Computing

Good

Nash

Equilibria

in

Graphical

Games

Edith

Elkind

Hebrew

University

of

Jerusalem,

Israel,

and

University

of

Southampton,

Southampton,

U.K.

Leslie

Ann

Goldberg

University

of

Liverpool

Liverpool

U.K.

Paul

Goldberg

University

of

Liverpool

Liverpool

U.K.

ABSTRACT

This

paper

addresses

the

problem

of

fair

equilibrium

selection

in

graphical

games.

Our

approach

is

based

on

the

data

structure

called

the

best

response

policy,

which

was

proposed

by

Kearns

et

al.

as

a

way

to

represent

all

Nash

equilibria

of

a

graphical

game.

In

it

was

shown

that

the

best

response

policy

has

polynomial

size

as

long

as

the

underlying

graph

is

a

path.

In

this

paper,

we

show

that

if

the

underlying

graph

is

a

bounded-degree

tree

and

the

best

response

policy

has

polynomial

size

then

there

is

an

efficient

algorithm

which

constructs

a

Nash

equilibrium

that

guarantees

certain

payoffs

to

all

participants.

Another

attractive

solution

concept

is

a

Nash

equilibrium

that

maximizes

the

social

welfare.

We

show

that,

while

exactly

computing

the

latter

is

infeasible

(we

prove

that

solving

this

problem

may

involve

algebraic

numbers

of

an

arbitrarily

high

degree),

there

exists

an

FPTAS

for

finding

such

an

equilibrium

as

long

as

the

best

response

policy

has

polynomial

size.

These

two

algorithms

can

be

combined

to

produce

Nash

equilibria

that

satisfy

various

fairness

criteria.

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

In

a

large

community

of

agents,

an

agent"s

behavior

is

not

likely

to

have

a

direct

effect

on

most

other

agents:

rather,

it

is

just

the

agents

who

are

close

enough

to

him

that

will

be

affected.

However,

as

these

agents

respond

by

adapting

their

behavior,

more

agents

will

feel

the

consequences

and

eventually

the

choices

made

by

a

single

agent

will

propagate

throughout

the

entire

community.

This

is

the

intuition

behind

graphical

games,

which

were

introduced

by

Kearns,

Littman

and

Singh

in

as

a

compact

representation

scheme

for

games

with

many

players.

In

an

n-player

graphical

game,

each

player

is

associated

with

a

vertex

of

an

underlying

graph

G,

and

the

payoffs

of

each

player

depend

on

his

action

as

well

as

on

the

actions

of

his

neighbors

in

the

graph.

If

the

maximum

degree

of

G

is

Δ,

and

each

player

has

two

actions

available

to

him,

then

the

game

can

be

represented

using

numbers.

In

contrast,

we

need

numbers

to

represent

a

general

n-player

game,

which

is

only

practical

for

small

values

of

n.

For

graphical

games

with

constant

Δ,

the

size

of

the

game

is

linear

in

n.

One

of

the

most

natural

problems

for

a

graphical

game

is

that

of

finding

a

Nash

equilibrium,

the

existence

of

which

follows

from

Nash"s

celebrated

theorem

(as

graphical

games

are

just

a

special

case

of

n-player

games).

The

first

attempt

to

tackle

this

problem

was

made

in

where

the

authors

consider

graphical

games

with

two

actions

per

player

in

which

the

underlying

graph

is

a

boundeddegree

tree.

They

propose

a

generic

algorithm

for

finding

Nash

equilibria

that

can

be

specialized

in

two

ways:

an

exponential-time

algorithm

for

finding

an

(exact)

Nash

equilibrium,

and

a

fully

polynomial

time

approximation

scheme

(FPTAS)

for

finding

an

approximation

to

a

Nash

equilibrium.

For

any

this

algorithm

outputs

an

-Nash

equilibrium,

which

is

a

strategy

profile

in

which

no

player

can

improve

his

payoff

by

more

than

by

unilaterally

changing

his

strategy.

While

-Nash

equilibria

are

often

easier

to

compute

than

exact

Nash

equilibria,

this

solution

concept

has

several

drawbacks.

First,

the

players

may

be

sensitive

to

a

small

loss

in

payoffs,

so

the

strategy

profile

that

is

an

-Nash

equilibrium

will

not

be

stable.

This

will

be

the

case

even

if

there

is

only

a

small

subset

of

players

who

are

extremely

price-sensitive,

and

for

a

large

population

of

players

it

may

be

difficult

to

choose

a

value

of

that

will

satisfy

everyone.

Second,

the

strategy

profiles

that

are

close

to

being

Nash

equilibria

may

be

much

better

with

respect

to

the

properties

under

consideration

than

exact

Nash

equilibria.

Therefore,

the

(approximation

to

the)

value

of

the

best

solution

that

corresponds

to

an

-Nash

equilibrium

may

not

be

indicative

of

what

can

be

achieved

under

an

exact

Nash

equilibrium.

This

is

especially

important

if

the

purpose

of

the

approximate

solution

is

to

provide

a

good

benchmark

for

a

system

of

selfish

agents,

as

the

benchmark

implied

by

an

-Nash

equilibrium

may

be

unrealistic.

For

these

reasons,

in

this

paper

we

focus

on

the

problem

of

computing

exact

Nash

equilibria.

Building

on

ideas

of

Elkind

et

al.

showed

how

to

find

an

(exact)

Nash

equilibrium

in

polynomial

time

when

the

underlying

graph

has

degree

(that

is,

when

the

graph

is

a

collection

of

paths

and

cycles).

By

contrast,

finding

a

Nash

equilibrium

in

a

general

degree-bounded

graph

appears

to

be

computationally

intractable:

it

has

been

shown

(see

to

be

complete

for

the

complexity

class

PPAD.

extends

this

hardness

result

to

the

case

in

which

the

underlying

graph

has

bounded

pathwidth.

A

graphical

game

may

not

have

a

unique

Nash

equilibrium,

indeed

it

may

have

exponentially

many.

Moreover,

some

Nash

equilibria

are

more

desirable

than

others.

Rather

than

having

an

algorithm

which

merely

finds

some

Nash

equilibrium,

we

would

like

to

have

algorithms

for

finding

Nash

equilibria

with

various

sociallydesirable

properties,

such

as

maximizing

overall

payoff

or

distributing

profit

fairly.

A

useful

property

of

the

data

structure

of

is

that

it

simultaneously

represents

the

set

of

all

Nash

equilibria

of

the

underlying

game.

If

this

representation

has

polynomial

size

(as

is

the

case

for

paths,

as

shown

in

one

may

hope

to

extract

from

it

a

Nash

equilibrium

with

the

desired

properties.

In

fact,

in

the

authors

mention

that

this

is

indeed

possible

if

one

is

interested

in

finding

an

(approximate)

-Nash

equilibrium.

The

goal

of

this

paper

is

to

extend

this

to

exact

Nash

equilibria.

Our

Results

In

this

paper,

we

study

n-player

graphical

games

on

bounded-degree

trees

for

which

the

data

structure

of

has

size

poly(n).

We

focus

on

the

problem

of

finding

exact

Nash

equilibria

with

certain

socially-desirable

properties.

In

particular,

we

show

how

to

find

a

Nash

equilibrium

that

(nearly)

maximizes

the

social

welfare,

i.e.,

the

sum

of

the

players"

payoffs,

and

we

show

how

to

find

a

Nash

equilibrium

that

(nearly)

satisfies

prescribed

payoff

bounds

for

all

players.

Graphical

games

on

bounded-degree

trees

have

a

simple

algebraic

structure.

One

attractive

feature,

which

follows

from

is

that

every

such

game

has

a

Nash

equilibrium

in

which

the

strategy

of

every

player

is

a

rational

number.

Section

studies

the

algebraic

structure

of

those

Nash

equilibria

that

maximize

social

welfare.

We

show

(Theorems

and

that,

surprisingly,

the

set

of

Nash

equilibria

that

maximize

social

welfare

is

more

complex.

In

fact,

for

any

algebraic

number

α

with

degree

at

most

n,

we

exhibit

a

graphical

game

on

a

path

of

length

O(n)

such

that,

in

the

unique

social

welfare-maximizing

Nash

equilibrium

of

this

game,

one

of

the

players

plays

the

mixed

strategy

This

result

shows

that

it

may

be

difficult

to

represent

an

optimal

Nash

equilibrium.

It

seems

to

be

a

novel

feature

of

the

setting

we

consider

here,

that

an

optimal

Nash

equilibrium

is

hard

to

represent,

in

a

situation

where

it

is

easy

to

find

and

represent

a

Nash

equilibrium.

As

the

social

welfare-maximizing

Nash

equilibrium

may

be

hard

to

represent

efficiently,

we

have

to

settle

for

an

approximation.

However,

the

crucial

difference

between

our

approach

and

that

of

previous

papers

is

that

we

require

our

algorithm

to

output

an

exact

Nash

equilibrium,

though

not

necessarily

the

optimal

one

with

respect

to

our

criteria.

In

Section

we

describe

an

algorithm

that

satisfies

this

requirement.

Namely,

we

propose

an

algorithm

that

for

any

finds

a

Nash

equilibrium

whose

total

payoff

is

within

of

optimal.

It

runs

in

polynomial

time

(Theorem

for

any

graphical

game

on

a

bounded-degree

tree

for

which

the

data

structure

proposed

by

(the

so-called

best

response

policy,

defined

below)

is

of

size

poly(n)

(note

that,

as

shown

in

this

is

always

the

case

when

the

underlying

graph

is

a

path).

More

A

related

result

in

a

different

context

was

obtained

by

Datta

who

shows

that

n-player

games

are

universal

in

the

sense

that

any

real

algebraic

variety

can

be

represented

as

the

set

of

totally

mixed

Nash

equilibria

of

such

games.

cisely,

the

running

time

of

our

algorithm

is

polynomial

in

n,

Pmax,

and

where

Pmax

is

the

maximum

absolute

value

of

an

entry

of

a

payoff

matrix,

i.e.,

it

is

a

pseudopolynomial

algorithm,

though

it

is

fully

polynomial

with

respect

to

We

show

(Section

that

under

some

restrictions

on

the

payoff

matrices,

the

algorithm

can

be

transformed

into

a

(truly)

polynomial-time

algorithm

that

outputs

a

Nash

equilibrium

whose

total

payoff

is

within

a

factor

from

the

optimal.

In

Section

we

consider

the

problem

of

finding

a

Nash

equilibrium

in

which

the

expected

payoff

of

each

player

Vi

exceeds

a

prescribed

threshold

Ti.

Using

the

idea

from

Section

we

give

(Theorem

a

fully

polynomial

time

approximation

scheme

for

this

problem.

The

running

time

of

the

algorithm

is

bounded

by

a

polynomial

in

n,

Pmax,

and

If

the

instance

has

a

Nash

equilibrium

satisfying

the

prescribed

thresholds

then

the

algorithm

constructs

a

Nash

equilibrium

in

which

the

expected

payoff

of

each

player

Vi

is

at

least

Ti

In

Section

we

introduce

other

natural

criteria

for

selecting

a

good

Nash

equilibrium

and

we

show

that

the

algorithms

described

in

the

two

previous

sections

can

be

used

as

building

blocks

in

finding

Nash

equilibria

that

satisfy

these

criteria.

In

particular,

in

Section

we

show

how

to

find

a

Nash

equilibrium

that

approximates

the

maximum

social

welfare,

while

guaranteeing

that

each

individual

payoff

is

close

to

a

prescribed

threshold.

In

Section

we

show

how

to

find

a

Nash

equilibrium

that

(nearly)

maximizes

the

minimum

individual

payoff.

Finally,

in

Section

we

show

how

to

find

a

Nash

equilibrium

in

which

the

individual

payoffs

of

the

players

are

close

to

each

other.

Related

Work

Our

approximation

scheme

(Theorem

and

Theorem

shows

a

contrast

between

the

games

that

we

study

and

two-player

n-action

games,

for

which

the

corresponding

problems

are

usually

intractable.

For

two-player

n-action

games,

the

problem

of

finding

Nash

equilibria

with

special

properties

is

typically

NP-hard.

In

particular,

this

is

the

case

for

Nash

equilibria

that

maximize

the

social

welfare

Moreover,

it

is

likely

to

be

intractable

even

to

approximate

such

equilibria.

In

particular,

Chen,

Deng

and

Teng

show

that

there

exists

some

inverse

polynomial

in

n,

for

which

computing

an

-Nash

equilibrium

in

games

with

n

actions

per

player

is

PPAD-complete.

Lipton

and

Markakis

study

the

algebraic

properties

of

Nash

equilibria,

and

point

out

that

standard

quantifier

elimination

algorithms

can

be

used

to

solve

them.

Note

that

these

algorithms

are

not

polynomial-time

in

general.

The

games

we

study

in

this

paper

have

polynomial-time

computable

Nash

equilibria

in

which

all

mixed

strategies

are

rational

numbers,

but

an

optimal

Nash

equilibrium

may

necessarily

include

mixed

strategies

with

high

algebraic

degree.

A

correlated

equilibrium

(CE)

(introduced

by

Aumann

is

a

distribution

over

vectors

of

players"

actions

with

the

property

that

if

any

player

is

told

his

own

action

(the

value

of

his

own

component)

from

a

vector

generated

by

that

distribution,

then

he

cannot

increase

his

expected

payoff

by

changing

his

action.

Any

Nash

equilibrium

is

a

CE

but

the

converse

does

not

hold

in

general.

In

contrast

with

Nash

equilibria,

correlated

equilibria

can

be

found

for

low-degree

graphical

games

(as

well

as

other

classes

of

conciselyrepresented

multiplayer

games)

in

polynomial

time

But,

for

graphical

games

it

is

NP-hard

to

find

a

correlated

equilibrium

that

maximizes

total

payoff

However,

the

NP-hardness

results

apply

to

more

general

games

than

the

one

we

consider

here,

in

particular

the

graphs

are

not

trees.

From

it

is

also

known

that

there

exist

games

for

which

the

expected

total

payoff

of

the

best

correlated

equilibrium

is

higher

than

the

best

Nash

equilibrium,

and

we

discuss

this

issue

further

in

Section

PRELIMINARIES

AND

NOTATION

We

consider

graphical

games

in

which

the

underlying

graph

G

is

an

n-vertex

tree,

in

which

each

vertex

has

at

most

Δ

children.

Each

vertex

has

two

actions,

which

are

denoted

by

and

A

mixed

strategy

of

a

player

V

is

represented

as

a

single

number

v

which

denotes

the

probability

that

V

selects

action

For

the

purposes

of

the

algorithm,

the

tree

is

rooted

arbitrarily.

For

convenience,

we

assume

without

loss

of

generality

that

the

root

has

a

single

child,

and

that

its

payoff

is

independent

of

the

action

chosen

by

the

child.

This

can

be

achieved

by

first

choosing

an

arbitrary

root

of

the

tree,

and

then

adding

a

dummy

parent

of

this

root,

giving

the

new

parent

a

constant

payoff

function,

e.g.,

Given

an

edge

(V,

W

of

the

tree

G,

and

a

mixed

strategy

w

for

W

let

G(V,W

),W

=w

be

the

instance

obtained

from

G

by

deleting

all

nodes

Z

which

are

separated

from

V

by

W

(i.e.,

all

nodes

Z

such

that

the

path

from

Z

to

V

passes

through

W

and

restricting

the

instance

so

that

W

is

required

to

play

mixed

strategy

w.

Definition

Suppose

that

(V,

W

is

an

edge

of

the

tree,

that

v

is

a

mixed

strategy

for

V

and

that

w

is

a

mixed

strategy

for

W

We

say

that

v

is

a

potential

best

response

to

w

(denoted

by

v

pbrV

(w))

if

there

is

an

equilibrium

in

the

instance

G(V,W

),W

=w

in

which

V

has

mixed

strategy

v.

We

define

the

best

response

policy

for

V

given

W

as

B(W,

V

{(w,

v)

v

pbrV

(w),

w

The

upstream

pass

of

the

generic

algorithm

of

considers

every

node

V

(other

than

the

root)

and

computes

the

best

response

policy

for

V

given

its

parent.

With

the

above

assumptions

about

the

root,

the

downstream

pass

is

straightforward.

The

root

selects

a

mixed

strategy

w

for

the

root

W

and

a

mixed

strategy

v

B(W,

V

for

each

child

V

of

W

It

instructs

each

child

V

to

play

v.

The

remainder

of

the

downward

pass

is

recursive.

When

a

node

V

is

instructed

by

its

parent

to

adopt

mixed

strategy

v,

it

does

the

following

for

each

child

U

It

finds

a

pair

(v,

u)

B(V,

U)

(with

the

same

v

value

that

it

was

given

by

its

parent)

and

instructs

U

to

play

u.

The

best

response

policy

for

a

vertex

U

given

its

parent

V

can

be

represented

as

a

union

of

rectangles,

where

a

rectangle

is

defined

by

a

pair

of

closed

intervals

(IV

IU

and

consists

of

all

points

in

IV

IU

it

may

be

the

case

that

one

or

both

of

the

intervals

IV

and

IU

consists

of

a

single

point.

In

order

to

perform

computations

on

B(V,

U),

and

to

bound

the

number

of

rectangles,

used

the

notion

of

an

event

point,

which

is

defined

as

follows.

For

any

set

A

that

is

represented

as

a

union

of

a

finite

number

of

rectangles,

we

say

that

a

point

u

on

the

U-axis

is

a

Uevent

point

of

A

if

u

or

u

or

the

representation

of

A

contains

a

rectangle

of

the

form

IV

IU

and

u

is

an

endpoint

of

IU

V

-event

points

are

defined

similarly.

For

many

games

considered

in

this

paper,

the

underlying

graph

is

an

n-vertex

path,

i.e.,

a

graph

G

(V,

E)

with

V

Vn}

and

E

Vn)}.

In

it

was

shown

that

for

such

games,

the

best

response

policy

has

only

polynomially-many

rectangles.

The

proof

that

the

number

of

rectangles

in

Vj)

is

polynomial

proceeds

by

first

showing

that

the

number

of

event

points

in

Vj

cannot

exceed

the

number

of

event

points

in

B(Vj,

by

more

than

and

using

this

fact

to

bound

the

number

of

rectangles

in

Vj

Let

(V

and

(V

be

the

expected

payoffs

to

V

when

it

plays

and

respectively.

Both

(V

and

(V

are

multilinear

functions

of

the

strategies

of

V

"s

neighbors.

In

what

follows,

we

will

frequently

use

the

following

simple

observation.

CLAIM

For

a

vertex

V

with

a

single

child

U

and

parent

W

given

any

A,

B,

C,

D

Q,

A

B

C

D

Q,

one

can

select

the

payoffs

to

V

so

that

(V

Auw

Bu

Cw

D,

(V

A

uw

B

u

C

w

D

Moreover,

if

all

A,

B,

C,

D,

A

B

C

D

are

integer,

the

payoffs

to

V

are

integer

as

well.

PROOF.

We

will

give

the

proof

for

(V

the

proof

for

(V

is

similar.

For

i,

j

let

Pij

be

the

payoff

to

V

when

U

plays

i,

V

plays

and

W

plays

j.

We

have

(V

w)

w)

u)w

We

have

to

select

the

values

of

Pij

so

that

A,

B,

C,

D.

It

is

easy

to

see

that

the

unique

solution

is

given

by

D,

C

D,

B

D,

A

B

C

D.

The

input

to

all

algorithms

considered

in

this

paper

includes

the

payoff

matrices

for

each

player.

We

assume

that

all

elements

of

these

matrices

are

integer.

Let

Pmax

be

the

greatest

absolute

value

of

any

element

of

any

payoff

matrix.

Then

the

input

consists

of

at

most

numbers,

each

of

which

can

be

represented

using

log

Pmax

bits.

NASH

EQUILIBRIA

THAT

MAXIMIZE

THE

SOCIAL

WELFARE:

SOLUTIONS

IN

R

Q

From

the

point

of

view

of

social

welfare,

the

best

Nash

equilibrium

is

the

one

that

maximizes

the

sum

of

the

players"

expected

payoffs.

Unfortunately,

it

turns

out

that

computing

such

a

strategy

profile

exactly

is

not

possible:

in

this

section,

we

show

that

even

if

all

players"

payoffs

are

integers,

the

strategy

profile

that

maximizes

the

total

payoff

may

have

irrational

coordinates;

moreover,

it

may

involve

algebraic

numbers

of

an

arbitrary

degree.

Warm-up:

quadratic

irrationalities

We

start

by

providing

an

example

of

a

graphical

game

on

a

path

of

length

with

integer

payoffs

such

that

in

the

Nash

equilibrium

that

maximizes

the

total

payoff,

one

of

the

players

has

a

strategy

in

R

Q.

In

the

next

subsection,

we

will

extend

this

example

to

algebraic

numbers

of

arbitrary

degree

n;

to

do

so,

we

have

to

consider

paths

of

length

O(n).

THEOREM

There

exists

an

integer-payoff

graphical

game

G

on

a

path

UV

W

such

that,

in

any

Nash

equilibrium

of

G

that

maximizes

social

welfare,

the

strategy,

u,

of

the

player

U

and

the

total

payoff,

p,

satisfy

u,

p

R

Q.

PROOF.

The

payoffs

to

the

players

in

G

are

specified

as

follows.

The

payoff

to

U

is

identically

i.e.,

(U)

(U)

Using

Claim

we

select

the

payoffs

to

V

so

that

(V

−uw

and

(V

(V

w(u

(u

where

u

and

w

are

the

(mixed)

strategies

of

U

and

W

respectively.

It

follows

that

V

is

indifferent

between

playing

and

if

and

only

if

w

f(u)

Observe

that

for

any

u

we

have

f(u)

The

payoff

to

W

is

if

it

selects

the

same

action

as

V

and

otherwise.

CLAIM

All

Nash

equilibria

of

the

game

G

are

of

the

form

(u,

f(u)).

That

is,

in

any

Nash

equilibrium,

V

plays

v

and

W

plays

w

f(u).

Moreover,

for

any

value

of

u,

the

vector

of

strategies

(u,

f(u))

constitutes

a

Nash

equilibrium.

PROOF.

It

is

easy

to

check

that

for

any

u

the

vector

(u,

f(u))

is

a

Nash

equilibrium.

Indeed,

U

is

content

to

play

any

mixed

strategy

u

no

matter

what

V

and

W

do.

Furthermore,

V

is

indifferent

between

and

as

long

as

w

f(u),

so

it

can

play

Finally,

if

V

plays

and

with

equal

probability,

W

is

indifferent

between

and

so

it

can

play

f(u).

Conversely,

suppose

that

v

Then

W

strictly

prefers

to

play

i.e.,

w

Then

for

V

we

have

(V

(V

(u

i.e.,

(V

(V

which

implies

v

a

contradiction.

Similarly,

if

v

player

W

prefers

to

play

so

we

have

w

Hence,

(V

(V

(u

(u

i.e.,

(V

(V

which

implies

v

a

contradiction.

Finally,

if

v

but

w

f(u),

player

V

is

not

indifferent

between

and

so

he

would

deviate

from

playing

This

completes

the

proof

of

Claim

By

Claim

the

total

payoff

in

any

Nash

equilibrium

of

this

game

is

a

function

of

u.

More

specifically,

the

payoff

to

U

is

the

payoff

to

V

is

−uf(u)

and

the

payoff

to

W

is

Therefore,

the

Nash

equilibrium

with

the

maximum

total

payoff

corresponds

to

the

value

of

u

that

maximizes

g(u)

−u

(u

u

u

u

(u

u

To

find

extrema

of

g(u),

we

compute

h(u)

d

du

g(u).

We

have

h(u)

(u

(u

(u

Hence,

h(u)

if

and

only

if

u

−2

Note

that

−2

The

function

g(u)

changes

sign

at

and

We

have

g(u)

for

g

g(u)

for

u

so

the

extremum

of

g(u)

that

lies

between

and

i.e.,

u

−2

is

a

local

maximum.

We

conclude

that

the

social

welfare-maximizing

Nash

equilibrium

for

this

game

is

given

by

the

vector

of

strategies

The

respective

total

payoff

is

This

concludes

the

proof

of

Theorem

Strategies

of

arbitrary

degree

We

have

shown

that

in

the

social

welfare-maximizing

Nash

equilibrium,

some

players"

strategies

can

be

quadratic

irrationalities,

and

so

can

the

total

payoff.

In

this

subsection,

we

will

extend

this

result

to

show

that

we

can

construct

an

integer-payoff

graphical

game

on

a

path

whose

social

welfare-maximizing

Nash

equilibrium

involves

arbitrary

algebraic

numbers

in

THEOREM

For

any

degree-n

algebraic

number

α

there

exists

an

integer

payoff

graphical

game

on

a

path

of

length

O(n)

such

that,

in

all

social

welfare-maximizing

Nash

equilibria

of

this

game,

one

of

the

players

plays

α.

PROOF.

Our

proof

consists

of

two

steps.

First,

we

construct

a

rational

expression

R(x)

and

a

segment

[x

x

such

that

x

x

Q

and

α

is

the

only

maximum

of

R(x)

on

[x

x

Second,

we

construct

a

graphical

game

whose

Nash

equilibria

can

be

parameterized

by

u

[x

x

so

that

at

the

equilibrium

that

corresponds

to

u

the

total

payoff

is

R(u)

and,

moreover,

some

player"s

strategy

is

u.

It

follows

that

to

achieve

the

payoff-maximizing

Nash

equilibrium,

this

player

has

to

play

α.

The

details

follow.

LEMMA

Given

an

algebraic

number

α

deg(α)

n,

there

exist

Q

and

x

x

Q

such

that

α

is

the

only

maximum

of

R(x)

x

x

on

[x

x

PROOF.

Let

P(x)

be

the

minimal

polynomial

of

α,

i.e.,

a

polynomial

of

degree

n

with

rational

coefficients

whose

leading

coefficient

is

such

that

P(α)

Let

A

αn}

be

the

set

of

all

roots

of

P(x).

Consider

the

polynomial

(x).

It

has

the

same

roots

as

P(x),

and

moreover,

for

any

x

A

we

have

Hence,

A

is

the

set

of

all

maxima

of

Now,

set

R(x)

Observe

that

R(x)

for

all

x

and

R(x)

if

and

only

if

Hence,

the

set

A

is

also

the

set

of

all

maxima

of

R(x)

on

Let

d

min{|αi

α|

αi

A,

αi

α},

and

set

α

max{α

α

min{α

Clearly,

α

is

the

only

zero

(and

hence,

the

only

maximum)

of

R(x)

on

[α

α

Let

x

and

x

be

some

rational

numbers

in

(α

α)

and

(α,

α

respectively;

note

that

by

excluding

the

endpoints

of

the

intervals

we

ensure

that

x

x

As

[x

x

[α

α

we

have

that

α

is

the

only

maximum

of

R(x)

on

[x

x

As

R(x)

is

a

proper

rational

expression

and

all

roots

of

its

denominator

are

simple,

by

partial

fraction

decomposition

theorem,

R(x)

can

be

represented

as

R(x)

x

x

where

are

rational

numbers.

Consider

a

graphical

game

on

the

path

where

k

Intuitively,

we

want

each

triple

Ui)

to

behave

similarly

to

the

players

U,

V

and

W

from

the

game

described

in

the

previous

subsection.

More

precisely,

we

define

the

payoffs

to

the

players

in

the

following

way.

The

payoff

to

is

no

matter

what

everyone

else

does.

The

expected

payoff

to

is

if

it

plays

and

(x

x

−x

if

it

plays

where

and

are

the

strategies

of

and

respectively.

The

expected

payoff

to

is

if

it

plays

and

if

it

plays

where

and

are

the

strategies

of

and

respectively.

For

each

i

k

the

expected

payoff

to

Vi

when

it

plays

is

(Vi)

and

the

expected

payoff

to

Vi

when

it

plays

is

(Vi)

(Vi)

ui)

where

Ai

and

and

ui

are

the

strategies

of

and

Ui,

respectively.

For

each

i

k,

the

payoff

to

Ui

does

not

depend

on

Vi

and

is

if

Ui

and

select

different

actions

and

otherwise.

We

will

now

characterize

the

Nash

equilibria

of

this

game

using

a

sequence

of

claims.

CLAIM

In

all

Nash

equilibria

of

this

game

plays

and

the

strategies

of

and

satisfy

(x

x

x

Consequently,

in

all

Nash

equilibria

we

have

[x

x

PROOF.

The

proof

is

similar

to

that

of

Claim

Let

(x

x

x

Clearly,

the

player

is

indifferent

between

playing

and

if

and

only

if

Suppose

that

Then

strictly

prefers

to

play

i.e.,

so

we

have

(x

x

x

As

x

(x

x

x

x

for

and

x

we

have

so

prefers

to

play

a

contradiction.

Similarly,

if

the

player

strictly

prefers

to

play

i.e.,

so

we

have

(x

x

x

As

x

x

x

we

have

so

prefers

to

play

a

contradiction.

Finally,

if

plays

but

player

is

not

indifferent

between

and

so

he

would

deviate

from

playing

Also,

note

that

x

x

and,

moreover,

[x

x

if

and

only

if

Hence,

in

all

Nash

equilibria

of

this

game

we

have

[x

x

CLAIM

In

all

Nash

equilibria

of

this

game

for

each

i

k

we

have

vi

and

the

strategies

of

the

players

Ui

and

satisfy

fi(ui),

where

u/(u

and

fi(u)

u)

for

i

PROOF.

The

proof

of

this

claim

is

also

similar

to

that

of

Claim

We

use

induction

on

i

to

prove

that

the

statement

of

the

claim

is

true

and,

additionally,

ui

for

i

For

the

base

case

i

note

that

by

the

previous

claim

(recall

that

x

x

are

selected

so

that

x

x

and

consider

the

triple

Let

be

the

strategy

of

First,

suppose

that

Then

strictly

prefers

to

play

i.e.,

Then

for

we

have

As

we

have

which

implies

a

contradiction.

Similarly,

if

player

prefers

to

play

so

we

have

Hence,

It

follows

that

which

implies

a

contradiction.

Finally,

if

but

player

is

not

indifferent

between

and

so

he

would

deviate

from

playing

Moreover,

as

and

we

have

The

argument

for

the

inductive

step

is

similar.

Namely,

suppose

that

the

statement

is

proved

for

all

i

i

and

consider

the

triple

(Ui,

Vi,

Let

vi

be

the

strategy

of

Vi.

First,

suppose

that

vi

Then

strictly

prefers

to

play

i.e.,

Then

for

Vi

we

have

(Vi)

i.e.,

(Vi)

(Vi),

which

implies

vi

a

contradiction.

Similarly,

if

vi

player

prefers

to

play

so

we

have

Hence,

(Vi)

(Vi)

ui.

By

inductive

hypothesis,

we

have

ui

Consequently,

(Vi)

(Vi),

which

implies

vi

a

contradiction.

Finally,

if

vi

but

ui),

player

Vi

is

not

indifferent

between

and

so

he

would

deviate

from

playing

Moreover,

as

ui)

and

ui

we

have

CLAIM

Any

strategy

profile

of

the

form

uk),

where

(x

x

x

and

ui)

for

i

constitutes

a

Nash

equilibrium.

PROOF.

First,

the

player

payoffs

do

not

depend

on

other

players"

actions,

so

he

is

free

to

play

any

strategy

in

As

long

as

(x

−x

+x

player

is

indifferent

between

and

so

he

is

content

to

play

a

similar

argument

applies

to

players

Finally,

for

each

i

k,

the

payoffs

of

player

Ui

only

depend

on

the

strategy

of

player

In

particular,

as

long

as

player

Ui

is

indifferent

between

playing

and

so

he

can

play

any

mixed

strategy

ui

To

complete

the

proof,

note

that

(x

x

x

for

all

for

all

and

ui)

for

all

ui

so

we

have

ui

for

all

i

k.

Now,

let

us

compute

the

total

payoff

under

a

strategy

profile

of

the

form

given

in

Claim

The

payoff

to

is

and

the

expected

payoff

to

each

of

the

Ui,

i

k,

is

The

expected

payoffs

to

and

are

Finally,

for

any

i

k

the

expected

payoff

to

Vi

is

Ti

It

follows

that

to

find

a

Nash

equilibrium

with

the

highest

total

payoff,

we

have

to

maximize

Ti

subject

to

conditions

(x

−x

and

for

i

k

We

would

like

to

express

Ti

as

a

function

of

To

simplify

notation,

set

u

LEMMA

For

i

k,

we

have

ui

u+i

PROOF.

The

proof

is

by

induction

on

i.

For

i

we

have

u/(u

Now,

for

i

suppose

that

(u

i

i

We

have

ui

(u

i

u

i

(u

i

i).

It

follows

that

for

i

k

we

have

Ti

Ai

u

i

u

i

u

i

u

i

Ai

u

i

u

i

−Ai

u

i

u

i

Observe

that

as

varies

from

to

u

varies

from

x

to

x

Therefore,

to

maximize

the

total

payoff,

we

have

to

choose

u

[x

x

so

as

to

maximize

u

Kk

u

k

R(u).

By

construction,

the

only

maximum

of

R(u)

on

[x

x

is

α.

It

follows

that

in

the

payoff-maximizing

Nash

equilibrium

of

our

game

plays

α.

Finally,

note

that

the

payoffs

in

our

game

are

rational

rather

than

integer.

However,

it

is

easy

to

see

that

we

can

multiply

all

payoffs

to

a

player

by

their

greatest

common

denominator

without

affecting

his

strategy.

In

the

resulting

game,

all

payoffs

are

integer.

This

concludes

the

proof

of

Theorem

APPROXIMATING

THE

SOCIALLY

OPTIMAL

NASH

EQUILIBRIUM

We

have

seen

that

the

Nash

equilibrium

that

maximizes

the

social

welfare

may

involve

strategies

that

are

not

in

Q.

Hence,

in

this

section

we

focus

on

finding

a

Nash

equilibrium

that

is

almost

optimal

from

the

social

welfare

perspective.

We

propose

an

algorithm

that

for

any

finds

a

Nash

equilibrium

whose

total

payoff

is

within

from

optimal.

The

running

time

of

this

algorithm

is

polynomial

in

n

and

|Pmax|

(recall

that

Pmax

is

the

maximum

absolute

value

of

an

entry

of

a

payoff

matrix).

While

the

negative

result

of

the

previous

section

is

for

graphical

games

on

paths,

our

algorithm

applies

to

a

wider

range

of

scenarios.

Namely,

it

runs

in

polynomial

time

on

bounded-degree

trees

as

long

as

the

best

response

policy

of

each

vertex,

given

its

parent,

can

be

represented

as

a

union

of

a

polynomial

number

of

rectangles.

Note

that

path

graphs

always

satisfy

this

condition:

in

we

showed

how

to

compute

such

a

representation,

given

a

graph

with

maximum

degree

Consequently,

for

path

graphs

the

running

time

of

our

algorithm

is

guaranteed

to

be

polynomial.

(Note

that

exhibits

a

family

of

graphical

games

on

bounded-degree

trees

for

which

the

best

response

policies

of

some

of

the

vertices,

given

their

parents,

have

exponential

size,

when

represented

as

unions

of

rectangles.)

Due

to

space

restrictions,

in

this

version

of

the

paper

we

present

the

algorithm

for

the

case

where

the

graph

underlying

the

graphical

game

is

a

path.

We

then

state

our

result

for

the

general

case;

the

proof

can

be

found

in

the

full

version

of

this

paper

Suppose

that

s

is

a

strategy

profile

for

a

graphical

game

G.

That

is,

s

assigns

a

mixed

strategy

to

each

vertex

of

G.

let

EPV

(s)

be

the

expected

payoff

of

player

V

under

s

and

let

EP(s)

=P

V

EPV

(s).

Let

M(G)

max{EP(s)

s

is

a

Nash

equilibrium

for

G}.

THEOREM

Suppose

that

G

is

a

graphical

game

on

an

nvertex

path.

Then

for

any

there

is

an

algorithm

that

constructs

a

Nash

equilibrium

s

for

G

that

satisfies

EP(s

M(G)−

The

running

time

of

the

algorithm

is

max/

PROOF.

Let

Vn}

be

the

set

of

all

players.

We

start

by

constructing

the

best

response

policies

for

all

Vi,

i

n

As

shown

in

this

can

be

done

in

time

Let

N

be

a

parameter

to

be

selected

later,

set

δ

and

define

X

{jδ

j

N}.

We

say

that

vj

is

an

event

point

for

a

player

Vi

if

it

is

a

Vi-event

point

for

B(Vi,

or

Vi).

For

each

player

Vi,

consider

a

finite

set

of

strategies

Xi

given

by

Xi

X

{vj

|vj

is

an

event

point

for

Vi}.

It

has

been

shown

in

that

for

any

i

n,

the

best

response

policy

B(Vi,

has

at

most

Vi-event

points.

As

we

require

N

we

have

|Xi|

assume

without

loss

of

generality

that

|Xi|

Order

the

elements

of

Xi

in

increasing

order

as

i

i

i

We

will

refer

to

the

strategies

in

Xi

as

discrete

strategies

of

player

Vi;

a

strategy

profile

in

which

each

player

has

a

discrete

strategy

will

be

referred

to

as

a

discrete

strategy

profile.

We

will

now

show

that

even

we

restrict

each

player

Vi

to

strategies

from

Xi,

the

players

can

still

achieve

a

Nash

equilibrium,

and

moreover,

the

best

such

Nash

equilibrium

(with

respect

to

the

social

welfare)

has

total

payoff

at

least

M(G)

as

long

as

N

is

large

enough.

Let

s

be

a

strategy

profile

that

maximizes

social

welfare.

That

is,

let

s

sn)

where

si

is

the

mixed

strategy

of

player

Vi

and

EP(s)

M(G).

For

i

n,

let

ti

max{xj

i

xj

i

si}.

First,

we

will

show

that

the

strategy

profile

t

tn)

is

a

Nash

equilibrium

for

G.

Fix

any

i,

i

n,

and

let

R

be

the

rectangle

in

B(Vi,

that

contains

(si,

As

is

a

Vi-event

point

of

B(Vi,

we

have

ti,

so

the

point

(ti,

is

inside

R.

Similarly,

the

point

is

a

point

of

B(Vi,

so

we

have

and

therefore

the

point

(ti,

is

inside

R.

This

means

that

for

any

i,

i

n,

we

have

(ti),

which

implies

that

t

tn)

is

a

Nash

equilibrium

for

G.

Now,

let

us

estimate

the

expected

loss

in

social

welfare

caused

by

playing

t

instead

of

s.

LEMMA

For

any

pair

of

strategy

profiles

t,

s

such

that

|ti

si|

δ

we

have

|EPVi

(s)

EPVi

(t)|

for

any

i

n.

PROOF.

Let

Pi

klm

be

the

payoff

of

the

player

Vi,

when

he

plays

k,

plays

l,

and

plays

m.

Fix

i

n

and

for

k,

l,

m

set

tklm

tk

tl

tm

sklm

sk

sl

sm

We

have

|EPVi

(s)

EPVi

(t)|

X

|Pi

klm(tklm

sklm

max

klm

|tklm

sklm

We

will

now

show

that

for

any

k,

l,

m

we

have

|tklm

sklm

clearly,

this

implies

the

lemma.

Indeed,

fix

k,

l,

m

Set

x

tk

x

sk

y

tl

y

sl

z

tm

z

sm

Observe

that

if

k

then

x

x

and

if

k

then

x

x

so

|x

x

δ.

A

similar

argument

shows

|y

y

δ,

|z

z

δ.

Also,

we

have

x,

x

y,

y

z,

z

Hence,

|tklm

−sklm

|xyz−x

y

z

|xyz

x

yz

x

yz

x

y

z

x

y

z

x

y

z

|x

x

|yz

|y

y

|x

z

|z

z

|x

y

Lemma

implies

Pn

|EPVi

(s)

EPVi

(t)|

so

by

choosing

δ

or,

equivalently,

setting

N

we

can

ensure

that

the

total

expected

payoff

for

the

strategy

profile

t

is

within

from

optimal.

We

will

now

show

that

we

can

find

the

best

discrete

Nash

equilibrium

(with

respect

to

the

social

welfare)

using

dynamic

programming.

As

t

is

a

discrete

strategy

profile,

this

means

that

the

strategy

profile

found

by

our

algorithm

will

be

at

least

as

good

as

t.

Define

ml,k

i

to

be

the

maximum

total

payoff

that

can

achieve

if

each

Vj

j

i,

chooses

a

strategy

from

Xj

for

each

j

i

the

strategy

of

Vj

is

a

potential

best

response

to

the

strategy

of

and,

moreover,

plays

xl

Vi

plays

xk

i

If

there

is

no

way

to

choose

the

strategies

for

to

satisfy

these

conditions,

we

set

ml,k

i

The

values

ml,k

i

i

n;

k,

l

N,

can

be

computed

inductively,

as

follows.

We

have

ml,k

for

k,

l

N.

Now,

suppose

that

we

have

already

computed

ml,k

j

for

all

j

i;

k,

l

N.

To

compute

mk,l

i

we

first

check

if

(xk

i

xl

B(Vi,

If

this

is

not

the

case,

we

have

ml,k

i

Otherwise,

consider

the

set

Y

(xl

i.e.,

the

set

of

all

discrete

strategies

of

that

are

potential

best

responses

to

xl

The

proof

of

Theorem

in

implies

that

the

set

(xl

is

non-empty:

the

player

has

a

potential

best

response

to

any

strategy

of

in

particular,

xl

By

construction

of

the

set

this

implies

that

Y

is

not

empty.

For

each

xj

Y

let

pjlk

be

the

payoff

that

receives

when

plays

xj

plays

xl

and

Vi

plays

xk

i

Clearly,

pjlk

can

be

computed

in

constant

time.

Then

we

have

ml,k

i

max{mj,l

pjlk

xj

Y

Finally,

suppose

that

we

have

computed

ml,k

n

for

l,

k

N.

We

still

need

to

take

into

account

the

payoff

of

player

Vn.

Hence,

we

consider

all

pairs

(xk

n,

xl

that

satisfy

xl

(xk

n),

and

pick

the

one

that

maximizes

the

sum

of

mk,l

n

and

the

payoff

of

Vn

when

he

plays

xk

n

and

plays

xl

This

results

in

the

maximum

total

payoff

the

players

can

achieve

in

a

Nash

equilibrium

using

discrete

strategies;

the

actual

strategy

profile

that

produces

this

payoff

can

be

reconstructed

using

standard

dynamic

programming

techniques.

It

is

easy

to

see

that

each

ml,k

i

can

be

computed

in

time

O(N),

i.e.,

all

of

them

can

be

computed

in

time

Recall

that

we

have

to

select

N

to

ensure

that

the

strategy

profile

we

output

has

total

payoff

that

is

within

from

optimal.

We

conclude

that

we

can

compute

an

-approximation

to

the

best

Nash

equilibrium

in

time

max/

This

completes

the

proof

of

Theorem

To

state

our

result

for

the

general

case

(i.e.,

when

the

underlying

graph

is

a

bounded-degree

tree

rather

than

a

path),

we

need

additional

notation.

If

G

has

n

players,

let

q(n)

be

an

upper

bound

on

the

number

of

event

points

in

the

representation

of

any

best

response

policy.

That

is,

we

assume

that

for

any

vertex

U

with

parent

V

B(V,

U)

has

at

most

q(n)

event

points.

We

will

be

interested

in

the

situation

in

which

q(n)

is

polynomial

in

n.

THEOREM

Let

G

be

an

n-player

graphical

game

on

a

tree

in

which

each

node

has

at

most

Δ

children.

Suppose

we

are

given

a

set

of

best-response

policies

for

G

in

which

each

best-response

policy

B(V,

U)

is

represented

by

a

set

of

rectangles

with

at

most

q(n)

event

points.

For

any

there

is

an

algorithm

that

constructs

a

Nash

equilibrium

s

for

G

that

satisfies

EP(s

M(G)

The

running

time

of

the

algorithm

is

polynomial

in

n,

Pmax

and

−1

provided

that

the

tree

has

bounded

degree

(that

is,

Δ

and

q(n)

is

a

polynomial

in

n.

In

particular,

if

N

max((Δ

(Δ

−1

and

Δ

then

the

running

time

is

For

the

proof

of

this

theorem,

see

A

polynomial-time

algorithm

for

multiplicative

approximation

The

running

time

of

our

algorithm

is

pseudopolynomial

rather

than

polynomial,

because

it

includes

a

factor

which

is

polynomial

in

Pmax,

the

maximum

(in

absolute

value)

entry

in

any

payoff

matrix.

If

we

are

interested

in

multiplicative

approximation

rather

than

additive

one,

this

can

be

improved

to

polynomial.

First,

note

that

we

cannot

expect

a

multiplicative

approximation

for

all

inputs.

That

is,

we

cannot

hope

to

have

an

algorithm

that

computes

a

Nash

equilibrium

with

total

payoff

at

least

)M(G).

If

we

had

such

an

algorithm,

then

for

graphical

games

G

with

M(G)

the

algorithm

would

be

required

to

output

the

optimal

solution.

To

show

that

this

is

infeasible,

observe

that

we

can

use

the

techniques

of

Section

to

construct

two

integercoefficient

graphical

games

on

paths

of

length

O(n)

such

that

for

some

X

R

the

maximal

total

payoff

in

the

first

game

is

X,

the

maximal

total

payoff

in

the

second

game

is

−X,

and

for

both

games,

the

strategy

profiles

that

achieve

the

maximal

total

payoffs

involve

algebraic

numbers

of

degree

n.

By

combining

the

two

games

so

that

the

first

vertex

of

the

second

game

becomes

connected

to

the

last

vertex

of

the

first

game,

but

the

payoffs

of

all

players

do

not

change,

we

obtain

a

graphical

game

in

which

the

best

Nash

equilibrium

has

total

payoff

yet

the

strategies

that

lead

to

this

payoff

have

high

algebraic

complexity.

However,

we

can

achieve

a

multiplicative

approximation

when

all

entries

of

the

payoff

matrices

are

positive

and

the

ratio

between

any

two

entries

is

polynomially

bounded.

Recall

that

we

assume

that

all

payoffs

are

integer,

and

let

Pmin

be

the

smallest

entry

of

any

payoff

matrix.

In

this

case,

for

any

strategy

profile

the

payoff

to

player

i

is

at

least

Pmin,

so

the

total

payoff

in

the

social-welfare

maximizing

Nash

equilibrium

s

satisfies

M(G)

nPmin.

Moreover,

Lemma

implies

that

by

choosing

δ

we

can

ensure

that

the

Nash

equilibrium

t

produced

by

our

algorithm

satisfies

nX

EPVi

(s)

nX

EPVi

(t)

nPmin

M(G),

i.e.,

for

this

value

of

δ

we

have

Pn

EPVi

(t)

)M(G).

Recall

that

the

running

time

of

our

algorithm

is

where

N

has

to

be

selected

to

satisfy

N

N

It

follows

that

if

Pmin

Pmax/Pmin

poly(n),

we

can

choose

N

so

that

our

algorithm

provides

a

multiplicative

approximation

guarantee

and

runs

in

time

polynomial

in

n

and

BOUNDED

PAYOFF

NASH

EQUILIBRIA

Another

natural

way

to

define

what

is

a

good

Nash

equilibrium

is

to

require

that

each

player"s

expected

payoff

exceeds

a

certain

threshold.

These

thresholds

do

not

have

to

be

the

same

for

all

players.

In

this

case,

in

addition

to

the

payoff

matrices

of

the

n

players,

we

are

given

n

numbers

Tn,

and

our

goal

is

to

find

a

Nash

equilibrium

in

which

the

payoff

of

player

i

is

at

least

Ti,

or

report

that

no

such

Nash

equilibrium

exists.

It

turns

out

that

we

can

design

an

FPTAS

for

this

problem

using

the

same

techniques

as

in

the

previous

section.

THEOREM

Given

a

graphical

game

G

on

an

n-vertex

path

and

n

rational

numbers

Tn,

suppose

that

there

exists

a

strategy

profile

s

such

that

s

is

a

Nash

equilibrium

for

G

and

EPVi

(s)

Ti

for

i

n.

Then

for

any

we

can

find

in

time

max/

a

strategy

profile

s

such

that

s

is

a

Nash

equilibrium

for

G

and

EPVi

(s

Ti

for

i

n.

PROOF.

The

proof

is

similar

to

that

of

Theorem

First,

we

construct

the

best

response

policies

for

all

players,

choose

N

and

construct

the

sets

Xi,

i

n,

as

described

in

the

proof

of

Theorem

Consider

a

strategy

profile

s

such

that

s

is

a

Nash

equilibrium

for

G

and

EPVi

(s)

Ti

for

i

n.

We

construct

a

strategy

profile

ti

max{xj

i

xj

i

si}

and

use

the

same

argument

as

in

the

proof

of

Theorem

to

show

that

t

is

a

Nash

equilibrium

for

G.

By

Lemma

we

have

|EPVi

(s)

EPVi

(t)|

so

choosing

δ

or,

equivalently,

N

we

can

ensure

EPVi

(t)

Ti

for

i

n.

Now,

we

will

use

dynamic

programming

to

find

a

discrete

Nash

equilibrium

that

satisfies

EPVi

(t)

Ti

for

i

n.

As

t

is

a

discrete

strategy

profile,

our

algorithm

will

succeed

whenever

there

is

a

strategy

profile

s

with

EPVi

(s)

Ti−

for

i

n.

Let

zl,k

i

if

there

is

a

discrete

strategy

profile

such

that

for

any

j

i

the

strategy

of

the

player

Vj

is

a

potential

best

response

to

the

strategy

of

the

expected

payoff

of

Vj

is

at

least

Tj

and,

moreover,

plays

xl

Vi

plays

xk

i

Otherwise,

let

zl,k

i

We

can

compute

zl,k

i

i

n;

k,

l

N

inductively,

as

follows.

We

have

zl,k

for

k,

l

N.

Now,

suppose

that

we

have

already

computed

zl,k

j

for

all

j

i;

k,

l

N.

To

compute

zk,l

i

we

first

check

if

(xk

i

xl

B(Vi,

If

this

is

not

the

case,

clearly,

zk,l

i

Otherwise,

consider

the

set

Y

(xl

i.e.,

the

set

of

all

discrete

strategies

of

that

are

potential

best

responses

to

xl

It

has

been

shown

in

the

proof

of

Theorem

that

Y

For

each

xj

Y

let

pjlk

be

the

payoff

that

receives

when

plays

xj

plays

xl

and

Vi

plays

xk

i

Clearly,

pjlk

can

be

computed

in

constant

time.

If

there

exists

an

xj

Y

such

that

zj,l

and

pjlk

set

zl,k

i

Otherwise,

set

zl,k

i

Having

computed

zl,k

n

l,

k

N,

we

check

if

zl,k

n

for

some

pair

(l,

k).

if

such

a

pair

of

indices

exists,

we

instruct

Vn

to

play

xk

n

and

use

dynamic

programming

techniques

(or,

equivalently,

the

downstream

pass

of

the

algorithm

of

to

find

a

Nash

equilibrium

s

that

satisfies

EPVi

(s

Ti

for

i

n

(recall

that

Vn

is

a

dummy

player,

i.e.,

we

assume

Tn

EPn(s

for

any

choice

of

s

If

zl,k

n

for

all

l,

k

N,

there

is

no

discrete

Nash

equilibrium

s

that

satisfies

EPVi

(s

Ti

for

i

n

and

hence

no

Nash

equilibrium

s

(not

necessarily

discrete)

such

that

EPVi

(s)

Ti

for

i

n.

The

running

time

analysis

is

similar

to

that

for

Theorem

we

conclude

that

the

running

time

of

our

algorithm

is

max/

REMARK

Theorem

can

be

extended

to

trees

of

bounded

degree

in

the

same

way

as

Theorem

Exact

Computation

Another

approach

to

finding

Nash

equilibria

with

bounded

payoffs

is

based

on

inductively

computing

the

subsets

of

the

best

response

policies

of

all

players

so

as

to

exclude

the

points

that

do

not

provide

sufficient

payoffs

to

some

of

the

players.

Formally,

we

say

that

a

strategy

v

of

the

player

V

is

a

potential

best

response

to

a

strategy

w

of

its

parent

W

with

respect

to

a

threshold

vector

T

Tn),

(denoted

by

v

pbrV

(w,

T))

if

there

is

an

equilibrium

in

the

instance

G(V,W

),W

=w

in

which

V

plays

mixed

strategy

v

and

the

payoff

to

any

player

Vi

downstream

of

V

(including

V

is

at

least

Ti.

The

best

response

policy

for

V

with

respect

to

a

threshold

vector

T

is

defined

as

B(W,

V,

T)

{(w,

v)

v

pbrV

(w,

T),

w

It

is

easy

to

see

that

if

any

of

the

sets

B(Vj,

T),

j

n,

is

empty,

it

means

that

it

is

impossible

to

provide

all

players

with

expected

payoffs

prescribed

by

T.

Otherwise,

one

can

apply

the

downstream

pass

of

the

original

algorithm

of

to

find

a

Nash

equilibrium.

As

we

assume

that

Vn

is

a

dummy

vertex

whose

payoff

is

identically

the

Nash

equilibrium

with

these

payoffs

exists

as

long

as

Tn

and

B(Vn,

T)

is

not

empty.

Using

the

techniques

developed

in

it

is

not

hard

to

show

that

for

any

j

n,

the

set

B(Vj

T)

consists

of

a

finite

number

of

rectangles,

and

one

can

compute

Vj

T)

given

B(Vj

T).

The

advantage

of

this

approach

is

that

it

allows

us

to

represent

all

Nash

equilibria

that

provide

required

payoffs

to

the

players.

However,

it

is

not

likely

to

be

practical,

since

it

turns

out

that

the

rectangles

that

appear

in

the

representation

of

B(Vj

T)

may

have

irrational

coordinates.

CLAIM

There

exists

a

graphical

game

G

on

a

path

UV

W

and

a

vector

T

such

that

B(V,

W,

T)

cannot

be

represented

as

a

union

of

a

finite

number

of

rectangles

with

rational

coordinates.

PROOF.

We

define

the

payoffs

to

the

players

in

G

as

follows.

The

payoff

to

U

is

identically

i.e.,

(U)

(U)

Using

Claim

we

select

the

payoffs

to

V

so

that

(V

uw,

(V

(V

w

where

u

and

w

are

the

(mixed)

strategies

of

U

and

W

respectively.

It

follows

that

V

is

indifferent

between

playing

and

if

and

only

if

w

f(u)

observe

that

for

any

u

we

have

f(u)

It

is

not

hard

to

see

that

we

have

B(W,

V

The

payoffs

to

W

are

not

important

for

our

construction;

for

example,

set

Now,

set

T

i.e.,

we

are

interested

in

Nash

equilibria

in

which

V

"s

expected

payoff

is

at

least

Suppose

w

The

player

V

can

play

a

mixed

strategy

v

when

W

is

playing

w

as

long

as

U

plays

u

(w)

(to

ensure

that

V

is

indifferent

between

and

and

(V

(V

uw

The

latter

condition

is

satisfied

if

w

or

w

Note

that

we

have

For

any

other

value

of

w,

any

strategy

of

U

either

makes

V

prefer

one

of

the

pure

strategies

or

does

not

provide

it

with

a

sufficient

expected

payoff.

There

are

also

some

values

of

w

for

which

V

can

play

a

pure

strategy

or

as

a

potential

best

response

to

W

and

guarantee

itself

an

expected

payoff

of

at

least

it

can

be

shown

that

these

values

of

w

form

a

finite

number

of

segments

in

We

conclude

that

any

representation

of

B(W,

V,

T)

as

a

union

of

a

finite

number

of

rectangles

must

contain

a

rectangle

of

the

form

w

]×[v

v

for

some

w

v

v

On

the

other

hand,

it

can

be

shown

that

for

any

integer

payoff

matrices

and

threshold

vectors

and

any

j

n

the

sets

Vj,

T)

contain

no

rectangles

of

the

form

[u

u

]×{v}

or

{v}×[w

w

where

v

R\Q.

This

means

that

if

B(Vn,

T)

is

non-empty,

i.e.,

there

is

a

Nash

equilibrium

with

payoffs

prescribed

by

T,

then

the

downstream

pass

of

the

algorithm

of

can

always

pick

a

strategy

profile

that

forms

a

Nash

equilibrium,

provides

a

payoff

of

at

least

Ti

to

the

player

Vi,

and

has

no

irrational

coordinates.

Hence,

unlike

in

the

case

of

the

Nash

equilibrium

that

maximizes

the

social

welfare,

working

with

irrational

numbers

is

not

necessary,

and

the

fact

that

the

algorithm

discussed

in

this

section

has

to

do

so

can

be

seen

as

an

argument

against

using

this

approach.

OTHER

CRITERIA

FOR

SELECTING

A

NASH

EQUILIBRIUM

In

this

section,

we

consider

several

other

criteria

that

can

be

useful

in

selecting

a

Nash

equilibrium.

Combining

welfare

maximization

with

bounds

on

payoffs

In

many

real

life

scenarios,

we

want

to

maximize

the

social

welfare

subject

to

certain

restrictions

on

the

payoffs

to

individual

players.

For

example,

we

may

want

to

ensure

that

no

player

gets

a

negative

expected

payoff,

or

that

the

expected

payoff

to

player

i

is

at

least

Pi

max

ξ,

where

Pi

max

is

the

maximum

entry

of

i"s

payoff

matrix

and

ξ

is

a

fixed

parameter.

Formally,

given

a

graphical

game

G

and

a

vector

Tn,

let

S

be

the

set

of

all

Nash

equilibria

s

of

G

that

satisfy

Ti

EPVi

(s)

for

i

n,

and

let

ˆs

argmaxs∈S

EP(s).

If

the

set

S

is

non-empty,

we

can

find

a

Nash

equilibrium

ˆs

that

is

-close

to

satisfying

the

payoff

bounds

and

is

within

from

ˆs

with

respect

to

the

total

payoff

by

combining

the

algorithms

of

Section

and

Section

Namely,

for

a

given

choose

δ

as

in

the

proof

of

Theorem

and

let

Xi

be

the

set

of

all

discrete

strategies

of

player

Vi

(for

a

formal

definition,

see

the

proof

of

Theorem

Combining

the

proofs

of

Theorem

and

Theorem

we

can

see

that

the

strategy

profile

ˆt

given

by

ˆti

max{xj

i

xj

i

ˆsi}

satisfies

EPVi

(ˆt)

Ti

|EP(ˆs)

EP(ˆt)|

Define

ˆml,k

i

to

be

the

maximum

total

payoff

that

can

achieve

if

each

Vj,

j

i,

chooses

a

strategy

from

Xj

for

each

j

i

the

strategy

of

Vj

is

a

potential

best

response

to

the

strategy

of

and

the

payoff

to

player

Vj

is

at

least

Tj

and,

moreover,

plays

xl

Vi

plays

xk

i

If

there

is

no

way

to

choose

the

strategies

for

to

satisfy

these

conditions,

we

set

ml,k

i

The

ˆml,k

i

can

be

computed

by

dynamic

programming

similarly

to

the

ml,k

i

and

zl,k

i

in

the

proofs

of

Theorems

and

Finally,

as

in

the

proof

of

Theorem

we

use

ml,k

n

to

select

the

best

discrete

Nash

equilibrium

subject

to

the

payoff

constraints.

Even

more

generally,

we

may

want

to

maximize

the

total

payoff

to

a

subset

of

players

(who

are

assumed

to

be

able

to

redistribute

the

profits

fairly

among

themselves)

while

guaranteeing

certain

expected

payoffs

to

(a

subset

of)

the

other

players.

This

problem

can

be

handled

similarly.

A

minimax

approach

A

more

egalitarian

measure

of

the

quality

of

a

Nash

equilibrium

is

the

minimal

expected

payoff

to

a

player.

The

optimal

solution

with

respect

to

this

measure

is

a

Nash

equilibrium

in

which

the

minimal

expected

payoff

to

a

player

is

maximal.

To

find

an

approximation

to

such

a

Nash

equilibrium,

we

can

combine

the

algorithm

of

Section

with

binary

search

on

the

space

of

potential

lower

bounds.

Note

that

the

expected

payoff

to

any

player

Vi

given

a

strategy

s

always

satisfies

−Pmax

EPVi

(s)

Pmax.

For

a

fixed

we

start

by

setting

T

−Pmax,

T

Pmax,

T∗

(T

T

We

then

run

the

algorithm

of

Section

with

Tn

T∗

If

the

algorithm

succeeds

in

finding

a

Nash

equilibrium

s

that

satisfies

EPVi

(s

T∗

for

all

i

n,

we

set

T

T∗

T∗

(T

T

otherwise,

we

set

T

T∗

T∗

(T

T

and

loop.

We

repeat

this

process

until

|T

T

It

is

not

hard

to

check

that

for

any

p

R,

if

there

is

a

Nash

equilibrium

s

such

that

EPVi

(s)

p,

then

our

algorithm

outputs

a

Nash

equilibrium

s

that

satisfies

EPVi

(s)

The

running

time

of

our

algorithm

is

max

log

−1

log

−1

Equalizing

the

payoffs

When

the

players"

payoff

matrices

are

not

very

different,

it

is

reasonable

to

demand

that

the

expected

payoffs

to

the

players

do

not

differ

by

much

either.

We

will

now

show

that

Nash

equilibria

in

this

category

can

be

approximated

in

polynomial

time

as

well.

Indeed,

observe

that

the

algorithm

of

Section

can

be

easily

modified

to

deal

with

upper

bounds

on

individual

payoffs

rather

than

lower

bounds.

Moreover,

we

can

efficiently

compute

an

approximation

to

a

Nash

equilibrium

that

satisfies

both

the

upper

bound

and

the

lower

bound

for

each

player.

More

precisely,

suppose

that

we

are

given

a

graphical

game

G,

rational

numbers

Tn,

Tn

and

Then

if

there

exists

a

strategy

profile

s

such

that

s

is

a

Nash

equilibrium

for

G

and

Ti

EPVi

(s)

Ti

for

i

n,

we

can

find

a

strategy

profile

s

such

that

s

is

a

Nash

equilibrium

for

G

and

Ti

EPVi

(s

Ti

for

i

n.

The

modified

algorithm

also

runs

in

time

max/

This

observation

allows

us

to

approximate

Nash

equilibria

in

which

all

players"

expected

payoffs

differ

by

at

most

ξ

for

any

fixed

ξ

Given

an

we

set

Tn

−Pmax,

Tn

−Pmax

ξ

and

run

the

modified

version

of

the

algorithm

of

Section

If

it

fails

to

find

a

solution,

we

increment

all

Ti,

Ti

by

and

loop.

We

continue

until

the

algorithm

finds

a

solution,

or

Ti

Pmax.

Suppose

that

there

exists

a

Nash

equilibrium

s

that

satisfies

|EPVi

(s)

EPVj

(s)|

ξ

for

all

i,

j

n.

Set

r

EPVi

(s);

we

have

r

EPVi

(s)

r

ξ

for

all

i

n.

There

exists

a

k

such

that

−Pmax

(k

r

−Pmax

k

During

the

kth

step

of

the

algorithm,

we

set

Tn

−Pmax

i.e.,

we

have

r−

Ti

r,

r

ξ

Ti

r

ξ

That

is,

the

Nash

equilibrium

s

satisfies

Ti

r

EPVi

(s)

r

ξ

Ti

which

means

that

when

Ti

is

set

to

−Pmax

(k

our

algorithm

is

guaranteed

to

output

a

Nash

equilibrium

t

that

satisfies

r

Ti

EPVi

(t)

Ti

r

+ξ

We

conclude

that

whenever

such

a

Nash

equilibrium

s

exists,

our

algorithm

outputs

a

Nash

equilibrium

t

that

satisfies

|EPVi

(t)

EPVj

(t)|

ξ

for

all

i,

j

n.

The

running

time

of

this

algorithm

is

max/

Note

also

that

we

can

find

the

smallest

ξ

for

which

such

a

Nash

equilibrium

exists

by

combining

this

algorithm

with

binary

search

over

the

space

ξ

This

identifies

an

approximation

to

the

fairest

Nash

equilibrium,

i.e.,

one

in

which

the

players"

expected

payoffs

differ

by

the

smallest

possible

amount.

Finally,

note

that

all

results

in

this

section

can

be

extended

to

bounded-degree

trees.

CONCLUSIONS

We

have

studied

the

problem

of

equilibrium

selection

in

graphical

games

on

bounded-degree

trees.

We

considered

several

criteria

for

selecting

a

Nash

equilibrium,

such

as

maximizing

the

social

welfare,

ensuring

a

lower

bound

on

the

expected

payoff

of

each

player,

etc.

First,

we

focused

on

the

algebraic

complexity

of

a

social

welfare-maximizing

Nash

equilibrium,

and

proved

strong

negative

results

for

that

problem.

Namely,

we

showed

that

even

for

graphical

games

on

paths,

any

algebraic

number

α

may

be

the

only

strategy

available

to

some

player

in

all

social

welfaremaximizing

Nash

equilibria.

This

is

in

sharp

contrast

with

the

fact

that

graphical

games

on

trees

always

possess

a

Nash

equilibrium

in

which

all

players"

strategies

are

rational

numbers.

We

then

provided

approximation

algorithms

for

selecting

Nash

equilibria

with

special

properties.

While

the

problem

of

finding

approximate

Nash

equilibria

for

various

classes

of

games

has

received

a

lot

of

attention

in

recent

years,

most

of

the

existing

work

aims

to

find

-Nash

equilibria

that

satisfy

(or

are

-close

to

satisfying)

certain

properties.

Our

approach

is

different

in

that

we

insist

on

outputting

an

exact

Nash

equilibrium,

which

is

-close

to

satisfying

a

given

requirement.

As

argued

in

the

introduction,

there

are

several

reasons

to

prefer

a

solution

that

constitutes

an

exact

Nash

equilibrium.

Our

algorithms

are

fully

polynomial

time

approximation

schemes,

i.e.,

their

running

time

is

polynomial

in

the

inverse

of

the

approximation

parameter

though

they

may

be

pseudopolynomial

with

respect

to

the

input

size.

Under

mild

restrictions

on

the

inputs,

they

can

be

modified

to

be

truly

polynomial.

This

is

the

strongest

positive

result

one

can

derive

for

a

problem

whose

exact

solutions

may

be

hard

to

represent,

as

is

the

case

for

many

of

the

problems

considered

here.

While

we

prove

our

results

for

games

on

a

path,

they

can

be

generalized

to

any

tree

for

which

the

best

response

policies

have

compact

representations

as

unions

of

rectangles.

In

the

full

version

of

the

paper

we

describe

our

algorithms

for

the

general

case.

Further

work

in

this

vein

could

include

extensions

to

the

kinds

of

guarantees

sought

for

Nash

equilibria,

such

as

guaranteeing

total

payoffs

for

subsets

of

players,

selecting

equilibria

in

which

some

players

are

receiving

significantly

higher

payoffs

than

their

peers,

etc.

At

the

moment

however,

it

is

perhaps

more

important

to

tigate

whether

Nash

equilibria

of

graphical

games

can

be

computed

in

a

decentralized

manner,

in

contrast

to

the

algorithms

we

have

introduced

here.

It

is

natural

to

ask

if

our

results

or

those

of

can

be

generalized

to

games

with

three

or

more

actions.

However,

it

seems

that

this

will

make

the

analysis

significantly

more

difficult.

In

particular,

note

that

one

can

view

the

bounded

payoff

games

as

a

very

limited

special

case

of

games

with

three

actions

per

player.

Namely,

given

a

two-action

game

with

payoff

bounds,

consider

a

game

in

which

each

player

Vi

has

a

third

action

that

guarantees

him

a

payoff

of

Ti

no

matter

what

everyone

else

does.

Then

checking

if

there

is

a

Nash

equilibrium

in

which

of

the

players

assigns

a

nonzero

probability

to

his

third

action

is

equivalent

to

checking

if

there

exists

a

Nash

equilibrium

that

satisfies

the

payoff

bounds

in

the

original

game,

and

Section

shows

that

finding

an

exact

solution

to

this

problem

requires

new

ideas.

Alternatively

it

may

be

interesting

to

look

for

similar

results

in

the

context

of

correlated

equilibria

(CE),

especially

since

the

best

CE

may

have

higher

value

(total

expected

payoff)

than

the

best

NE.

The

ratio

between

these

values

is

called

the

mediation

value

in

It

is

known

from

that

the

mediation

value

of

games

with

non-negative

payoffs

is

at

most

and

they

exhibit

a

game

for

which

it

is

infinite.

Furthermore,

a

example

from

also

has

infinite

mediation

value.

REFERENCES

I.

Ashlagi,

D.

Monderer

and

M.

Tenneholtz,

On

the

Value

of

Correlation,

Proceedings

of

Dagstuhl

seminar

R.

Aumann,

Subjectivity

and

Correlation

in

Randomized

Strategies,

Journal

of

Mathematical

Economics

pp.

B.

Blum,

C.

R.

Shelton,

and

D.

Koller,

A

Continuation

Method

for

Nash

Equilibria

in

Structured

Games,

Proceedings

of

X.

Chen,

X.

Deng

and

S.

Teng,

Computing

Nash

Equilibria:

Approximation

and

Smoothed

Complexity,

Proceedings

of

X.

Chen,

X.

Deng,

Settling

the

Complexity

of

Nash-Equilibrium,

Proceedings

of

V.

Conitzer

and

T.

Sandholm,

Complexity

Results

about

Nash

Equilibria,

Proceedings

of

C.

Daskalakis,

P.

W.

Goldberg

and

C.

H.

Papadimitriou,

The

Complexity

of

Computing

a

Nash

Equilibrium,

Proceedings

of

R.

S.

Datta,

Universality

of

Nash

Equilibria,

Mathematics

of

Operations

Research

E.

Elkind,

L.

A.

Goldberg,

and

P.

W.

Goldberg,

Nash

Equilibria

in

Graphical

games

on

Trees

Revisited,

Proceedings

of

ACM

E.

Elkind,

L.

A.

Goldberg,

and

P.

W.

Goldberg,

Computing

Good

Nash

Equilibria

in

Graphical

Games,

I.

Gilboa

and

E.

Zemel,

Nash

and

Correlated

Equilibria:

Some

Complexity

Considerations,

Games

and

Economic

Behavior,

pp.

P.

W.

Goldberg

and

C.

H.

Papadimitriou,

Reducibility

Among

Equilibrium

Problems,

Proceedings

of

M.

Kearns,

M.

Littman,

and

S.

Singh,

Graphical

Models

for

Game

Theory,

Proceedings

of

M.

Littman,

M.

Kearns,

and

S.

Singh,

An

Efficient

Exact

Algorithm

for

Singly

Connected

Graphical

Games,

Proceedings

of

R.

Lipton

and

E.

Markakis,

Nash

Equilibria

via

Polynomial

Equations,

Proceedings

of

L.

Ortiz

and

M.

Kearns,

Nash

Propagation

for

Loopy

Graphical

Games,

Proceedings

of

C.H.

Papadimitriou,

Computing

Correlated

Equilibria

in

Multi-Player

Games,

Proceedings

of

C.H.

Papadimitriou

and

T.

Roughgarden,

Computing

Equilibria

in

Multi-Player

Games,

Proceedings

of

D.

Vickrey

and

D.

Koller,

Multi-agent

Algorithms

for

Solving

Graphical

Games,

Proceedings

of

exponential-time algorithm
various sociallydesirable property
integer-payoff graphical game g
several drawback
degree-bounded graph

graphical game
nash equilibrium
approximation scheme
approximation
overall payoff
distributing profit
social welfare
strategy profile