Combinatorial

Agency

[Extended

Abstract]

Moshe

Babaioff

School

of

Information

Management

and

Systems

UC

Berkeley

Berkeley,

CA,

USA

moshe@sims.berkeley.edu

Michal

Feldman

School

of

Engineering

and

Computer

Science

The

Hebrew

University

of

Jerusalem

Jerusalem,

Israel

mfeldman@cs.huji.ac.il

Noam

Nisan

School

of

Engineering

and

Computer

Science

The

Hebrew

University

of

Jerusalem

Jerusalem,

Israel

noam@cs.huji.ac.il

ABSTRACT

Much

recent

research

concerns

systems,

such

as

the

Internet,

whose

components

are

owned

and

operated

by

different

parties,

each

with

his

own

selfish

goal.

The

field

of

Algorithmic

Mechanism

Design

handles

the

issue

of

private

information

held

by

the

different

parties

in

such

computational

settings.

This

paper

deals

with

a

complementary

problem

in

such

settings:

handling

the

hidden

actions

that

are

performed

by

the

different

parties.

Our

model

is

a

combinatorial

variant

of

the

classical

principalagent

problem

from

economic

theory.

In

our

setting

a

principal

must

motivate

a

team

of

strategic

agents

to

exert

costly

effort

on

his

behalf,

but

their

actions

are

hidden

from

him.

Our

focus

is

on

cases

where

complex

combinations

of

the

efforts

of

the

agents

influence

the

outcome.

The

principal

motivates

the

agents

by

offering

to

them

a

set

of

contracts,

which

together

put

the

agents

in

an

equilibrium

point

of

the

induced

game.

We

present

formal

models

for

this

setting,

suggest

and

embark

on

an

analysis

of

some

basic

issues,

but

leave

many

questions

open.

Categories

and

Subject

Descriptors

[Social

and

Behavioral

Sciences]:

Economics;

[Electronic

Commerce]:

Payment

schemes;

[ComputerCommunication

Networks]:

Distributed

Systems

General

Terms

Design,

Economics,

Theory

INTRODUCTION

Background

One

of

the

most

striking

characteristics

of

modern

computer

networks

in

particular

the

Internet

is

that

different

parts

of

it

are

owned

and

operated

by

different

individuals,

firms,

and

organizations.

The

analysis

and

design

of

protocols

for

this

environment

thus

naturally

needs

to

take

into

account

the

different

selfish

economic

interests

of

the

different

participants.

Indeed,

the

last

few

years

have

seen

much

work

addressing

this

issue

using

game-theoretic

notions

(see

for

an

influential

survey).

A

significant

part

of

the

difficulty

stems

from

underlying

asymmetries

of

information:

one

participant

may

not

know

everything

that

is

known

or

done

by

another.

In

particular,

the

field

of

algorithmic

mechanism

design

uses

appropriate

incentives

to

extract

the

private

information

from

the

participants.

This

paper

deals

with

the

complementary

lack

of

knowledge,

that

of

hidden

actions.

In

many

cases

the

actual

behaviors

actions

of

the

different

participants

are

hidden

from

others

and

only

influence

the

final

outcome

indirectly.

Hidden

here

covers

a

wide

range

of

situations

including

not

precisely

measurable,

costly

to

determine,

or

even

non-contractible

meaning

that

it

can

not

be

formally

used

in

a

legal

contract.

An

example

that

was

discussed

in

is

Quality

of

Service

routing

in

a

network:

every

intermediate

link

or

router

may

exert

a

different

amount

of

effort

(priority,

bandwidth,

when

attempting

to

forward

a

packet

of

information.

While

the

final

outcome

of

whether

a

packet

reached

its

destination

is

clearly

visible,

it

is

rarely

feasible

to

monitor

the

exact

amount

of

effort

exerted

by

each

intermediate

link

how

can

we

ensure

that

they

really

do

exert

the

appropriate

amount

of

effort?

Many

other

complex

resource

allocation

problems

exhibit

similar

hidden

actions,

e.g.,

a

task

that

runs

on

a

collection

of

shared

servers

may

be

allocated,

by

each

server,

an

unknown

percentage

of

the

CPU"s

processing

power

or

of

the

physical

memory.

How

can

we

ensure

that

the

right

combination

of

allocations

is

actually

made

by

the

different

servers?

A

related

class

of

examples

concerns

security

issues:

each

link

in

a

complex

system

may

exert

different

levels

of

effort

for

protecting

some

desired

security

property

of

the

system.

How

can

we

ensure

that

the

desired

level

of

collective

security

is

obtained?

Our

approach

to

this

problem

is

based

on

the

well

studied

principal-agent

problem

in

economic

theory:

How

can

a

principal

motivate

a

rational

agent

to

exert

costly

effort

towards

the

welfare

of

the

principal?

The

crux

of

the

model

is

that

the

agent"s

action

(i.e.

whether

he

exerts

effort

or

not)

is

invisible

to

the

principal

and

only

the

final

outcome,

which

is

probabilistic

and

also

influenced

by

other

factors,

is

visible.

This

problem

is

well

studied

in

many

contexts

in

classical

economic

theory

and

we

refer

the

readers

to

introductory

texts

on

economic

theory

such

as

Chapter

The

solution

is

based

on

the

observation

that

a

properly

designed

contract,

in

which

the

payments

are

contingent

upon

the

final

outcome,

can

influence

a

rational

agent

to

exert

the

required

effort.

In

this

paper

we

initiate

a

general

study

of

handling

combinations

of

agents

rather

than

a

single

agent.

While

much

work

was

already

done

on

motivating

teams

of

agents

our

emphasis

is

on

dealing

with

the

complex

combinatorial

structure

of

dependencies

between

agents"

actions.

In

the

general

case,

each

combination

of

efforts

exerted

by

the

n

different

agents

may

result

in

a

different

expected

gain

for

the

principal.

The

general

question

asks

which

conditional

payments

should

the

principal

offer

to

which

agents

as

to

maximize

his

net

utility?

In

our

setting

and

unlike

in

previous

work

(see,

e.g.,

the

main

challenge

is

to

determine

the

optimal

amount

of

effort

desired

from

each

agent.

This

paper

suggest

models

for

and

provides

some

interesting

initial

results

about

this

combinatorial

agency

problem.

We

believe

that

we

have

only

scratched

the

surface

and

leave

many

open

questions,

conjectures,

and

directions

for

further

research.

We

believe

that

this

type

of

analysis

may

also

find

applications

in

regular

economic

activity.

Consider

for

example

a

firm

that

sub-contracts

a

family

of

related

tasks

to

many

individuals

(or

other

firms).

It

will

often

not

be

possible

to

exactly

monitor

the

actual

effort

level

of

each

sub-contractor

(e.g.,

in

cases

of

public-relations

activities,

consulting

activities,

or

any

activities

that

require

cooperation

between

different

sub-contractors.)

When

the

dependencies

between

the

different

subtasks

are

complex,

we

believe

that

combinatorial

agency

models

can

offer

a

foundation

for

the

design

of

contracts

with

appropriate

incentives.

It

may

also

be

useful

to

view

our

work

as

part

of

a

general

research

agenda

stemming

from

the

fact

that

all

types

of

economic

activity

are

increasingly

being

handled

with

the

aid

of

sophisticated

computer

systems.

In

general,

in

such

computerized

settings,

complex

scenarios

involving

multiple

agents

and

goods

can

naturally

occur,

and

they

need

to

be

algorithmically

handled.

This

calls

for

the

study

of

the

standard

issues

in

economic

theory

in

new

complex

settings.

The

principal-agent

problem

is

a

prime

example

where

such

complex

settings

introduce

new

challenges.

Our

Models

We

start

by

presenting

a

general

model:

in

this

model

each

of

n

agents

has

a

set

of

possible

actions,

the

combination

of

actions

by

the

players

results

in

some

outcome,

where

this

happens

probabilistically.

The

main

part

of

the

specification

of

a

problem

in

this

model

is

a

function

that

specifies

this

distribution

for

each

n-tuple

of

agents"

actions.

Additionally,

the

problem

specifies

the

principal"s

utility

for

each

possible

outcome,

and

for

each

agent,

the

agent"s

cost

for

each

possible

action.

The

principal

motivates

the

agents

by

offering

to

each

of

them

a

contract

that

specifies

a

payment

for

each

possible

outcome

of

the

whole

Key

here

is

that

the

actions

of

the

players

are

non-observable

and

thus

the

contract

cannot

make

the

payments

directly

contingent

on

the

actions

of

the

players,

but

rather

only

on

the

outcome

of

the

whole

project.

Given

a

set

of

contracts,

the

agents

will

each

optimize

his

own

utility:

i.e.

will

choose

the

action

that

maximizes

his

expected

payment

minus

the

cost

of

his

action.

Since

the

outcome

depends

on

the

actions

of

all

players

together,

the

agents

are

put

in

a

game

and

are

assumed

to

reach

a

Nash

The

principal"s

problem,

our

problem

in

this

paper,

is

of

designing

an

optimal

set

of

contracts:

i.e.

contracts

that

maximize

his

expected

utility

from

the

outcome,

minus

his

expected

total

payment.

The

main

difficulty

is

that

of

determining

the

required

Nash

equilibrium

point.

In

order

to

focus

on

the

main

issues,

the

rest

of

the

paper

deals

with

the

basic

binary

case:

each

agent

has

only

two

possible

actions

exert

effort

and

shirk

and

there

are

only

two

possible

outcomes

success

and

failure.

It

seems

that

this

case

already

captures

the

main

interesting

In

this

case,

each

agent"s

problem

boils

down

to

whether

to

exert

effort

or

not,

and

the

principal"s

problem

boils

down

to

which

agents

should

be

contracted

to

exert

effort.

This

model

is

still

pretty

abstract,

and

every

problem

description

contains

a

complete

table

specifying

the

success

probability

for

each

subset

of

the

agents

who

exert

effort.

We

then

consider

a

more

concrete

model

which

concerns

a

subclass

of

problem

instances

where

this

exponential

size

table

is

succinctly

represented.

This

subclass

will

provide

many

natural

types

of

problem

instances.

In

this

subclass

every

agent

performs

a

subtask

which

succeeds

with

a

low

probability

γ

if

the

agent

does

not

exert

effort

and

with

a

higher

probability

δ

γ,

if

the

agent

does

exert

effort.

The

whole

project

succeeds

as

a

deterministic

Boolean

function

of

the

success

of

the

subtasks.

This

Boolean

function

can

now

be

represented

in

various

ways.

Two

basic

examples

are

the

AND

function

in

which

the

project

succeeds

only

if

all

subtasks

succeed,

and

the

OR

function

which

succeeds

if

any

of

the

subtasks

succeeds.

A

more

complex

example

considers

a

communication

network,

where

each

agent

controls

a

single

edge,

and

success

of

the

subtask

means

that

a

message

is

forwarded

by

that

edge.

Effort

by

the

edge

increases

this

success

probability.

The

complete

project

succeeds

if

there

is

a

complete

path

of

successful

edges

between

a

given

source

and

sink.

Complete

definitions

of

the

models

appear

in

Section

Our

Results

One

could

think

of

a

different

model

in

which

the

agents

have

intrinsic

utility

from

the

outcome

and

payments

may

not

be

needed,

as

in

In

this

paper

our

philosophy

is

that

the

principal

can

suggest

a

Nash

equilibrium

point

to

the

agents,

thus

focusing

on

the

best

Nash

equilibrium.

One

may

alternatively

study

the

worst

case

equilibrium

as

in

or

alternatively,

attempt

modeling

some

kind

of

an

extensive

game

between

the

agents,

as

in

However,

some

of

the

more

advanced

questions

we

ask

for

this

case

can

be

viewed

as

instances

of

the

general

model.

We

address

a

host

of

questions

and

prove

a

large

number

of

results.

We

believe

that

despite

the

large

amount

of

work

that

appears

here,

we

have

only

scratched

the

surface.

In

many

cases

we

were

not

able

to

achieve

the

general

characterization

theorems

that

we

desired

and

had

to

settle

for

analyzing

special

cases

or

proving

partial

results.

In

many

cases,

simulations

reveal

structure

that

we

were

not

able

to

formally

prove.

We

present

here

an

informal

overview

of

the

issues

that

we

studied,

what

we

were

able

to

do,

and

what

we

were

not.

The

full

treatment

of

most

of

our

results

appears

only

in

the

extended

version

and

only

some

are

discussed,

often

with

associated

simulation

results,

in

the

body

of

the

paper.

Our

first

object

of

study

is

the

structure

of

the

class

of

sets

of

agents

that

can

be

contracted

for

a

given

problem

instance.

Let

us

fix

a

given

function

describing

success

probabilities,

fix

the

agent"s

costs,

and

let

us

consider

the

set

of

contracted

agents

for

different

values

of

the

principal"s

associated

value

from

success.

For

very

low

values,

no

agent

will

be

contracted

since

even

a

single

agent"s

cost

is

higher

than

the

principal"s

value.

For

very

high

values,

all

agents

will

always

be

contracted

since

the

marginal

contribution

of

an

agent

multiplied

by

the

principal"s

value

will

overtake

any

associated

payment.

What

happens

for

intermediate

principal"s

values?

We

first

observe

that

there

is

a

finite

number

of

transitions

between

different

sets,

as

the

principal"s

project

value

increases.

These

transitions

behave

very

differently

for

different

functions.

For

example,

we

show

that

for

the

AND

function

only

a

single

transition

occurs:

for

low

enough

values

no

agent

will

be

contracted,

while

for

higher

values

all

agents

will

be

contracted

there

is

no

intermediate

range

for

which

only

some

of

the

agents

are

contracted.

For

the

OR

function,

the

situation

is

opposite:

as

the

principal"s

value

increases,

the

set

of

contracted

agents

increases

one-by-one.

We

are

able

to

fully

characterize

the

types

of

functions

for

which

these

two

extreme

types

of

transitions

behavior

occur.

However,

the

structure

of

these

transitions

in

general

seems

quite

complex,

and

we

were

not

able

to

fully

analyze

them

even

in

simple

cases

like

the

Majority

function

(the

project

succeeds

if

a

majority

of

subtasks

succeeds)

or

very

simple

networks.

We

do

have

several

partial

results,

including

a

construction

with

an

exponential

number

of

transitions.

During

the

previous

analysis

we

also

study

what

we

term

the

price

of

unaccountability:

How

much

is

the

social

utility

achieved

under

the

optimal

contracts

worse

than

what

could

be

achieved

in

the

non-strategic

where

the

socially

optimal

actions

are

simply

dictated

by

the

principal?

We

are

able

to

fully

analyze

this

price

for

the

AND

function,

where

it

is

shown

to

tend

to

infinity

as

the

number

of

agents

tends

to

infinity.

More

general

analysis

remains

an

open

problem.

Our

analysis

of

these

questions

sheds

light

on

the

difficulty

of

the

various

natural

associated

algorithmic

problems.

In

particular,

we

observe

that

the

optimal

contract

can

be

found

in

time

polynomial

in

the

explicit

representation

of

the

probability

function.

We

prove

a

lower

bound

that

shows

that

the

optimal

contract

cannot

be

found

in

number

of

queries

that

is

polynomial

just

in

the

number

of

agents,

in

a

general

black-box

model.

We

also

show

that

when

the

probability

function

is

succinctly

represented

as

The

non-strategic

case

is

often

referred

to

as

the

case

with

contractible

actions

or

the

principal"s

first-best

solution.

a

read-once

network,

the

problem

becomes

#P-hard.

The

status

of

some

algorithmic

questions

remains

open,

in

particular

that

of

finding

the

optimal

contract

for

technologies

defined

by

serial-parallel

networks.

In

a

follow-up

paper

we

deal

with

equilibria

in

mixed

strategies

and

show

that

the

principal

can

gain

from

inducing

a

mixed-Nash

equilibrium

between

the

agents

rather

than

a

pure

one.

We

also

show

cases

where

the

principal

can

gain

by

asking

agents

to

reduce

their

effort

level,

even

when

this

effort

comes

for

free.

Both

phenomena

can

not

occur

in

the

non-strategic

setting.

MODEL

AND

PRELIMINARIES

The

General

Setting

A

principal

employs

a

set

of

agents

N

of

size

n.

Each

agent

i

N

has

a

possible

set

of

actions

Ai,

and

a

cost

(effort)

ci(ai)

for

each

possible

action

ai

Ai

(ci

Ai

The

actions

of

all

players

determine,

in

a

probabilistic

way,

a

contractible

outcome

o

O,

according

to

a

success

function

t

An

Δ(O)

(where

Δ(O)

denotes

the

set

of

probability

distributions

on

O).

A

technology

is

a

pair,

(t,

c),

of

a

success

function,

t,

and

cost

functions,

c

cn).

The

principal

has

a

certain

value

for

each

possible

outcome,

given

by

the

function

v

O

As

we

will

only

consider

risk-neutral

players

in

this

we

will

also

treat

v

as

a

function

on

Δ(O),

by

taking

simple

expected

value.

Actions

of

the

players

are

invisible,

but

the

final

outcome

o

is

visible

to

him

and

to

others

(in

particular

the

court),

and

he

may

design

enforceable

contracts

based

on

the

final

outcome.

Thus

the

contract

for

agent

i

is

a

function

(payment)

pi

O

again,

we

will

also

view

pi

as

a

function

on

Δ(O).

Given

this

setting,

the

agents

have

been

put

in

a

game,

where

the

utility

of

agent

i

under

the

vector

of

actions

a

an)

is

given

by

ui(a)

pi(t(a))−ci(ai).

The

agents

will

be

assumed

to

reach

Nash

equilibrium,

if

such

equilibrium

exists.

The

principal"s

problem

(which

is

our

problem

in

this

paper)

is

how

to

design

the

contracts

pi

as

to

maximize

his

own

expected

utility

u(a)

v(t(a))

P

i

pi(t(a)),

where

the

actions

an

are

at

Nash-equilibrium.

In

the

case

of

multiple

Nash

equilibria

we

let

the

principal

choose

the

equilibrium,

thus

focusing

on

the

best

Nash

equilibrium.

A

variant,

which

is

similar

in

spirit

to

strong

implementation

in

mechanism

design

would

be

to

take

the

worst

Nash

equilibrium,

or

even,

stronger

yet,

to

require

that

only

a

single

equilibrium

exists.

Finally,

the

social

welfare

for

a

A

is

u(a)

P

i∈N

ui(a)

v(t(a))

P

i∈N

ci(ai).

The

Binary-Outcome

Binary-Action

Model

We

wish

to

concentrate

on

the

complexities

introduced

by

the

combinatorial

structure

of

the

success

function

t,

we

restrict

ourselves

to

a

simpler

setting

that

seems

to

focus

more

clearly

on

the

structure

of

t.

A

similar

model

was

used

in

We

first

restrict

the

action

spaces

to

have

only

two

states

(binary-action):

(low

effort)

and

(high

effort).

The

cost

function

of

agent

i

is

now

just

a

scalar

ci

denoting

the

cost

of

exerting

high

effort

(where

the

low

effort

has

cost

The

vector

of

costs

is

c

cn),

The

risk-averse

case

would

obviously

be

a

natural

second

step

in

the

research

of

this

model,

as

has

been

for

noncombinatorial

scenarios.

and

we

use

the

notation

(t,

c)

to

denote

a

technology

in

such

a

binary-outcome

model.

We

then

restrict

the

outcome

space

to

have

only

two

states

(binary-outcome):

(project

failure)

and

(project

success).

The

principal"s

value

for

a

successful

project

is

given

by

a

scalar

v

(where

the

value

of

project

failure

is

We

assume

that

the

principal

can

pay

the

agents

but

not

fine

them

(known

as

the

limited

liability

constraint).

The

contract

to

agent

i

is

thus

now

given

by

a

scalar

value

pi

that

denotes

the

payment

that

i

gets

in

case

of

project

success.

If

the

project

fails,

the

agent

gets

When

the

lowest

cost

action

has

zero

cost

(as

we

assume),

this

immediately

implies

that

the

participation

constraint

holds.

At

this

point

the

success

function

t

becomes

a

function

t

where

an)

denotes

the

probability

of

project

success

where

players

with

ai

do

not

exert

effort

and

incur

no

cost,

and

players

with

ai

do

exert

effort

and

incur

a

cost

of

ci.

As

we

wish

to

concentrate

on

motivating

agents,

rather

than

on

the

coordination

between

agents,

we

assume

that

more

effort

by

an

agent

always

leads

to

a

better

probability

of

success,

i.e.

that

the

success

function

t

is

strictly

monotone.

Formally,

if

we

denote

by

a−i

A−i

the

(n

vector

of

the

actions

of

all

agents

excluding

agent

i.

i.e.,

a−i

an),

then

a

success

function

must

satisfy:

∀i

N,

∀a−i

A−i

a−i)

a−i)

Additionally,

we

assume

that

t(a)

for

any

a

A

(or

equivalently,

Definition

The

marginal

contribution

of

agent

i,

denoted

by

Δi,

is

the

difference

between

the

probability

of

success

when

i

exerts

effort

and

when

he

shirks.

Δi(a−i)

a−i)

a−i)

Note

that

since

t

is

monotone,

Δi

is

a

strictly

positive

function.

At

this

point

we

can

already

make

some

simple

observations.

The

best

action,

ai

Ai,

of

agent

i

can

now

be

easily

determined

as

a

function

of

what

the

others

do,

a−i

A−i,

and

his

contract

pi.

Claim

Given

a

profile

of

actions

a−i,

agent

i"s

best

strategy

is

ai

if

pi

ci

Δi(a−i)

and

is

ai

if

pi

ci

Δi(a−i)

(In

the

case

of

equality

the

agent

is

indifferent

between

the

two

alternatives.)

As

pi

ci

Δi(a−i)

if

and

only

if

a−i)

pi

a−i)−ci

pi

a−i)

a−i),

i"s

best

strategy

is

to

choose

ai

in

this

case.

This

allows

us

to

specify

the

contracts

that

are

the

principal"s

optimal,

for

inducing

a

given

equilibrium.

Observation

The

best

contracts

(for

the

principal)

that

induce

a

A

as

an

equilibrium

are

pi

for

agent

i

who

exerts

no

effort

(ai

and

pi

ci

Δi(a−i)

for

agent

i

who

exerts

effort

(ai

In

this

case,

the

expected

utility

of

agent

i

who

exerts

effort

is

ci

Δi(a−i)

and

for

an

agent

who

shirk.

The

principal"s

expected

utility

is

given

by

u(a,

v)

(v−P)·t(a),

where

P

is

the

total

payment

in

case

of

success,

given

by

P

P

ci

Δi(a−i)

We

say

that

the

principal

contracts

with

agent

i

if

pi

(and

ai

in

the

equilibrium

a

A).

The

principal"s

goal

is

to

maximize

his

utility

given

his

value

v,

i.e.

to

determine

the

profile

of

actions

a∗

A,

which

gives

the

highest

value

of

u(a,

v)

in

equilibrium.

Choosing

a

A

corresponds

to

choosing

a

set

S

of

agents

that

exert

effort

(S

{i|ai

We

call

the

set

of

agents

S∗

that

the

principal

contracts

with

in

a∗

(S∗

{i|a∗

i

an

optimal

contract

for

the

principal

at

value

v.

We

sometimes

abuse

notation

and

denote

t(S)

instead

of

t(a),

when

S

is

exactly

the

set

of

agents

that

exert

effort

in

a

A.

A

natural

yardstick

by

which

to

measure

this

decision

is

the

non-strategic

case,

i.e.

when

the

agents

need

not

be

motivated

but

are

rather

controlled

directly

by

the

principal

(who

also

bears

their

costs).

In

this

case

the

principal

will

simply

choose

the

profile

a

A

that

optimizes

the

social

welfare

(global

efficiency),

t(a)

v

P

ci.

The

worst

ratio

between

the

social

welfare

in

this

non-strategic

case

and

the

social

welfare

for

the

profile

a

A

chosen

by

the

principal

in

the

agency

case,

may

be

termed

the

price

of

unaccountability.

Given

a

technology

(t,

c),

let

S∗

(v)

denote

the

optimal

contract

in

the

agency

case

and

let

S∗

ns(v)

denote

an

optimal

contract

in

the

non-strategic

case,

when

the

principal"s

value

is

v.

The

social

welfare

for

value

v

when

the

set

S

of

agents

is

contracted

is

t(S)

v

P

i∈S

ci

(in

both

the

agency

and

non-strategic

cases).

Definition

The

price

of

unaccountability

POU(t,

c)

of

a

technology

(t,

c)

is

defined

as

the

worst

ratio

(over

v)

between

the

total

social

welfare

in

the

non-strategic

case

and

the

agency

case:

POU(t,

c)

t(S∗

ns(v))

v

P

i∈S∗

ns(v)

ci

t(S∗(v))

v

P

i∈S∗(v)

ci

In

cases

where

several

sets

are

optimal

in

the

agency

case,

we

take

the

worst

set

(i.e.,

the

set

that

yields

the

lowest

social

welfare).

When

the

technology

(t,

c)

is

clear

in

the

context

we

will

use

POU

to

denote

the

price

of

unaccountability

for

technology

(t,

c).

Note

that

the

POU

is

at

least

for

any

technology.

As

we

would

like

to

focus

on

results

that

derived

from

properties

of

the

success

function,

in

most

of

the

paper

we

will

deal

with

the

case

where

all

agents

have

an

identical

cost

c,

that

is

ci

c

for

all

i

N.

We

denote

a

technology

(t,

c)

with

identical

costs

by

(t,

c).

For

the

simplicity

of

the

presentation,

we

sometimes

use

the

term

technology

function

to

refer

to

the

success

function

of

the

technology.

Structured

Technology

Functions

In

order

to

be

more

concrete,

we

will

especially

focus

on

technology

functions

whose

structure

can

be

described

easily

as

being

derived

from

independent

agent

tasks

we

call

these

structured

technology

functions.

This

subclass

will

first

give

us

some

natural

examples

of

technology

function,

and

will

also

provide

a

succinct

and

natural

way

to

represent

the

technology

functions.

In

a

structured

technology

function,

each

individual

succeeds

or

fails

in

his

own

task

independently.

The

project"s

success

or

failure

depends,

possibly

in

a

complex

way,

on

the

set

of

successful

sub-tasks.

Thus

we

will

assume

a

monotone

Boolean

function

f

which

denotes

whether

the

project

succeeds

as

a

function

of

the

success

of

the

n

agents"

tasks

(and

is

not

determined

by

any

set

of

agents).

Additionally

there

are

constants

γi

δi

where

γi

denotes

the

probability

of

success

for

agent

i

if

he

does

not

exert

effort,

and

δi

γi)

denotes

the

probability

of

success

if

he

does

exert

effort.

In

order

to

reduce

the

number

of

parameters,

we

will

restrict

our

attention

to

the

case

where

γ1

γn

γ

and

δ1

δn

γ

thus

leaving

ourselves

with

a

single

parameter

γ

s.t.

γ

Under

this

structure,

the

technology

function

t

is

defined

by

an)

being

the

probability

that

xn)

where

the

bits

xn

are

chosen

according

to

the

following

distribution:

if

ai

then

xi

with

probability

γ

and

xi

with

probability

γ;

otherwise,

i.e.

if

ai

then

xi

with

probability

γ

and

xi

with

probability

γ.

We

denote

x

xn).

The

question

of

the

representation

of

the

technology

function

is

now

reduced

to

that

of

representing

the

underlying

monotone

Boolean

function

f.

In

the

most

general

case,

the

function

f

can

be

given

by

a

general

monotone

Boolean

circuit.

An

especially

natural

sub-class

of

functions

in

the

structured

technologies

setting

would

be

functions

that

can

be

represented

as

a

read-once

network

a

graph

with

a

given

source

and

sink,

where

every

edge

is

labeled

by

a

different

player.

The

project

succeeds

if

the

edges

that

belong

to

player"s

whose

task

succeeded

form

a

path

between

the

source

and

the

A

few

simple

examples

should

be

in

order

here:

The

AND

technology:

xn)

is

the

logical

conjunction

of

xi

(f(x)

V

i∈N

xi).

Thus

the

project

succeeds

only

if

all

agents

succeed

in

their

tasks.

This

is

shown

graphically

as

a

read-once

network

in

Figure

If

m

agents

exert

effort

P

i

ai

m),

then

t(a)

tm

γn−m

γ)m

E.g.

for

two

players,

the

technology

function

is

given

by

γ2

γ),

and

The

OR

technology:

xn)

is

the

logical

disjunction

of

xi

(f(x)

W

i∈N

xi).

Thus

the

project

succeeds

if

at

least

one

of

the

agents

succeed

in

their

tasks.

This

is

shown

graphically

as

a

read-once

network

in

Figure

If

m

agents

exert

effort,

then

tm

γm

γ)n−m

.E.g.

for

two

players,

the

technology

function

is

given

by

γ),

and

γ2

The

Or-of-Ands

(OOA)

technology:

f(x)

is

the

logical

disjunction

of

conjunctions.

In

the

simplest

case

of

equal-length

clauses

(denote

by

nc

the

number

of

clauses

and

by

nl

their

length),

f(x)

Wnc

Vnl

xj

k).

Thus

the

project

succeeds

if

in

at

least

one

clause

all

agents

succeed

in

their

tasks.

This

is

shown

graphically

as

a

read-once

network

in

Figure

If

mi

agents

on

path

i

exert

effort,

then

mnc

Q

γnl−mi

γ)mi

E.g.

for

four

players,

the

technology

function

is

given

by

γ2

γ2

and

so

on.

One

may

view

this

representation

as

directly

corresponding

to

the

project

of

delivering

a

message

from

the

source

to

the

sink

in

a

real

network

of

computers,

with

the

edges

being

controlled

by

selfish

agents.

Figure

Graphical

representations

of

(a)

AND

and

(b)

OR

technologies.

Figure

Graphical

representations

of

(a)

OOA

and

(b)

AOO

technologies.

The

And-of-Ors

(AOO)

technology:

f(x)

is

the

logical

conjunction

of

disjunctions.

In

the

simplest

case

of

equal-length

clauses

(denote

by

nl

the

number

of

clauses

and

by

nc

their

length),

f(x)

Vnl

Wnc

xj

k).

Thus

the

project

succeeds

if

at

least

one

agent

from

each

disjunctive-form-clause

succeeds

in

his

tasks.

This

is

shown

graphically

as

a

read-once

network

in

Figure

If

mi

agents

on

clause

i

exert

effort,

then

mnc

Q

γmi

γ)nc−mi

E.g.

for

four

players,

the

technology

function

is

given

by

and

so

on.

The

Majority

technology:

f(x)

is

if

a

majority

of

the

values

xi

are

Thus

the

project

succeeds

if

most

players

succeed.

The

majority

function,

even

on

inputs,

can

not

be

represented

by

a

read-once

network,

but

is

easily

represented

by

a

monotone

Boolean

formula

maj(x,

y,

z)

xy+yz+xz.

In

this

case

the

technology

function

is

given

by

γ)

γ3

γ3

γ

etc.

ANALYSIS

OF

SOME

ANONYMOUS

TECHNOLOGIES

A

success

function

t

is

called

anonymous

if

it

is

symmetric

with

respect

to

the

players.

I.e.

an)

depends

only

on

P

i∈N

ai

(the

number

of

agents

that

exert

effort).

A

technology

(t,

c)

is

anonymous

if

t

is

anonymous

and

the

cost

c

is

identical

to

all

agents.

Of

the

examples

presented

above,

the

AND,

OR,

and

majority

technologies

were

anonymous

(but

not

AOO

and

OOA).

As

for

an

anonymous

t

only

the

number

of

agents

that

exert

effort

is

important,

we

can

shorten

the

notations

and

denote

tm

Δm

tm,

pm

c

Δm−1

and

um

tm

(v

m

pm),

for

the

case

of

identical

cost

c.

v

gamma

gamma

Figure

Number

of

agents

in

the

optimal

contract

of

the

AND

(left)

and

OR

(right)

technologies

with

players,

as

a

function

of

γ

and

v.

AND

technology:

either

or

agents

are

contracted,

and

the

transition

value

is

monotonic

in

γ.

OR

technology:

for

any

γ

we

can

see

all

transitions.

AND

and

OR

Technologies

Let

us

start

with

a

direct

and

full

analysis

of

the

AND

and

OR

technologies

for

two

players

for

the

case

γ

and

c

Example

AND

technology

with

two

agents,

c

γ

we

have

γ2

γ)

and

thus

Δ0

and

Δ1

The

principal

has

possibilities:

contracting

with

or

agents.

Let

us

write

down

the

expressions

for

his

utility

in

these

cases:

Agents:

No

agent

is

paid

thus

and

the

principal"s

utility

is

v

Agent:

This

agent

is

paid

on

success

and

the

principal"s

utility

is

Agents:

each

agent

is

paid

on

success,

and

the

principal"s

utility

is

Notice

that

the

option

of

contracting

with

one

agent

is

always

inferior

to

either

contracting

with

both

or

with

none,

and

will

never

be

taken

by

the

principal.

The

principal

will

contract

with

no

agent

when

v

with

both

agents

whenever

v

and

with

either

non

or

both

for

v

This

should

be

contrasted

with

the

non-strategic

case

in

which

the

principal

completely

controls

the

agents

(and

bears

their

costs)

and

thus

simply

optimizes

globally.

In

this

case

the

principal

will

make

both

agents

exert

effort

whenever

v

Thus

for

example,

for

v

the

globally

optimal

decision

(non-strategic

case)

would

give

a

global

utility

of

while

the

principal"s

decision

(in

the

agency

case)

would

give

a

global

utility

of

giving

a

ratio

of

It

turns

out

that

this

is

the

worst

price

of

unaccountability

in

this

example,

and

it

is

obtained

exactly

at

the

transition

point

of

the

agency

case,

as

we

show

below.

Example

OR

technology

with

two

agents,

c

γ

we

have

γ)

and

γ2

thus

Δ0

and

Δ1

Let

us

write

down

the

expressions

for

the

principal"s

utility

in

these

three

cases:

Agents:

No

agent

is

paid

and

the

principal"s

utility

is

v

Agent:

This

agent

is

paid

on

success

and

the

principal"s

utility

is

Agents:

each

agent

is

paid

on

success,

and

the

principal"s

utility

is

Now

contracting

with

one

agent

is

better

than

contracting

with

whenever

v

(and

is

equivalent

for

v

and

contracting

with

both

agents

is

better

than

contracting

with

one

agent

whenever

v

(and

is

equivalent

for

v

thus

the

principal

will

contract

with

no

agent

for

v

with

one

agent

for

v

and

with

both

agents

for

v

In

the

non-strategic

case,

in

comparison,

the

principal

will

make

a

single

agent

exert

effort

for

v

and

the

second

one

exert

effort

as

well

when

v

It

turns

out

that

the

price

of

unaccountability

here

is

and

is

achieved

at

v

which

is

exactly

the

transition

point

from

to

contracted

agents

in

the

agency

case.

This

is

not

a

coincidence

that

in

both

the

AND

and

OR

technologies

the

POU

is

obtained

for

v

that

is

a

transition

point

(see

full

proof

in

Lemma

For

any

given

technology

(t,

c)

the

price

of

unaccountability

POU(t,

c)

is

obtained

at

some

value

v

which

is

a

transition

point,

of

either

the

agency

or

the

non-strategic

cases.

Proof

sketch:

We

look

at

all

transition

points

in

both

cases.

For

any

value

lower

than

the

first

transition

point,

agents

are

contracted

in

both

cases,

and

the

social

welfare

ratio

is

Similarly,

for

any

value

higher

than

the

last

transition

point,

n

agents

are

contracted

in

both

cases,

and

the

social

welfare

ratio

is

Thus,

we

can

focus

on

the

interval

between

the

first

and

last

transition

points.

Between

any

pair

of

consecutive

points,

the

social

welfare

ratio

is

between

two

linear

functions

of

v

(the

optimal

contracts

are

fixed

on

such

a

segment).

We

then

show

that

for

each

segment,

the

suprimum

ratio

is

obtained

at

an

end

point

of

the

segment

(a

transition

point).

As

there

are

finitely

many

such

points,

the

global

suprimum

is

obtained

at

the

transition

point

with

the

maximal

social

welfare

ratio.

We

already

see

a

qualitative

difference

between

the

AND

and

OR

technologies

(even

with

agents):

in

the

first

case

either

all

agents

are

contracted

or

none,

while

in

the

second

case,

for

some

intermediate

range

of

values

v,

exactly

one

agent

is

contracted.

Figure

shows

the

same

phenomena

for

AND

and

OR

technologies

with

players.

Theorem

For

any

anonymous

AND

there

exists

a

v∗

such

that

for

any

v

v∗

it

is

optimal

to

contract

with

no

agent,

for

v

v∗

it

is

optimal

to

contract

with

all

n

agents,

and

for

v

v∗,

both

contracts

n)

are

optimal.

AND

technology

with

any

number

of

agents

n

and

any

γ,

and

any

identical

cost

c.

v∗

is

a

function

of

n,

γ,

c.

the

price

of

unaccountability

is

obtained

at

the

transition

point

of

the

agency

case,

and

is

POU

γ

´n−1

γ

γ

Proof

sketch:

For

any

fixed

number

of

contracted

agents,

k,

the

principal"s

utility

is

a

linear

function

in

v,

where

the

slope

equals

the

success

probability

under

k

contracted

agents.

Thus,

the

optimal

contract

corresponds

to

the

maximum

over

a

set

of

linear

functions.

Let

v∗

denote

the

point

at

which

the

principal

is

indifferent

between

contracting

with

or

n

agents.

In

we

show

that

at

v∗,

the

principal"s

utility

from

contracting

with

(or

n)

agents

is

higher

than

his

utility

when

contracting

with

any

number

of

agents

k

n

As

the

number

of

contracted

agents

is

monotonic

non-decreasing

in

the

value

(due

to

Lemma

for

any

v

v∗,

contracting

with

agents

is

optimal,

and

for

any

v

v∗,

contracting

with

n

agents

is

optimal.

This

is

true

for

both

the

agency

and

the

non-strategic

cases.

As

in

both

cases

there

is

a

single

transition

point,

the

claim

about

the

price

of

unaccountability

for

AND

technology

is

proved

as

a

special

case

of

Lemma

below.

For

AND

technology

·γ

γn

γ

and

tn

·γ

γ

1−γ

and

the

expressions

for

the

POU

follows.

In

we

present

a

general

characterization

of

technologies

with

a

single

transition

in

the

agency

and

the

non-strategic

cases,

and

provide

a

full

proof

of

Theorem

as

a

special

case.

The

property

of

a

single

transition

occurs

in

both

the

agency

and

the

non-strategic

cases,

where

the

transition

occurs

at

a

smaller

value

of

v

in

the

non-strategic

case.

Notice

that

the

POU

is

not

bounded

across

the

AND

family

of

technologies

(for

various

n,

γ)

as

POU

either

if

γ

(for

any

given

n

or

n

(for

any

fixed

γ

Next

we

consider

the

OR

technology

and

show

that

it

exhibits

all

n

transitions.

Theorem

For

any

anonymous

OR

technology,

there

exist

finite

positive

values

vn

such

that

for

any

v

s.t.

vk

v

contracting

with

exactly

k

agents

is

optimal

(for

v

no

agent

is

contracted,

and

for

v

vn,

all

n

agents

are

contracted).

For

v

vk,

the

principal

is

indifferent

between

contracting

with

k

or

k

agents.

Proof

sketch:

To

prove

the

claim

we

define

vk

to

be

the

value

for

which

the

principal

is

indifferent

between

contracting

with

k

agents,

and

contracting

with

k

agents.

We

then

show

that

for

any

k,

vk

As

the

number

of

contracted

agents

is

monotonic

non-decreasing

in

the

value

(due

to

Lemma

vn

is

a

sufficient

condition

for

the

theorem

to

hold.

The

same

behavior

occurs

in

both

the

agency

and

the

nonstrategic

case.

This

characterization

is

a

direct

corollary

of

a

more

general

characterization

given

in

While

in

the

AND

technology

we

were

able

to

fully

determine

the

POU

analytically,

the

OR

technology

is

more

difficult

to

analyze.

Open

Question

What

is

the

POU

for

OR

with

n

agents?

Is

it

bounded

by

a

constant

for

every

n?

We

are

only

able

to

determine

the

POU

of

the

OR

technology

for

the

case

of

two

agents

Even

for

the

agents

case

we

already

observe

a

qualitative

difference

between

the

POU

in

the

AND

and

OR

technologies.

Observation

While

in

the

AND

technology

the

POU

for

n

is

not

bounded

from

above

(for

γ

the

highest

POU

in

OR

technology

with

two

agents

is

(for

γ

What

Determines

the

Transitions?

Theorems

and

say

that

both

the

AND

and

OR

technologies

exhibit

the

same

transition

behavior

(changes

of

the

optimal

contract)

in

the

agency

and

the

non-strategic

cases.

However,

this

is

not

true

in

general.

In

we

provide

a

full

characterization

of

the

sufficient

and

necessary

conditions

for

general

anonymous

technologies

to

have

a

single

transition

and

all

n

transitions.

We

find

that

the

conditions

in

the

agency

case

are

different

than

the

ones

in

the

non-strategic

case.

We

are

able

to

determine

the

POU

for

any

anonymous

technology

that

exhibits

a

single

transition

in

both

the

agency

and

the

non-strategic

cases

(see

full

proof

in

Lemma

For

any

anonymous

technology

that

has

a

single

transition

in

both

the

agency

and

the

non-strategic

cases,

the

POU

is

given

by:

POU

tn

and

it

is

obtained

at

the

transition

point

of

the

agency

case.

Proof

sketch:

Since

the

payments

in

the

agency

case

are

higher

than

in

the

non-strategic

case,

the

transition

point

in

the

agency

case

occurs

for

a

higher

value

than

in

the

non-strategic

case.

Thus,

there

exists

a

region

in

which

the

optimal

numbers

of

contracted

agents

in

the

agency

and

the

non-strategic

cases

are

and

n,

respectively.

By

Lemma

the

POU

is

obtained

at

a

transition

point.

As

the

social

welfare

ratio

is

decreasing

in

v

in

this

region,

the

POU

is

obtained

at

the

higher

value,

that

is,

at

the

transition

point

of

the

agency

case.

The

transition

point

in

the

agency

case

is

the

point

at

which

the

principal

is

indifferent

between

contracting

with

and

with

n

agents,

v∗

c·n

tn

Substituting

the

transition

point

of

the

agency

case

into

the

POU

expression

yields

the

required

expression.

POU

v∗

tn

c

n

v∗

tn

The

MAJORITY

Technology

The

project

under

the

MAJORITY

function

succeeds

if

the

majority

of

the

agents

succeed

in

their

tasks

(see

Section

We

are

unable

to

characterize

the

transition

behavior

of

the

MAJORITY

technology

analytically.

Figure

presents

the

optimal

number

of

contracted

agents

as

a

function

of

v

and

γ,

for

n

The

phenomena

that

we

observe

in

this

example

(and

others

that

we

looked

at)

leads

us

to

the

following

conjecture.

Conjecture

For

any

Majority

technology

(any

n,

γ

and

c),

there

exists

l,

l

such

that

the

first

transition

is

from

to

l

agents,

and

then

all

the

remaining

n

l

transitions

exist.

gamma

v

Figure

Simulations

results

showing

the

number

of

agents

in

the

optimal

contract

of

the

MAJORITY

technology

with

players,

as

a

function

of

γ

and

v.

As

γ

decreases

the

first

transition

is

at

a

lower

value

and

to

a

higher

number

of

agents.

For

any

sufficiently

small

γ,

the

first

transition

is

to

agents,

and

for

any

sufficiently

large

γ,

the

first

transition

is

to

agents.

For

any

γ,

the

first

transition

is

never

to

more

than

agents,

and

after

the

first

transition

we

see

all

following

possible

transitions.

Moreover,

for

any

fixed

c,

n,

l

when

γ

is

close

enough

to

l

is

a

non-decreasing

function

of

γ

(with

image

and

l

when

γ

is

close

enough

to

NON-ANONYMOUS

TECHNOLOGIES

In

non-anonymous

technologies

(even

with

identical

costs),

we

need

to

talk

about

the

contracted

set

of

agents

and

not

only

about

the

number

of

contracted

agents.

In

this

section,

we

identify

the

sets

of

agents

that

can

be

obtained

as

the

optimal

contract

for

some

v.

These

sets

construct

the

orbit

of

a

technology.

Definition

For

a

technology

t,

a

set

of

agents

S

is

in

the

orbit

of

t

if

for

some

value

v,

the

optimal

contract

is

exactly

with

the

set

S

of

agents

(where

ties

between

different

S"s

are

broken

according

to

a

lexicographic

The

korbit

of

t

is

the

collection

of

sets

of

size

exactly

k

in

the

orbit.

Observe

that

in

the

non-strategic

case

the

k-orbit

of

any

technology

with

identical

cost

c

is

of

size

at

most

(as

all

sets

of

size

k

has

the

same

cost,

only

the

one

with

the

maximal

probability

can

be

on

the

orbit).

Thus,

the

orbit

of

any

such

technology

in

the

non-strategic

case

is

of

size

at

most

n

We

show

that

the

picture

in

the

agency

case

is

very

different.

A

basic

observation

is

that

the

orbit

of

a

technology

is

actually

an

ordered

list

of

sets

of

agents,

where

the

order

is

determined

by

the

following

lemma.

Lemma

Monotonicity

lemma)

For

any

technology

(t,

c),

in

both

the

agency

and

the

non-strategic

cases,

the

This

implies

that

there

are

no

two

sets

with

the

same

success

probability

in

the

orbit.

expected

utility

of

the

principal

at

the

optimal

contracts,

the

success

probability

of

the

optimal

contracts,

and

the

expected

payment

of

the

optimal

contract,

are

all

monotonically

nondecreasing

with

the

value.

Proof.

Suppose

the

sets

of

agents

and

are

optimal

in

and

respectively.

Let

Q(S)

denote

the

expected

total

payment

to

all

agents

in

S

in

the

case

that

the

principal

contracts

with

the

set

S

and

the

project

succeeds

(for

the

agency

case,

Q(S)

t(S)

P

i∈S

ci

t(S)−t(S\i)

while

for

the

non-strategic

case

Q(S)

P

i∈S

ci).

The

principal"s

utility

is

a

linear

function

of

the

value,

u(S,

v)

t(S)·v−Q(S).

As

is

optimal

at

and

as

and

We

conclude

that

thus

the

utility

is

monotonic

non-decreasing

in

the

value.

Next

we

show

that

the

success

probability

is

monotonic

non-decreasing

in

the

value.

is

optimal

at

thus:

is

optimal

at

thus:

Summing

these

two

equations,

we

get

that

which

implies

that

if

than

Finally

we

show

that

the

expected

payment

is

monotonic

non-decreasing

in

the

value.

As

is

optimal

at

and

we

observe

that:

or

equivalently,

which

is

what

we

wanted

to

show.

AOO

and

OOA

Technologies

We

begin

our

discussion

of

non-anonymous

technologies

with

two

examples;

the

And-of-Ors

(AOO)

and

Or-of-Ands

(OOA)

technologies.

The

AOO

technology

(see

figure

is

composed

of

multiple

OR-components

that

are

Anded

together.

Theorem

Let

h

be

an

anonymous

OR

technology,

and

let

f

Vnc

h

be

the

AOO

technology

that

is

obtained

by

a

conjunction

of

nc

of

these

OR-components

on

disjoint

inputs.

Then

for

any

value

v,

an

optimal

contract

contracts

with

the

same

number

of

agents

in

each

OR-component.

Thus,

the

orbit

of

f

is

of

size

at

most

nl

where

nl

is

the

number

of

agents

in

h.

Part

of

the

proof

of

the

theorem

(for

the

complete

proof

see

is

based

on

such

AOO

technology

being

a

special

case

of

a

more

general

family

of

technologies,

in

which

disjoint

anonymous

technologies

are

And-ed

together,

as

explained

in

the

next

section.

We

conjecture

that

a

similar

result

holds

for

the

OOA

technology.

Conjecture

In

an

OOA

technology

which

is

a

disjunction

of

the

same

anonymous

paths

(with

the

same

number

of

agents,

γ

and

c,

but

over

disjoint

inputs),

for

any

value

v

the

optimal

contract

is

constructed

from

some

number

of

fully-contracted

paths.

Moreover,

there

exist

vnl

such

that

for

any

v,

vi

v

exactly

i

paths

are

contracted.

We

are

unable

to

prove

it

in

general,

but

can

prove

it

for

the

case

of

an

OOA

technology

with

two

paths

of

length

two

(see

Orbit

Characterization

The

AOO

is

an

example

of

a

technology

whose

orbit

size

is

linear

in

its

number

of

agents.

If

conjecture

is

true,

the

same

holds

for

the

OOA

technology.

What

can

be

said

about

the

orbit

size

of

a

general

non-anonymous

technology?

In

case

of

identical

costs,

it

is

impossible

for

all

subsets

of

agents

to

be

on

the

orbit.

This

holds

by

the

observation

that

the

(a

single

agent

that

exerts

effort)

is

of

size

at

most

Only

the

agent

that

gives

the

highest

success

probability

(when

only

he

exerts

effort)

can

be

on

the

orbit

(as

he

also

needs

to

be

paid

the

least).

Nevertheless,

we

next

show

that

the

orbit

can

have

exponential

size.

A

collection

of

sets

of

k

elements

(out

of

n)

is

admissible,

if

every

two

sets

in

the

collection

differ

by

at

least

elements

(e.g.

for

and

can

not

be

together

in

the

collection,

but

and

can

be).

Theorem

Every

admissible

collection

can

be

obtained

as

the

k

orbit

of

some

t.

Proof

sketch:

The

proof

is

constructive.

Let

S

be

some

admissible

collection

of

k-size

sets.

For

each

set

S

S

in

the

collection

we

pick

S,

such

that

for

any

two

admissible

sets

Si

Sj,

Si

Sj

We

then

define

the

technology

function

t

as

follows:

for

any

S

S,

t(S)

S

and

∀i

S,

t(S

i)

S.

Thus,

the

marginal

contribution

of

every

i

S

is

S.

Note

that

since

S

is

admissible,

t

is

well

defined,

as

for

any

two

sets

S,

S

S

and

any

two

agents

i,

j,

S

i

S

j.

For

any

other

set

Z,

we

define

t(Z)

in

a

way

that

ensures

that

the

marginal

contribution

of

each

agent

in

Z

is

a

very

small

(the

technical

details

appear

in

the

full

version).

This

completes

the

definition

of

t.

We

show

that

each

admissible

set

S

S

is

optimal

at

the

value

vS

ck

S

We

first

show

that

it

is

better

than

any

other

S

S.

At

the

value

vS

ck

S

the

set

S

that

corresponds

to

S

maximizes

the

utility

of

the

principal.

This

result

is

obtained

by

taking

the

derivative

of

u(S,

v).

Therefore

S

yields

a

higher

utility

than

any

other

S

S.

We

also

pick

the

range

of

S

to

ensure

that

at

vS,

S

is

better

than

any

other

set

S

i

s.t.

S

S.

Now

we

are

left

to

show

that

at

vS,

the

set

S

yields

a

higher

utility

than

any

other

set

Z

S.

The

construction

of

t(Z)

ensures

this

since

the

marginal

contribution

of

each

agent

in

Z

is

such

a

small

that

the

payment

is

too

high

for

the

set

to

be

optimal.

In

we

present

the

full

proof

of

the

theorem,

as

well

as

the

full

proofs

of

all

other

claims

presented

in

this

section

without

such

a

proof.

We

next

show

that

there

exist

very

large

admissible

collections.

Lemma

For

any

n

k,

there

exists

an

admissible

collection

of

k-size

sets

of

size

Ω(

n

`n

k

Proof

sketch:

The

proof

is

based

on

an

error

correcting

code

that

corrects

one

bit.

Such

a

code

has

a

distance

thus

admissible.

It

is

known

that

there

are

such

codes

with

/n)

code

words.

To

ensure

that

an

appropriate

fraction

of

these

code

words

have

weight

k,

we

construct

a

new

code

by

XOR-ing

each

code

word

with

a

random

word

r.

The

properties

of

XOR

ensure

that

the

new

code

remains

admissible.

Each

code

word

is

now

uniformly

mapped

to

the

whole

cube,

and

thus

its

probability

of

having

weight

k

is

`n

k

Thus

the

expected

number

of

weight

k

words

is

Ω(

`n

k

/n),

and

for

some

r

this

expectation

is

achieved

or

exceeded.

For

k

we

can

construct

an

exponential

size

admissible

collection,

which

by

Theorem

can

be

used

to

build

a

technology

with

exponential

size

orbit.

Corollary

There

exists

a

technology

(t,

c)

with

orbit

of

size

Ω(

n

n

Thus,

we

are

able

to

construct

a

technology

with

exponential

orbit,

but

this

technology

is

not

a

network

technology

or

a

structured

technology.

Open

Question

Is

there

a

Read

Once

network

with

exponential

orbit?

Is

there

a

structured

technology

with

exponential

orbit?

Nevertheless,

so

far,

we

have

not

seen

examples

of

seriesparallel

networks

whose

orbit

size

is

larger

than

n

Open

Question

How

big

can

the

orbit

size

of

a

seriesparallel

network

be?

We

make

the

first

step

towards

a

solution

of

this

question

by

showing

that

the

size

of

the

orbit

of

a

conjunction

of

two

disjoint

networks

(taking

the

two

in

a

serial)

is

at

most

the

sum

of

the

two

networks"

orbit

sizes.

Let

g

and

h

be

two

Boolean

functions

on

disjoint

inputs

and

let

f

g

V

h

(i.e.,

take

their

networks

in

series).

The

optimal

contract

for

f

for

some

v,

denoted

by

S,

is

composed

of

some

agents

from

the

h-part

and

some

from

the

g-part,

call

them

T

and

R

respectively.

Lemma

Let

S

be

an

optimal

contract

for

f

g

V

h

on

v.

Then,

T

is

an

optimal

contract

for

h

on

v

tg(R),

and

R

is

an

optimal

contract

for

g

on

v

th(T).

Proof

sketch:

We

exress

the

pricipal"s

utility

u(S,

v)

from

contracting

with

the

set

S

when

his

value

is

v.

We

abuse

notation

and

use

the

function

to

denote

the

technology

as

well.

Let

Δf

i

(S

i)

denote

the

marginal

contribution

of

agent

i

S.

Then,

for

any

i

T,

Δf

i

(S

i)

g(R)

Δh

i

(T

i),

and

for

any

i

R,

Δf

i

(S

i)

h(T)

Δg

i

(R

i).

By

substituting

these

expressions

and

f(S)

h(T)

g(R),

we

derive

that

u(S,

v)

h(T)

g(R)

v

P

i∈T

ci

Δh

i

(T

\i)

g(R)

P

i∈R

ci

Δ

g

i

(R\i)

The

first

term

is

maximized

at

a

set

T

that

is

optimal

for

h

on

the

value

g(R)

v,

while

the

second

term

is

independent

of

T

and

h.

Thus,

S

is

optimal

for

f

on

v

if

and

only

if

T

is

an

optimal

contract

for

h

on

v

tg(R).

Similarly,

we

show

that

R

is

an

optimal

contract

for

g

on

v

th(T).

Lemma

The

real

function

v

th(T),

where

T

is

the

h

part

of

an

optimal

contract

for

f

on

v,

is

monotone

non-decreasing

(and

similarly

for

the

function

v

tg(R)).

Proof.

Let

be

the

optimal

contract

for

f

on

and

let

be

the

optimal

contract

for

f

on

By

Lemma

and

since

f

g

h,

Assume

in

contradiction

that

then

since

this

implies

that

By

Lemma

is

optimal

for

h

on

and

is

optimal

for

h

on

As

and

is

optimal

for

h

on

a

larger

value

than

thus

by

Lemma

a

contradiction.

Based

on

Lemma

and

Lemma

we

obtain

the

following

Lemma.

For

the

full

proof,

see

Lemma

Let

g

and

h

be

two

Boolean

functions

on

disjoint

inputs

and

let

f

g

V

h

(i.e.,

take

their

networks

in

series).

Suppose

x

and

y

are

the

respective

orbit

sizes

of

g

and

h;

then,

the

orbit

size

of

f

is

less

or

equal

to

x

y

By

induction

we

get

the

following

corollary.

Corollary

Assume

that

{(gj,

cj

)}m

is

a

set

of

anonymous

technologies

on

disjoint

inputs,

each

with

identical

agent

cost

(all

agents

of

technology

gj

has

the

same

cost

cj).

Then

the

orbit

of

f

Vm

gj

is

of

size

at

most

Pm

nj

where

nj

is

the

number

of

agents

in

technology

gj

(the

orbit

is

linear

in

the

number

of

agents).

In

particular,

this

holds

for

AOO

technology

where

each

OR-component

is

anonymous.

It

would

also

be

interesting

to

consider

a

disjunction

of

two

Boolean

functions.

Open

Question

Does

Lemma

hold

also

for

the

Boolean

function

f

g

W

h

(i.e.,

when

the

networks

g,

h

are

taken

in

parallel)?

We

conjecture

that

this

is

indeed

the

case,

and

that

the

corresponding

Lemmas

and

exist

for

the

OR

case

as

well.

If

this

is

true,

this

will

show

that

series-parallel

networks

have

polynomial

size

orbit.

ALGORITHMIC

ASPECTS

Our

analysis

throughout

the

paper

sheds

some

light

on

the

algorithmic

aspects

of

computing

the

best

contract.

In

this

section

we

state

these

implications

(for

the

proofs

see

We

first

consider

the

general

model

where

the

technology

function

is

given

by

an

arbitrary

monotone

function

t

(with

rational

values),

and

we

then

consider

the

case

of

structured

technologies

given

by

a

network

representation

of

the

underlying

Boolean

function.

Binary-Outcome

Binary-Action

Technologies

Here

we

assume

that

we

are

given

a

technology

and

value

v

as

the

input,

and

our

output

should

be

the

optimal

contract,

i.e.

the

set

S∗

of

agents

to

be

contracted

and

the

contract

pi

for

each

i

S∗

In

the

general

case,

the

success

function

t

is

of

size

exponential

in

n,

the

number

of

agents,

and

we

will

need

to

deal

with

that.

In

the

special

case

of

anonymous

technologies,

the

description

of

t

is

only

the

numbers

tn,

and

in

this

case

our

analysis

in

section

completely

suffices

for

computing

the

optimal

contract.

Proposition

Given

as

input

the

full

description

of

a

technology

(the

values

tn

and

the

identical

cost

c

for

an

anonymous

technology,

or

the

value

t(S)

for

all

the

possible

subsets

S

N

of

the

players,

and

a

vector

of

costs

c

for

non-anonymous

technologies),

the

following

can

all

be

computed

in

polynomial

time:

The

orbit

of

the

technology

in

both

the

agency

and

the

non-strategic

cases.

An

optimal

contract

for

any

given

value

v,

for

both

the

agency

and

the

non-strategic

cases.

The

price

of

unaccountability

POU(t,

c).

Proof.

We

prove

the

claims

for

the

non-anonymous

case,

the

proof

for

the

anonymous

case

is

similar.

We

first

show

how

to

construct

the

orbit

of

the

technology

(the

same

procedure

apply

in

both

cases).

To

construct

the

orbit

we

find

all

transition

points

and

the

sets

that

are

in

the

orbit.

The

empty

contract

is

always

optimal

for

v

Assume

that

we

have

calculated

the

optimal

contracts

and

the

transition

points

up

to

some

transition

point

v

for

which

S

is

an

optimal

contract

with

the

highest

success

probability.

We

show

how

to

calculate

the

next

transition

point

and

the

next

optimal

contract.

By

Lemma

the

next

contract

on

the

orbit

(for

higher

values)

has

a

higher

success

probability

(there

are

no

two

sets

with

the

same

success

probability

on

the

orbit).

We

calculate

the

next

optimal

contract

by

the

following

procedure.

We

go

over

all

sets

T

such

that

t(T)

t(S),

and

calculate

the

value

for

which

the

principal

is

indifferent

between

contracting

with

T

and

contracting

with

S.

The

minimal

indifference

value

is

the

next

transition

point

and

the

contract

that

has

the

minimal

indifference

value

is

the

next

optimal

contract.

Linearity

of

the

utility

in

the

value

and

monotonicity

of

the

success

probability

of

the

optimal

contracts

ensure

that

the

above

works.

Clearly

the

above

calculation

is

polynomial

in

the

input

size.

Once

we

have

the

orbit,

it

is

clear

that

an

optimal

contract

for

any

given

value

v

can

be

calculated.

We

find

the

largest

transition

point

that

is

not

larger

than

the

value

v,

and

the

optimal

contract

at

v

is

the

set

with

the

higher

success

probability

at

this

transition

point.

Finally,

as

we

can

calculate

the

orbit

of

the

technology

in

both

the

agency

and

the

non-strategic

cases

in

polynomial

time,

we

can

find

the

price

of

unaccountability

in

polynomial

time.

By

Lemma

the

price

of

unaccountability

POU(t)

is

obtained

at

some

transition

point,

so

we

only

need

to

go

over

all

transition

points,

and

find

the

one

with

the

maximal

social

welfare

ratio.

A

more

interesting

question

is

whether

if

given

the

function

t

as

a

black

box,

we

can

compute

the

optimal

contract

in

time

that

is

polynomial

in

n.

We

can

show

that,

in

general

this

is

not

the

case:

Theorem

Given

as

input

a

black

box

for

a

success

function

t

(when

the

costs

are

identical),

and

a

value

v,

the

number

of

queries

that

is

needed,

in

the

worst

case,

to

find

the

optimal

contract

is

exponential

in

n.

Proof.

Consider

the

following

family

of

technologies.

For

some

small

and

k

we

define

the

success

probability

for

a

given

set

T

as

follows.

If

|T|

k,

then

t(T)

|T|

If

|T|

k,

then

t(T)

(n

|T|)

For

each

set

of

agents

ˆT

of

size

k,

the

technology

t

ˆT

is

defined

by

t(

ˆT)

(n

ˆT|)

and

t(T)

|T|

for

any

T

ˆT

of

size

k.

For

the

value

v

c·(k

the

optimal

contract

for

t

ˆT

is

ˆT

(for

the

contract

ˆT

the

utility

of

the

principal

is

about

v

−c·k

while

for

any

other

contract

the

utility

is

negative).

If

the

algorithm

queries

about

at

most

n

sets

of

size

k,

then

it

cannot

always

determine

the

optimal

contract

(as

any

of

the

sets

that

it

has

not

queried

about

might

be

the

optimal

one).

We

conclude

that

n

queries

are

needed

to

determine

the

optimal

contract,

and

this

is

exponential

in

n.

Structured

Technologies

In

this

section

we

will

consider

the

natural

representation

of

read-once

networks

for

the

underlying

Boolean

function.

Thus

the

problem

we

address

will

be:

The

Optimal

Contract

Problem

for

Read

Once

Networks:

Input:

A

read-once

network

G

(V,

E),

with

two

specific

vertices

s,

t;

rational

values

γe,

δe

for

each

player

e

E

(and

ce

and

a

rational

value

v.

Output:

A

set

S

of

agents

who

should

be

contracted

in

an

optimal

contract.

Let

t(E)

denote

the

probability

of

success

when

each

edge

succeeds

with

probability

δe.

We

first

notice

that

even

computing

the

value

t(E)

is

a

hard

problem:

it

is

called

the

network

reliability

problem

and

is

known

to

be

#P

hard

Just

a

little

effort

will

reveal

that

our

problem

is

not

easier:

Theorem

The

Optimal

Contract

Problem

for

Read

Once

Networks

is

#P-hard

(under

Turing

reductions).

Proof.

We

will

show

that

an

algorithm

for

this

problem

can

be

used

to

solve

the

network

reliability

problem.

Given

an

instance

of

a

network

reliability

problem

G,

{ζe}e∈E

(where

ζe

denotes

e"s

probability

of

success),

we

define

an

instance

of

the

optimal

contract

problem

as

follows:

first

define

a

new

graph

G

which

is

obtained

by

Anding

G

with

a

new

player

x,

with

γx

very

close

to

and

δx

γx.

For

the

other

edges,

we

let

δe

ζe

and

γe

By

choosing

γx

close

enough

to

we

can

make

sure

that

player

x

will

enter

the

optimal

contract

only

for

very

large

values

of

v,

after

all

other

agents

are

contracted

(if

we

can

find

the

optimal

contract

for

any

value,

it

is

easy

to

find

a

value

for

which

in

the

original

network

the

optimal

contract

is

E,

by

keep

doubling

the

value

and

asking

for

the

optimal

contract.

Once

we

find

such

a

value,

we

choose

γx

s.t.

c

is

larger

than

that

value).

Let

us

denote

βx

The

critical

value

of

v

where

player

x

enters

the

optimal

contract

of

G

can

be

found

using

binary

search

over

the

algorithm

that

supposedly

finds

the

optimal

contract

for

any

network

and

any

value.

Note

that

at

this

critical

value

v,

the

principal

is

indifferent

between

the

set

E

and

E

{x}.

Now

when

we

write

the

expression

for

this

indifference,

in

terms

of

t(E)

and

Δt

i(E)

we

observe

the

following.

t(E)

γx

v

X

i∈E

c

γx

Δt

i(E

i)

γx)

v

X

i∈E

c

γx)

Δt

i(E

i)

c

t(E)

βx

if

and

only

if

t(E)

γx)

c

v

thus,

if

we

can

always

find

the

optimal

contract

we

are

also

able

to

compute

the

value

of

t(E).

In

conclusion,

computing

the

optimal

contract

in

general

is

hard.

These

results

suggest

two

natural

research

directions.

The

first

avenue

is

to

study

families

of

technologies

whose

optimal

contracts

can

be

computed

in

polynomial

time.

The

second

avenue

is

to

explore

approximation

algorithms

for

the

optimal

contract

problem.

A

possible

candidate

for

the

first

direction

is

the

family

of

series-parallel

networks,

for

which

the

network

reliability

problem

(computing

the

value

of

t)

is

polynomial.

Open

Question

Can

the

optimal

contract

problem

for

Read

Once

series-parallel

networks

be

solved

in

polynomial

time?

We

can

only

handle

the

non-trivial

level

of

AOO

networks:

Lemma

Given

a

Read

Once

AND-of-OR

network

such

that

each

OR-component

is

an

anonymous

technology,

the

optimal

contract

problem

can

be

solved

in

polynomial

time.

Acknowledgments.

This

work

is

supported

by

the

Israel

Science

Foundation,

the

USA-Israel

Binational

Science

Foundation,

the

Lady

Davis

Fellowship

Trust,

and

by

a

National

Science

Foundation

grant

number

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optimal set of contract
contract optimal set
service quality
contractible action
k-orbit
series-parallel network
unaccountability price
agency theory
principal-agent model
incentive
con

classical principalagent
quality of service
combinatorial agency
nash equilibrium
anonymous technology
price of unaccountability