ON THE IMPOSSIBILITY OF A N ORTHODOX SOCIAL
THEORY A ND OF A N ORTHODOX SOLUTION TO
E NV IRONME NTA L PROBLEMS
R. ROUTLEY
Let us begin on wh a t ma y be castigated as a foolish enterprise:
On ly those wh o f a il to appreciate [the present imperfect]
condition [ o f economics] a re l i k e l y t o a tte mp t t h e co n struction o f universal systems (vo n Neumann & Mo rg e n stern, [1], p. 2).
Von Neumann and Morgenstern appear to have missed the
importance o f g e n e ra l (a n d mo s t ly inapplicable) mo d e ls i n
establishing negative results, f o r example, th a t such-and-such
is impossible, t h a t th e so-and-so p ro b le m is unsolvable. Th e
design o f u n ive rsa l systems i s imp o rta n t i n t h e p ro g ra m o f
showing th a t certain general so cia l problems a re unsolvable,
for example th a t a n o rth o d o x economic so lu tio n o f e n viro n mental problems is impossible, that a general assessment p ro cedure in education, o r sociology, is impossible.
A so cia l o r economic system, like an ethical system, is f irst
of a ll a system. A system in the general sense, ma y be represented b y a relational structure (
1
system
S i s represented b y a re la tio n a l stru ctu re o r mo d e l
). = T
S
(K,hR)a wh
t e re i K iss a se t o f elements (o f alternatives o r
states
t
o o r worlds,
s inathe cases
y
, to be studied) and R { R
iindexed
i sclass ao f relations
n
wh e re each re la tio n R
a}
i( iswga i relation
t eh n i onen K,dri.e.eaa general
xl
structure is e ffe ctive ly a set
with
a
batch
of
relations
on
it
(cf.
Bertalanffy's
'sets of elements
)
standing in interaction', [8], p. 38). Thus algebras, wh ich are sets
(I) Th e ex pos ition of general systems theory , general opt imis at ion theory ,
and t he synthesis o f dec is ion t heory methods sketches t h a t elaborat ed i n
[7], A ppendix 6.
146
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RO UTLEY
with ce rta in two-place relations defined on them, and geometries, w h i c h a re sets w i t h three-place re la tio n s su ch a s b e tweenness on them, are systems. The expressive power of systems is rich, e.g. whatever can be said in quantificational logic,
and much more, can be expressed. Ma n y f a milia r notions can
be defined f o r systems, a s w o rk i n mo d e l th e o ry reveals. I n
particular S' R ' ) i s a subsystem o f S ( K , R) i f f
is a subset o f K and R' consists o f the restrictions o f the re la tions o f R to K'. Thus e ve ry subset of K generates a subsystem
of (K, R).
General systems th e o ry, a s envisaged b y Bertalanffy, ca n
begin a t this point, and its success i n u n ifyin g wo rk in b io lo gical sciences is a lre a d y substantial enough t o deal a h e a vy
blow t o th e vo n Neumann-Morgenstern contention as t o th e
point of constructing universal systems. Some o f the ma in distinctions Bertalanffy is concerned to emphasize can be developed almost at once. Fo r example, an unorganised system is a
system such that the (extensional) properties o f the system can
be derived f ro m the properties o f (proper) components o f the
system (g ive n ju st intersystemic relations such as spatio-temporal ones and general laws). A n organised system is one that
is not thus unorganised, i.e. where the whole is more than the
la wlike consequences o f the distributed components. I t is immediate th a t a n organised syste m cannot b e d e d u ctive ly re duced to features o f its components. If, as appears certain, o rganised systems a ct u a lly o ccu r, t h e n d e d u ctive re d u ct ivism
(the ma in fo rm o f mechanism) is bound to fa il.
The systems co mmo n ly considered i n syste m th e o ry, e.g.
input-output models, automata, feedback models, f o rma l systems, equational models, a re a ll special so rts o f systems o b tained b y considering certain specific relations, e.g. in p u t and
output functions, d e riva b ilit y relations, e tc. De cisio n t h e o ry
and optimisation theory, branches o f general systems theory,
also u su a lly begin wit h a mu ch mo re specifc structure. A n d
some stru ctu ra l elaboration is essential i n o rd e r to introduce
optimisation notions. A General optimisation system OS is re presented by a structure (K, R, f ) wh e re (K, R) is a system and
f is an n-place function (the objective function o f OS) defined
SOLUTION TO E NV I RO NME NTA L PROBLEMS
1
4
7
on a subset of K. The objective in OS is to ma ximize values o f
f s u b j e c t to constraints imposed, i n effect, b y relations R.
In decision th e o ry the problem is u su a lly mo re circumscribed
(in wa ys indicated, e.g., in [7]).
Economic systems — there is much evidence fo r cla imin g —
are special sorts o f optimisation systems. Fo r though the systems studied in va ria b ly concern optimisation, o r the d e te rmination o f e q u ilib riu m states, the factors optimised are specialised; they va ry from some mix of profits, efficiency, states, etc.,
to some mix of GDP and welfare and amenity factors fo r example. Fu rth e rmo re t h e syste ms w i l l b e expected t o in clu d e
characteristically economic elements o r relations, such as p ro ducers, consumers, b u d g e ta ry a n d co mmo d it y re stra in ts, o r
such like.
No w t h e co n d itio n s imp o se d o n t h e o b je ct ive f u n ct io n
through the systematic relations in economic systems ma y render optimisation impossible; indeed the conditions ma y render
an o ve ra ll consistent ranking, on wh ich optimisation depends,
impossible. Th is is th e genesis o f imp o ssib ility results. Mo re over t h e co n d itio n s imp o se d o n t h e o b je ct ive f u n ct io n o r
overall ra n kin g ma y be re a listic o r reasonable ones — wh ich
can thus be reconstituted as requirements o f ra tio n a lity — in
which case th e imp o ssib ility results ca n h u rt. A rro w' s re su lt
(of [3]), f o r instance, h a s done considerable damage — t h e
extent has still not been fu lly assessed — to welfare economics.
The main imp o ssib ility result to be discussed — o n ly the first
of se ve ra l limit a t iv e re su lt s t h a t me ri t in ve stig a tio n — i s
obtained b y an elementary relocation o f A rro w's imp o ssib ility
theorem t o a mo re g e n e ra l se ttin g . F o r A rro w ' s t h e o re m
applies to much mo re than the determination o f a social we lfare o r co lle ctive choice recipe. Freed f ro m it s conventional
setting it is n o t ju st a problem f o r we lfa re economics, i t is a
problem f o r so cia l a n d economic assessment generally. Th e
transposition — to a general imp o ssib ility result in the so cia l
sciences — is achieved simp ly b y replacing the social welfare
or co lle ctive choice function to be determined fro m in d ivid u a l
preference rankings, b y an o ve ra ll ra n kin g to be constrained
by f a ct o r ra n kin g s, a n d o b se rvin g t h a t t h e co n d itio n s i m posed o n a so cia l we lf a re ra n kin g ca n b e replaced b y a p -
148
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pealing conditions o n th e o ve ra ll ra n kin g ru le . I n short, i n dividuals ca n b e co n vin cin g ly replaced b y factors, a n d i n dividual reduction assumptions removed. Th e wo rkin g examples should cla rif y the scope and importance o f ra n kin g p ro blems — a ll o f wh ich are subject to the general imp o ssib ility
result.
*
*
*
§1. Ra n kin g Problems. (1 ) En viro n me n ta l d e cisio n p ro b le ms
with a fo re stry case as wo rkin g example. Consider a 3-alternative 3 -fa cto r case. Th e f u t u re management o f a p ie ce o f
wet scle ro p h yll eucalypt forest o ve r the n e xt 100 years is t o
be determined. Three alternatives a re considered in th is re a l
life example.
a: Th e forest is conservatively managed as a eucalypt forest
and retained mo re o r less in its natural state.
b: Th e fo re st is in te n sive ly managed as a e u ca lyp t forest,
with species a n d u n d e rsto re y reduction, p la n tin g , e tc.,
and also wit h special recreational facilities.
c: Th e forest is converted to a pine plantation.
Suppose, f o r simp licit y, t h e f o llo win g th re e factors o n ly a re
considered.
i: t imb e r (value)
j: recreation
k: wild lif e . A lmo s t a n y o t h e r e n viro n me n ta l f a ct o r co u ld
supplant wild lif e i n th e example, e.g. so il, f lo ra . A l t e rnatively, a n d b e tte r f o r subsequent purposes, k ca n b e
considered a s a n amalgam o f these e n viro n me n ta l f a ctors.
The rankings o f the alternatives wit h respect to each factor
can be represented in the fo llo win g table in wh ich the columns
show t h e placement (first, second o r t h ird ) o f each a lte rn a tive wit h respect to the factor wh ich controls the column.
SOLUTION T O E NV I RO NME NTA L PROBLEMS
149
Ranking table 1
a lte rn a tive \fa cto r
a
b
c
3
2
1
2
1
3
1
2
3
Where t h e s t ric t ra n kin g w i t h respect t o f a ct o r f i s re p re sented b y P
t
may
be expressed linearly: c P
i, The
a n pdro b le m f o r f o re st ry d e cisio n -ma kin g sh o u ld b e t o
x PP
b
determine
an o ve ra ll ra n kin g P o f the alternative set { a, b,
ifwith one factor ranked above the other two, so that a ra tio n a l
ychoice
a
,P b Po f management alternatives can be made. For, as elsewhere
(e.g. [21, p . 24), a necessary co n d itio n f o r a ra t io n a l
if
is th a t th e mo st h ig h ly ranked (preferred) a lte rn a tive
zchoice
a
P
iabe chosen.
b b
cr ,The
e avp ro
Pi b le m is th u s e xa ct ly p a ra lle l t o t h e so cia l ch o ice
problem
ka t e wh e re a social preference o rd e rin g P is t o be detersmined
b
P in terms o f in d ivid u a l rankings Pt f o r each in d ivid u a l f.
kx A n y cost-benefit analysis ca n b e recast i n a s imila r fo rm,
once
cP
. rankings o r placements g ive wa y to net benefit (benefit—
cost)
figures. I n the fo re stry example the table wo u ld be supf
planted
b y a ma t rix o f the fo llo win g form, wit h ma t rix values
y
giving
net
benefits, e.g. in d o lla r terms:
a
n
dA lt e rn a t ive \ Fa ct o r
y
a
P
i
b
t
iac
z
ii
b
,(Thus, fo r instance, the foiaC
re stry cost-benefit analysis tabulated
tin [7], p. 84a, is re a d ily transcribed
ki
b
to th is form, on ta kin g net
hbenefits a n d interchanging
kC ro ws a n d columns.) G i v e n t h a t
eties a re a llo we d f o r, b y kh a vin g mo re t h a n o n e a lte rn a tive
thaving t h e sa me p la ce (e.g. t h ird equal) w i t h re sp e ct t o a
agiven factor, e ve ry cost-benefit analysis w i l l yie ld a ra n kin g
b
l
e
150
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RO UTLEY
table wit h rankings representable in te rms o f a ra n kin g re la tion P. Thus cost-benefit analysis — such as those f o r supersonic transport, dams, n u cle a r in sta lla tio n s — y ie ld f u rt h e r
examples, a n d these examples w i l l be subject t o th e general
impossibility theorem to be adduced.
(2) Educational assessment problems. I t should su ffice t o in dicate the range o f examples b y specifying kinds o f a lte rnatives and factors.
El. Ca re e r alternatives: e xe cu t ive , manual wo rke r, academic,...
Factors: s a l a r y , hours, job satisfaction,...
Objective: d e t e r m i n a t i o n o f career o r o ve ra ll
ranking o f ca re e r opportunities.
E2. Course combination alternatives: arts, science, la w,....
Factors: c h a n c e o f passing, ease, interest,
Objective: c o u r s e determination.
E3. Alte rn a tive s: m e m b e r s o f c la s s , e . g . A v r i l ,
Anne, Isobel,...
Factors: s u b j e c t s , e.g. mathematics, music,
Maori, conduct,...
Objective: o v e r a l l cla ss ra n kin g , a n d cla ss
prizes.
It is amusing that vo tin g paradoxes extend to such examples
if a ma jo rity decision method is used, e.g. to determine o ve ra ll
class ranking in case E3. Fo r consider the fo llo win g placement
table.
Ranking table 2.
A v ril
Anne
Isobel
Music
Mathematics
Ma o ri
1
2
3
3
1
2
2
3
1
SOLUTION TO E NV I RO NME NTA L PROBLEMS
1
5
1
Then under the ma jo rit y test f o r class p rize : A v r i l P An n e P
Isobel P A v ril P... i.e. tra n sitivity of P fails and it is impossible
to determine best choice.
Wit h these cases at hand it is easy to mu lt ip ly decision problem examples and extend the range into other areas, e.g. wo rk
value, and ethics (1, where conspicuous d ifficu ltie s emerge f o r
utilitarians — since u tilita ria n s have somehow to devise o ve rall happiness o r net pleasure rankings on the basis o f simila r
rankings f o r each member o f the base class, e.g. persons, sentient creatures, o r wh a te ve r (and f o r f u rt h e r difficulties, se e
[15]).
§ 2. Imp o ssib ility Results. I n order to show that the results app ly to a wid e range o f ranking problems we abstract the re le vant fo rma l structure that the examples g ive n a ll display.
A d e cisio n (mo d e l) stru ctu re (d.s.) i s a syste m C = (K , F)
where K is a non-null set o f alternatives (o r wo rld s) a n d F a
non-null se t o f fa cto rs ( o r subjects). Th u s C i s a table, o r
matrix, structure. A d.s. is u su a l i f f K has mo re than 2 e le ments (
alises
on standard expositions of A rro w's theorem b y removing
3
the
interpretational
re strictio n to so cia l we lfa re contexts, b u t
).
also
T h removes standard limita tio n s, e.g. t o f in it e se ts o f (e xclusive)
alternatives and to fin ite sets o f subjects.
e
A
f
u
ll
ranking p on a d.s. C is a function wh ich assigns, f o r
b a
each
factor
i in F, a 2-place re la tio n P
s i
ic —
ranking,
pdefined
( i ) , on KxK,
c a and
l lalso
e an
d o ve ra ll ranking P = P —
1:1)
a
t
o
r
fWI )r o n KxK, fwh eare (I)cis an
a rb itra ry ite m not in F U K . Thus
a
p is defined on F* F U f(1)1 wit h values in K
m
the
2 fo llo win g (ranking) requirements:
e
. P
i s
s u b j e c t
w
t (
o
o
r in terms of whic h one may t ry to et hic ally rank , as bet t er o r wors e
virtues,
8
kpersons,
members o f a c ommunity .
)
a (O
not8
n where K has 2 elements: see [3], p. 48,
d
o )e
T
p ih
n
t et
e re
d er
t se
h us
lt
u ti
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B. Fo r each factor j in F, P
transitive
i i s
a and asymmetric, on K; that is,
s t ifr xi Pc t
i xr P t i a l
p aif
iy d e r i n
o rmetry).
A
setn o f factor orderings is admissible (consistent) i f it cong ,ya
to requirement
B.
tdforms
h.
i
e
D.
yve1nra ll ranking P is stro n g ly transitive and a symme tric on
. Oe
K,
i.e.
i3
i
t
t3if xP y and
i
s — zPy then xPz, f o r x,y,z in K (strong tra n sitivity).
iz
st
h
n
Strong
t ra n sit ivit y a n d a symme t ry guarantee t ra n sit ivit y,
e — Suppose otherwise, f o r x,y,z i n K , x P y and yP z b u t
thus:o
tn Then b y asymmetry, y P x , but, b y strong tra n sitivity,
—xPz.
yPx wh
tx ich is impossible, b y a symme try again.
Strong
P
h
tra n sitivity, n o t required f o r fa cto r rankings, in co rporates
a
connectedness, requirement, and so enables th e re i
e
covery
cz of the usual postulates fo r relation R (of «preference o r
indifference»;
see, e.g., [3 ] pp. 12-13, [4], pp. 2-4); n a me ly
,
a
sf
o
e
R-connectedness:
Fo r x,y in K, e ith e r xRy o r yRx; and
tr
R-tra
n sitivity: Fo r x,y,z in K, x Ry and yRz st rict ly imp ly
xRz.
x
h
,
a
For, tyu p o n d e fin in g x R y a s y P x , R-connectedness f o llo ws
fromy, a symme try and R-tra n sitivity fo llo ws f ro m strong transit ivit ylz using th e p rin cip le o f An tilo g ism. Re la tio n R w i l l n o t
be used
in what follows. No r is it required that factor rankings
i
p
meetinsuch implausible requirements as that x P
ior that
xK x P
thereby
eylimin
yi &,(
P a tin g one f a milia r source o f criticism o f A rro wstyle
theorems.
iy &ft
zy It
i o
. x P mo
i zre illu min a tin g t o separate th e rankings
risD sometimes
into
trawo classes, factor rankings, P
i
iz overall
an
ranking, P, wh ich is supposed to be in some measure
xn
,x f Po
determined
r
ei n ate rms
c h o f th e f a ct o r ra n kin g s c o m m o n l y a
,s
function
o
f
these.
fi a yi c t o r
iz
i
n
it
F
,
(
w
i
n
ah K
d
ve n
r
e
(i
'
t
a
I
sy
'
y)
r
e
;
m
p
r
SOLUTION T O E NV I RO NME NTA L PROBLEMS
1
5
3
A decision table M = ( K , F , { P
d.s.i C wit h fa cto r and o ve ra ll rankings o n it. A lt e rn a t ive ly i t
may
e }represented
b y ot h e nstru ctu re (K , F, p ) wh e re p i s a
: i Eb F
,
P )
ranking.
Th
u
s
d
e
cisio
C
= n ta b le s fo rma lise t h e ra n kin g tables,
with
o
ve
ra
ll
rankings,
(
K
,
F
) given in the examples above. No te that
M ican be expressed
in the in it ia l systemic f o rm b y recasting
s
asathe structure (K , { P
the
} general character of the relations admitted that makes the
structure
) w h eo fr economic
e
interest — though it s interest is b y no
means
confined
j
i
sto economics.
table should sa tisfy conditions wh ich — u n like
i A decision
n
t
conditions
h
Be a n d D — re la te t h e o ve ra ll ra n kin g t o f a ct o r
rankings.
The
i
n
d most
e straightforward, and plausible, o f the conditions
to be imposed are these: —
x
s
e
t T. (Pareto, o r Factor Unanimity). Fo r any x,y in K, if x P
every
Fi
i in *F, then xPy. That is, if every factor ranks alternative
x. above
y f o yr then so does the o ve ra ll ranking.
I
t M. (Mu lt ip le Value, o r Non-Dominance). There is no i in F
i
such
that, f o r e ve ry f u ll ra n kin g p and e ve ry x
is
then
xPy. Fa cto r j is dominant if f f o r e ve ry x , y in K i f )(P
then
According
toi Mf no factor
iy i xPy.
n
K ,
x Pis dominant, i.e. there is no
factor
which
determines
overall
rankings
in every case contrary
i
),
to
y mu ltip le use principles. Fo r mu ltip le use imp lie s ta kin g due
account of all relevant factors in decision making, wh ich in turn
means there w i l l be some cases a t least where a single fa cto r
does not dominate. Mu ltip le use, wh ich is le g isla tive ly required
of fo re stry (and land) management practices in Un ite d States,
and wh ich can be explicated, in a wa y the legislation suggests,
in te rms o f a n optimisation mo d e llin g (cf. [7], p . 225 ff), th u s
requires as a min ima l condition non-dominance. O f course in
a single factor decision structure one factor will dominate, but
American legislation on, and optimisation modellings of, mu ltiple use both prescribe several factors that are to be taken into
account. M is thus no ad hoc condition, but fundamental in environmental, and multifactor, decision making.
An u n d e rlyin g assumption, to be expected fro m analyses o f
154
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social we lfa re and co lle ctive choice, is that P is a function o f
factor rankings, th a t is
E. P i s a function of elements of {P,:j F L i.e. in finite factor
cases P f ( P
1
, . assumption
..,P„)
This
— made i n effect in conventional exposir w' s th e o re m (e.g. {4], 13. 41) — is h o we ve r n o tions fo f oA rro
wheres re qou ire d in th e generalised theorem. N o r is i t a lwa ys
desirable
m since
e
o ve ra ll rankings ma y not (e.g. in holistic cases)
be simp
f ly
u an
matter of, reducible to, o r analysable in terms of,
factorc rankings;
t i
a l l t h a t is re q u ire d i s t h a t P a n d {P , : j
standoin some
relationship, just restricted by conditions such as
n
T andf M. Part of the ro le intended fo r E is absorbed in the fo llowing
. weak reduction requirement:
I. (Relevant Fa ct o r Completeness, o r Independence o f I r relevant Alternatives). Let p be a ra n kin g (o f a given sort) o n
d.s. C = (K ,F) and let p' be any other ranking (of the same sort)
on C, and let P
for
i a i Fn ' d
P
1x,y
in S and e ve ry j in F, x P
i
'S,xPy
i' ;
if f xP'yi
ayb ni fe f x P
logical formalism:
dor
it inhclassical
e
f'c yi o, r tr h e n ,
nfe as(x,y
op So) (rj F ) ( x P
le
r
y
n l dyi vi n e
lIxg fo llo
y ws
x , fro
P m they p rin cip le
eid
i e
n
I+.
tf (Extensionality).
' iy )
D
S
. e
n
(j
bd
( F ' x) ( x P,
iye
er
S )
y( a gxxive
PPn the
al This
y =classical content o f the usual statement that
xPy
depends
o
n ly on the factor rankings x P
i
nt
xi 1
iyo
'3 yn /)
j
F
,
sys ( f D
yo )r
uw 4 .(h i c h
b ) x4
T
s h P
)
e i y
t s x
o i P
f s '
t
Kh y
,
.
SOLUTION TO E NV I RO NME NTA L PROBLEMS
1
5
5
implies the fa milia r statement (e.g. 121, p, 79) th a t the ordering
of a set S o f alternatives should depend o n ly on the o rd e rin g
of the factors over S regardless of orderings over K-S.
The critica l — and disputable — condition I has a ke y ro le
in sp e cifyin g h o w o ve ra ll ra n kin g s a re determined th ro u g h
factor rankings. The fo rma l statements o f I and M already in volve — since they consider alternative o r va ria n t rankings to
a g ive n ra n kin g — sets o f rankings o v e r d.s. A n d wh a t th e
remaining f a milia r co n d itio n (la b e lle d U ) o f u n ive rsa lit y o r
unrestricted domain re a lly requires is that a ll sets o f rankings,
conforming to the given conditions, be considered — in short,
consideration o f the class o f a ll decision tables conforming to
T, I and M.
A general ranking method (GRM) o n a d.s. C is a procedure
(relation o r rule) wh ich relates an o ve ra ll ra n kin g to each admissible set of factor orderings on Ci hence it determines a fu ll
ranking p on C given a set o f factor rankings on C. Thus too
a GRM on C satisfying conditions TI M should define some subject — a n o rth o d o x decision ( o r assessment) subset (u n d e r
GRM) on C o f the class of all decision (o r assessment) tables
which contain a ll admissible fa cto r rankings o n C. I n the important special case discussed in choice lite ra tu re a G RM is a
function (device o r machine) wh ich g ive n a n y admissible set
of f a ct o r ra n kin g s d e live rs a n o v e ra ll ra n kin g . A g e n e ra l
ranking method should, l i k e a decision procedure, w o r k f o r
every admissible f a ct o r ra n kin g , n o t ju s t f o r sp e cia l cases.
Moreover a general social o r economic th e o ry should — so it
is not unreasonable to require — wo rk fo r e ve ry case, not ju st
for specially selected cases. But, according to the main impossib ility re su lt to be adduced, n o such general procedure ca n
succeed.
Impossibility Theorem. Th e re i s n o general ra n kin g method
satisfying conditions T, I and M on a n y usual decision stru cture. I n other words, there is no orthodox decision (o r assessment) set on a usual d.s.
The proof of the theorem is but a (somewhat tortuous) re ctification and extension — designed to make the role of condition
I and other conditions transparent — o f a proof given b y A r-
156
R
.
RO UTLEY
ro w (A rro w's p ro o f is set out in [3], p. 97ff.; the corrected e xtended version appears in [ il] ).
Since t h e co n d itio n s T I M a re o rt h o d o x re q u ire me n ts o n
ranking o r assessment p ro ce d u re s i n t h e s o c ia l scie n ce s
(whether e mp iricist o r Ma rxist i n orientation), t h e upshot o f
the theorem is th a t there is n o o rth o d o x solution t o ra n kin g
and assessment problems in the social sciences.
§ 3. Speculative Corollaries. The prototype f o r a ll these co ro llaries is the f a irly pervasive economic assumption (
row's
theorem somehow renders we lfa re economic th e o ry im5
possible
) t h a—
t or,Ain rwe a- ke r more metaphorical fo rm, creates a n
(impenetrable) b a rrie r to we lfa re aggregation and group decision making.
1. A n orthodox economic theory is impossible. The essence o f
the argument is simp ly this: an economic th e o ry wo u ld imp ly
a general ranking method satisfying — when orthodox — TI M
on decision structures, wh ic h i s impossible. Th e speculative
details concern the character of an orthodox economic theory,
and h o w i t is thereby required th a t there be such a general
ranking method.
Economic t h e o ry in vo lve s t h e d e te rmin a tio n o f t h e b e st
choices between alternatives in each economic system. I t is a
commonplace of economic textbooks (whether true o r not need
not concern us) t h a t economic t h e o ry arises o u t o f sca rcity
(or mo re g e n e ra lly constraints). S ca rcity forces decisions. So
economic t h e o ry e me rg e s a s a g e n e ra l t h e o ry o f ch o ice ,
making superior choices among competing alternatives (thus,
Boulding, Samelson, Bergson) ('). A s s u c h t h e t h e o ry
must in vo lve t h e equivalent o f a n o ve ra ll ra n kin g o f a lt e rnatives (o r a n o b je ctive fu n ctio n o n wh ich t o ma ximize ) o n
(
Arrow's
remark s in [3].
5
(°)
) A c c ording t o Bergs on [12], lit e problem o f economics is «the opt imum
alloc
T ation o f resources»,
h
e
a
s
s
u
m
p
t
i
SOLUTION T O E NV I RO NME NTA L PROBLEMS
1
5
7
economic systems. Th is f irs t p o in t ma y b e argued f o r i n a
number o f wa ys. On e f a milia r w a y is th is: Economic t h e o ry
implies t h e d e te rmin a tio n o f b e st ch o ice i n e a ch e co n o mic
system, o f maximisation on the objective function — choice as
determined through maximisation constitutes the foundation of
economic th e o ry (thus, e.g., He ilb ro n e r POI, and sources cited
therein: a cco rd in g ly economic th e o ry is but a branch o f o p timisation th e o ry). Since subsystems o f economic systems a re
economic systems, economic th e o ry imp lie s me rit rankings in
each 2 a lte rn a tive system. Hence, sin ce each syste m ca n b e
decomposed into a string of 2 alternative subsystems (as argued
in A rro w [3]), economic th e o ry imp lie s an o ve ra ll ranking, in
terms o f merit, o f alternatives in each system, that is, it implies
a general o ve ra ll ranking method.
Another f a milia r w a y o f a rg u in g t h e f irs t p o in t (A rro w' s
succinct way, [3], p. 22, p. 104) is this: — The aim, o f economic
theory, is to maximize on what is desired (e.g. net value, u tility,
social we lfa re ) su b je ct t o re le va n t constraints (e.g. resource,
technological, ethical, etc.), th a t is again, the a im is optimisation (over rankings) in economic systems. But a necessary condition for such optimisation is, in each case, an o ve ra ll ranking
satisfying ordering conditions D.
Secondly, an o ve ra ll ra n kin g o f alternatives is n o t independent of, b u t related to, and ch a ra cte ristica lly d e riva tive fro m,
rankings o f the factors taken in to account in the ra n kin g assessment (fro m wh ich factors the o ve ra ll ranking is t yp ica lly a
composite). Th e ra n kin g and assessment problems o f § 1 illu strate t h e point. A n d a t least i n orthodox economic cases —
in a ll reasonable cases most would want to cla im — the o ve ra ll
ranking and factor rankings conform to conditions T, I and M.
Indeed the central economic example o f a competitive ma rke t
does, according to A rro w ([3], p. 110): where such a ma rke t allocation system fails is as regards generality (
7
). (
a Paret
7
o proc edure — prov ides s atis fac tory rank ings wh e r e i t does, b u t
whether
i t prov ides a general proc edure c onf orming t o TI M. Th a t i t does
)
notT of f er a s atis fac tory proc edure wh e re i t does w o r k is a l l t o o ev ident,
especially
f rom env ironment al cases.
h
e
q
u
e
s
t
i
o
n
i
158
R
.
RO UTLEY
Hence, t h ird ly , a n e co n o mic t h e o ry ch a ra cte ristica lly i n volves, a t bottom, f u l l ra n kin g s o n e co n o mic syste ms co n strained b y orthodox (o r A rro w-ra tio n a lity) conditions. An d so
an economic theory, wh ich has to wo rk fo r e ve ry case, imp lie s
a general ranking method on economic systems. But the impossib ility theorem renders th is impossible; t h a t is a n o rth o d o x
economic theory is impossible.
Though p a rt o f th e argument resembles th e in it ia l p a rt o f
A rro w's a rg u me n t ( [ 4 p . 22 ff; a lso p . 104) t h a t a we lf a re
theory t yp ica lly in vo lve s ra tio n a l determination, a n d o p t imisation, o f a so cia l we lfa re function o n the basis o f in d ivid u a l
rankings subject to technological and resource constraints, i t
is wo rt h re ma rkin g th a t th e argument does n o t re q u ire t h e
further, a n d dubious, moves A r r o w makes i n defence o f h is
position, notably,
(i) t h e strong reduction assumption, th a t social rankings re duce to, o r depend o n ly on, the rankings of the individuals
that make up the society, and, connected wit h this,
(ii) t h e assumption of methodological individualism, that social
wholes a re always analysable in terms o f th e ir in d ivid u a l
components,
a thesis defended by imp licit appeal to
(iii) t h e ve rif ica t io n p rin cip le a n d o t h e r ri g h t l y questioned
tenets of positivism.
For example, th e argument given does n o t exclude organic
or h o listic features o f o ve ra ll rankings — it is simp ly required
that such rankings are appropriately related to factor rankings
(and perhaps not ju st to in d ivid u a l rankings even in the social
case since there ma y be other factors to take account of) A n d
while th e independence co n d itio n I comes clo se i n A rro w' s
social welfare context to a reduction assumption (compare, e.g.
the statement and defence o f the reduction assumption, in the
form Th e o ve ra ll re la tive ra n kin g o f a p a ir o f alternatives is
dependent o n ly on the ranking o f the p a ir fo r each in d ivid u a l'
SOLUTION TO E NV I RO NME NTA L PROBLEMS
1
5
9
p. 22 ff, wit h that o f 1, p. 26 ff.), I and the strong reduction assumption diverge in the larger context where factors other than
individual preferences ma y ma ke a difference t o th e o ve ra ll
(social) ra n kin g . So though a strong reduction assumption o f
this typ e 'serves as a ju stifica tio n o f both p o litica l democracy
and laissez-faire economics ([3], p. 23), it is not incorporated in
the arguments offered. No r should it be adopted as a basic assumption. I n vie w of the g ro win g opposition to methodological
individualism in the social sciences such an assumption can no
longer be expected to gain such widespread acceptance as in
the heydays o f positivism. Mo re o ve r there are straightforward
environmental reasons f o r believing th a t the strong reduction
assumption is mistaken, in particular (where the social rankings
are tied in fa milia r ways wit h wh a t the society should do, see
e.g. [15]).
What is being canvassed is the imp o ssib ility o f an orthodox
(and general) economic theory, n o t the imp o ssib ility o f economics (
8 theoretical economics are unaffected. But though descriptive
of
).
economics,
wh ic h i s concerned me re ly w i t h t h e d e scrip tio n
of
and ma rke t decisions a ctu a lly made and such-like,
N choices
e
is
e ndo t impeded b y the imp o ssib ility result, economics, we re it
to
l ereduce to the descriptive, wo u ld re a lly vanish into a branch
of
s ssociology. No r is it being denied that economic theory could
wo
t rk i n limit e d areas (it s cu rre n t la c k o f success i n ma n y
areas
is again evident), o r even that a good th e o ry ma y someo
times
a ccu ra te ly p re d ict future economic decisions and behas
viour.
Here o n ly the generality of such an economic theory (as
a
a
general
th e o ry o f choice and economic behaviour) is being
y
denied
—
a d e n ia l t h a t has a special b e a rin g o n economic
,
issues
conspicuously
beyond th e re a ch o f cu rre n t economic
d
theory,
such as environmental economics, wh e re n e w factors,
e
that
are
notoriously hard to quantify o r to compare in terms o f
s
the
cu
rre
n t fra me wo rk, fig u re p ro min e n tly, and wh e re reducc
tion
condition
I appears to fa il.
r
i
(
p
a different
8
t )
i I
v t
ei
es
p
c a
or
nt
ol
y
argument and thesis — is ent it led I s ec onomic t heory possible ?'.
160
R
.
RO UTLE Y
2. A n orthodox economic solution o f environmental problems
is impossible. Fo r the argument applying in the case o f orthodox economic th e o ry ce rta in ly applies i n the special case o f
environmental problems, wh e re questions o f in te rfa cto r co mparison become cru cia l. Th a t is t o say, t h e e a rlie r argument
is re ru n and i t is observed th a t th e in co n tro ve rtib ility o f the
premisses is not diminished in the environmental case — as the
environmental ra n kin g problems o f § 1 a re intended to show.
3. A n orthodox optimisation theory is impossible. Fo r if it were,
so wo u ld an orthodox theory o f economics be possible (').
4. Th e re is no orthodox engineering o r technological solution
to environmental problems. Fo r there is no orthodox theoretical solution — wh ich these presuppose — b y 2 and 3. Thus
the f a milia r unqualified technological o p timism, according t o
which there are no insoluble developmental o r environmental
problems, i s unjustified, a n d ca n be demolished lo g ica lly (").
Simila r results a p p ly to mo re specific areas such as la n d use
planning.
5. A n orthodox social th e o ry is impossible. Mo re specifically,
there is no orthodox solution to ra n kin g and assessment p ro blems in such disciplines as psychology, education and sociology. Th e co ro lla ry puts another ma jo r obstacle in the wa y o f
typical unified science programs since such programs characte ristica lly presuppose an orthodox social theory.
6. Th e re is no general method o f factor analysis. Fo r suppose
one starts wit h an o ve ra ll ranking o r assessment, e.g. o f group
or in d ivid u a l in te llig e n ce , and a ims t o u n co ve r t h e f a ct o r
rankings o n wh ich th e o ve ra ll ra n kin g is based. Because the
respective rankings will commonly stand in relations satisfying
A rro w-ra tio n a lity conditions, and so w i l l in general include a ll
cases required fo r the application of the imp o ssib ility theorem,
(g) I t is ant ic ipat ed t hat int eres t ing a n d mo re s pec ific res ult s as t o t h e
uns olv ability o f classes o f opt imis at ion problems w i l l emerge f ro m rec ursion t heory . A n d thes e w i l l p ro v id e t h e genes is o f f u rt h e r impos s ibilit y
results in economics.
(
optimism,
t hat there are no ins oluble problems is refuted by theorems f rom
0
recursion
t heory . O pt imis m o f t his k in d is , ac c ordingly , irrat ional.
)
T
h
e
e
x
t
r
a
o
r
SOLUTION TO E NV I RO NME NTA L PROBLEMS
1
6
1
the general p ro b le m o f determining factors succumbs t o t h e
imp o ssib ility theorem. F o r o th e rwise th e re wo u ld b e a f u l l
Arro w-ra tio n a l ranking on e ve ry usual decision structure, contradicting the theorem.
This co ro lla ry emphasizes th a t there re a lly a re t wo classes
of limitations emerging fro m the general imp o ssib ility result —
a limit on amalgamation, as in the o rig in a l A rro w result, and a
limita tio n on analysis. There is a wa y up — composition, syn thesis o r aggregation fro m factor rankings to o ve ra ll rankings
— a n d a w a y d o wn — decomposition o r a n a lysis f ro m a n
overall ra n kin g to fa cto r rankings. Social choice th e o ry illu strates t h e se rio u s limit a t io n s t h e re a re o n t h e f irst , f a c t o r
analysis on the second.
7. Th e general p ro b le m o f assigning card inals t o a l l ra n ke d
alternatives, a n d so o f absolute in te rfa cto r comparisons, i s
unsolvable. Hence, f irs t ly th e re is n o general p ricin g mechanism wh ich w i l l co ve r a ll factors (assigning cardinals). Fo r i f
there wh e re su ch a mechanism th e re wo u ld b e a n absolute
interfactor method, co n t ra ry t o 2 a n d vio la t in g 1. A second
outcome is that cost-benefit analyses are not o f general a p p licability.
A critica l issue of course — on which much of environmental
economics hinges — is the ma tte r o f interfactor comparisons.
The ma tte r o f interpersonal comparison has had a protracted
discussion i n th e lite ra tu re ; b u t ma n y o f th e mo re fa vo u re d
alternatives f o r d e live rin g ca rd in a l comparisons i n th e in t e rpersonal case, e.g. methods u sin g min ima l o r ju st noticeable
dicriminable differences — none o f wh ich wo rk in the in te n ded context — are either inapplicable o r do not extend plausib ly to in te rfa cto r comparisons wh e re persons ma y n o t be (a t
all d ire ct ly) in vo lve d . O f co u rse i f , i n defiance o f A r r o w
rationality conditions, a me th o d o f in t e rf actor comparisons
which d e live re d ca rd in a l assigments co u ld b e devised, th e n
the unsolvable problems cited wo u ld no longer be unsolvable
on the score argued: but several theoretical and e mp irica l considerations (th e la tte r set o u t in Robbins [14]) co me together
to rule out this possibility.
*
*
*
162
R
.
RO UTLEY
§ 4. Enforcing orthodox requirements? Th e speculative co ro llaries come to nought if orthodox (ra tio n a lity) conditions such
as B, D, T, M, and especially I, o n wh ich th e u n d e rlyin g impossibility theorem is premissed, cannot be made to st ick appropriately.
With o u t B there are no factor rankings to relate to the o ve rall rankings and decision and assessment procedures characteristic of the social sciences cannot even begin — rational assessment procedures a r e v i rt u a l l y d isso lve d , sin ce o v e ra ll
rankings, wh e n challenged a re a lmo st in v a ria b ly defended,
and often can o n ly be defended, b y appeal back to the factors
or considerations on which they are based. The asymmetry and
tra n sitivity requirements o f B o n rankings ma y o f course be
questioned, b u t i f a f a ct o r ra n kin g i s g ive n a n a symme tric
ranking can always be defined (b y excluding cases of equality)
and its transitive closure taken. In short, B can be guaranteed.
Requirement D is a necessary condition fo r optimisation and
for best o r better solutions t o problems. Th u s i n th e case o f
economics t h e v e ry characterisation o f economic th e o ry, i n
terms o f optimum choice in constraint-imposing economic systems, has a consequence re q u ire me n t D. F o r th e dimensionlimitin g conditions f o r an o p timu m to be even defined ensure
an appropriately connected strict partial ordering. I n any case
strong t ra n sit ivit y co u ld b e a g a in e n fo rce d b y t a k in g t h e
strongly transitive closure of any putative overall ranking relation th a t is g ive n (and th e subsequent method o f fa cto r e n richment need n o t clash wit h th e in it ia l procedure o f ta kin g
transitive closures o f rankings).
But it is not re a lly B and D, o r fo r that matter T and M, that
are at issue as ra tio n a lity requirements. I t is I that has rig h t ly
been u n d e r challenge. Th e re a re h o we ve r wa ys o f enforcing
conditions I, T and M on a ranking p on d.s. C, so that a general
solution t o ra n kin g problems in vo lve s a general so lu tio n t o
ranking problems satisfying I, T and M. A common, b u t quite
unsatisfactory, wa y of tryin g to enforce orthodox requirements
— especially I and such supporting doctrines as th a t there is
no (empirical) method fo r assigning interpersonally viable cardinal preference rankings — is a hard-line way, e.g. appeal to
SOLUTION TO E NV I RO NME NTA L PROBLEMS
1
6
3
the ve rifica tio n p rin cip le a n d o th e r methodological assumptions o f p o sit ivism (a lo n g lin e s commonplace i n t h e so cia l
sciences, especially economics, and we ll exhibited in Robbins
[14], p. 139 ff). Bu t a less assumption-loaded, and mo re direct,
wa y of tryin g to enforce orthodox requirements I, T and M, as
thoroughly reasonable in the enlarged context, is th e method
of extra factors. The method is simp ly to go on inserting additional fa cto rs u n t il conditions T I M a re satisfied ("). Th e i n tuitive considerations at wo rk are these: —
ad T. Suppose f o r some ranking p on some d.s. C ( K , F), fo r
some x, y in K, x P
y,
i i.e. —xPy. In this case there must be some factor h, not in F,
which
t h e difference, a n d f o r wh ich x i s n o t ra n ke d
y f o makes
r
above
e v ey, i.e.
r y —xPhy; that is there must be such a fa cto r wh ich
i
accounts
f o r th e discrepancy. Th u s i t should b e possible t o
extend
i
n F t o a class F+ o f factors, w i t h c F + , su ch th a t f o r
rankings
F
, on C+ = (K , F+ ), condition T is satisfied.
ad
fo r some environment S c K, P
b I. Suppose
u
iP
t
'ix c fro
ges
o m
i nP,csay
i d xP'y
e s but —
w xPyi fort some
h x, y E S. Then, insofar
as
fi P' andsP a re controlled b y the respective fa cto r rankings,
there
mu st
o
n
o be some fa cto r rankings, e.g. th a t f o r h 0 F such
that
uP
rt
he
in
r this acase the addition o f further relevant factors to F, wh ich
'vnv
ensure
kcompleteness, should guarantee condition I. I n short,
ae n o ndly because of the incompleteness o f factors taken in to
fails
in the assessment.
daccount
ra
b
uyo There
P
ivs presumably then, a w a y o f enlarging d.s. — n o t
generally
available in the social we lfa re case, where the class
hfe
of
individuals
is fixed b y adding factors so as to ensure that
va
(c
ot (
sroolution
1
o f s oc ial rank ing problems b y as s ignment of c ardinals t o eac h fac tor
1
ra
n
k
in
g t h e re b y p ro v id in g a me t h o d o f int erf ac t or c omparis on a n d
vr
si imple
)
arit hmet ic al met hods f o r o b t a in in g o v e r a ll rank ings . F o r o v e r a ll
i T ings are not inv ariant in t he wa y t hey s hould be, wh e n ex t ra relev ant
rank
cfactors
( h
a re introduc ed, under pres umed c ardinal s olutions , s inc e t he ex t ra
e
factors
i e
may alt er ov erall orderings .
v. m
ee e
r. t
sf h
o
ao d
)r o
fx f
o, e
ry x
164
R
.
RO UTLEY
conditions T and I hold ("). I n fa ct T can be guaranteed in a
rather t riv ia l wa y, b u t a t the cost o f vio la tin g M. Fo r simp ly
define (fo r x, y E K and h e F U K)Ph thus: xPhy if f xPy, and
set F+ F U O a Th e n T is a u to ma tica lly satisfied, sin ce i f
—xPy then — xPhy. Ho we ve r the definition makes h dominant,
violating M. Th is defect can be avoided b y d e fin in g Ph i n a
zig-zag o r diagonal fashion, b u t f irs t l e t u s guarantee I . A
simple wa y to do this is to replace F b y F*. Fo r then
(j F * ) ( x 1
Ii
D xP'y x P y
21
i
T is guaranteed
as we ll, but, since factor (1) is dominant, M is
'y x P
violated.
i
It appears however that M can be ensured along with
T andy I )by having the further factors dominant only over proper
subsets
D o f K. I n particular, fa cto r j is lo ca lly dominant w rt x
and yx if f 1if xP,y then xPy. I t suffices then to introduce into F+
a set3of factors (I)(x,y), f o r each x and y in K, wh ich are lo ca lly
dominant.
Set xP
y =----- xP y , a n d u P
—
uPv, f o r
0
4)(x,y) c D ( x , y )
{u , v}' distinct from x , y 1 . Then (1) is represented in F+ f o r each
y K
pair of
2motivate the
bExtra
u t Factor Thesis. Fo r any ranking p on usual d.s. C = ( K Y )
nwhich
o satisfies M, there is a set of factors F+ wh ich includes F
tand a ra n kin g p+ o n (K , F+) wh ich satisfies TI M. i.e.. there is
orthodox
ra n kin g on KY+.
uan n
i
fCall
o (K
r , ) a n extension o f (K , F) wh e re F Ç F. Th e imp o rt
of the
m
l yextra factor thesis is that even where there is a general
franking method satisfying M on a usual d.s. C there are extenosions o f C fo r wh ich there are no general ranking methods. So
rthe damaging effects o f the imp o ssib ility theorem, e.g. f o r environmental
economics, a re a p p a re n tly n o t re mo ve d s imp ly
e
by
v q u a lifyin g or abandoning I o r T: given the thesis, the effects
e
r
y (
pscope
1 o f ex tens ional-s ty le s emantic al analy s is .
a 2
i )
r T
h
. e
Sm
u e
c t
h h
c o
SOLUTION TO E NV I RO NME NTA L PROBLEMS
1
6
5
will reappear in a la rg e r context, thereby destroying, so i t is
suggested, prospects fo r a general theory.
There are simila r strategies, o f wide applicability, wh ich appear to weaken the force o f objections t o o th e r quite rig h t ly
questioned assumptions used i n th e u n d e rlyin g imp o ssib ilit y
theorem. If, fo r instance, the overall ranking fails to conform to
strong t ra n sit ivit y, th e n i t w i l l b e possible t o define a n e w
overall ra n kin g wh ich does co n fo rm (e.g., b y ta kin g closure)
and then to increase the class o f factors in o rd e r to reinstate
conditions th a t are th ro wn out b y strengthening tra n sitivity.
li
or tanother, there is, i t thus appears, n o easy escape fro m the
imp
h o ssib ility theorem o r its co ro lla rie s f o r the social and environmental
sciences. B u t appearances a re h e re deceptive.
e
Even
if
ranking
methods on an extension of a usual d.s. C suco
cumb
t o t h e imp o ssib ilit y theorem, methods o n C itse lf, i f
r
unorthodox,
ma y we ll not. To argue t o imp o ssib ility o n th is
t
basis
h is scarcely better than arguing to undecidability of a system
o o n th e basis th a t i t has undecidable extensions. Th e re quirements
o f o rth o d o xy cannot b e enforced i n u n o rth o d o x
d
procedures
b y the extra factor method o r other such methods.
o
The
x imp o ssib ility result fo r orthodox social and environmental
theories
does n o t extend — a t least wit h o u t fu rth e r and d if r
ferent
e ado — to unorthodox theories, a n y mo re than Gôdel's
impossibility
theorems f o r classical theories extend, wit h o u t
q
further
ado, t o nonclassical theories. O n t h e co n t ra ry, t h e
u
prospects
f o r unorthodox so cia l a n d e n viro n me n ta l th e o rie s
i
which
make
due allowance fo r interdependence and in te rre la r
tions
o
f
factors,
and a cco rd in g ly abandon requirement I , a re
e
bright.
m
e
Research
School of Social Sciences
n
Australian
National University.
1 3t
) s
R
.
ROUTLEY (
c(
tralas
1
a ian As s oc iation o f Philos ophy s y mpos ium o n problems i n t h e f o u n 3
n
)
b
T
e
h
s
e
m
o
a
i
i
m
n
p
c
o
o
s
n
t
166
R
.
RO UTLEY
REFERENCES
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1111 R. ROUTLEY, 'Repairing proofs of A rro w's general impos s ibilit y theorem
and e n la rg in g t h e s c ope o f t h e t heorem', N o t r e Da me J o u rn a l o f
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1121 A . BERGSON, 'Soc ialis t economics', in A Surv ey of Cont emporary Economics (ed. H. S. Ellis ), Blak is ton, Philadelphia (1948).
[131 1. M. D. Lrrri.x , 'Soc ial c hoic e and indiv idual values', J ournal of Polit ic al
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[14] L . ROBBINS, A n Es s ay o n t h e Na t u r e a n d Signific anc e o f Ec onomic
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[15] R. ROUTLEY, 'I s t here a need f o r a new, a n env ironmental, et hic ?' P roceedings o f t h e X V I I I W o r l d Congres s o f Philos ophy , (1973), V o l. 1.
pp. 205-10.
dations o f env ironment al economics, h e ld a t Mac quarie Univ ers it y , Augus t
1975. I a m muc h indebt ed t o t he o t h e r symposiast, C. Thomas Rogers , f o r
many v aluable discussions o f A r r o w problems i n t h e s oc ial sciences, f o r
the us ef ul c ons olidat ing t e rm ' A r r o w problems ' it s elf , a n d f o r s t imulat ion
to res earc h in t his area.