on the possibility of rational policy evaluation

THOMAS SCHWARTZ

O N T H E P O S S I B I L I T Y O F

R A T I O N A L P O L I C Y E V A L U A T I O N *

ABSTRACT. Arrow proved the inconsistency of a set of reasonable looking conditions on a social decision rule. These conditions are stated and their rationale explained. It is argued that the blame for the inconsistency must lie with Arrow's Collective Rationality condition. Arrow's abstract problem is generalized and interpreted in terms of individual as well as collective decision-making. His conditions are revised so that (1) cardinal - even interpersonal - utility comparisons are allowed and (2) the Collective Rationality condition - which formulates the traditional conception of rational choice as maximizing choice - is weakened to its bare bones. The revised set of conditions is still inconsistent. Once again the culprit is the Collective Rationality condition, now drastically weakened: even the bare bones of Arrow's conception of rational choice as maximizing choice is untenable. An alternative conception is proposed.

I. ARROW'S PARADOX AND BEYOND

In his classic s tudy o f social dec is ion-making , Kenne th A r r o w p roved

tha t cer tain reasonable look ing condi t ions canno t be satisfied by any rule

for guiding social decisions - for eva lua t ing social pol icy opt ions . 1 To

d e m a n d tha t a social decis ion rule D meet these condi t ions amoun t s to

impos ing three requi rements on D :

MINIMUM DEMOCRACY REQUIREMENT. Collect ive d e c i s i o n s - socialdecisions

prescribed by D - should reflect people's preferences at least to a certain

limited extent.

F o r instance, i f everyone prefers x to y, x should be collectively preferred

to y in this sense: D enjoins society to choose x when x and y exhaust the

avai lab le opt ions .

F o r convenience I fo l low A r r o w in speaking o f preferences. But

'p reference ' m a y be in te rpre ted as you please. Y o u might t ake ' Jones

prefers x to y ' to mean Jones would p ick x ra ther than y, given a choice

between the two. Y o u might take i t to mean x is p referab le to y accord ing

to Jones ' s tastes, values, goals, desires or interests. Or you might t ake i t

to mean Jones would der ive more happiness , sat isfact ion, benefit , 'u t i l i ty '

o r wha tno t f rom x than f rom y.

Theory and Decision 1 (1970) 89-106. All Rights Reserved Copyright �9 1970 by D. Reidel Publishing Company, Dordrecht-Holland

90 THOMAS SCHWARTZ

NONCARDINALITY REQUIREMENT. Collective decisions shouM not depend upon preference intensities (cardinal utilities).

To see the force of this requirement, imagine a society comprising in-

dividuals A 1 . . . . . A s, B 1, ..., B 4. The As prefer x to y; the Bs, y to x. So far it seems reasonable for x to be collectively preferred to y. But while

the As would be quite satisfied with y and barely more satisfied with x, the Bs would be very dissatisfied with x but highly satisfied with y. Given

this additional data, it seems reasonable for y to be collectively preferred to x. But the additional data concerns preference intensities. It consists of cardinal 'utility' comparisons - interpersonal ones at that.

COLLECTIVE RATIONALITY REQUIREMENT. D shouM satisfy the usual conditions of 'rationality'.

An example is the transitivity of collective preference. The inconsistency of Arrow's conditions is symptomatic of a more

widespread, disconcerting problem. To show this, I will extend Arrow's result in two directions:

(i) I will tamper with the Minimum Democracy Requirement in in-

tuitively inessential ways, weaken the Collective Rationality Requirement to its bare bones and discard the Noncardinality Requirement entirely, thereby allowing cardinal utility comparisons - even interpersonal ones - plus a good deal of what Arrow would regard as 'irrationality'. The resulting set of conditions is still inconsistent. ~

(ii) I will show how the revised conditions can be interpreted as assumptions about a wide variety of rules for guiding decisions by in- dividuals as well as societies. And I will argue that if it is reasonable to impose these conditions on social decision rules, it is reasonable to impose them on individual decision rules too. z

I think the trouble is caused by the weakened Collective Rationality Requirement. This captures the minimum kernel of the traditional conception of rational choice as maximizing choice. If I 'm right, this conception is mistaken.

II. ARROW'S CONCEPTION OF THE PROBLEM

D is used by a society comprising, say, n people, Messrs. 1, 2 .... , n.

P O S S I B I L I T Y OF R A T I O N A L P O L I C Y E V A L U A T I O N 91

Arrow assumes

(A1) n > 2.

Let S be the set of all conceivable policy options of the type which D is used to evaluate. If the problem is one of economic planning, S might comprise the infinitude of conceivable resource allocations. Or if the problem is to fill a political office, S might comprise every complete description that could conceivably describe a holder of that office. S is not necessarily the set of options available to society at any particular time. Rather, every set of options from which society might ever have to choose is a subset of S. So interpreted, S is infinite. But Arrow just assumes:

(A2) S has at least three elements.

x is collectively preferred to y if x, yeS , x ~ y and D enjoins society to choose x on condition that x and y exhaust the available options, x is collectively quasi-preferred to y if x, y e S and y is not collectively preferred to x.

If R is a relation, let IIR be the relation of x to y when xRy but not yRx. Then if R is the relation of collective quasi-preference, IIR is that of collective preference.

Collective decisions may depend upon how Messrs. 1, 2,. . . , n (would) rank the elements of S in order of preference. Mr. i's ranking of S may be identified with the relation R of x to y when Mr. i ranks x above or at

the same level as y. Mr. i prefers x to y if he ranks x above y, i.e. if xRy but not yRx, i.e. if xHRy.

Preferential rankings of S are weak orderings of S - binary relations on S that are transitive and strongly connected in S. 4 A profile is an ordered n-tuple (R1, ..., R,) of weak orderings of S. It represents a possible situation in which R~ is Mr. i's ranking of S. Any two profiles represent situations that differ only in people's preferences (plus, of course, any features that depend upon such preferences).

If (RI,..., Rn) is a profile, F (R~,..., R,) is the relation of x to y when x is collectively quasi-preferred to y in the situation represented by (Rt .. . . . R.). So

(Aa) F is a mapping of the set of ordered n-tuples of weak orderings of S into the set of binary relations on S.

92 T H O M A S S C H W A R T Z

I I I . M I N I M U M D E M O C R A C Y

One of Arrow's 'democracy' conditions, Pareto Optimality, says x is collectively preferred to y if everyone prefers x to y:

(A4) I f R1, .... Rn are weak orderings of S and xrlR~y (i = 1, 2, ..., n) then xrlF(R1,.. . , R,) y.

You are a dictator for x vs. y if x is collectively preferred to y in every possible situation in which you prefer x to y.

DEFINITION. i is a dictator for x vs. y just in case i = 1, 2 .. . . , n, x, yeS and for all weak orderings R1,..., R, of S,/J'xHRiy then xIIF(R1, .... R.) y.

Arrow's other 'democracy' condition is Nondictatorship: (As) There is no dictator for every x in S vs. every other y in S.

IV. NONCARDINALITY

Let (R1, .... R,) and (R~,..., R'~) represent situations such that those who prefer x to y (y to x) in one situation do so in the other:

(1) RI . . . . . R,, RI . . . . , R~ are weak orderings of S and

(2) R, n {x, y}2 = R;n (x, y}2 (i = 1, 2 . . . . , n).

Surely, then, if any preferential differences between the two situations are relevant to the collective decision on the issue of x vs. y, these can only be differences in preference intensities (cardinal utilities). But the only differences between the two situations are preferential ones: two profiles represent situations that differ only in people's preferences. So if the collective decision on the issue of x vs. y is different in the two situations, this must be due to differences in preference intensities.

By the Noncardinality Requirement, then, the collective decision on this issue should be the same in both situations. So if x is collectively preferred to y in the first situation (xlIF(R1 ..... R,) y), x should be collectively preferred to y in the second (xIIF (R'~,..., R'~) y).

What we've arrived at is Arrow's Independence condition:

(As) I f (1)-(2) hold and xHF(RD ..., Rn) y, x/IF(R~, ..., R ') y.


V. C O L L E C T I V E R A T I O N A L I T Y

Arrow's 'rationality' condition says the relation of collective quasi- preference must be a weak ordering of S :

(AT) I f R1 ... . . Rn are weak orderings of S, so is F(R1, ..., Rn).

Applied to a profile, F issues in a relation of collective quasi-preference. This tells us how to compare options-to choose between them two at a time. What happens, though, when society has to choose among three or more options? Arrow's answer gives some insight into (Av).

He says that if R is the relation of collective quasi-preference and S ' the set of live options, D allows society to choose x just in case x is an R-maximal element of S'. This means x ~ S ' and xRy for all y in S'. So an optimal element of S' is any element that is collectively quasi-preferred to every element.

(A7) implies that it is possible to choose from any finite range of alternatives in this way - that every finite set of options has a maximal element with respect to collective quasi-preference:

(MA) I f Rl, ...Rn are weak orderings of S and S' a finite, non-empty subset of S, there is an x in S' such that xF(R1, ..., R . ) y for all y in S'.5

VI. TH E L E S S O N OF A R R O W ' S T H E O R E M

Arrow proved (Ax)-(A7) inconsistent. Where to pin the blame for this disturbing result?

(A1) and (A2) are scarcely open to question. They just limit the scope of Arrow's analysis to the virtually general case where there are at least two people and three conceivable policy options.

(A3) is unobjectionable too. It just formulates a condition of the problem: it should be possible to use D no matter how people rank the options in S. In other words, Arrow investigated the general problem of evaluat- ing social policy under the constraints listed in Section I.

Neither is (A4) or (As) a likely culprit. As 'democracy' assumptions go, they are so weak they are satisfied by all extant forms of government, however 'undemocratic'.

If cardinal (perhaps interpersonal) utility comparisons were possible,


(A6) would not be universally acceptable, and Arrow's result would loose some of its punch. But only some. For it is at least often the case that preference intensities are unknown - e.g. when preferential data is got by taking a vote. And it is paradoxical to hold that a rational evaluation of social policy is impossible in the absence of a type of data that sometimes doesn't exist.

This leaves (A7) as the only possible bone of contention. It has been argued that (A7) should be replaced by the weaker con-

dition: I f R1, ..., Rn are weak orderh~gs o f S then F(R1 .... , Rn) is strongly connected in S and HF (R 1 .... , R~) is transitive. 6

(MA) is weaker still. This is important because Arrow wanted to formalize the traditional conception of rational choice as maximizing

choice, and (MA) by itself captures the crux of this conception. A virtue of (MA) is that it yields a simple reduction of the task of

choosing from any finite set of options to that of comparing options two at a time. But the same job is done by an even weaker condition:

(MA') I f Rl , ... , R n are weak orderings o f S and S' a finite, non-empty

subset o f S, then for some x in S' there is no y in S' such that

yI IF(R t, ..., Rn) x.

This means every finite set of options has at least one element to which none other is collectively preferred.

So we can weaken (AT) a good deal while preserving what seems es- sential to Arrow's conception of rational choice.

VII . THE G E N E R A L P R O B L E M

Arrow assumed n~>2, ignoring the Robinson Crusoe case. Let's ignore the Crusoe-Friday case too:

(G1) n >i 3.

Since S is infinite by interpretation, we may assume:

(Gz) S has at least n elements.

Let's represent people's 'preferences' by 'utility' measures instead of preferential rankings. While utility indices may have only ordinal significance, they may have cardinal and even interpersonal significance.


Let M be the set of all possible utility measures on S. Then:

(G3) I f u s M , u is a real-valued fitnction on S.

Can we also say that every real-valued function on S is a possible

utility measure? No t necessarily. On some conceivable methods of

measuring 'utility', not every real number is a possible utility index.

Possible utility indices might all be rational, or even integral. They might

all be positive. Or they might all lie on some bounded interval. But on any plausible method of measuring 'utility', surely, there will be at

least n + 1 possible utility indices, say to, r , , . . . , r , in order of magnitude,

where rl - ro, r2 - r~, r 3 - r2 etc. are all equal. Then if b = r~ - ro,

(G4) b > 0,

and if a = ro, then r o, rx, r2, ..., r , are equal to a, a+b, a+2b, ..., a+nb, respectively. So the latter are possible utility indices. Hence:

(Gs) I f u is a function on S into {a, a + b , a + 2 b , ..., a + n b } , uEM.

I f ul,. . . , u ,~M, (u~,..., u,) represents a possible situation in which

us is Mr. i 's utility measure on S, and P (ul, ..., u,,) is the relation of x to y

when x is collectively preferred to y in that situation. So

(G6) P is a function on M". 7

V I I I . I N T E R P R E T A T I O N S

The abstract problem I ' m discussing admits of several interpretations:

(i) Collective Decision-Making. This is a generalization of the inter-

pretation already given. Messrs. 1, 2 . . . . . n can be the members of any group, not just a society. Nor need they be people. They might themselves

be groups. In that case the group to which they belong is a group of

groups, like the U.N. (ii) Moral or Altruistic Decision-Making. Bertha wants her decisions

to satisfy the interests of Messrs. 1, 2, ..., n at least to a certain limited extent. These n people might constitute all of mankind. Or they might be Bertha's customers, friends, relatives, countrymen or the like. The difference between this and the last interpretation is that D is used by an individual agent rather than a collective or corporate agent, s

96 T H O MA S S C H W A R T Z

(iii) Decision-Making under Uncertainty. Now Bertha's problem is to reach decisions without knowing which of n eventualities, El,..., E,, will obtain; she cannot even judge their relative likelihood. The E~ may be actions by others, weather conditions, stock prices, etc. Any (ul .... , u,) in M" represents a situation in which u~(x) is the 'utility' x would have for Bertha if E i obtained, xP (ul,..., u,)y holds if, in the (u: .... , u,)- situation, x is preferable to y for Bertha, all eventualities considered.

(iv) Decision-MakhTg under Risk. This is like (iii), only Bertha can now assign probabilities (in some sense) to the El. Any (u~,..., u,) in M ~ represents a situation in which ui(x) is the product of (1) the 'raw' utility x would have for Bertha if E~ obtained and (2) the probability that E~ will obtain.

(v) Simple Multi-Criterial Decision-Making. 9 Bertha's task now is to reach decisions on the basis of n criteria, C1,..., C,. S might comprise cars, which Bertha chooses on the basis of speed, safety, comfort, resale value, etc. Bertha does not attach more importance to some criteria than to others. Any (u,, ..., u,) in M" represents a situation in which us(x) is x's value according to Ct. x e (ul ..... u,) y holds if, in the (u, . . . . , u,,)- situation, x is preferable to y, all criteria considered.

(vi) Weighted Multi-Criterial Decision-Making. This is like (v), except Bertha may now attach more importance to some criteria than to others. Any (Ul,..., u,) in M n represents a situation in which u~(x) is the product of (1) the 'raw' value of x according to Ci and (2) an index of the im-

portance of Cv 1~ On each interpretation the problem is one of multi-dimensional de-

cision-making - of reaching decisions by (perhaps among other things) 'combining' n evaluations of the available options.

IX. D E C I S I V E N E S S

Let (Ul .... , u,) represent a situation in which Mr. i prefers x to y while everyone else prefers y to x. This means

(1) x, yeS , i = l , 2 . . . . . n and Ul . . . . , u n e M ,

(2) ui(x) > ui(y) and

(3) uj(y)>uj(x) ( j = l .... , i - l , i + l , . . . , n ) .


It seems reasonable for y to be collectively preferred to x in this

situation:

(*) yP (ul . . . . , it.) x .

If nothing else it is unrealistic to imagine a society so constituted that someone could get his way though he were unanimously opposed. And many would condemn such a society as 'undemocratic'.

You might object all the same that while everyone but Mr. i prefers

y to x, it could be that no one would benefit significantly more from y than from x. This means that if Mr. j is anyone other than Mr. i, the utility difference zg(y)-u~(x ) is not significant - not big enough for Mr. j to complain if x were chosen instead of y.

Let's rule out this possibility by supposing uj(y)-uj(x) is significant whenever j # i . What is a significant utility difference? It is certainly compatible with the conditions so far imposed upon b to suppose a utility difference of b units or better is significant. Then to guarantee that uj(y)-uj(x) is significant when j r it suffices to assume this difference is at least b:

(3') u j ( y ) - u j ( x ) / > b ( j = l , . . . , i - l , i + l , . . . , n ) .

Given (3') can we infer (*)? Since b>0 , (3') implies (3). So to ask whether (*) can be inferred from (1)-(3) plus (3') is equivalent to asking whether (*) can be inferred from (1), (2) and (3'). You might object to this inference on the following ground. Though Mr. i is the only one who would be better off if x were chosen instead of y, it could be that Mr. i would suffer if), were chosen while no one would suffer i f x were chosen.

Let's rule out this possibility by supposing Mr. i would be no worse off if y were chosen than someone else would be if x were chosen:

(4) ui(y )>1 uj(x) for some j = 1, 2 . . . . . n.

Can we now conclude that (*) holds? You might object as follows. Perhaps Mr. i would benefit enormously from x while everyone else, though benefitting significantly more from y than from x, would still benefit very little from y.

Let's rule out this possibility by supposing Mr. i would be no better off if x were chosen than someone else would be if y were chosen:

(5) ui(x ) <~ uj(y) for some j = 1,2 . . . . . n.


Given (4) and (5) is there still any reason not to conclude that (*) holds? Perhaps you will object that the aggregate benefit produced by y might be far less than that produced by x.

Let's rule out this possibility too by supposing the aggregate benefit produced by y is no less than that produced by x:

(6) ~ uj(y)>>, ~ uj(x). j = l j = l

On the third, fourth, fifth and sixth interpretations of Section VIII, it is reasonable to infer (*) from (1), (2), (3') and (4)-(6).

Take Decision-Making under Uncertainty. In view of (1), (2), (3') and (4)-(6), there is no reason for Bertha to pick x rather than y, given a choice between the two. But there are two good reasons for Bertha to pick y: First, she would be playing it safe because y is preferable to x under virtually every eventuality. Second, the hypothesis that y is the better choice has a far greater variety of support than the hypothesis that x is the better choice. In the case of Decision-Making under Uncertainty as well as under Risk, the principle that (1), (2), (3') and (4)-(6) imply (*) amounts to the maxim: Other things being equal, don't put all your eggs in one basket.

Or consider Weighted Multi-Criterial Decision-Making. Say Bertha has to choose between a Ferrari (x) and a Jaguar (y). Though the Ferrari is faster (criterion Ci), the Jaguar is significantly superior in all other respects; such is the force of (2) and (3'). Weighing the greater speed of the Ferrari along with the importance of speed against the superiority of the Jaguar according to the other criteria along with the importance of those criteria, the Jaguar is at least as good as the Ferrari, in view of (4)-(6). What's more, since the Jaguar is superior to the Ferrari in every respect but one, it is by far the more versatile car, the better all-around car. It has by far the greater variety of virtues. It is therefore the better choice, i.e. (*) holds.

Though the inference of (*) from (1), (2), (3') and (4)-(6) is plausible on the first and second interpretations too, one might conceivably object to this inference for the following reason. It may be that the issue of x vs. y is one for which Mr. i should be given veto power, either because it is expedient to treat him as boss on this issue, or because he has a right to decide this issue. In the first case, the issue might require technical


expertise which Mr. i possesses to a far greater degree than the rest of society put together. In the second, the issue might concern the dis- position of Mr. i ' s property.

But if Mr. i has veto power, x is collectively preferred to y. So D still prescribes a definite decision one way or the other on the issue of x vs. y,

i.e.

(**) either xP(ul , . . . ,Un) Y or ye(u~, . . . ,Un) X.

What we've arrived at is the Decisiveness condition:

(GT) I f ( l ) , (2), (3') and (4)-(6) hold, so does (**).

It would have been plausible to adopt a much stronger condition: I f

(I) and (3') hold, so does (**). For (3') certainly provides a sufficiently strong presumption in y's favor so that, at least in the absence of any other grounds for the collective preferability of x to y or y to x, we may insist that y be collectively preferred to x.

Here is a still stronger condition that is also plausible: I f (1) holds,

so does (**). For if society has to choose between two courses of action and its decision process prescribes neither course, what can it do? It can't abstain from choosing. By hypothesis, it must choose one of the two options. But neither can it reach a decision by means of its decision process. It is faced with a political deadlock - a paralysis of governement. These are generally precluded, of course, by decision processes used in ongoing societies.

X. M I N I M U M I N D E P E N D E N C E

Though (1), (2), (3') and (4)-(6) provide strong grounds for the collective preferability o f y to x in the (u 1 .... , Un)-Situation, suppose x is collectively preferred to y:

(7) xP(u l , ..., u.) v.

Let (vl, ..., v,) represent any situation in which Mr. i prefers x to y while everyone else prefers y to x:

(8) vl . . . . . v , ~ M and

(9) vi(x) > v,(y) but v j ( y ) > v2(x) ( j = l . . . . , i - l ,

i + 1 . . . . . n).

100 THOMAS SCHWARTZ

Those who prefer x to y ( y to x) in one situation do so in the other. So (7) and (A6) imply that x is collectively preferred to y in the second situation:

(***) xP(v , . . . . , v,) y .

Of course, having discarded the Noncardinality Requirement, I can't appeal to (A6) to infer (***) from (1), (2), (3') and (4)-(9). But you can reject (A6) while accepting the weaker Minimum Independence condition:

(Gs) I f ( l ) , (2), (3') and (4)-(9) hold, so does (***).

As I argued in the last section, on the third, fourth, fifth and sixth interpretations of Section VIII, if (1), (2), (3') and (4)-(6) hold, (7) cannot hold. So (Gs) is acceptable on these interpretations.

(Gs) is acceptable on the first and second interpretations too. For, if(7) holds despite (1), (2), (3') and (4)-(6), Mr. i must have veto power for reasons having nothing to do with preference intensities (cardinal utilities). Maybe it was expedient to treat him as boss. Or perhaps his rights would have been violated had he not been given veto power. In either case, since the only differences, if any, between the two situations that could be relevant to the issue of x vs. y are differences in preference intensities, and since Mr. i has veto power in the first situation for reasons that are independent of preference intensities, Mr. i must have veto power in the second situation too. But this just means that (***) holds if (1), (2), (3') and (4)-(9) hold.

Like (G7), (Gs) is part of a slightly enlarged Minimum Democracy Requirement. It is the result of radically weakening (A6) to a point where it no longer imposes a constraint on the available preferential data. It merely demands a kind of consistency in the way D reflects or fails to reflect such data. For it requires that if someone is given veto power, and if preference intensities provide no basis for giving him veto power, his veto power is independent of preference intensities.

XL NONDESPOTISM AND PARETO OPTIMALITY

While a dictator for x vs. y must insure the collective preferability o f x to y in every situation in which he prefers x to y, a despot for x vs. y need only insure the collective preferability of x to y in all those situations in which he prefers x to y while everyone else prefers y to x.


DEFINITION. i is a despot for x vs. y just h~ case x, yeS, i= 1, 2 , . . , n and for all u j , . . . , u, in M, i fu i (x) > u i (y) while u j (y) > u j(x) (j = 1, ..., i - 1, i + 1, ..., n) then xP(ul, ..., Un)Y.

In place of Nondictatorship let's adopt the slightly stronger but equally unobjectionable condition of Nondespotism:

(Gg) There is no despot for every x in S vs. every other y in S.

This is obviously acceptable on all the interpretations of Section VIII. So is Pareto Optimality:

(Glo) l f x , y e S , u~ , . . . ,uneM a n d u i ( x ) > u i ( y ) ( i = l , 2 , . . . , n ) , t h e n

xP(ul, ..., u,) y.

XII . M A X I M A L I T Y

In place of ( A 7 ) let's adopt (MA') and call it Maximality:

(Gil) I f S' is a finite, non-empty subset of S and u l, ..., u, eM, then for some x in S' there is 17o y h~ S' such that yP(ut , . . . , un) x.

This represents the bare bones of Arrow's conception of rational choice. On each of the interpretations of Section VIII, (G1,) says it is possible to choose from any finite set of options so that no other choice would have been preferable.

XII I . A C O N T R A D I C T I O N

Despite their plausibility, (Gi)-(Gll ) are inconsistent. I will deduce from (G1)-(Gll) that someone is a despot for every pair of options, contrary to (a9).

(GT) and (G8) yield a sufficient condition for despotism that is easier to work with than the defining condition:

CONSEQUENCE 1. i is a despot for x vs. y i f

(1) x, ye S, i= 1,..., n and ul, . . . , u~EM; (2) ui(x)-- ui(y) = n b - b a n d u j ( y ) - u j ( x ) = b ( l < ~ j < ~ n , j r (3) u j (y)=ui (x) and uk(x)=ui(y) for some j, k = 1, ..., n; and (4) not yP(u 1 .... , un) x.

Proof. L e t N = {1, ..., n}. By (G1), (G4) and (2),

(5) u,(x)>u,(y).


But

E "Ay)- E j e N j e N

= (u, (y ) - u, ( x ) ) + ~, (uj (y ) - u j ( x ) ) jeN-{i}

= - (nb - b) + (n - 1) b = 0by (2), whence

(6) ,,j(y)= j ~ N j e N

So by (1)-(3), (5) and Decisiveness, either xP (up. . . , u,) y or yP (ul,... , u,) x. By (4), then, xP (ul,.. . , u,) y, whence it follows by (1)-(3), (5), (6) and Minimum Independence that for all v t . . . . . v, e M, if vi (x) > vi (y) while v j (y )>vj (x ) ( j = l , . . . , i - l , i + l , . . . , n ) then xP(va, . . . ,v,)y. That is, i is a despot for x vs. y.

To show that someone is a despot for every pair of options, it suffices to show he is a despot for some pair or other. A despot for any pair of

options is a despot for every pair. To prove this I need two lemmas.

LEMMA 1. l f i is a despot for x vs. y and z eS - {x}, i is a despot for x vs. z.

Proof. Trivial if z = y. Let z # y. By hypothesis,

(1) x , y , z e S and i = 1 . . . . , n .

Since n>~3, let j, k e{ l , ..., n } - {i} and j # k . Define functions u: .... , u, on S by setting um(w)=a for all w e S - {x, y, z} and m = 1, ..., n; u , (x)=

a+ n b - b , u , ( y )=a + b and u,(z)=a; u j (x )=a + n b - 2b, ua(y)=a + nb and u j ( z ) = a + n b - b ; and U,n(X)=a, Um(y)=a+nb and Um(Z)=a+b for all me{ l , ..., n } - { i , j } . Since the values of u 1 .... , u, belong to {a, a+b, ..., a+nb},

(2) u 1 . . . . . u, e M .

But by construction we have

(3) u , ( x ) - u,(z) = nb - b and urn(z ) - urn(x) = b ( l ~ m < ~ n , m # i ) ,

(4) u,(x) = u j (z) and ui(z) = uk(x), (5) u,(x)>u,(y) but Um(y)>Hm(X ) ( l<~m<~n,m#i) (6) urn(y) > urn(z) (1 ~< m ~< n).

and

POSSIBILITY OF RATIONAL POLICY EVALUATION 103

By (1), (2) and Maximality, for some we{x, y, z} there is no te{x, y, z} sucht that tP (ul,..., u,)w. But by hypothesis and (5), xP (up..., u , )y , whence w e y. And by (1), (2), (6) and Pareto Optimality, yP (ul, ..., u,) z, whence wCz. So w=x. Hence not zP (ul,..., u,) x, and thus, by (1)-(4) and Consequence 1, i is a despot for x vs. z.

LEMMA 2. I f i is a despot for x vs. y and z e S - {y}, i is a despot for z vs. y.

Proof. Trivial if z=x. Let z r Since n~>3, let j , ke{1,. . . , n } - { i } and j r Define functions ul, . . . , u, on S by setting u,,(w)=a for all w e S - {x, y, z} and m = I, ..., n; u,(x)=a+2b, ui(y)=a+b and ut(z)= a+nb; uj(x)=a, uj(y)=a+nb and uj (z )=a+nb-b; and urn(x)= a, u,,(y)=a+2b and u,,(z)=a+b for all rn~{l, ..., n } - {i,j}. As in the

preceding argument,

(1) x ,y , z eS , i = l , . . . , n and u 1 . . . . . u .eM,

(2) u ~ ( z ) - u , ( y ) = n b - b and um(y)--Um(Z) = b (l <~m<<.n,m#i),

(3) ui(z) = uj(y) and ul(y) = Uk(Z),

(4) Ui(X) > u,(y) but u,,(y) > u,,(x) (1 ~< m ~< n, m ~ i)

(5) U m (Z ) > U m ( X ) (1 ~.~ n"/~,~ n).

and

But xP (ui,..., u,) y by hypothesis and (4), and zP (ul,..., u,) x by (1), (5) and Pareto Optimality. From this plus (1) and Maximality we can deduce, as in the preceding argument, that not yP (ui .... u,) z whence, by (I)-(3) and Consequence 1, i is a despot for z s. y.

CONSEQUENCE 2. I f i is a despot for x vs. y and i f z, w6S and z # w , i is a despot for z vs. w.

Proof: By (G1) and (G2), there is a t ~ S - {x, z}. Then i is a despot for x vs. t by hypothesis and Lemma 1, whence i is a despot for z vs. t by Lemma 2, so i is a despot for z vs. w by hypothesis and Lemma I.

Here is the consequence advertised earlier:

CONSEQUENCE 3. There is a despot for every element of S vs. every other element.


Proof. Let p ( j ) = n i f j = l and p ( j ) = j - 1 i f j > l ( j = l .... ,n). By (G2), there is an n-element subset S * = {xl,... , x,} of S. Define functions ul, . . . , u, on S by setting u2(y)=a if y e S - S* (1 ~<j ~<n) and

(1) u j ( x k ) = a + j b - k b if l ~ < k < j ~ < n and

(2) uj(xk) = a + jb - kb + nb if l <. j ~ Ic <. n.

Since the values of up. . , , u, belong to {a, a+b,.. . , a+nb},

(3) ut . . . . , u, e M .

So by Maximality, for some x,eS*,

(4) not xp(o P(ul, ..., u,) xi.

But ui(xi)= up( o (xp(i))=a+ nb by (2), so

(5) u,(xi) = uj(xp(i) ) for some j = 1 . . . . , n .

If l<j<~n, uj(xpo))=uj(xj_1)=a+b by (1). But u~(xpo))=ux(x,)= a + b by (2). So a + b = uj (xpo)) = u, (Xp(0), J = 1, ..., n. Hence, since i =p ( j ) for some j,

(6) ui(xp(o) = uj(xi) for some j = 1, ..., n.

Also, since ui(xl)=a+nb by (2),

(7) ui(xi) - ui(xp(o) =nb - b.

If l<k<~n, Uk(Xl)=a+kb-b by (1) and uk(xv(~))=Uk(X,)=a+kb-nb+ +nb by (2). So

(8) UK(Xp(,))--Uk(X~)=b if l < k ~ < n .

If l<i<k<.n , then l<~i<k<~n and l ~ i - l < k < . n , whence Uk(X~)= a + k b - i b and Uk(Xv(o)=uk(xi_x)=a+kb-(i - 1)b by (1). So

(9) uk(xp(i))--ug(x~)=b if l < i < k ~ < n .

If l<~k<i, then l<~k<~i~n and l<~k<~i-l~n, whence Uk(Xi)= a+kb- ib+nb and uk(xp(o)=uk(x,_x)=a+kb-(i-1 ) b+nb by (2). So Uk(Xp(O)--Uk(X,)=b if l~<k<i. Thus, by (8) and (9), Uk(Xp(o)--Xk(X~)= b, k = 1,..., i - 1 , i + 1 , . . , n. So by (3)-(7) and Consequence 1, i is a despot for x~ vs. xv(~ ). By Consequence 2, then, i is a despot for every element of S vs. every other element.


Since Consequence 3 contradicts Nondespotism, no decision rule can satisfy (G1)-(G11).

XIV. THE D E C L I N E A N D F A L L OF THE M A X I M U M

Arrow had a glimpse of a difficulty more widespread than has been sup- posed. It arises when cardinal and even interpersonal utility comparisons are allowed. And it is not limited to collective as opposed to individual decision-making. It is a general feature of multi-dimensional decision- making, and most if not all decision-making is multi-dimensional, often in more ways than one.

(AT) was to blame for Arrow's paradox. For much the same reasons, the weaker condition of Maximality is to blame for the inconsistency of (G1)-(G11). Even the bare bones of the traditional conception of rational choice as maximizing choice - of optimization as maximization - is untenable. On each of the interpretations of Section VIII, the agent in question cannot reasonally be enjoined to choose in all cases so that no other choice would have been 'preferable'. Afort ior i the agent cannot reasonably be enjoined in all cases to maximize 'benefits', 'welfare', 'utility' or whatnot, or to minimize 'costs', 'harm', 'disutility' or the like.

The significance of this for decision theory, value theory, economics, ethics and political philosophy is obvious. In particular, utilitarianism is untenable since in all its forms it enjoins either the individual or the state to choose in all cases so as to maximize 'good' or minimize 'evil'. lz

The traditional conception of rational choice had a virtue: it reduced the problem of choosing from any finite set of options to that of comparing options two at a time. Here is a slightly different conception that has this virtue too without running afoul of the difficulty I've discussed:

On the traditional view, an optimal element of a finite set A of options is an element to which none other is 'preferable.' Trouble is, A may have no such element. To suppose otherwise is to impose excessive constraints on the 'preferability' relation.

My alternative idea is that an optimal element of A is an element of any subset B of A such that (1) nothing in A - B is 'preferable' to anything in B and (2) no nonempty proper subset of B satisfies (1). A must have an optimal element in this sense - independently of any properties of the "preferability' relation. And the proposed definition of optimality follows

106 THOMAS S C H W A R T Z

from some quite uncontroversial axioms of 'rational choice.' But that is a topic for another occasion.

Department o f Philosophy,

S ta l ford University

N O T E S

* I read an earlier version of this paper at the University of Pittsburgh, December 18, 1969. I 've profited from discussing various matters treated here with Kurt Baler, Nuel D. Belnap, Jr., Alex C. Michalos, Joseph Sneed and Michael Tooley. 1 K . J . Arrow, Social Choice and Individual Values, 2nd ed., Wiley, New York, 1966. I base my discussion on the argument in the Appendix to the second edition. The argument in the main body of the text, which is the same as the first edition (1951), is defective.

Frederick Schick, in 'Arrow's Proof and the Logic of Preference', Philosophy of Science 36 (1969), proposes to resolve Arrow's paradox by weakening his Collective Rationality Requirement. But my version of this requirement is weaker even than Schick's. That we'd automatically be free of the Arrow difficulty if only we could make sense of interpersonal utility comparisons is one of the most widely held dogmas about Arrow's Theorem. Arrow himself adopts it in 'Public and Private Values', in Human Values and Economic Policy (ed. by S. Hook), New York University Press, New York, 1967. a j . M. Buchanan, in Buchanan and Tnilock, The Calculus of Consent, The University of Michigan Press, Ann Arbor, Mich., 1962, Appendix 2, says Arrow's Paradox is caused by demanding of social decision-making a kind of 'rationality' that is properly attributable only to individual decision-making. 4 R is transitive just in case, for all x, y, z, if xRy and yRz, xRz. R is strongly connected in S just in case, for all x, y in S either xRy or yRx. 5 This doesn't hold when the finitude condition is dropped. e Schick, op. cit. Schick thinks replacing (AT) by this weaker condition resolves Arrow's Paradox. But his argument is remarkably nonconstructive: He doesn't give a single example of a social decision rule fulfilling the weakened set of conditions. Not that these conditions are inconsistent. But their consistency doesn't prove they are satisfied by some reasonable rule. v While there's no need to postulate that the values of P are relations (no less binary relations on S), I will still write 'xP (ul . . . . . un) y" as short for '(x, y )~P (ul . . . . , un)'. s Whether D really represents 'moral ' decision-making depends, inter alia, upon who Messrs. 1, 2, ..., n are taken to be and what their 'utility' measures are taken to measure. 9 K .O . May gave a similar interpretation of Arrow's problem in 'Intransitivity, Utility, and the Aggregation of Preference Patterns', Econometrica 22 (1954). t0 Using 'criterion' broadly, each problem just sketched is a version of multicriterial decision-making. By the way, the Ci may be moral criteria, corresponding to prima facie duties. 11 The choices in question may be choices of rules or of kinds of action rather than of particular actions.

on the possibility of rational policy evaluation

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