Complexity

of

(Iterated)

Dominance∗

Vincent

Conitzer

Carnegie

Mellon

University

Forbes

Avenue

Pittsburgh,

PA

USA

conitzer@cs.cmu.edu

Tuomas

Sandholm

Carnegie

Mellon

University

Forbes

Avenue

Pittsburgh,

PA

USA

sandholm@cs.cmu.edu

ABSTRACT

We

study

various

computational

aspects

of

solving

games

using

dominance

and

iterated

dominance.

We

first

study

both

strict

and

weak

dominance

(not

iterated),

and

show

that

checking

whether

a

given

strategy

is

dominated

by

some

mixed

strategy

can

be

done

in

polynomial

time

using

a

single

linear

program

solve.

We

then

move

on

to

iterated

dominance.

We

show

that

determining

whether

there

is

some

path

that

eliminates

a

given

strategy

is

NP-complete

with

iterated

weak

dominance.

This

allows

us

to

also

show

that

determining

whether

there

is

a

path

that

leads

to

a

unique

solution

is

NP-complete.

Both

of

these

results

hold

both

with

and

without

dominance

by

mixed

strategies.

(A

weaker

version

of

the

second

result

(only

without

dominance

by

mixed

strategies)

was

already

known

Iterated

strict

dominance,

on

the

other

hand,

is

path-independent

(both

with

and

without

dominance

by

mixed

strategies)

and

can

therefore

be

done

in

polynomial

time.

We

then

study

what

happens

when

the

dominating

strategy

is

allowed

to

place

positive

probability

on

only

a

few

pure

strategies.

First,

we

show

that

finding

the

dominating

strategy

with

minimum

support

size

is

NP-complete

(both

for

strict

and

weak

dominance).

Then,

we

show

that

iterated

strict

dominance

becomes

path-dependent

when

there

is

a

limit

on

the

support

size

of

the

dominating

strategies,

and

that

deciding

whether

a

given

strategy

can

be

eliminated

by

iterated

strict

dominance

under

this

restriction

is

NP-complete

(even

when

the

limit

on

the

support

size

is

Finally,

we

study

Bayesian

games.

We

show

that,

unlike

in

normal

form

games,

deciding

whether

a

given

pure

strategy

is

dominated

by

another

pure

strategy

in

a

Bayesian

game

is

NP-complete

(both

with

strict

and

weak

dominance);

however,

deciding

whether

a

strategy

is

dominated

by

some

mixed

strategy

can

still

be

done

in

polynomial

time

with

a

single

linear

program

solve

(both

with

strict

and

weak

This

material

is

based

upon

work

supported

by

the

National

Science

Foundation

under

ITR

grants

and

and

a

Sloan

Fellowship.

dominance).

Finally,

we

show

that

iterated

dominance

using

pure

strategies

can

require

an

exponential

number

of

iterations

in

a

Bayesian

game

(both

with

strict

and

weak

dominance).

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

multiagent

systems

with

self-interested

agents,

the

optimal

action

for

one

agent

may

depend

on

the

actions

taken

by

other

agents.

In

such

settings,

the

agents

require

tools

from

game

theory

to

rationally

decide

on

an

action.

Game

theory

offers

various

formal

models

of

strategic

settings-the

best-known

of

which

is

a

game

in

normal

(or

matrix)

form,

specifying

a

utility

(payoff)

for

each

agent

for

each

combination

of

strategies

that

the

agents

choose-as

well

as

solution

concepts,

which,

given

a

game,

specify

which

outcomes

are

reasonable

(under

various

assumptions

of

rationality

and

common

knowledge).

Probably

the

best-known

(and

certainly

the

most-studied)

solution

concept

is

that

of

Nash

equilibrium.

A

Nash

equilibrium

specifies

a

strategy

for

each

player,

in

such

a

way

that

no

player

has

an

incentive

to

(unilaterally)

deviate

from

the

prescribed

strategy.

Recently,

numerous

papers

have

studied

computing

Nash

equilibria

in

various

settings

and

the

complexity

of

constructing

a

Nash

equilibrium

in

normal

form

games

has

been

labeled

one

of

the

two

most

important

open

problems

on

the

boundary

of

P

today

The

problem

of

computing

solutions

according

to

the

perhaps

more

elementary

solution

concepts

of

dominance

and

iterated

dominance

has

received

much

less

attention.

(After

an

early

short

paper

on

an

easy

case

the

main

computational

study

of

these

concepts

has

actually

taken

place

in

a

paper

in

the

game

theory

community

A

strategy

strictly

dominates

another

strategy

if

it

performs

strictly

This

is

not

to

say

that

computer

scientists

have

ignored

better

against

all

vectors

of

opponent

strategies,

and

weakly

dominates

it

if

it

performs

at

least

as

well

against

all

vectors

of

opponent

strategies,

and

strictly

better

against

at

least

one.

The

idea

is

that

dominated

strategies

can

be

eliminated

from

consideration.

In

iterated

dominance,

the

elimination

proceeds

in

rounds,

and

becomes

easier

as

more

strategies

are

eliminated:

in

any

given

round,

the

dominating

strategy

no

longer

needs

to

perform

better

than

or

as

well

as

the

dominated

strategy

against

opponent

strategies

that

were

eliminated

in

earlier

rounds.

Computing

solutions

according

to

(iterated)

dominance

is

important

for

at

least

the

following

reasons:

it

can

be

computationally

easier

than

computing

(for

instance)

a

Nash

equilibrium

(and

therefore

it

can

be

useful

as

a

preprocessing

step

in

computing

a

Nash

equilibrium),

and

(iterated)

dominance

requires

a

weaker

rationality

assumption

on

the

players

than

(for

instance)

Nash

equilibrium,

and

therefore

solutions

derived

according

to

it

are

more

likely

to

occur.

In

this

paper,

we

study

some

fundamental

computational

questions

concerning

dominance

and

iterated

dominance,

including

how

hard

it

is

to

check

whether

a

given

strategy

can

be

eliminated

by

each

of

the

variants

of

these

notions.

The

rest

of

the

paper

is

organized

as

follows.

In

Section

we

briefly

review

definitions

and

basic

properties

of

normal

form

games,

strict

and

weak

dominance,

and

iterated

strict

and

weak

dominance.

In

the

remaining

sections,

we

study

computational

aspects

of

dominance

and

iterated

dominance.

In

Section

we

study

one-shot

(not

iterated)

dominance.

In

Section

we

study

iterated

dominance.

In

Section

we

study

dominance

and

iterated

dominance

when

the

dominating

strategy

can

only

place

probability

on

a

few

pure

strategies.

Finally,

in

Section

we

study

dominance

and

iterated

dominance

in

Bayesian

games.

DEFINITIONS

AND

BASIC

PROPERTIES

In

this

section,

we

briefly

review

normal

form

games,

as

well

as

dominance

and

iterated

dominance

(both

strict

and

weak).

An

n-player

normal

form

game

is

defined

as

follows.

Definition

A

normal

form

game

is

given

by

a

set

of

players

n};

and,

for

each

player

i,

a

(finite)

set

of

pure

strategies

Σi

and

a

utility

function

ui

Σ1

Σ2

Σn

R

(where

σn)

denotes

player

i"s

utility

when

each

player

j

plays

action

σj).

The

two

main

notions

of

dominance

are

defined

as

follows.

Definition

Player

i"s

strategy

σi

is

said

to

be

strictly

dominated

by

player

i"s

strategy

σi

if

for

any

vector

of

strategies

σ−i

for

the

other

players,

ui(σi,

σ−i)

ui(σi,

σ−i).

Player

i"s

strategy

σi

is

said

to

be

weakly

dominated

by

player

i"s

strategy

σi

if

for

any

vector

of

strategies

σ−i

for

the

other

players,

ui(σi,

σ−i)

ui(σi,

σ−i),

and

for

at

least

one

vector

of

strategies

σ−i

for

the

other

players,

ui(σi,

σ−i)

ui(σi,

σ−i).

In

this

definition,

it

is

sometimes

allowed

for

the

dominating

strategy

σi

to

be

a

mixed

strategy,

that

is,

a

probability

distribution

over

pure

strategies.

In

this

case,

the

utilities

in

dominance

altogether.

For

example,

simple

dominance

checks

are

sometimes

used

as

a

subroutine

in

searching

for

Nash

equilibria

the

definition

are

the

expected

There

are

other

notions

of

dominance,

such

as

very

weak

dominance

(in

which

no

strict

inequality

is

required,

so

two

strategies

can

dominate

each

other),

but

we

will

not

study

them

here.

When

we

are

looking

at

the

dominance

relations

for

player

i,

the

other

players

(−i)

can

be

thought

of

as

a

single

Therefore,

in

the

rest

of

the

paper,

when

we

study

one-shot

(not

iterated)

dominance,

we

will

focus

without

loss

of

generality

on

two-player

In

two-player

games,

we

will

generally

refer

to

the

players

as

r

(row)

and

c

(column)

rather

than

and

In

iterated

dominance,

dominated

strategies

are

removed

from

the

game,

and

no

longer

have

any

effect

on

future

dominance

relations.

Iterated

dominance

can

eliminate

more

strategies

than

dominance,

as

follows.

σr

may

originally

not

dominate

σr

because

the

latter

performs

better

against

σc;

but

then,

once

σc

is

removed

because

it

is

dominated

by

σc,

σr

dominates

σr,

and

the

latter

can

be

removed.

For

example,

in

the

following

game,

R

can

be

removed

first,

after

which

D

is

dominated.

L

R

U

D

Either

strict

or

weak

dominance

can

be

used

in

the

definition

of

iterated

dominance.

We

note

that

the

process

of

iterated

dominance

is

never

helped

by

removing

a

dominated

mixed

strategy,

for

the

following

reason.

If

σi

gives

player

i

a

higher

utility

than

σi

against

mixed

strategy

σj

for

player

j

i

(and

strategies

σ−{i,j}

for

the

other

players),

then

for

at

least

one

pure

strategy

σj

that

σj

places

positive

probability

on,

σi

must

perform

better

than

σi

against

σj

(and

strategies

σ−{i,j}

for

the

other

players).

Thus,

removing

the

mixed

strategy

σj

does

not

introduce

any

new

dominances.

More

detailed

discussions

and

examples

can

be

found

in

standard

texts

on

microeconomics

or

game

theory

We

are

now

ready

to

move

on

to

the

core

of

this

paper.

DOMINANCE

(NOT

ITERATED)

In

this

section,

we

study

the

notion

of

one-shot

(not

iterated)

dominance.

As

a

first

observation,

checking

whether

a

given

strategy

is

strictly

(weakly)

dominated

by

some

pure

strategy

is

straightforward,

by

checking,

for

every

pure

strategy

for

that

player,

whether

the

latter

strategy

performs

strictly

better

against

all

the

opponent"s

strategies

(at

least

as

well

against

all

the

opponent"s

strategies,

and

strictly

The

dominated

strategy

σi

is,

of

course,

also

allowed

to

be

mixed,

but

this

has

no

technical

implications

for

the

paper:

when

we

study

one-shot

dominance,

we

ask

whether

a

given

strategy

is

dominated,

and

it

does

not

matter

whether

the

given

strategy

is

pure

or

mixed;

when

we

study

iterated

dominance,

there

is

no

use

in

eliminating

mixed

strategies,

as

we

will

see

shortly.

This

player

may

have

a

very

large

strategy

space

(one

pure

strategy

for

every

vector

of

pure

strategies

for

the

players

that

are

being

replaced).

Nevertheless,

this

will

not

result

in

an

increase

in

our

representation

size,

because

the

original

representation

already

had

to

specify

utilities

for

each

of

these

vectors.

We

note

that

a

restriction

to

two-player

games

would

not

be

without

loss

of

generality

for

iterated

dominance.

This

is

because

for

iterated

dominance,

we

need

to

look

at

the

dominated

strategies

of

each

individual

player,

so

we

cannot

merge

any

players.

better

against

at

least

Next,

we

show

that

checking

whether

a

given

strategy

is

dominated

by

some

mixed

strategy

can

be

done

in

polynomial

time

by

solving

a

single

linear

program.

(Similar

linear

programs

have

been

given

before

we

present

the

result

here

for

completeness,

and

because

we

will

build

on

the

linear

programs

given

below

in

Theorem

Proposition

Given

the

row

player"s

utilities,

a

subset

Dr

of

the

row

player"s

pure

strategies

Σr,

and

a

distinguished

strategy

σ∗

r

for

the

row

player,

we

can

check

in

time

polynomial

in

the

size

of

the

game

(by

solving

a

single

linear

program

of

polynomial

size)

whether

there

exists

some

mixed

strategy

σr,

that

places

positive

probability

only

on

strategies

in

Dr

and

dominates

σ∗

r

both

for

strict

and

for

weak

dominance.

Proof.

Let

pdr

be

the

probability

that

σr

places

on

dr

Dr.

We

will

solve

a

single

linear

program

in

each

of

our

algorithms;

linear

programs

can

be

solved

in

polynomial

time

For

strict

dominance,

the

question

is

whether

the

pdr

can

be

set

so

that

for

every

pure

strategy

for

the

column

player

σc

Σc,

dr∈Dr

pdr

ur(dr,

σc)

ur(σ∗

r

σc).

Because

the

inequality

must

be

strict,

we

cannot

solve

this

directly

by

linear

programming.

We

proceed

as

follows.

Because

the

game

is

finite,

we

may

assume

without

loss

of

generality

that

all

utilities

are

positive

(if

not,

simply

add

a

constant

to

all

utilities.)

Solve

the

following

linear

program:

minimize

dr∈Dr

pdr

such

that

for

any

σc

Σc,

dr∈Dr

pdr

ur(dr,

σc)

ur(σ∗

r

σc).

If

σ∗

r

is

strictly

dominated

by

some

mixed

strategy,

this

linear

program

has

a

solution

with

objective

value

(The

dominating

strategy

is

a

feasible

solution

with

objective

value

exactly

Because

no

constraint

is

binding

for

this

solution,

we

can

reduce

one

of

the

probabilities

slightly

without

affecting

feasibility,

thereby

obtaining

a

solution

with

objective

value

Moreover,

if

this

linear

program

has

a

solution

with

objective

value

there

is

a

mixed

strategy

strictly

dominating

σ∗

r

which

can

be

obtained

by

taking

the

LP

solution

and

adding

the

remaining

probability

to

any

strategy

(because

all

the

utilities

are

positive,

this

will

add

to

the

left

side

of

any

inequality,

so

all

inequalities

will

become

strict).

For

weak

dominance,

we

can

solve

the

following

linear

program:

maximize

σc∈Σc

dr∈Dr

pdr

ur(dr,

σc))

ur(σ∗

r

σc))

such

that

for

any

σc

Σc,

dr∈Dr

pdr

ur(dr,

σc)

ur(σ∗

r

σc);

dr∈Dr

pdr

If

σ∗

r

is

weakly

dominated

by

some

mixed

strategy,

then

that

mixed

strategy

is

a

feasible

solution

to

this

program

with

objective

value

because

for

at

least

one

strategy

σc

Σc

we

have

dr∈Dr

pdr

ur(dr,

σc))

ur(σ∗

r

σc)

On

the

other

hand,

if

this

program

has

a

solution

with

objective

value

then

for

at

least

one

strategy

σc

Σc

we

Recall

that

the

assumption

of

a

single

opponent

(that

is,

the

assumption

of

two

players)

is

without

loss

of

generality

for

one-shot

dominance.

must

have

dr∈Dr

pdr

ur(dr,

σc))

ur(σ∗

r

σc)

and

thus

the

linear

program"s

solution

is

a

weakly

dominating

mixed

strategy.

ITERATED

DOMINANCE

We

now

move

on

to

iterated

dominance.

It

is

well-known

that

iterated

strict

dominance

is

path-independent

is,

if

we

remove

dominated

strategies

until

no

more

dominated

strategies

remain,

in

the

end

the

remaining

strategies

for

each

player

will

be

the

same,

regardless

of

the

order

in

which

strategies

are

removed.

Because

of

this,

to

see

whether

a

given

strategy

can

be

eliminated

by

iterated

strict

dominance,

all

that

needs

to

be

done

is

to

repeatedly

remove

strategies

that

are

strictly

dominated,

until

no

more

dominated

strategies

remain.

Because

we

can

check

in

polynomial

time

whether

any

given

strategy

is

dominated

(whether

or

not

dominance

by

mixed

strategies

is

allowed,

as

described

in

Section

this

whole

procedure

takes

only

polynomial

time.

In

the

case

of

iterated

dominance

by

pure

strategies

with

two

players,

Knuth

et

al.

slightly

improve

on

(speed

up)

the

straightforward

implementation

of

this

procedure

by

keeping

track

of,

for

each

ordered

pair

of

strategies

for

a

player,

the

number

of

opponent

strategies

that

prevent

the

first

strategy

from

dominating

the

second.

Hereby

the

runtime

for

an

m

n

game

is

reduced

from

O((m

to

O((m

(Actually,

they

only

study

very

weak

dominance

(for

which

no

strict

inequalities

are

required),

but

the

approach

is

easily

extended.)

In

contrast,

iterated

weak

dominance

is

known

to

be

For

example,

in

the

following

game,

using

iterated

weak

dominance

we

can

eliminate

M

first,

and

then

D,

or

R

first,

and

then

U.

L

M

R

U

D

Therefore,

while

the

procedure

of

removing

weakly

dominated

strategies

until

no

more

weakly

dominated

strategies

remain

can

certainly

be

executed

in

polynomial

time,

which

strategies

survive

in

the

end

depends

on

the

order

in

which

we

remove

the

dominated

strategies.

We

will

investigate

two

questions

for

iterated

weak

dominance:

whether

a

given

strategy

is

eliminated

in

some

path,

and

whether

there

is

a

path

to

a

unique

solution

(one

pure

strategy

left

per

player).

We

will

show

that

both

of

these

problems

are

computationally

hard.

Definition

Given

a

game

in

normal

form

and

a

distinguished

strategy

σ∗

IWD-STRATEGY-ELIMINATION

asks

whether

there

is

some

path

of

iterated

weak

dominance

that

eliminates

σ∗

Given

a

game

in

normal

form,

IWDUNIQUE-SOLUTION

asks

whether

there

is

some

path

of

iterated

weak

dominance

that

leads

to

a

unique

solution

(one

strategy

left

per

player).

The

following

lemma

shows

a

special

case

of

normal

form

games

in

which

allowing

for

weak

dominance

by

mixed

strategies

(in

addition

to

weak

dominance

by

pure

strategies)

does

There

is,

however,

a

restriction

of

weak

dominance

called

nice

weak

dominance

which

is

path-independent

For

an

overview

of

path-independence

results,

see

Apt

not

help.

We

will

prove

the

hardness

results

in

this

setting,

so

that

they

will

hold

whether

or

not

dominance

by

mixed

strategies

is

allowed.

Lemma

Suppose

that

all

the

utilities

in

a

game

are

in

Then

every

pure

strategy

that

is

weakly

dominated

by

a

mixed

strategy

is

also

weakly

dominated

by

a

pure

strategy.

Proof.

Suppose

pure

strategy

σ

is

weakly

dominated

by

mixed

strategy

σ∗

If

σ

gets

a

utility

of

against

some

opponent

strategy

(or

vector

of

opponent

strategies

if

there

are

more

than

players),

then

all

the

pure

strategies

that

σ∗

places

positive

probability

on

must

also

get

a

utility

of

against

that

opponent

strategy

(or

else

the

expected

utility

would

be

smaller

than

Moreover,

at

least

one

of

the

pure

strategies

that

σ∗

places

positive

probability

on

must

get

a

utility

of

against

an

opponent

strategy

that

σ

gets

against

(or

else

the

inequality

would

never

be

strict).

It

follows

that

this

pure

strategy

weakly

dominates

σ.

We

are

now

ready

to

prove

the

main

results

of

this

section.

Theorem

IWD-STRATEGY-ELIMINATION

is

NPcomplete,

even

with

players,

and

with

and

being

the

only

utilities

occurring

in

the

matrix-whether

or

not

dominance

by

mixed

strategies

is

allowed.

Proof.

The

problem

is

in

NP

because

given

a

sequence

of

strategies

to

be

eliminated,

we

can

easily

check

whether

this

is

a

valid

sequence

of

eliminations

(even

when

dominance

by

mixed

strategies

is

allowed,

using

Proposition

To

show

that

the

problem

is

NP-hard,

we

reduce

an

arbitrary

satisfiability

instance

(given

by

a

nonempty

set

of

clauses

C

over

a

nonempty

set

of

variables

V

with

corresponding

literals

L

{+v

v

V

{−v

v

V

to

the

following

IWD-STRATEGY-ELIMINATION

instance.

(In

this

instance,

we

will

specify

that

certain

strategies

are

uneliminable.

A

strategy

σr

can

be

made

uneliminable,

even

when

and

are

the

only

allowed

utilities,

by

adding

another

strategy

σr

and

another

opponent

strategy

σc,

so

that:

σr

and

σr

are

the

only

strategies

that

give

the

row

player

a

utility

of

against

σc.

σr

and

σr

always

give

the

row

player

the

same

utility.

σc

is

the

only

strategy

that

gives

the

column

player

a

utility

of

against

σr,

but

otherwise

σc

always

gives

the

column

player

utility

This

makes

it

impossible

to

eliminate

any

of

these

three

strategies.

We

will

not

explicitly

specify

the

additional

strategies

to

make

the

proof

more

legible.)

In

this

proof,

we

will

denote

row

player

strategies

by

s,

and

column

player

strategies

by

t,

to

improve

legibility.

Let

the

row

player"s

pure

strategy

set

be

given

as

follows.

For

every

variable

v

V

the

row

player

has

corresponding

strategies

+v,

+v,

−v,

−v.

Additionally,

the

row

player

has

the

following

strategies:

and

where

σ∗

r

(that

is,

it

is

the

strategy

we

seek

to

eliminate).

Finally,

for

every

clause

c

C,

the

row

player

has

corresponding

strategies

c

(uneliminable)

and

c.

Let

the

column

player"s

pure

strategy

set

be

given

as

follows.

For

every

variable

v

V

the

column

player

has

a

corresponding

strategy

tv.

For

every

clause

c

C,

the

column

player

has

a

corresponding

strategy

tc,

and

additionally,

for

every

literal

l

L

that

occurs

in

c,

a

strategy

tc,l.

For

every

variable

v

V

the

column

player

has

corresponding

strategies

t+v,

t−v

(both

uneliminable).

Finally,

the

column

player

has

three

additional

strategies:

(uneliminable),

and

The

utility

function

for

the

row

player

is

given

as

follows:

+v,

tv)

for

all

v

V

+v,

tv)

for

all

v

V

−v,

tv)

for

all

v

V

−v,

tv)

for

all

v

V

+v,

for

all

v

V

+v,

for

all

v

V

−v,

for

all

v

V

−v,

for

all

v

V

ur(sb

+v,

t+v)

for

all

v

V

and

b

ur(sb

−v,

t−v)

for

all

v

V

and

b

ur(sl,

t)

otherwise

for

all

l

L

and

t

tc)

for

all

c

C;

tc)

for

all

c

C;

ur(sb

for

all

b

ur(sb

t)

otherwise

for

all

b

and

t

ur(sb

c,

t)

otherwise

for

all

c

C

and

b

and

the

row

player"s

utility

is

in

every

other

case.

The

utility

function

for

the

column

player

is

given

as

follows:

uc(s,

tv)

for

all

v

V

and

s

uc(s,

for

all

s

l

tc)

for

all

c

C

and

l

L

where

l

c

(literal

l

occurs

in

clause

c);

for

all

c

C

and

L,

where

c;

c,

tc)

for

all

c

C;

c,

tc)

for

all

c

C;

uc(sb

c,

tc,l)

for

all

c

C,

l

L,

and

b

tc)

tc,l)

otherwise

for

all

c

C

and

l

L;

and

the

column

player"s

utility

is

in

every

other

case.

We

now

show

that

the

two

instances

are

equivalent.

First,

suppose

there

is

a

solution

to

the

satisfiability

instance:

that

is,

a

truth-value

assignment

to

the

variables

in

V

such

that

all

clauses

are

satisfied.

Then,

consider

the

following

sequence

of

eliminations

in

our

game:

For

every

variable

v

that

is

set

to

true

in

the

assignment,

eliminate

tv

(which

gives

the

column

player

utility

everywhere).

Then,

for

every

variable

v

that

is

set

to

true

in

the

assignment,

eliminate

+v

using

+v

(which

is

possible

because

tv

has

been

eliminated,

and

because

has

not

been

eliminated

(yet)).

Now

eliminate

(which

gives

the

column

player

utility

everywhere).

Next,

for

every

variable

v

that

is

set

to

false

in

the

assignment,

eliminate

−v

using

−v

(which

is

possible

because

has

been

eliminated,

and

because

tv

has

not

been

eliminated

(yet)).

For

every

clause

c

which

has

the

variable

corresponding

to

one

of

its

positive

literals

l

+v

set

to

true

in

the

assignment,

eliminate

tc

using

tc,l

(which

is

possible

because

l

has

been

eliminated,

and

c

has

not

been

eliminated

(yet)).

For

every

clause

c

which

has

the

variable

corresponding

to

one

of

its

negative

literals

l

−v

set

to

false

in

the

assignment,

eliminate

tc

using

tc,l

(which

is

possible

because

l

has

been

eliminated,

and

c

has

not

been

eliminated

(yet)).

Because

the

assignment

satisfied

the

formula,

all

the

tc

have

now

been

eliminated.

Thus,

we

can

eliminate

σ∗

r

using

It

follows

that

there

is

a

solution

to

the

IWD-STRATEGY-ELIMINATION

instance.

Now

suppose

there

is

a

solution

to

the

IWD-STRATEGYELIMINATION

instance.

By

Lemma

we

can

assume

that

all

the

dominances

are

by

pure

strategies.

We

first

observe

that

only

can

eliminate

σ∗

r

because

it

is

the

only

other

strategy

that

gets

the

row

player

a

utility

of

against

and

is

uneliminable.

However,

because

performs

better

than

against

the

tc

strategies,

it

follows

that

all

of

the

tc

strategies

must

be

eliminated.

For

each

c

C,

the

strategy

tc

can

only

be

eliminated

by

one

of

the

strategies

tc,l

(with

the

same

c),

because

these

are

the

only

other

strategies

that

get

the

column

player

a

utility

of

against

c,

and

c

is

uneliminable.

But,

in

order

for

some

tc,l

to

eliminate

tc,

l

must

be

eliminated

first.

Only

l

can

eliminate

l

because

it

is

the

only

other

strategy

that

gets

the

row

player

a

utility

of

against

tl,

and

tl

is

uneliminable.

We

next

show

that

for

every

v

V

only

one

of

+v,

−v

can

be

eliminated.

This

is

because

in

order

for

+v

to

eliminate

+v,

tv

needs

to

have

been

eliminated

and

not

(so

tv

must

be

eliminated

before

but

in

order

for

−v

to

eliminate

−v,

needs

to

have

been

eliminated

and

tv,

not

(so

must

be

eliminated

before

tv).

So,

set

v

to

true

if

+v

is

eliminated,

and

to

false

otherwise

Because

by

the

above,

for

every

clause

c,

one

of

the

l

with

l

c

must

be

eliminated,

it

follows

that

this

is

a

satisfying

assignment

to

the

satisfiability

instance.

Using

Theorem

it

is

now

(relatively)

easy

to

show

that

IWD-UNIQUE-SOLUTION

is

also

NP-complete

under

the

same

restrictions.

Theorem

IWD-UNIQUE-SOLUTION

is

NP-complete,

even

with

players,

and

with

and

being

the

only

utilities

occurring

in

the

matrix-whether

or

not

dominance

by

mixed

strategies

is

allowed.

Proof.

Again,

the

problem

is

in

NP

because

we

can

nondeterministically

choose

the

sequence

of

eliminations

and

verify

whether

it

is

correct.

To

show

NP-hardness,

we

reduce

an

arbitrary

IWD-STRATEGY-ELIMINATION

instance

to

the

following

IWD-UNIQUE-SOLUTION

instance.

Let

all

the

strategies

for

each

player

from

the

original

instance

remain

part

of

the

new

instance,

and

let

the

utilities

resulting

from

the

players

playing

a

pair

of

these

strategies

be

the

same.

We

add

three

additional

strategies

σ1

r

σ2

r

σ3

r

for

the

row

player,

and

three

additional

strategies

σ1

c

σ2

c

σ3

c

for

the

column

player.

Let

the

additional

utilities

be

as

follows:

ur(σr,

σj

c)

for

all

σr

r

σ2

r

σ3

r

and

j

ur(σi

r,

σc)

for

all

i

and

σc

c

σ3

c

ur(σi

r,

σ2

c

for

all

i

r

σ3

c

and

the

row

player"s

utility

is

in

all

other

cases

involving

a

new

strategy.

r

σc)

for

all

σc

c

σ2

c

σ3

c

uc(σ∗

r

σj

c)

for

all

j

(σ∗

r

is

the

strategy

to

be

eliminated

in

the

original

instance);

uc(σi

r,

σ1

c

for

all

i

r

σ2

c

r

σ3

c

and

the

column

player"s

utility

is

in

all

other

cases

involving

a

new

strategy.

We

proceed

to

show

that

the

two

instances

are

equivalent.

First

suppose

there

exists

a

solution

to

the

original

IWDSTRATEGY-ELIMINATION

instance.

Then,

perform

the

same

sequence

of

eliminations

to

eliminate

σ∗

r

in

the

new

IWD-UNIQUE-SOLUTION

instance.

(This

is

possible

because

at

any

stage,

any

weak

dominance

for

the

row

player

in

the

original

instance

is

still

a

weak

dominance

in

the

new

instance,

because

the

two

strategies"

utilities

for

the

row

player

are

the

same

when

the

column

player

plays

one

of

the

new

strategies;

and

the

same

is

true

for

the

column

player.)

Once

σ∗

r

is

eliminated,

let

σ1

c

eliminate

σ2

c

(It

performs

better

against

σ2

r

Then,

let

σ1

r

eliminate

all

the

other

remaining

strategies

for

the

row

player.

(It

always

performs

better

against

either

σ1

c

or

σ3

c

Finally,

σ1

c

is

the

unique

best

response

against

σ1

r

among

the

column

player"s

remaining

strategies,

so

let

it

eliminate

all

the

other

remaining

strategies

for

the

column

player.

Thus,

there

exists

a

solution

to

the

IWD-UNIQUE-SOLUTION

instance.

Now

suppose

there

exists

a

solution

to

the

IWD-UNIQUESOLUTION

instance.

By

Lemma

we

can

assume

that

all

the

dominances

are

by

pure

strategies.

We

will

show

that

of

the

new

strategies

r

σ2

r

σ3

r

σ1

c

σ2

c

σ3

c

can

either

eliminate

another

strategy,

or

be

eliminated

before

σ∗

r

is

eliminated.

Thus,

there

must

be

a

sequence

of

eliminations

ending

in

the

elimination

of

σ∗

r

which

does

not

involve

any

of

the

new

strategies,

and

is

therefore

a

valid

sequence

of

eliminations

in

the

original

game

(because

all

original

strategies

perform

the

same

against

each

new

strategy).

We

now

show

that

this

is

true

by

exhausting

all

possibilities

for

the

first

elimination

before

σ∗

r

is

eliminated

that

involves

a

new

strategy.

of

the

σi

r

can

be

eliminated

by

a

σr

r

σ2

r

σ3

r

because

the

σi

r

perform

better

against

σ1

c

σ1

r

cannot

eliminate

any

other

strategy,

because

it

always

performs

poorer

against

σ2

c

σ2

r

and

σ3

r

are

equivalent

from

the

row

player"s

perspective

(and

thus

cannot

eliminate

each

other),

and

cannot

eliminate

any

other

strategy

because

they

always

perform

poorer

against

σ3

c

of

the

σj

c

can

be

eliminated

by

a

σc

c

σ2

c

σ3

c

because

the

σj

c

always

perform

better

against

either

σ1

r

or

σ2

r

σ1

c

cannot

eliminate

any

other

strategy,

because

it

always

performs

poorer

against

either

σ∗

r

or

σ3

r

σ2

c

cannot

eliminate

any

other

strategy,

because

it

always

performs

poorer

against

σ2

r

or

σ3

r

σ3

c

cannot

eliminate

any

other

strategy,

because

it

always

performs

poorer

against

σ1

r

or

σ3

r

Thus,

there

exists

a

solution

to

the

IWDSTRATEGY-ELIMINATION

instance.

A

slightly

weaker

version

of

the

part

of

Theorem

concerning

dominance

by

pure

strategies

only

is

the

main

result

of

Gilboa

et

al.

(Besides

not

proving

the

result

for

dominance

by

mixed

strategies,

the

original

result

was

weaker

because

it

required

utilities

rather

than

just

(and

because

of

this,

our

Lemma

cannot

be

applied

to

it

to

get

the

result

for

mixed

strategies).)

(ITERATED)

DOMINANCE

USING

MIXED

STRATEGIES

WITH

SMALL

SUPPORTS

When

showing

that

a

strategy

is

dominated

by

a

mixed

strategy,

there

are

several

reasons

to

prefer

exhibiting

a

dominating

strategy

that

places

positive

probability

on

as

few

pure

strategies

as

possible.

First,

this

will

reduce

the

number

of

bits

required

to

specify

the

dominating

strategy

(and

thus

the

proof

of

dominance

can

be

communicated

quicker):

if

the

dominating

mixed

strategy

places

positive

probability

on

only

k

strategies,

then

it

can

be

specified

using

k

real

numbers

for

the

probabilities,

plus

k

log

m

(where

m

is

the

number

of

strategies

for

the

player

under

consideration)

bits

to

indicate

which

strategies

are

used.

Second,

the

proof

of

dominance

will

be

cleaner:

for

a

dominating

mixed

strategy,

it

is

typically

(always

in

the

case

of

strict

dominance)

possible

to

spread

some

of

the

probability

onto

any

unused

pure

strategy

and

still

have

a

dominating

strategy,

but

this

obscures

which

pure

strategies

are

the

ones

that

are

key

in

making

the

mixed

strategy

dominating.

Third,

because

(by

the

previous)

the

argument

for

eliminating

the

dominated

strategy

is

simpler

and

easier

to

understand,

it

is

more

likely

to

be

accepted.

Fourth,

the

level

of

risk

neutrality

required

for

the

argument

to

work

is

reduced,

at

least

in

the

extreme

case

where

dominance

by

a

single

pure

strategy

can

be

exhibited

(no

risk

neutrality

is

required

here).

This

motivates

the

following

problem.

Definition

(MINIMUM-DOMINATING-SET).

We

are

given

the

row

player"s

utilities

of

a

game

in

normal

form,

a

distinguished

strategy

σ∗

for

the

row

player,

a

specification

of

whether

the

dominance

should

be

strict

or

weak,

and

a

number

k.

We

are

asked

whether

there

exists

a

mixed

strategy

σ

for

the

row

player

that

places

positive

probability

on

at

most

k

pure

strategies,

and

dominates

σ∗

in

the

required

sense.

Unfortunately,

this

problem

is

NP-complete.

Theorem

MINIMUM-DOMINATING-SET

is

NPcomplete,

both

for

strict

and

for

weak

dominance.

Proof.

The

problem

is

in

NP

because

we

can

nondeterministically

choose

a

set

of

at

most

k

strategies

to

give

positive

probability,

and

decide

whether

we

can

dominate

σ∗

with

these

k

strategies

as

described

in

Proposition

To

show

NP-hardness,

we

reduce

an

arbitrary

SET-COVER

instance

(given

a

set

S,

subsets

Sr,

and

a

number

t,

can

all

of

S

be

covered

by

at

most

t

of

the

subsets?)

to

the

following

MINIMUM-DOMINATING-SET

instance.

For

every

element

s

S,

there

is

a

pure

strategy

σs

for

the

column

player.

For

every

subset

Si,

there

is

a

pure

strategy

σSi

for

the

row

player.

Finally,

there

is

the

distinguished

pure

strategy

σ∗

for

the

row

player.

The

row

player"s

utilities

are

as

follows:

ur(σSi

σs)

t

if

s

Si;

ur(σSi

σs)

if

s

Si;

ur(σ∗

σs)

for

all

s

S.

Finally,

we

let

k

t.

We

now

proceed

to

show

that

the

two

instances

are

equivalent.

First

suppose

there

exists

a

solution

to

the

SET-COVER

instance.

Without

loss

of

generality,

we

can

assume

that

there

are

exactly

k

subsets

in

the

cover.

Then,

for

every

Si

that

is

in

the

cover,

let

the

dominating

strategy

σ

place

exactly

k

probability

on

the

corresponding

pure

strategy

σSi

Now,

if

we

let

n(s)

be

the

number

of

subsets

in

the

cover

containing

s

(we

observe

that

that

n(s)

then

for

every

strategy

σs

for

the

column

player,

the

row

player"s

expected

utility

for

playing

σ

when

the

column

player

is

playing

σs

is

u(σ,

σs)

n(s)

k

(k

k

u(σ∗

σs).

So

σ

strictly

(and

thus

also

weakly)

dominates

σ∗

and

there

exists

a

solution

to

the

MINIMUM-DOMINATING-SET

instance.

Now

suppose

there

exists

a

solution

to

the

MINIMUMDOMINATING-SET

instance.

Consider

the

(at

most

k)

pure

strategies

of

the

form

σSi

on

which

the

dominating

mixed

strategy

σ

places

positive

probability,

and

let

T

be

the

collection

of

the

corresponding

subsets

Si.

We

claim

that

T

is

a

cover.

For

suppose

there

is

some

s

S

that

is

not

in

any

of

the

subsets

in

T

Then,

if

the

column

player

plays

σs,

the

row

player

(when

playing

σ)

will

always

receive

utility

opposed

to

the

utility

of

the

row

player

would

receive

for

playing

σ∗

contradicting

the

fact

that

σ

dominates

σ∗

(whether

this

dominance

is

weak

or

strict).

It

follows

that

there

exists

a

solution

to

the

SET-COVER

instance.

On

the

other

hand,

if

we

require

that

the

dominating

strategy

only

places

positive

probability

on

a

very

small

number

of

pure

strategies,

then

it

once

again

becomes

easy

to

check

whether

a

strategy

is

dominated.

Specifically,

to

find

out

whether

player

i"s

strategy

σ∗

is

dominated

by

a

strategy

that

places

positive

probability

on

only

k

pure

strategies,

we

can

simply

check,

for

every

subset

of

k

of

player

i"s

pure

strategies,

whether

there

is

a

strategy

that

places

positive

probability

only

on

these

k

strategies

and

dominates

σ∗

using

Proposition

This

requires

only

O(|Σi|k

such

checks.

Thus,

if

k

is

a

constant,

this

constitutes

a

polynomial-time

algorithm.

A

natural

question

to

ask

next

is

whether

iterated

strict

dominance

remains

computationally

easy

when

dominating

strategies

are

required

to

place

positive

probability

on

at

most

k

pure

strategies,

where

k

is

a

small

constant.

(We

have

already

shown

in

Section

that

iterated

weak

dominance

is

hard

even

when

k

that

is,

only

dominance

by

pure

strategies

is

allowed.)

Of

course,

if

iterated

strict

dominance

were

path-independent

under

this

restriction,

computational

easiness

would

follow

as

it

did

in

Section

However,

it

turns

out

that

this

is

not

the

case.

Observation

If

we

restrict

the

dominating

strategies

to

place

positive

probability

on

at

most

two

pure

strategies,

iterated

strict

dominance

becomes

path-dependent.

Proof.

Consider

the

following

game:

Let

(i,

j)

denote

the

outcome

in

which

the

row

player

plays

the

ith

row

and

the

column

player

plays

the

jth

column.

Because

and

are

all

Nash

equilibria,

of

the

column

player"s

pure

strategies

will

ever

be

eliminated,

and

neither

will

rows

and

We

now

observe

that

randomizing

uniformly

over

rows

and

dominates

row

and

randomizing

uniformly

over

rows

and

dominates

row

However,

if

we

eliminate

row

first,

it

becomes

impossible

to

dominate

row

without

randomizing

over

at

least

pure

strategies.

Indeed,

iterated

strict

dominance

turns

out

to

be

hard

even

when

k

Theorem

If

we

restrict

the

dominating

strategies

to

place

positive

probability

on

at

most

three

pure

strategies,

it

becomes

NP-complete

to

decide

whether

a

given

strategy

can

be

eliminated

using

iterated

strict

dominance.

Proof.

The

problem

is

in

NP

because

given

a

sequence

of

strategies

to

be

eliminated,

we

can

check

in

polynomial

time

whether

this

is

a

valid

sequence

of

eliminations

(for

any

strategy

to

be

eliminated,

we

can

check,

for

every

subset

of

three

other

strategies,

whether

there

is

a

strategy

placing

positive

probability

on

only

these

three

strategies

that

dominates

the

strategy

to

be

eliminated,

using

Proposition

To

show

that

the

problem

is

NP-hard,

we

reduce

an

arbitrary

satisfiability

instance

(given

by

a

nonempty

set

of

clauses

C

over

a

nonempty

set

of

variables

V

with

corresponding

literals

L

{+v

v

V

{−v

v

V

to

the

following

two-player

game.

For

every

variable

v

V

the

row

player

has

strategies

s+v,

s−v,

v,

v,

v,

v,

and

the

column

player

has

strategies

v,

v,

v,

v.

For

every

clause

c

C,

the

row

player

has

a

strategy

sc,

and

the

column

player

has

a

strategy

tc,

as

well

as,

for

every

literal

l

occurring

in

c,

an

additional

strategy

tl

c.

The

row

player

has

two

additional

strategies

and

is

the

strategy

that

we

are

seeking

to

eliminate.)

Finally,

the

column

player

has

one

additional

strategy

The

utility

function

for

the

row

player

is

given

as

follows

(where

is

some

sufficiently

small

number):

ur(s+v,

tj

v)

if

j

for

all

v

V

ur(s+v,

tj

v)

if

j

for

all

v

V

ur(s−v,

tj

v)

if

j

for

all

v

V

ur(s−v,

tj

v)

if

j

for

all

v

V

ur(s+v,

t)

ur(s−v,

t)

for

all

v

V

and

t

v,

v,

v,

v};

ur(si

v,

ti

v)

for

all

v

V

and

i

ur(si

v,

t)

for

all

v

V

i

and

t

ti

v;

ur(sc,

tc)

for

all

c

C;

ur(sc,

t)

for

all

c

C

and

t

tc;

t)

for

all

t

tc)

for

all

c

C;

t)

for

all

t

{tc

c

C}.

The

utility

function

for

the

column

player

is

given

as

follows:

uc(si

v,

ti

v)

for

all

v

V

and

i

uc(s,

ti

v)

for

all

v

V

i

and

s

si

v;

uc(sc,

tc)

for

all

c

C;

uc(sl,

tc)

for

all

c

C

and

l

L

occurring

in

c;

uc(s,

tc)

for

all

c

C

and

s

{sc}

{sl

l

c};

uc(sc,

tl

c)

for

all

c

C;

uc(sl

tl

c)

for

all

c

C

and

l

l

occurring

in

c;

uc(s,

tl

c)

for

all

c

C

and

s

{sc}

{sl

l

c,

l

l

uc(s,

for

all

s

We

now

show

that

the

two

instances

are

equivalent.

First,

suppose

that

there

is

a

solution

to

the

satisfiability

instance.

Then,

consider

the

following

sequence

of

eliminations

in

our

game:

For

every

variable

v

that

is

set

to

true

in

the

satisfying

assignment,

eliminate

s+v

with

the

mixed

strategy

σr

that

places

probability

on

s−v,

probability

on

v,

and

probability

on

v.

(The

expected

utility

of

playing

σr

against

v

or

v

is

against

v

or

v,

it

is

and

against

anything

else

it

is

Hence

the

dominance

is

valid.)

Similarly,

for

every

variable

v

that

is

set

to

false

in

the

satisfying

assignment,

eliminate

s−v

with

the

mixed

strategy

σr

that

places

probability

on

s+v,

probability

on

v,

and

probability

on

v.

(The

expected

utility

of

playing

σr

against

v

or

v

is

against

v

or

v,

it

is

and

against

anything

else

it

is

Hence

the

dominance

is

valid.)

For

every

c

C,

eliminate

tc

with

any

tl

c

for

which

l

was

set

to

true

in

the

satisfying

assignment.

(This

is

a

valid

dominance

because

tl

c

performs

better

than

tc

against

any

strategy

other

than

sl,

and

we

eliminated

sl

in

step

or

in

step

Finally,

eliminate

with

(This

is

a

valid

dominance

because

performs

better

than

against

any

strategy

other

than

those

in

{tc

c

C},

which

we

eliminated

in

step

Hence,

there

is

an

elimination

path

that

eliminates

Now,

suppose

that

there

is

an

elimination

path

that

eliminates

The

strategy

that

eventually

dominates

must

place

most

of

its

probability

on

because

is

the

only

other

strategy

that

performs

well

against

which

cannot

be

eliminated

before

But,

performs

significantly

worse

than

against

any

strategy

tc

with

c

C,

so

it

follows

that

all

these

strategies

must

be

eliminated

first.

Each

strategy

tc

can

only

be

eliminated

by

a

strategy

that

places

most

of

its

weight

on

the

corresponding

strategies

tl

c

with

l

c,

because

they

are

the

only

other

strategies

that

perform

well

against

sc,

which

cannot

be

eliminated

before

tc.

But,

each

strategy

tl

c

performs

significantly

worse

than

tc

against

sl,

so

it

follows

that

for

every

clause

c,

for

one

of

the

literals

l

occurring

in

it,

sl

must

be

eliminated

first.

Now,

strategies

of

the

form

tj

v

will

never

be

eliminated

because

they

are

the

unique

best

responses

to

the

corresponding

strategies

sj

v

(which

are,

in

turn,

the

best

responses

to

the

corresponding

tj

v).

As

a

result,

if

strategy

s+v

(respectively,

s−v)

is

eliminated,

then

its

opposite

strategy

s−v

(respectively,

s+v)

can

no

longer

be

eliminated,

for

the

following

reason.

There

is

no

other

pure

strategy

remaining

that

gets

a

significant

utility

against

more

than

one

of

the

strategies

v,

v,

v,

v,

but

s−v

(respectively,

s+v)

gets

significant

utility

against

all

and

therefore

cannot

be

dominated

by

a

mixed

strategy

placing

positive

probability

on

at

most

strategies.

It

follows

that

for

each

v

V

at

most

one

of

the

strategies

s+v,

s−v

is

eliminated,

in

such

a

way

that

for

every

clause

c,

for

one

of

the

literals

l

occurring

in

it,

sl

must

be

eliminated.

But

then

setting

all

the

literals

l

such

that

sl

is

eliminated

to

true

constitutes

a

solution

to

the

satisfiability

instance.

In

the

next

section,

we

return

to

the

setting

where

there

is

no

restriction

on

the

number

of

pure

strategies

on

which

a

dominating

mixed

strategy

can

place

positive

probability.

(ITERATED)

DOMINANCE

IN

BAYESIAN

GAMES

So

far,

we

have

focused

on

normal

form

games

that

are

flatly

represented

(that

is,

every

matrix

entry

is

given

plicitly).

However,

for

many

games,

the

flat

representation

is

too

large

to

write

down

explicitly,

and

instead,

some

representation

that

exploits

the

structure

of

the

game

needs

to

be

used.

Bayesian

games,

besides

being

of

interest

in

their

own

right,

can

be

thought

of

as

a

useful

structured

representation

of

normal

form

games,

and

we

will

study

them

in

this

section.

In

a

Bayesian

game,

each

player

first

receives

privately

held

preference

information

(the

player"s

type)

from

a

distribution,

which

determines

the

utility

that

that

player

receives

for

every

outcome

of

(that

is,

vector

of

actions

played

in)

the

game.

After

receiving

this

type,

the

player

plays

an

action

based

on

Definition

A

Bayesian

game

is

given

by

a

set

of

players

n};

and,

for

each

player

i,

a

(finite)

set

of

actions

Ai,

a

(finite)

type

space

Θi

with

a

probability

distribution

πi

over

it,

and

a

utility

function

ui

Θi

An

R

(where

ui(θi,

an)

denotes

player

i"s

utility

when

i"s

type

is

θi

and

each

player

j

plays

action

aj).

A

pure

strategy

in

a

Bayesian

game

is

a

mapping

from

types

to

actions,

σi

Θi

Ai,

where

σi(θi)

denotes

the

action

that

player

i

plays

for

type

θi.

Any

vector

of

pure

strategies

in

a

Bayesian

game

defines

an

(expected)

utility

for

each

player,

and

therefore

we

can

translate

a

Bayesian

game

into

a

normal

form

game.

In

this

normal

form

game,

the

notions

of

dominance

and

iterated

dominance

are

defined

as

before.

However,

the

normal

form

representation

of

the

game

is

exponentially

larger

than

the

Bayesian

representation,

because

each

player

i

has

|Ai||Θi|

distinct

pure

strategies.

Thus,

any

algorithm

for

Bayesian

games

that

relies

on

expanding

the

game

to

its

normal

form

will

require

exponential

time.

Specifically,

our

easiness

results

for

normal

form

games

do

not

directly

transfer

to

this

setting.

In

fact,

it

turns

out

that

checking

whether

a

strategy

is

dominated

by

a

pure

strategy

is

hard

in

Bayesian

games.

Theorem

In

a

Bayesian

game,

it

is

NP-complete

to

decide

whether

a

given

pure

strategy

σr

Θr

Ar

is

dominated

by

some

other

pure

strategy

(both

for

strict

and

weak

dominance),

even

when

the

row

player"s

distribution

over

types

is

uniform.

Proof.

The

problem

is

in

NP

because

it

is

easy

to

verify

whether

a

candidate

dominating

strategy

is

indeed

a

dominating

strategy.

To

show

that

the

problem

is

NP-hard,

we

reduce

an

arbitrary

satisfiability

instance

(given

by

a

set

of

clauses

C

using

variables

from

V

to

the

following

Bayesian

game.

Let

the

row

player"s

action

set

be

Ar

{t,

f,

and

let

the

column

player"s

action

set

be

Ac

{ac

c

C}.

Let

the

row

player"s

type

set

be

Θr

{θv

v

V

with

a

distribution

πr

that

is

uniform.

Let

the

row

player"s

utility

function

be

as

follows:

ur(θv,

ac)

for

all

v

V

and

c

C;

ur(θv,

b,

ac)

|V

for

all

v

V

c

C,

and

b

{t,

f}

such

that

setting

v

to

b

satisfies

c;

ur(θv,

b,

ac)

−1

for

all

v

V

c

C,

and

b

{t,

f}

such

that

setting

v

to

b

does

not

satisfy

c.

In

general,

a

player

can

also

receive

a

signal

about

the

other

players"

preferences,

but

we

will

not

concern

ourselves

with

that

here.

Let

the

pure

strategy

to

be

dominated

be

the

one

that

plays

for

every

type.

We

show

that

the

strategy

is

dominated

by

a

pure

strategy

if

and

only

if

there

is

a

solution

to

the

satisfiability

instance.

First,

suppose

there

is

a

solution

to

the

satisfiability

instance.

Then,

let

σd

r

be

given

by:

σd

r

(θv)

t

if

v

is

set

to

true

in

the

solution

to

the

satisfiability

instance,

and

σd

r

(θv)

f

otherwise.

Then,

against

any

action

ac

by

the

column

player,

there

is

at

least

one

type

θv

such

that

either

+v

c

and

σd

r

(θv)

t,

or

−v

c

and

σd

r

(θv)

f.

Thus,

the

row

player"s

expected

utility

against

action

ac

is

at

least

|V

|V

|V

|V

|V

So,

σd

r

is

a

dominating

strategy.

Now,

suppose

there

is

a

dominating

pure

strategy

σd

r

This

dominating

strategy

must

play

t

or

f

for

at

least

one

type.

Thus,

against

any

ac

by

the

column

player,

there

must

at

least

be

some

type

θv

for

which

ur(θv,

σd

r

(θv),

ac)

That

is,

there

must

be

at

least

one

variable

v

such

that

setting

v

to

σd

r

(θv)

satifies

c.

But

then,

setting

each

v

to

σd

r

(θv)

must

satisfy

all

the

clauses.

So

a

satisfying

assignment

exists.

However,

it

turns

out

that

we

can

modify

the

linear

programs

from

Proposition

to

obtain

a

polynomial

time

algorithm

for

checking

whether

a

strategy

is

dominated

by

a

mixed

strategy

in

Bayesian

games.

Theorem

In

a

Bayesian

game,

it

can

be

decided

in

polynomial

time

whether

a

given

(possibly

mixed)

strategy

σr

is

dominated

by

some

other

mixed

strategy,

using

linear

programming

(both

for

strict

and

weak

dominance).

Proof.

We

can

modify

the

linear

programs

presented

in

Proposition

as

follows.

For

strict

dominance,

again

assuming

without

loss

of

generality

that

all

the

utilities

in

the

game

are

positive,

use

the

following

linear

program

(in

which

pσr

r

(θr,

ar)

is

the

probability

that

σr,

the

strategy

to

be

dominated,

places

on

ar

for

type

θr):

minimize

θr∈Θr

ar∈Ar

pr(ar)

such

that

for

any

ac

Ac,

θr∈Θr

ar∈Ar

π(θr)ur(θr,

ar,

ac)pr(θr,

ar)

θr∈Θr

ar∈Ar

π(θr)ur(θr,

ar,

ac)pσr

r

(θr,

ar);

for

any

θr

Θr,

ar∈Ar

pr(θr,

ar)

Assuming

that

π(θr)

for

all

θr

Θr,

this

program

will

return

an

objective

value

smaller

than

|Θr|

if

and

only

if

σr

is

strictly

dominated,

by

reasoning

similar

to

that

done

in

Proposition

For

weak

dominance,

use

the

following

linear

program:

maximize

ac∈Ac

θr∈Θr

ar∈Ar

π(θr)ur(θr,

ar,

ac)pr(θr,

ar)−

θr∈Θr

ar∈Ar

π(θr)ur(θr,

ar,

ac)pσr

r

(θr,

ar))

such

that

for

any

ac

Ac,

θr∈Θr

ar∈Ar

π(θr)ur(θr,

ar,

ac)pr(θr,

ar)

θr∈Θr

ar∈Ar

π(θr)ur(θr,

ar,

ac)pσr

r

(θr,

ar);

for

any

θr

Θr,

ar∈Ar

pr(θr,

ar)

This

program

will

return

an

objective

value

greater

than

if

and

only

if

σr

is

weakly

dominated,

by

reasoning

similar

to

that

done

in

Proposition

We

now

turn

to

iterated

dominance

in

Bayesian

games.

Na¨ıvely,

one

might

argue

that

iterated

dominance

in

Bayesian

games

always

requires

an

exponential

number

of

steps

when

a

significant

fraction

of

the

game"s

pure

strategies

can

be

eliminated,

because

there

are

exponentially

many

pure

strategies.

However,

this

is

not

a

very

strong

argument

because

oftentimes

we

can

eliminate

exponentially

many

pure

strategies

in

one

step.

For

example,

if

for

some

type

θr

Θr

we

have,

for

all

ac

Ac,

that

u(θr,

r,

ac)

u(θr,

r,

ac),

then

any

pure

strategy

for

the

row

player

which

plays

action

r

for

type

θr

is

dominated

(by

the

strategy

that

plays

action

r

for

type

θr

instead)-and

there

are

exponentially

many

such

strategies.

It

is

therefore

conceivable

that

we

need

only

polynomially

many

eliminations

of

collections

of

a

player"s

strategies.

However,

the

following

theorem

shows

that

this

is

not

the

case,

by

giving

an

example

where

an

exponential

number

of

iterations

(that

is,

alternations

between

the

players

in

eliminating

strategies)

is

required.

(We

emphasize

that

this

is

not

a

result

about

computational

complexity.)

Theorem

Even

in

symmetric

Bayesian

games,

iterated

dominance

by

pure

strategies

can

require

an

exponential

number

of

iterations

(both

for

strict

and

weak

dominance),

even

with

only

three

actions

per

player.

Proof.

Let

each

player

i

have

n

types

θ1

i

θ2

i

i

Let

each

player

i

have

actions

ai,

bi,

ci,

and

let

the

utility

function

of

each

player

be

defined

as

follows.

(In

the

below,

i

and

i

are

shorthand

for

i

and

i

when

used

as

player

indices.

Also,

can

be

replaced

by

a

sufficiently

negative

number.

Finally,

δ

and

should

be

chosen

to

be

very

small

(even

compared

to

and

should

be

more

than

twice

as

large

as

δ.)

i

ai,

i

ai,

for

or

i

bi,

for

and

i

bi,

for

or

i

ci,

for

all

ui(θj

i

ai,

for

all

when

j

ui(θj

i

bi,

for

all

when

j

ui(θj

i

ci,

δ

for

all

when

j

ui(θj

i

ci,

δ−

for

all

and

when

j

Let

the

distribution

over

each

player"s

types

be

given

by

p(θj

i

(with

the

exception

that

i

We

will

be

interested

in

eliminating

strategies

of

the

following

form:

play

bi

for

type

θ1

i

and

play

one

of

bi

or

ci

otherwise.

Because

the

utility

function

is

the

same

for

any

type

θj

i

with

j

these

strategies

are

effectively

defined

by

the

total

probability

that

they

place

on

which

is

i

tj

i

where

tj

i

if

player

i

Note

that

the

strategies

are

still

pure

strategies;

the

probability

placed

on

an

action

by

a

strategy

here

is

simply

the

sum

of

the

probabilities

of

the

types

for

which

the

strategy

chooses

that

action.

plays

ci

for

type

θj

i

and

otherwise.

This

probability

is

different

for

any

two

different

strategies

of

the

given

form,

and

we

have

exponentially

many

different

strategies

of

the

given

form.

For

any

probability

q

which

can

be

expressed

as

(with

all

tj

let

σi(q)

denote

the

(unique)

strategy

of

the

given

form

for

player

i

which

places

a

total

probability

of

q

on

ci.

Any

strategy

that

plays

ci

for

type

θ1

i

or

ai

for

some

type

θj

i

with

j

can

immediately

be

eliminated.

We

will

show

that,

after

that,

we

must

eliminate

the

strategies

σi(q)

with

high

q

first,

slowly

working

down

to

those

with

lower

q.

Claim

If

and

have

not

yet

been

eliminated,

and

q

q

then

σi(q)

cannot

yet

be

eliminated.

Proof:

First,

we

show

that

no

strategy

σi(q

can

eliminate

σi(q).

Against

the

utility

of

playing

σi(p)

is

p

δ

p

q

Thus,

when

q

it

is

best

to

set

p

as

high

as

possible

(and

we

note

that

and

have

not

been

eliminated),

but

when

q

it

is

best

to

set

p

as

low

as

possible

because

δ

q

Thus,

whether

q

q

or

q

q

σi(q)

will

always

do

strictly

better

than

σi(q

against

some

remaining

opponent

strategies.

Hence,

no

strategy

σi(q

can

eliminate

σi(q).

The

only

other

pure

strategies

that

could

dominate

σi(q)

are

strategies

that

play

ai

for

type

θ1

i

and

bi

or

ci

for

all

other

types.

Let

us

take

such

a

strategy

and

suppose

that

it

plays

c

with

probability

p.

Against

(which

have

not

yet

been

eliminated),

the

utility

of

playing

this

strategy

is

−(q

p

δ

p

q

On

the

other

hand,

playing

σi(q)

gives

q

δ

q

q

Because

q

q,

we

have

−(q

−q

q

and

because

δ

and

are

small,

it

follows

that

σi(q)

receives

a

higher

utility.

Therefore,

no

strategy

dominates

σi(q),

proving

the

claim.

Claim

If

for

all

q

q,

and

have

been

eliminated,

then

σi(q)

can

be

eliminated.

Proof:

Consider

the

strategy

for

player

i

that

plays

ai

for

type

θ1

i

and

bi

for

all

other

types

(call

this

strategy

σi);

we

claim

σi

dominates

σi(q).

First,

if

either

of

the

other

players

k

plays

ak

for

θ1

k,

then

σi

performs

better

than

σi(q)

(which

receives

in

some

cases).

Because

the

strategies

for

player

k

that

play

ck

for

type

θ1

k,

or

ak

for

some

type

θj

k

with

j

have

already

been

eliminated,

all

that

remains

to

check

is

that

σi

performs

better

than

σi(q)

whenever

both

of

the

other

two

players

play

strategies

of

the

following

form:

play

bk

for

type

θ1

k,

and

play

one

of

bk

or

ck

otherwise.

We

note

that

among

these

strategies,

there

are

left

that

place

probability

greater

than

q

on

ck.

Letting

qk

denote

the

probability

with

which

player

k

plays

ck,

the

expected

utility

of

playing

σi

is

On

the

other

hand,

the

utility

of

playing

σi(q)

is

q

δ

q

Because

q,

the

difference

between

these

two

expressions

is

at

least

δ,

which

is

positive.

It

follows

that

σi

dominates

σi(q).

From

Claim

it

follows

that

all

strategies

of

the

form

σi(q)

will

eventually

be

eliminated.

However,

Claim

shows

that

we

cannot

go

ahead

and

eliminate

multiple

such

strategies

for

one

player,

unless

at

least

one

other

player

simultaneously

keeps

up

in

the

eliminated

strategies:

every

time

a

σi(q)

is

eliminated

such

that

and

have

not

yet

been

eliminated,

we

need

to

eliminate

one

of

the

latter

two

strategies

before

any

σi(q

with

q

q

can

be

eliminated-that

is,

we

need

to

alternate

between

players.

Because

there

are

exponentially

many

strategies

of

the

form

σi(q),

it

follows

that

iterated

elimination

will

require

exponentially

many

iterations

to

complete.

It

follows

that

an

efficient

algorithm

for

iterated

dominance

(strict

or

weak)

by

pure

strategies

in

Bayesian

games,

if

it

exists,

must

somehow

be

able

to

perform

(at

least

part

of)

many

iterations

in

a

single

step

of

the

algorithm

(because

if

each

step

only

performed

a

single

iteration,

we

would

need

exponentially

many

steps).

Interestingly,

Knuth

et

al.

argue

that

iterated

dominance

appears

to

be

an

inherently

sequential

problem

(in

light

of

their

result

that

iterated

very

weak

dominance

is

P-complete,

that

is,

apparently

not

efficiently

parallelizable),

suggesting

that

aggregating

many

iterations

may

be

difficult.

CONCLUSIONS

While

the

Nash

equilibrium

solution

concept

is

studied

more

and

more

intensely

in

our

community,

the

perhaps

more

elementary

concept

of

(iterated)

dominance

has

received

much

less

attention.

In

this

paper

we

studied

various

computational

aspects

of

this

concept.

We

first

studied

both

strict

and

weak

dominance

(not

iterated),

and

showed

that

checking

whether

a

given

strategy

is

dominated

by

some

mixed

strategy

can

be

done

in

polynomial

time

using

a

single

linear

program

solve.

We

then

moved

on

to

iterated

dominance.

We

showed

that

determining

whether

there

is

some

path

that

eliminates

a

given

strategy

is

NP-complete

with

iterated

weak

dominance.

This

allowed

us

to

also

show

that

determining

whether

there

is

a

path

that

leads

to

a

unique

solution

is

NP-complete.

Both

of

these

results

hold

both

with

and

without

dominance

by

mixed

strategies.

(A

weaker

version

of

the

second

result

(only

without

dominance

by

mixed

strategies)

was

already

known

Iterated

strict

dominance,

on

the

other

hand,

is

path-independent

(both

with

and

without

dominance

by

mixed

strategies)

and

can

therefore

be

done

in

polynomial

time.

We

then

studied

what

happens

when

the

dominating

strategy

is

allowed

to

place

positive

probability

on

only

a

few

pure

strategies.

First,

we

showed

that

finding

the

dominating

strategy

with

minimum

support

size

is

NP-complete

(both

for

strict

and

weak

dominance).

Then,

we

showed

that

iterated

strict

dominance

becomes

path-dependent

when

there

is

a

limit

on

the

support

size

of

the

dominating

strategies,

and

that

deciding

whether

a

given

strategy

can

be

eliminated

by

iterated

strict

dominance

under

this

restriction

is

NP-complete

(even

when

the

limit

on

the

support

size

is

Finally,

we

studied

dominance

and

iterated

dominance

in

Bayesian

games,

as

an

example

of

a

concise

representation

language

for

normal

form

games

that

is

interesting

in

its

own

right.

We

showed

that,

unlike

in

normal

form

games,

deciding

whether

a

given

pure

strategy

is

dominated

by

another

pure

strategy

in

a

Bayesian

game

is

NP-complete

(both

with

strict

and

weak

dominance);

however,

deciding

whether

a

strategy

is

dominated

by

some

mixed

strategy

can

still

be

done

in

polynomial

time

with

a

single

linear

program

solve

(both

with

strict

and

weak

dominance).

Finally,

we

showed

that

iterated

dominance

using

pure

strategies

can

require

an

exponential

number

of

iterations

in

a

Bayesian

game

(both

with

strict

and

weak

dominance).

There

are

various

avenues

for

future

research.

First,

there

is

the

open

question

of

whether

it

is

possible

to

complete

iterated

dominance

in

Bayesian

games

in

polynomial

time

(even

though

we

showed

that

an

exponential

number

of

alternations

between

the

players

in

eliminating

strategies

is

sometimes

required).

Second,

we

can

study

computational

aspects

of

(iterated)

dominance

in

concise

representations

of

normal

form

games

other

than

Bayesian

games-for

example,

in

graphical

games

or

local-effect/action

graph

games

(How

to

efficiently

perform

iterated

very

weak

dominance

has

already

been

studied

for

partially

observable

stochastic

games

Finally,

we

can

ask

whether

some

of

the

algorithms

we

described

(such

as

the

one

for

iterated

strict

dominance

with

mixed

strategies)

can

be

made

faster.

REFERENCES

Krzysztof

R.

Apt.

Uniform

proofs

of

order

independence

for

various

strategy

elimination

procedures.

Contributions

to

Theoretical

Economics,

Nivan

A.

R.

Bhat

and

Kevin

Leyton-Brown.

Computing

Nash

equilibria

of

action-graph

games.

In

UAI,

Ben

Blum,

Christian

R.

Shelton,

and

Daphne

Koller.

A

continuation

method

for

Nash

equilibria

in

structured

games.

In

IJCAI,

Vincent

Conitzer

and

Tuomas

Sandholm.

Complexity

results

about

Nash

equilibria.

In

IJCAI,

pages

Drew

Fudenberg

and

Jean

Tirole.

Game

Theory.

MIT

Press,

Itzhak

Gilboa,

Ehud

Kalai,

and

Eitan

Zemel.

On

the

order

of

eliminating

dominated

strategies.

Operations

Research

Letters,

Itzhak

Gilboa,

Ehud

Kalai,

and

Eitan

Zemel.

The

complexity

of

eliminating

dominated

strategies.

Mathematics

of

Operation

Research,

Eric

A.

Hansen,

Daniel

S.

Bernstein,

and

Shlomo

Zilberstein.

Dynamic

programming

for

partially

observable

stochastic

games.

In

AAAI,

pages

Michael

Kearns,

Michael

Littman,

and

Satinder

Singh.

Graphical

models

for

game

theory.

In

UAI,

Leonid

Khachiyan.

A

polynomial

algorithm

in

linear

programming.

Soviet

Math.

Doklady,

Donald

E.

Knuth,

Christos

H.

Papadimitriou,

and

John

N

Tsitsiklis.

A

note

on

strategy

elimination

in

bimatrix

games.

Operations

Research

Letters,

Kevin

Leyton-Brown

and

Moshe

Tennenholtz.

Local-effect

games.

In

IJCAI,

Richard

Lipton,

Evangelos

Markakis,

and

Aranyak

Mehta.

Playing

large

games

using

simple

strategies.

In

ACM-EC,

pages

Michael

Littman

and

Peter

Stone.

A

polynomial-time

Nash

equilibrium

algorithm

for

repeated

games.

In

ACM-EC,

pages

Leslie

M.

Marx

and

Jeroen

M.

Swinkels.

Order

independence

for

iterated

weak

dominance.

Games

and

Economic

Behavior,

Leslie

M.

Marx

and

Jeroen

M.

Swinkels.

Corrigendum,

order

independence

for

iterated

weak

dominance.

Games

and

Economic

Behavior,

Andreu

Mas-Colell,

Michael

Whinston,

and

Jerry

R.

Green.

Microeconomic

Theory.

Oxford

University

Press,

Roger

Myerson.

Game

Theory:

Analysis

of

Conflict.

Harvard

University

Press,

Cambridge,

Martin

J

Osborne

and

Ariel

Rubinstein.

A

Course

in

Game

Theory.

MIT

Press,

Christos

Papadimitriou.

Algorithms,

games

and

the

Internet.

In

STOC,

pages

Ryan

Porter,

Eugene

Nudelman,

and

Yoav

Shoham.

Simple

search

methods

for

finding

a

Nash

equilibrium.

In

AAAI,

pages

multiagent system
self-interested agent
iterated dominance

optimal action
game theory
normal form game
nash equilibrium
dominance
elimination
strategy
bayesian game