Black D. on the Rationale of Group Deicision-making

7/23/2019 Black D. on the Rationale of Group Deicision-making

http://slidepdf.com/reader/full/black-d-on-the-rationale-of-group-deicision-making 1/12

Lhe

whole

t

toward

rrnational

comprise

:ensifying

re

wants

nl collec-

uction;

at

ark of

re-

orizon. t

r

between

of collec-

lem

of the

i

the

task

:rful

ana-

ical explana-

I

to

shift

the

: the second

n'hoie heory

along

other

no

emphasis

so,

he elabo-

developedn

y

about

the

un, ndB. M.

toward the

le wants, al-

by which

he

nlly

interact

)n

the

litera-

xely

related,

;ory of

"eco-

Econont'ics

'n

i.

ed, Edwin

*,

r93rl,

pp .

ON THE RATIONALE OF'GROUP

DECISION-MAKING

DTINCAN

BLACK

dicate

the

course

of

the

argument'r

I. GENERAL

ASSUMPTIONS

To

d,evelop

ur

theorY,

we

must

make

some

urthei

assumptions'

Our

major

as-

sumption

willbe

that

each

member

of

the

.o**itt.e

ranks

the

motions

n

a

definite

order

of

preference,

whattYgt

that

order

*uy

be.

bo

take

a simple

illustration,

if

thete

are four motions denotedby'o', a"'

&t,

&4,say,

efore

a committee,

he

mem-

a""t

may

prefer

a'2

o

any

of

the

others'

-ut

Ue

ttdifferent

between

03

and

ao,

and

may

prefer

either

of

them

to

a,'

it io,

/.'s

valuation

of

the

motions

could

be

represented

y

the

schedutre

f

preferences

n

the

left-hand

side

of

Fig'

i, io which o" standshighest;ls a2d a,

next

highest,

each

at

the

same

evel;

and

o,

lo*Jut.

And

similar

scales

could

be

drawt

for

other

members

of

the'com-

mittee

with

(h

.

..

04

appearing

n some

defi.nite

order

on

each

scale,

hough

the

ordering

of

the

motions

might

be

differ-

ent

on

the

scale

of

each

member'

We are here using

the

theory

of

rela-

tive

valuation

of

orthodox

Economic

S.i*n..,

whether

the

theory

of

reiative

utility

or

the

theory

of

indifference

.nr.r.t.

The

only

points

which

have

sig-

,rifi.un..

on

the

-d'irected

straight

line

representing

a

member's

scheduie

of

pr.f.r.rr.es

are

those

at

which

motions

are

marLed.,

and

his

scale

really

consists

of

a

number

of points placed n a certain

order

in

relation

to

each

other'

No

sig-

nificance

attaches

to

the

distance

be-

tween

the

points

on

the

scale,

and

any

two

scales

would

be

equivalent

on

which

the

motions

occurred

n the

same

order'

Slhen

a

member

values

the

motions

before

a

committee

in

a

definite

order,

it is reasonable o assume hat, when

these

motions

are

put

against

each

other,

23

theory will

be set out at

greater

length

in

a

ring

book

on

The Pure

Sci,ence

Pol,itics.



o A

DUNCAN BLACK

he votes

in

accordance

with

his

valua-

tion,

i.e.,

n accordance

ith

his schedule

of

preferences.

Thus the

member

.4

would be assumed o vote for o, when it

was

put

in

a

vote

against

a,;

or

lI a3were

put against

oo-since

he is indifferent be-

tween

the two

and

it

would be

irrational

for him

to support

either

against the

other-he

would

be assumed

o

abstain

from voting.

A member's

ievel of

preference

be-

tween the

different

motions

may also be

shown

by denoting the motions put

forward

by

particular

points

on

a

hori-

zontal

axis, while

we

mark

level of

oPA€p

0F

PREF€FEITCE"

oz

o3,o4

O1

preference along the vertical axis. For

instance,

he same

set

of

valuations

of

the

individual

.4 is shown

in the

right

and left

parts

of

Fig.

r. The only

points

in the

diagram

having significance

would

be those

for the

values

e'r, a2,

a3,

e,4,

rr

the

horizontal

axis, corresponding

o

the

motions

actually

put

forward.

We

have

joined

these

points

standing

at various

levels of

preferenceby straight-line seg-

ments,

but this

is done merely to

assist

the eye, since

the curve

would

be

imaginary

except at the

four

points. In

this

dia,gram,

as n the case

of

the

prefer-

ence schedule,

it

is

only the reiative

heights

of different

points

which

have

meaning,

not

their

absolute

heights.'

"

C{.

F. H.

Knight, Ri,sh,

Uncertaintyand Prof.t,

pp.

68-7o.

While

a member's

preference

curve

may

be of

any

shapewhatever,

there

is

reason

o expect hat,

in some

mportant

practical

problems,

the valuations

actu-

ally carried

out will

tend to take

the

form

of

isolated

pointson

single-peaked urves.

This

would

be

particularly

likely

to

happen were

the committee

considering

different

possible

sizes

of a

numerical

quantity

and

choosing

one size n

prefer-

ence

to the others.

It

might be reaching

a decision,

say, with

regard

to the

price

of a product to bemarketedby a firm, or

the

output

for a future

period,

or the

wage

ate of

labor,or the

height

of

a

par-

PotNf

,SEr

AEPRESEIttrrlO t

ovopa

ticular tax, or the legal school-Ieaving

age,and

so

on.

In

such cases

he

committee

member,

in arriving at

an

opinion

on the matter,

would often

try

initially to

judge

which

size

s for

him

the

optimum.

Once

he

had

arrived

at his view of

the

optimum

size,

the

farther

any

proposal departed

rom

it on the

one side

or the other, the less

he would

favor it. The

valuations

carried

out by the

member would then take

the

form of

points

on a

single-peaked

or

O-shaped

urve.

In

working

out our theory we shall

de-

vote considerable

attention

to

this class

of curves

which slope continuously

up-

ward

to a

peak

and

slope continuously

downwardfrom that peak. We shall refer

to the motion corresponding

o the

peak

oRaEP

oF

PF€FEPE*CE

A

of

an

for

th

mum

An

occur

wher

wh

;i



e

curve

there

is

rportant

ns

actu-

bheorm

I

curves.

kely to

sidering

rmerical

r

prefer-

reaching

he price

, i.rm, or

,

or the

of a

par-

c

Eonops

[-leaving

member,

:

matter,

3e

which

:e

he

had

.um

size,

ted

from

t}le

less

s carried

take the

:aked

or

shall

de-.,,

:his class,,:

usly

up'

inuousll

hall refer

ON

TIIE

RATIONALE

OF

GROUP

DECISION-MAKING

25

of

any

curve-the

most-preferred

motion

for

the

member

concerned-as

his

opti-

mum.

Another

case

ikely

to

be

of

frequent

occurrencen practice-especially, again,

where

the

committee

is

selecting

a

par-

ticular

size

of

a

numer.{pal

Quantity-is

that

in

which

the valudtions

carried

out

by

a

member

take

the

form

of

points on

a

single-peaked

curve

with

a truncated

top.

Such

a case

would

arise

when

the

inclividual

feels

uncertain

as

to

which

of two or more numerical

quantities

pro-

posed

represents

his

optimum

choice'

He

cannot

discriminate

n choice

between

(say) two

of

these

numerical

quantities;

but

the

farther

the

proposal

made

falls

below

the

lower

of

these

vaiues,

or

the

higher

it rises

above

the

larger of

them,

the

less

he esteems

he

motion

concerned'

We

shall

work

out

the theory

first

for

,

the case

n which the members' prefer-

't

,,

ence

curves

ale

single-peaked,

nd,

after

'

'

that,

we shall

show

how

the

answer

to

'

any

problem

can be

obtained

no matter

I

r,

,

*nu, the

shape

of

the

members'

curves

..,,,,1

may be.

When

any

matter

is being

con-

sidered

n a committee,

nly

a

finile

num-

' r r - - - - - - ^ - l ^ * . J

motions

is

put

forward

and valued

by

the

members.

We

assume

hat

the

committee

with

which

we

are

concerned

makes

use

of

a

simplemajority in its voting' In practice,

voting

would

be

so

conducted

hat,

after

discussion,

ne

motion

would

be

made

and,

after

further

discussion,

another

-otioo

("tt

"amendment,"

that

is)

might

tte

moved.

If

so,

the

original

mo-

tion

and

amendment

would

be

Placed

against

each

other

in

a vote.

One

of

the

two

motions

having

been

disposed

of,

leaving a single motion'in the field, a

further

amendment

to

it

might

be

movedl

then

a

further

vote

would

be

taken

between

the

survivor

of

the

first

vote

and

the

new

motion;

and so

on'

If

z

motions

were

Put

forward,

r vote

would

be

taken;

if

3

motions,

z votesl

and,

in

general,

if.

m

motions

were

put

forward., herewould be

(m

-

r) votes'3

Now

it will

be

found

to

simPlifY

the

development

of

the

theory

if,

in the

flrst

instance,

we

suppose

that

the voting

procedure

s difierent

from

this.

We

wish

io

make

the

assumPtion

that

when

m

motions

a,,

a",

' .

.

atu

(saY)

have

been

put forward,

the

committee

places

each

of

th"t"

motions

against

every

other

in

a

vote and picks out that motion, if any,

which

is

able

to

get

a simple

majority

against

nery

otlter

motion.

The

motion

o,

it

to be

envisaged

sbeing

put

against

all the

other

motions

02

.

. ami a,

will

already

have

been

put

against

o,, and

we

assume

hat

it will

then

be

put against

a3 .

.

.

a*;

and

so

on'

a--,

finallY being

pitted

against

a^. On

this

assumption

the

number

of

votes

taken

will be the

number

of ways

of

choosing

z

things

out

be

supplied.

..,:;,

r.

lber

of motions

will

be

put forward

and

1s,.curvcs

nd-since

there

are

an

infi-

te number

of

points

on

a.nycontinu-

ls curve-we imply that the person or

hom

the

curve

s drawn

has carriedout

ion

of each

of an infinite

num-

of motions

in

regard"

to each

of

the

:rs.

This

is

unrealistic,

t is true,

but,

'n

the

theory

is

worked

out

for this

re

can

easily get

the

answer

for

;e n

which

only

a finite

number

of

i i ,

iii

l

the

peak



z6

DUNCAN

BLACK

of

m,i.e.,

m(m-t)

/z

votes,

nstead f the

(*

-

t) votes which

would

be taken

in

practice.

This assumption

enables

he

theory to

proceedmore

smoothly

and

quickly

than

the

assumption

that only

(m

-

r) votes

are

held. When

we'have

worked

out the

theory

on this

basis,

we

can

go

on

to

prove

that-in

the classof cases

n which

.

we

are

mainly

interested-the same an-

swer

would

be

given

whether

m(m-

r)

1

z

votes were

held,

as we assume,

or

oniy

the (rn - r) votes of reality. The as-

sumption

is a

kind

of

theoreticai

scaf-

folding

which

can be

discarded

once it

has

served

ts turn.

These,

b.en,

re our

assumptions:

hat

in

a

committee

m motions

are

put

for-

ward,

that each

member carries

out an

evaluation

of

each

motion

in

regard to

every other, that in the voting eachmo-

tion

is

put against

every other,

and that

the

committee

adopts as

its decision

("resolution")

that

motion,

f

any, which

is

able to

get

a simple

majority over

every other.

It

can

be shown

that,

at

most,

onlY

one motion

wiil

be

able to

get

a

sirnple

majority overevery other.To prove his,

let us assrrme

hat

or is

such a motion,

i.e., that

an carr

get

a simple

majority

over

every other.

And

let us assume

hat

this

is also rue

of some

other motion,

ar,.

By our first

assumption,

owevet,

a1 czrr

get

a simple

majority

over every

other

motion,

including

ar. Therefore

or csrl-

not get a simplemajority o.ver 6. {ence,

at

most, only

onemotion

can

get

a

simple

majority

over every other.

II.

TIiTF'MBERS'

PREFERNNCE CURVES

ALL

SINGLE-PEAKED

The method of reasoning

which

we

em-

ploy can

be

seenmost easily

from

a par-

ticular

example.

Figure z

shows

the

preferencecurves

of

the

5

members

of

a

committee.

Only

part of each

curve

has

been

drawn,

and the

curves

are supposed

to

extend

over

a common

range of"the

horizontaL

axis.

Then

if aa

is

put against oa (where

at

I

an

(O,),

the

preference urve

of

each

member-irrespective

of what

its

precise

shape

may

be-is

upsloping

rom

ah to

o,k;

and

azo,

tanding

at

a

higher

level

of

preferenceon the

curve

of each

member,

will

get a

5:5

(5

out of

5)

ma-

jority

against

aa.

f o7,s

put

against

oe,

(where an I at ( O,), at least 4 mem-

bers-viz.,

those

w-ith optimums

at

or

above O,-rvill

have

preference curves

relations

hold

for motions

corresponding

to values above Or. If two values above l

O,

are

placecl

aga,inst each other

in

a

l

vote,

the

nearer

of the two

values

o O"

wiil

get

a

majority

of at

least

3:5

against

he other.

ff a

value

a6

(where

an

I

Ot)

is

put

,

against

a

value

a1e

where

a6) O),

before

we

could find

which of the

values

would win in a vote, we would have to

draw

the courplete

preference curve

for

eaclr

member,

fi.nd whether

{Ln

ot

an

stood

higher

on

the preference

curve of

which

are

upsloping

from

a5 to

a1,; and

ar

will

get at least

a

4:

5

majority

against

ad.If an

{

or

(

Ou

at wil l

get

at

least

a

3:5

majority

against

on. And similar

i

i i ,

: t

i

i i r

i i

r

ii.l

i ,:.ti :

t:

:: ,

.l ;

I

i:,

:,

i'1,

i

'ii

,

,iL

. i i

i i

l i

i

1..

each

member

and

count up the votes

cast

;

tor

an and

an.

But even though

a value

:

below

the

median

optimum O, should

de-

"

feat

all values

to the

left

of

itself, and

,

should defeat someof the values above

Or,

this wouid

be without significance.

,

What we

are

looking

for is that

motion ''

which

can

defeat

every other by

at least.

,

a

simple

majority.

And

we

notice that ;

the

preference urvesof

at least

3

mem-

:

bers

are

downsloping

from O,

leftward,

.

and the

preference

curves

of at least

3

r;

members

are downsloping

rom

O,

right-

ward.

Therefore

O, can defeat

any other'

,

val

si

se

onl

ad

m

P



has

rsed

the

lere

: o f

its

:om

Jher

ach

ma-

Ak t

em-

. o r

'ves

and

inst

:ast

ilar

ting

ove

n a

, O s

3 : 5

put

) ' ) ,

Lues

: t o

for

A*

: o f

last

rlue

de-.

end

ove

1Ce.

:ion

:ast

hat

:rnrj

rrd,

l

I L

ON

THE

RATIONALE

OF

GROUP DECISION-MAKING

or 7

value

in

the

entire

range by

at least a

simple

majority.

And, aswe

have already

seen

(end

of

Sec.

),

this

can be

true

of

only

a single value. The resolution

adopted

by the

committee

must

be the

motion

corresponding

o

theuvalue-Or.

ORnEF oF

PACFERETICE.

the

horizontal axis corresponding

o the

members'

optimums are

named O,,

O",

Or,

.

.

., in the order

pf

their

occur-

rence. The middle or median ontimum

will be the

(n

*

r)/r",

and,

in Flgure

3,

only this median optimum,

the one

im-

PotNT

sE7

Re U RES n rrilq

ktu /Ottl6

ORaEQ

oF

PREFEQEilCE.

to/Atr

sE7

To give

the generai proof,

two

cases

tnust

be

worked

out-that

in which

the

mediately above t and the one immedi-

ately

below

it are

shown.

Then Op4,y7zwill be the motion

adopted

by the committee. Suppose

Opa,11.

were

placed against any lower

value,

sa,. ,

p.

Since

a

*

r)/z members

have optimums at

or

above Op141z1 s

'we

move from left to right from on to

Osq'y1",

t

least

@

*

r)

/z

curvesare

up-

sloprng, viz., those of

members

with

optimurns at or to the right of O14-.17,.t

of members n the committee is

tdd

and

that

in

which

it is

even. We will

'nsider

each

n

turn.

Let

there

be z

members

in

the com-

-tjee,

where

z

is odd. We

suppose hat

rrdering

of

the

points

on

the

horizon-

axis

representing

motions

exists,

{6ring

the preference

curves

of all

Fro. z

single-peaked.

The

points

on

tht



^ Y

DTINCAN

BLACK

Ieast

(n

*

r)/z

members

prefer Op4'11,

to at"and,inavote againstan,Og+r11"will

get a majority

of

at

Ieast

(n

I

r)/z:n,

and this

is sufficient

to.give it

at

least a

simple

majority.

{lherefore

Os+;y,

ca;r-

get

at

ieast

a

simple

majority

against

any

lower value

which

is

put

against

t.

Simi-.

larly

it

can

get at least

a

simple

majority

against

any

higher value.

Thus

it can

get a simple

majority

against

any other

value which can be proPosed.And bY

previous

argument,

it is the

only

value

which

can

do

so.

When

the

number

of

members,

n,

in

the

committee,

is even,

there maY

be

a

tie

in the

voting;

and

we will

suppose

that

an additional

personacting

as

chair-

man,

in

the event

of a

tie

has

the

right

to casta decidingvote.

Let us

suppose,

irst, that

this

mem-

ber

who

acts as

chairman

has

his

opti-

mum

at

Ony2

ot

at,

one

of the

lower

ootimums.

It can

be shown

that

the

motion

corresponding

o the value

Oo1"

wiil

be able

to defeat

any

lower value

(Fig.

+).

Let

an

be

such

a value,

that is,

an

l

On/2.

Then

(n/z

*

t)

members

have

optimums

at or

above

On1,;and at

least

(n/z

{

r)

preference

curves will be

upsloping

as we

move

from left to right

fiom

anto Onp.

At least

(n/,

*

r) mem-

bers

will

vote

for O^1"

against an,

and

this

is

suftcient

to

give On/" & simple

majority.

If Onf

is

put against

anY

value

o6

(where an ) O^n)-since there are nf z

optimums

at

or below

O,p-the

prefer-

ence

curves

of at

least nf

z

members

will

be

downsloping

from

Onp

to &4

3,\d

O,p vmTl

et

at

Ieast

nf z votes

against

o,10,

.e., wili

at least

tie with

or. In

the

event

of

a tie O*1,will

defeat

a*

with

the

aid

of

the

chairman's

decicling

vote be-

cause, by hypothesis, his optimum is

situated

at or below

Onp

a.ndhis

prefer-

ence

curve

must be

downsloping

from

On1"o a7r.

Thus, when

the

chairman's

oPtimum

is situated

at

or below

On/",

O,/"

will

be

able to

get

at

least

a simple

majority

against

any

other

value which

may be

proposed.

Similarly

when

rz, the

number

of

members

n the committee,

is even,

and

the chairman's

optimum

is

at

or

above

Op1"1y', t can be shown hat01"7a"',wili

be

able

to

get at least

a simple

majority

,

against

every

other

value.

One

cannot

leave

the

theorem

of

the

,

preceding

paragraphs

without

pointing

I

out its

analogy

with

the

central

principle

of

economics-that

showing

how

price is '

fixed

by

demand

and

suPPlY.

The

theoremwe

have

proved shows

hat

the

I

decision

adopted

by

the

committee

be-

'

comes

determinate

as soon

as the

posi-

tion

of

one

optimum-which

we can

I

refer

to conveniently

enough

as the ,,

median

optimum-is

given. No

matter

in

what

manner

the

preference

curves

or

l

optimums

of

the

other

members

alter

or

move

about,

f

it is

given

that

one

opti-',

mum remains the median oPtimum, the

decision

of

the committee

must

remain

fixed.

The

analogy

with

economic

science

is that,

in

the

determination

of

price

in

a.r

' l

: f '

. : .

'

:lr

: L .

i r

'

market,

price remains

unchanged

so

ong

as

the

point of

intersection

of the

demand'

and

supply

curves

s

fixed

and

given,

ir-u

respective

of

how

thesecurves

may

alter

their

shapes

above

and below

that

point'

Or,

in

the

version

of

the theorY due

Bcihm-Bawerk,

which

brings

out

the

point very

clearly,

price remains

un:

altered

o

io.tg

u,

th.

n*utginal

pairs"

of

buyers

and

sellers

and

their

price

attlr

tudes

remain

unchanged.

But

the

analogy

exists

only

betwee'

the two

theories;

here

is

a marked

dil

ference n the materials to which the

relate.

In the case

of market

Price,

the

price

of

mined,

&

s

part

of

the

existence

a

commodity,

purchase,

cance

at th

This

is

one

monies

runn

economic

if

one who un

pressed-an

of,the

iast

g

impressed.

decisions,

in

general)

grandharm

the persist

cord is as

the

certaint

In

reach

we

assume

mittee

vote

forward

in

on

his

sch

shown hat,

r,,

would.

defe

,,

voted

in thi

',,,,,

,ff1gfn g1,

9y

,.r,,

ng

in

con

,i.,r,that

some

preferred

;

':

,fesolution

:, :

]f oniy (z

".;1:

no

longer

hold

oPA

PREF

i:,



e

g

D

.e

is

IE

IC

ri-

tn

in

or

or

ti-

he

lin,

ICe,i

1 & r

'ngl

"E

the

price

of

a commodity

is being

deter-

mined,

a

series

of

adjustments

on.the

oart

of

the

consumers

will

bring

into

i"irt".t..

a state

of

affairs

in which

this

pressed-and

by

which

the

economists

of

the

last

generation

were

perhaps

over-

impressed.ln the material of committee

OPD€R

OT

PREFER€TICE.

ON

THE

RATIONALE

OF

GROUP

DECISION.MAKING

29

to

them,

however,

to

vote

in suclr

a way

that

no

motion

wiil

be

able

tb

get a

majority

over

all

the

others'

it ali

me-bers

voted

as

we

have

suP-

posed,

he

motion

adopted

by

the

com-

,nitt."

would be that corresponding o

the

median

optimum,

0*66,

S&]'

Let

us

suppose

now

that

one

or

more

members

with

optimums

above

O*"a-by

voting

otherw^ise

han

directly

in

accordance

with

their

schedule

of

preferences-at-

tempt

to

give

some

other

value'

sa'

an,

a

majority

over

all

the

others,

where

&n O^.a'

Pohlr

6Er

RePPESENtl'yq

,ronoils

2',

o+

dec i s i ons , ( o r o f po l i t i c a l phenom enaB u t w hen t hem em ber s v o t ed i r ec t l y

in

general)

on

the

othe,

itana,

no

such

in

accordance

with

their

preference

,

grand

harmony

.*i.it-

ift"

possiUility-

f

scales'

hose

who

have

anhigher

on

their

,.

the

persistence

r-Jirrr-i-i"y

and

dis-

scales

han

O-"a

would

already

be

sup-

i.,,

cord

is

as

striking

in

the

one

case

as is

porting-oa

against,of"u

Td'*ii::: ii

- ;#:ili'U "t tffi"- r"'tr'" rn.i. iould"bedefeatedv o-"a'Before

t

1.,

.

In reaching

th"

f;;;;ing

conclusions,

could.

defeat

O*a,

or,

would

require

the

r.r.,;

,

we assumed

hat

a

member

of

the com-

support

of

members

whose

optimums

iie

i',,,

llrlJi "ii"u

""'irt"

various

motions

ut

u"io*

o*"o.

The

only

members

ho-by

i,.r,.r forward in

accordance

with

their

order

voting

otherwise

than

in

accordance

i-.,;;;lJ:.illu"rl?il;;:;;;"fr;;1,1;;

witrr"their

cares

r

prererences-courd

i ;i1i1,,,

hown hat,

when

l;i;;;;ists

which

make

an he,resolution^1f,^*:.:."f-nlti'

iljfi

ffi;;;;t

other

f

the

members

are hose

with

optimums

elow

o'"4'

i'e''

l r l ^ + ^ J ^ . ^

. .

;otj

ilril

*"i,

it

is

not

open

o

any

those

gainstwfiose

nterest

t

is

to

do so'

;i:

;.i'i";

Jt^""t

i;*ber

of memb"r,u"i- It wouldbe possible, f course.,or a

.

. I r w I l I U L I ,

Vr

A U J

"^

* ^ - " _ "

- -

. 1 - _ _

ftg,

i"

concert,

o atrer

heir

.vo.ting

o

nu1ler

of

1e*":ti:,:::: ::,"T::"1:

tfi;";

"ifrli

-"tion

which

is

more

motion

would

get

a majority

over

every

j;.IJft;;i;;;;te

adopted

s

he

other'

f,

for

eTa,mpl:t

1ffi:i'1^":-:

$fuil;b""ir

ffi'ffiffi;ii;l'

;o;;

ber

ot

voters

ith

optimums

bove

*aa

' - l :-:*, '-

". -

."- ' .. , 1i 1r-:- ̂ -^,..^:^-

were

to

vote

against

O'ua

when

it

wa s

,iJj-:lt{"rk.

r)

votes re

held'

his

conclusion

pt"..a

against

,o*.

uulrr.

which

stood

. . :

tiri..,i

i : i

:

It,,



i : '

Iower on their scales f preferences'

O- 4

are

single-peaked,

s

we

suppose,

t can

be

shown

hat

voting

between

he

differ-

ent

motions

obeys

the

transitive

prop-

ertys

and

that

if-of any three valueso,,

a2,

a3-or

c&rr

defeat

o'

in

a

vote

and

a,

can

defeat

ar,

then,

of

necessity,

or can

defeat

ar.

This

can

be

proved

by

consideration

of

the

orderings

of

the

points

ar,

oz,

as,

in

relation

to

the

median

optimum'

It

can

be

shown

that

each

ordering

of

the

feats

or.

The

transitive

property

can

easily

be

extended

o

show

that,

if

o, can

defeat

azt a2 can defeat at, . .. and

aF

can

clefeat

41,

hen

at can

defeat

o1'

It

follows

rom

the

transitive

property

that

a,

can

defeat

or.

BY

hYPothesis,

,

can

defeat

ao.

Hence

ar

can

defeat

oo'

Proceeding

y

successive

pplications

we

can

see

hat

a, can

defeat

o7'

DUNCAN

BLACK

In

arriving

at

the

above-mentioned

e-

sults,

we

assumed

hat

every

motion

was

placed

against

every

other

and that in all

m(m

-

i

/z

votes

were

held.

We

can

now

remove

this

assumption

and

show

that

the same

motion

will

be

adopted

by

a

committee

when

onIY

(m

-

r) votes

are

taken

as

n

the

committee

practice

of

real

life.

For

the

case

when

n is

odd,

O1,ar17z

s

s

The

transitive

property

is

defined

in

L'

S'

Stebbing,

A

Modern

ntrod'uction

o

Logic,

pp'

rr2

and

168.

one

of

the

motions

put

forward

and

it

must

enter

into.

the seriesof votes at

some

point. When

it

does,

t

will

defeat

the

fiist

motion

which

it

meets'

t

will

likewise

defeat

the

second

and

every

other

motion

which

is

put

against

it'

That

is,

O1r+'172

rlust

enter

the

voting

process

at

some

stage,

and,

when

it

.lo.*,

it wiil

defeat

he

other

motions

put

against t and become he decisionof the

.J**itt.e.

The

conclusion

Me

eached

holds

good

not

only

for

the

imaginary

procedure

of

placing

everY

Pot19+

against

every

other

but

also

for

the

aitual

committee

procedure

of

real

life'

The

same

is

true

of the

conclusions

we

reached

or

the

case

n which

the

number

of members n the committeewas even'

In

the

committee

procedure

of

real

life

Onp

ot

Os1,1a'wiil

be

the

motion

actually

adopted.

The

assumPtion,

that

m(m

-

t)

/

z

votes

were

held,

enabled

us

to

give

a

mathematical

Proof

which

was

both

definite

and

short.

But

our

conclusions

are true independentiy of this assump-

tion.

As

an

examPle

of

the

use

of

this

:

technique,

we

may

suPPgry

that

the,i

three

directors

of

a

monopolistic

irm

are';

fixing

the

price

of

their

product

for

a';

forthloming

period.

Let

us

further

as--';

sume

hat

neither

future

sales

or

future.rl

costs can be calculated with certainti':

and

that

there

is

no

possibility

of

a.

choice

of

price

being

made

purely

by.

means

of

cost

accounting.

Subjective''

factors

enter,

and'

varying

estimates

ofj

the

future

position

are

formed

by

tbg

different

directors.

If, on

their

difierenti

views

of

the

situation,

the

directors'1

scales f preference reas shown

(Fig:,S)i

the

price

fixed

will

be

that

corresponding

to

the

rnotion

ar'

NI. WIIE

ENCE

When

th

are

not of

solution

to

arrived

at

number

of

ORo

PREFE,



ON

THE

RATIONALE

OF

rE. wrrEN

THE

MEMBERS'

pRETER-

ENCE

CITR\rES ARE

SUsJECT

TO

NO RESTRICTION

When the members'preference

curves

are not

of

the single-peaked

variety,

a

solution to

any

problem

can

always,ibe

arrived at

arithmetically,

provided"the

number

of motions

put

forward

is

finite.

::"1f::,f",.

4,2

0naen

or

PREFERENCE.

GROUP

DECISION-MAKING

is

voting

in

a committee

in

which,the

four

motions ar.

.

. a* have

6.sn,

put

forward.

Along

the

top row

and down,the

left-hand column are shown the rnotions

a\

. . . an.Tn

each

cell

of

the

matrix,

we

-'-record

the

individual's

vote for

one mo-

tion

when

it is placed

against

another.

Looking

to

the topmost

row

of figures,

Poutrset

3 r

44

Frc.

5

d3

otro*.

4t

4r

8z

(for)

43

O,a

(o,

t)

(o,

r)

(t,

o)

(t,

o)

(t, o)

(o,

t)

(against)

&z

ai

?4

(o,

t)

(o,

o)

(t,

o)

.,

o)

(o, o) (o, r)

A

Frc.

6

ii,-,.

ii"'i.r,',

To

begin

with,

we return

to

the as-

: , , . 1 , , r . :

- O - - -

1i;1,;Surnption

hat

every

motion is

placed.

. : ' i r ; ; ' , i - r r i { - ^ ^ - - i r m ,

ilim.e

vote

against

every

other. The

re-

iii.-q1{,tsf the seriesof votes can be shown

l#ry.

readily

by

the

construction

of

a

when

o,

is placed

against

a",

A

votes

for

a.,

and

we enter in

the cell

(a.,

a,)

the

figures

(o,

r).

When

o, is

placed

against

ar, he votes for a, and, in the cell (ar,

or),

we enter

the

figures

(o,

r)

standing

for

o

votes

for and

r

against.

The

other

cells are filled

in

the same

way.

SinceL

is

indifferent

n

choice

betweena"

andao,he

will abstain

rom

voting

when a,

is placed

against

ao,

and

the

cell

(ar,

ao)

will

show

(o,

o).The

figures

n

the

cell(oo,

,)

will

alsobe (o, o).

matrix.6

Construction

f

a

matrix

is illus-

in

Figure

6,

which gives

the

Fi,:c0rrqsponding

o

the

schedule

of

g19n..sof

the

single

member

L who

indebted

to

Dr.

R.

A.

Newing for

sug-

: useot a matrix notation.



3 2

DUNCAN

BLACK

Along

the

main

diagonal

of

the matrix,

instead

of

having

cells

of the

usual

type,

we

have

simply

placed

a series

of

zeros

and joined them by a straight line. This

is

to

indicate

that

the

cells

(4,,

a'),

(a,,

a,),

..

.

,

which

would,

denote

hat

a,

was

placed

against

art

ar&i gainst

a"t .

.

.

t

have

no

meaning.

In

constructing

the

matrix

in

practice,

t

is

usually

easiest

o

enter

these

zeros

along

he main

diagonal

first

and

join

them

by

a straight

line.

Each row to the right of the main

diagonal

is

a reflection

in the

diagonal

of

the

column

mmediately

beneath,

with

tJre

igures

n the

cells

reversed.

Thus

the

cell

(.a,",

er)

immediately

to

the rlght

of

the

diagonal

shows

(o,

t), the

reflec-

tion

in

the

diagonal

of the

figure

(r,

o)

immediately

below

the

diagonal.

The

cell (a,, oo), wo places o the right of the

diagonal,

is the

reflection

of

the

ceil

(oo,

a,) two

places

below

the

diagonal.

The

reason

or

this

is

that the

figures

in any

cell

(ar,,

ax)

must

be

those

of

the cell

(or,,

an)

on the

other

side

of

the

diagonal,

placed

n

the

reverse

order.

This

feature

roughly

halves

the

work

of

constructing

a

matrix:

we

can

filI

in the

figures

on one

side

of

the

diagonal and then complete

the

matrix

by

reflection

of these

figures

in the

diagonal.

The construction

of

an

individual

ma-

trix

would

be

gratuitous

labor

since

it

merely

gives, in

a clumsier

form,

infor-

mation

which

is shown

clearly

enough

in the

member's

schedule

of

preference.

When, however, we have a group of indi-

viduals

voting

on

a

particular

topic

and

the

preference

schedule

of

each

s

known,

the

matrix

for

the

group

presents

n

very

convenient

orm

the

information

that we

need.

For

instance,

for the

group of

schedules

hown

n

Figure

7,

the

accom-

panying

matrix

has been

constructed

preciselyas describedabove.Along the

main

diagonal,

as

before,

we enter

zeros

and

join

them

by

a straight

line.

In

the

cell

(a,,

a")

we

enter

the

figure

(t,

S)

because

on

the scales

of

z

members

or

standshigher than a,, and on the scales

of

the

remaining

3

members

a, stands

higher.

The

other

ceils

are

filled

in the'

same

way

and,

as before,

he

half

of

the

matrix

on

one

side

of

the

diagonal

can be

obtained

by

reversal

of

the

frequencies

n

the

corresponding

eils

on

the other

side.

From the

group

matrix

we

can

read

off

immediately that-when the motions

(h

. . .

a6 a,tE

placed

each

against

every

j

other,

as we

suppose-o,

wiil

be

able

to

get a

simple

majority

over

each

of

the

other

motions

put forward.

For

this

com-

,

mittee

a3

would

be

the

resolution ::

adopted.

If a. motion

exists

which

would

be

able to

get

a simple

majority

over

all

the

others

when

the

members

voted

di-

rectly

in

accordance

with

their

schedules

of

pieferences,

t

would

not be

open

to

any

member

or

group

of

members

bY

,

voting

in

some

other

fashion-to

bring

i

into

existence

as

the

resolution

of

the

committee

a

motion

which

stood

higher

on

the scales

f

all

of

them.

Proof

of

this

proposition s almost identical with that

::

bt

o,rt

earlier

analysis

see

above,

p.

z6).

,'

If,

when

m(m

-

r)/z

votes

are

held,

a motion

exists

which

is able

to

get

at

least

a

simple

majority

over

each

of

the

I

other

motions

put

forward,

it

can

be

i

proved,

as before,

tJrat,

when

the mem-

I

t.tt

vote

d.irectly

in

accordance

with

1

their schedules f preferences'his would

"i

be

bound

to be

the

motion

adopted

even

,i

though

only

(*

-

,) votes

had been

i:

held.

"i:

LLt.

i

But

when

the

members'

Preference

curves

are

not single-peaked,

o motion

'

need

exist

which

is

able

to

get at

least

a,

simple

majority

over

every

other.

ThiSr+

can be seenvery'quickly

from

the

ac*

companying

group bf

scheduies

Fig. 8).'

in

which t

(Lrt

Azt

A3,

metrical.

feated by

and

C;

w

feated

by

ORaee

ar

FREFERENCI

Q,

a2

ag

la+

ag

a6



ne.

In

t

(2,

3)

bers

o. .

: scales

stands

in

the

iof

the

canbe

ncies

n

er side.

rcad

off

notions

I

every

able

to

of

the

ris

com-

;olution

ruld

be

lver

all

rted di-

hedules

open

o

ers-by

o

bring

of

the

I higher

f

of

this

ith that

,

P '

26) '

re

held,

>

get

at

h

of

the

can

be

re mem-

ce

with

:

iswould,

:ed

even

,

ld been,.

eference

motion

t

least

a, i

er.

This,;i

the

ac=

(Fig. B)i

ON

THE

RATIONALE OF

in

which

the

arrangement

of the

motions

a,, a"t ar, on

the

members'

scales

s

syrn-

metrical.

When

a, is put forward,

t is de-

feated

by

or, which

gets the

votes

of

B

and

C;

when

a, is

put forward,

it is de-

feated

by

o,;

when

a, is

put forward,

t

qRoee

or

's?

PREF€REI{CE.

GROUP

DECISION-MAKING

a a

J J

number

of

motions

put forward

in a

committee of

any

given size,

he

greater

will be

the

percentage f the

total num-

ber of

possible

ases

n which

there exists

no motion which is able to get a simple

majority

over

each of the others.

a

,

a

2

as

as

a2

ag

as

a4

o4

4c

b, t)

43

(2,s)

(2,

g)

L4

Q,s)

k,

r)

k,

r)

?5

Q,s)

(2,

s)

k,

r)

Q,

s)

a6

Q, )

k,

")

G,

z)

(5,

o)

(q,

t)

a3

a

5

o4

o6

o6

?t

&z

a3

44

8.5

a6

k,

z)

k,

)

k,

r)

(g,

,)

b,

")

a4

42

o6

"s

4g

k,

r)

(2,

s)

G , ' )

b,

s)

(r,

g)

Q,

s)

(2,

s)

g

44

os

G'

2

oz

k,

z)

(o,

s)

( t ,

q)

a-

o

*6

4/

@t

6rl

Frc.

7

OnaaB

oe

PREFERENCE.

qt

oe

a

3

a2

ag

2,r

o.z

a3

2,r &z &3

-e-t.-

(2,

r)

(r,

r)

(r,

z)

\e---

(e,

r)

( r , t ) ( r , z ) \

oJ

al

oz

A

I

c

Frc.

B

of

the

simple

In

this

state

of

affairs, when

no one

motion

can

obtain

a simple majority

over eachof the others, he procedureof

a committee

which

holds only

(m

-

r)

votes

will arcive

at the

adoption

of a

particular motion,

whereas-if the

re-

quirement were

that

a motion should

be

able

to

get a

simple

majority over

every

other-no

motion

would

be

adopted.

The

particuiar motion which

is

adopted by

the committee using the proceclureof



34

DUNCAN

BLACK

practice

will

depend

on

chance-the

chance

of

particular

motions

coming

earlier

or

later

into

the

voting

process.

For

Figure 8,

if only

(m

-

,

:

z)

votes

were

taken,

that

motion,

a\ ot

a2or

a3,

would

be

adopted

which

was

introduced

last

into the

voting

process.

f, for ex-

ample,

or were

first

Put

against

a", a,

would

be

eliminated;

and, with

the field

thus

cleared,

o, would

defeat

o..

If, then,

only

(m

-

r)

votes

are

held

and

if

no motion

exists

which

is able to

get

a simple

majority

over

every

other,

we cannot read off directlY from the

matrix

the

decision

adopted

by

the com-

mittee.

But

when,

in addition

to

the

matrix,

we

know

the

order

in

which

the

motions

are

put against

one

another

n

a

vote,

again

we

can

deduce what

the

decision

of

the committee

must

be.

Reference

o

Figure 8 will

show

that,

when

the

shapes

of the

preference urves

are

subject to no restriction, the transi-

tive

property

does

not necessarily

hold

good.

IV, CONCLUSION

The

technique

of

this

paper applies

r-

respective

of

the topic

to which

the

mo-

tions

may

relate.

TheY

maY refer

to

price,

quantity,

or other

economic

phe-

nomena;they may relate to motions put

forward

in

regard

to

colonial

govern-

ment,

to

the

structure

of

a

college

cur-

riculum,

and so

on.'.The

tleory

applies

to a decision

aken

on

any topic

by

means

of

voting-so

far, of course,

as the

as-

sumptions

which

are

made correspond

o

reality.

And it

is

possible to widen

the

assumptions, for example, to include

cases

of complementary

valuation;

to

make

allowance

for the

time element;

and

to

cover

the

cases

of

committees

making

use

of special

majorities

of

any

stipulated

size.

With

these

extensions

n

the

assumptions

here

would

be

a widen-

ing

of

the

fie1d

of

phenomena

o which

the theory applies.

The theory,

indeed,

would

aPPear o

present the basis

or

the development

of

'

a

pure science

f

politics'

This

would

em-

ptoy the

same

theory

of

relative

valua-

tion

as

economic

science'

t

would

em-

ploy

a different

definition

of

equilibrium.

Equilibrium

would

now be

defined

in

terms

of

voting,

in

place of

the

type

of

definition employed n economicscience.

We could

move

from the

one

science

o

the

other

with

the

alteration

of

a single

defi.nition.

This,

in

the

view

of the

writer,

would

be the

main

function

of

the theory'

It

fairly

obviously,

too,

enables

solne

oarts

of economics-those

which

relate

to

clecisions

aken

by

groups-to

be

carried a stage beyond

their

present

development.

Gr-asoow

U*rvrnsrtv

tt.

t

r

l.

i:

Tffi;

I

tty-

under

th

Business

a

University

praisal

of

t

dustriai

o

able for a

economis

perfect

co

full

critiqu

business

dustry in

level of e

lotment

of

ficiency

of

distributio

Black D. on the Rationale of Group Deicision-making

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