv∗

and

∆v∗

pu∗

Since

each

of

them

has

perimeter

less

than

that

of

∆u∗

v∗

w∗

and

all

three

triangles

are

contained

in

T,

by

the

minimality

of

∆u∗

v∗

w∗

(a∗

d,

c∗

(c∗

d,

b∗

(b∗

d,

a∗

S.

Thus

ρa∗d

ρdc∗

ρc∗a∗

ρc∗d

ρdb∗

ρb∗c∗

ρb∗d

ρda∗

ρa∗b∗

Summing

up

the

three

equalities,

(ρa∗d

ρdc∗

ρc∗d

ρdb∗

ρb∗d

ρda∗

+(ρc∗a∗

ρb∗c∗

ρa∗b∗

which

yields

a

contradiction

ρa∗b∗

ρb∗c∗

ρc∗a∗

With

the

zero

two-cycle

and

three-cycle

properties,

the

zero

cycle

property

of

Gf

is

immediate.

As

noted

earlier,

this

completes

the

proof

of

theorem

Theorem

Every

directed

cycle

of

Gf

has

weight

zero.

Proof.

Clearly,

zero

two-cycle

and

three-cycle

properties

imply

triangle

equality

ρab

+ρbc

ρac

for

all

a,

b,

c

A.

For

a

directed

cycle

C

by

inductively

applying

triangle

equality,

we

have

Therefore,

the

weight

of

C

is

As

final

remarks,

we

note

that

our

result

implies

the

following

strengthenings

of

theorem

Corollary

For

any

b,

c

A,

every

admissible

(b,

c)path

has

the

same

δ-weight

ρbc.

Proof.

First

notice

that

for

any

b,

c

A,

if

Db

∩Dc

δbc

ρbc.

To

see

this,

pick

v

Db

Dc

arbitrarily.

Obviously,

path

−→a

(b,

c)

is

compatible

with

linear

sequence

−→u

(v,

v,

v)

and

is

thus

a

straight

(b,

c)-path.

Hence

ρbc

δ(−→a

δbc.

Now

for

any

b,

c

A

and

any

(b,

c)-path

−→a

with

C(−→a

let

−→u

C(−→a

Since

ui

Dai

for

i

[|−→a

δ(−→a

|−→a

X

|−→a

X

which

by

theorem

−ρa|−→a

ρbc.

Corollary

For

any

b,

c

A,

ρbc

is

equal

to

δ∗

bc,

the

minimum

δ-weight

over

all

(b,

c)-paths.

Proof.

Clearly

ρbc

δ∗

bc

by

corollary

On

the

other

hand,

for

every

(b,

c)-path

−→a

(b

ak

c),

by

lemma

δ(−→a

which

by

theorem

ρbc.

Hence

ρbc

δ∗

bc,

which

completes

the

proof.

COUNTEREXAMPLES

TO

STRONGER

FORMS

OF

THEOREM

Theorem

applies

to

social

choice

functions

with

convex

domain

and

finite

range.

We

now

show

that

neither

of

these

hypotheses

can

be

omitted.

Our

examples

are

single

player

functions.

The

first

example

illustrates

that

convexity

can

not

be

omitted.

We

present

an

untruthful

single

player

social

choice

function

with

three

outcomes

a,

b,

c

satisfying

W-MON

on

a

path-connected

but

non-convex

domain.

The

domain

is

the

boundary

of

a

triangle

whose

vertices

are

x

y

and

z

x

and

the

open

line

segment

zx

is

assigned

outcome

a,

y

and

the

open

line

segment

xy

is

assigned

outcome

b,

and

z

and

the

open

line

segment

yz

is

assigned

outcome

c.

Clearly,

δab

−δba

δbc

−δcb

δca

−δac

W-MON

(the

nonnegative

twocycle

property)

holds.

Since

there

is

a

negative

cycle

δab

δbc

δca

by

lemma

this

is

not

a

truthful

choice

function.

We

now

show

that

the

hypothesis

of

finite

range

can

not

be

omitted.

We

construct

a

family

of

single

player

social

choice

functions

each

having

a

convex

domain

and

an

infinite

number

of

outcomes,

and

satisfying

weak

monotonicity

but

not

truthfulness.

Our

examples

will

be

specified

by

a

positive

integer

n

and

an

n

n

matrix

M

satisfying

the

following

properties:

M

is

non-singular.

M

is

positive

semidefinite.

There

are

distinct

ik

[n]

satisfying

(M(ij,

ij)

M(ij

(M(ik,

ik)

M(ik,

Here

is

an

example

matrix

with

n

and

−1

−1

−1

A

Let

en

denote

the

standard

basis

of

Rn

Let

Sn

denote

the

convex

hull

of

en},

which

is

the

set

of

vectors

in

Rn

with

nonnegative

coordinates

that

sum

to

The

range

of

our

social

choice

function

will

be

the

set

Sn

and

the

domain

D

will

be

indexed

by

Sn,

that

is

D

{yλ

λ

Sn},

where

yλ

is

defined

below.

The

function

f

maps

yλ

to

λ.

Next

we

specify

yλ.

By

definition,

D

must

be

a

set

of

functions

from

Sn

to

R.

For

λ

Sn,

the

domain

element

yλ

Sn

R

is

defined

by

yλ(α)

λT

Mα.

The

nonsingularity

of

M

guarantees

that

yλ

yµ

for

λ

µ

Sn.

It

is

easy

to

see

that

D

is

a

convex

subset

of

the

set

of

all

functions

from

Sn

to

R.

The

outcome

graph

Gf

is

an

infinite

graph

whose

vertex

set

is

the

outcome

set

A

Sn.

For

outcomes

λ,

µ

A,

the

edge

weight

δλµ

is

equal

to

δλµ

inf{v(λ)

v(µ)

f(v)

λ}

yλ(λ)

yλ(µ)

λT

Mλ

λT

Mµ

λT

M(λ

µ).

We

claim

that

Gf

satisfies

the

nonnegative

two-cycle

property

(W-MON)

but

has

a

negative

cycle

(and

hence

is

not

truthful).

For

outcomes

λ,

µ

A,

δλµ

+δµλ

λT

M(λ−µ)+µT

M(µ−λ)

(λ−µ)T

M(λ−µ),

which

is

nonnegative

since

M

is

positive

semidefinite.

Hence

the

nonnegative

two-cycle

property

holds.

Next

we

show

that

Gf

has

a

negative

cycle.

Let

ik

be

a

sequence

of

indices

satisfying

property

of

M.

We

claim

eik

is

a

negative

cycle.

Since

δeiej

eT

i

M(ei

ej)

M(i,

i)

M(i,

j)

for

any

i,

j

[k],

the

weight

of

the

cycle

δeij

δeik

(M(ij

ij

M(ij,

(M(ik,

ik)

M(ik,

which

completes

the

proof.

Finally,

we

point

out

that

the

third

property

imposed

on

the

matrix

M

has

the

following

interpretation.

Let

R(M)

rn}

be

the

set

of

row

vectors

of

M

and

let

hM

be

the

single

player

social

choice

function

with

domain

R(M)

and

range

n}

mapping

ri

to

i.

Property

is

equivalent

to

the

condition

that

the

outcome

graph

GhM

has

a

negative

cycle.

By

lemma

this

is

equivalent

to

the

condition

that

hM

is

untruthful.

FUTURE

WORK

As

stated

in

the

introduction,

the

goal

underlying

the

work

in

this

paper

is

to

obtain

useful

and

general

characterizations

of

truthfulness.

Let

us

say

that

a

set

D

of

P

A

real

valuation

matrices

is

a

WM-domain

if

any

social

choice

function

on

D

satisfying

weak

monotonicity

is

truthful.

In

this

paper,

we

showed

that

for

finite

A,

any

convex

D

is

a

WM-domain.

Typically,

the

domains

of

social

choice

functions

considered

in

mechanism

design

are

convex,

but

there

are

interesting

examples

with

non-convex

domains,

e.g.,

combinatorial

auctions

with

unknown

single-minded

bidders.

It

is

intriguing

to

find

the

most

general

conditions

under

which

a

set

D

of

real

matrices

is

a

WM-domain.

We

believe

that

convexity

is

the

main

part

of

the

story,

i.e.,

a

WM-domain

is,

after

excluding

some

exceptional

cases,

essentially

a

convex

set.

Turning

to

parametric

representations,

let

us

say

a

set

D

of

P

A

matrices

is

an

AM-domain

if

any

truthful

social

choice

function

with

domain

D

is

an

affine

maximizer.

Roberts"

theorem

says

that

the

unrestricted

domain

is

an

AM-domain.

What

are

the

most

general

conditions

under

which

a

set

D

of

real

matrices

is

an

AM-domain?

Acknowledgments

We

thank

Ron

Lavi

for

helpful

discussions

and

the

two

anonymous

referees

for

helpful

comments.

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Israel,

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

individual preference
non-truthful function

social choice function
weak monotonicity
convex domain
truthful implementation
recognition algorithm
nonnegative cycle property
truthfulness
affine maximizer
dominant strategy
mechanism design
strategyproof
truthful