Quality and Quantity, 16 (1982) 493-505 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

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A Really General Impossibility Theorem * THOMAS SCHWARTZ

Department of Government, University of Texas at Austin

Introduction

According to Arrow’s celebrated impossibility theorem (e.g., Arrow, 1963, Chap. 7) the following six conditions governing a pairwise collective-choice process are jointly inconsistent: collective rationality, unrestricted domain, (binary) independence, (weak pairwise) Pareto condition, nondictatorship, and the assumption that the universal set A of alternatives has three or more members. The condition of collective rationality states that the relations of collective preference and collective indifference (the latter interpreted as the absence of collective preference in either direction between members of A) are both transitive. By contrast, the “rationality” condition involved in the classical voting paradox is the weaker one of acyclicity, which states nothing about collective indifference and does not state that collective preference is transitive but only that it is noncyclic (so Arrow’s theorem is not a generali- zation of the classical voting paradox). The first Arrow-style impossibility theorem based on acyclicity rather than collective rationality was that of Schwartz (1970). A different theorem was proved by Mas-Cole11 and Son- nenschein ( 1972).

The conditions proved inconsistent by Schwartz include acyclicity, unre- stricted domain, independence, Pareto condition, nondictatorship (strength- ened very slightly), and three non-Arrowian conditions: (1) the number n of individuals is at least three; (2) 1 A J 2 n; and (3) minimum decisiveness: if one individual prefers x toy and the others all prefer y to x, then x and y are not collectively indifferent [ 11.

The conditions proved inconsistent by Mas-Cole11 and Sonnenschein include acyclicity, unrestricted domain, independence, Pareto condition, the assumption that (Al 2 3, and three non-Arrowian conditions: (1) n k 4; (2) nonblocker (my term): there is no individual who, by preferring one alterna-

* Presented at the Annual Public Choice Society Meeting. San Francisco, March 1980. An earlier version was presented at the California Institute of Technology in November 1979.

0033-5177/82/0000-0000/$02.75 0 1982 Elsevier Scientific Publishing Company

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tive to another, can automatically ensure that the latter is not collectively preferred to the former-a reasonable strengthening of nondictatorship; and (3) positive responsiveness: if x is collectively preferred or indifferent to y and some individual changes his relative evaluation of x and y in x’s favor (other things remaining the same), then x will be collectively preferred to y. Mas-Cole11 and Sonnenschein did not use full acyclicity but only the weaker triple acyclicity, which proscribes three-element collective-preference cycles.

By assuming just that 1 A 1 2 3 rather than that J A 1 Z n, the Mas-Colell- Sonnenschein theorem improves upon the Schwartz theorem in one respect. However, its generality is severely limited in two other important respects, as follows.

(1) Whereas the Arrow and Schwartz proofs remain valid when the domain is restricted to linear preferences, the Mas-Colell-Sonnenschein proof does not: at almost every step, it makes essential use of the assump- tion, built into the unrestricted domain condition, that the domain contains nonlinear individual preferences. This limitation is a severe one because many collective-choice models postulate linearity and virtually all real-world voting procedures in which individuals rank alternatives on a ballot (prefer- ential voting) require linear rankings.

(2) Positive responsiveness requires perfect sensitivity to small changes in individual preferences: if x and y are collectively indifferent and a single individual switches from a preference for y over x to an indifference between x and y, or from the latter to a preference for x over y, that is sufficient to break the tie in x ‘s favor. Although a number of simple pairwise-voting rules meet this condition, it is hard to believe that many complex real-world institutions fulfill it completely. Certainly the condition is violated by any collective-choice process, such as the “ratings” system used to decide broad- cast advertising prices, which polls a sample rather than an entire popula- tion. Positive responsiveness may represent an unimpeachable ideal, but it is an ideal rather than a minimum standard of good collective-choice behavior. Yet so far as the theorem is concerned, it might take just one trivial deviation from the ideal-just one case, say, in which a single individual fails to break a tie when he switches from a preference to an indifference-to avoid collective-preference cycles altogether.

I here generalize the Mas-Colell-Sonnenschein theorem in three ways, as follows.

(1) I weaken the unrestricted domain condition by dropping the assump- tion that the domain contains nonlinear individual preferences; this does not mean that linearity is assumed: the domain condition is neutral on the issue (this makes it necessary, incidentally, to assume that n 2 5, inasmuch as the relation of simple majority preference is perforce noncyclic when n = 4 and individuals have only linear preferences).

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(2) I drop the condition of positive responsiveness, which says that a single individual can break any tie, in favor of the less demanding condition that a coalition comprising a fifth of the voting population can break any tie (by reversing a shared preference between the tied alternatives).

(3) In place of acyclicity or even triple acyclicity, I assume only that, for some possible cycle size (finite or infinite), there is no collective-preference cycle of that size having the following three features: (1) from any given alternative in the cycle, there is a path of at most four steps leading to any other given alternative (so that movement from any point in the cycle to any other point is relatively easy); (2) if the cycle contains more than four alternatives, it contains a Pareto-inefficient alternative; (3) if the given cycle size is 1 A 1 then the cycle encompasses all of A -a vacuous requirement when A is finite but not when A is infinite. To elaborate this point: the other conditions imply the existence (for some preference profile) of a cycle of any given possible size having features (l)-(3). In particular, they imply the existence of a cycle encompassing all of A having features ( 1) and (2). So if A is taken to be the feasible set of the moment, there is no stable (un- dominated) feasible alternative and the top-cycle set comprises all the feasible alternatives and is Pareto-inefficient if 1 A I> 4. Whereas the Schwartz and Mas-Colell-Sonnenschein theorems show only that some potential feasi- ble set has no stable outcome or proper top-cycle subset, the theorem proved below shows this (among other things) for a fixed feasible set (which is not even assumed to be finite).

Basic Terms and Definitions

The present theorem concerns structures consisting of a positive integer n, sets A and PROF, a function P, and a cardinal number y (which may be finite or infinite). Here n is the number of individuals, 1,2, . . ., n; A is the “universal” set of alternatives, those between which the pairwise-choice process under consideration is designed to choose; PROF is a set of preference profiles, each an ordered n-tuple of relations p,, . . . , p,, representing a possible situation in which p, is l’s (strict) preference relation, pZ 2’s, etc.; P is a function that assigns to each v in PROF the corresponding relation P” of collective (strict) preference in the v situation; and y is the size of cycle proscribed by the weakened acyclicity condition discussed above.

DEFINITION 1

N= { 1, 2,...,n)

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DEFINITION 2

f = the greatest integer G n/5

(by coalitions comprising a fifth of the voting population is meant subsets of N containing f or more members).

DEFINITION 3

xP,.“y iff, for some p1 ,..., p,, u=(p ,,..., p,) E PROF and either (1) rEN and xp,y, or (2) r c N and xp,y for every i E Y (thus xPrUy holds iff, in the u situation, Y is either an individual who prefers alternative x to alternative y, or a coalition whose members all prefer x to y).

DEFINITION 4

If (Y is a set and p a binary relation,

p/Cl=pn(u2

If (Y is a set and p a vector of binary relations p,, . . . , pk,

P/a= (P,/% PZ/%...~Pk/4

Otherwise, p/a is the empty set (for a set (Y and a binary relation or vector of binary relations p, p/a is the restriction of p to (Y).

DEFINITION 5

g is decisive for x versus y iff g c N, x, y E A, x # y, and for every u E PROF, if xP,“y but yPi_,x then xP”y.

DEFINITION 6

g is nearb decisive for x versus y iff g c N, x, y E A, x fy, and for every u E PROF, if xPg”r but yP&,x then not yP”x.

The definition of decisiveness is Arrow’s. Decisiveness implies near de- cisiveness, but not conversely: a nearly decisive coalition can dictate a collective preference or indifference (despite the unanimous opposition of nonmembers), but only a decisive coalition can dictate a (strict) collective preference.

The Conditions

The present theorem is that the following six conditions are inconsistent:

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DOMAIN

(a) n 2 5. (b) Every member of PROF is an ordered n-tuple of asymmetric binary

relations on A. (c) To every IJ E PROF, P assigns an asymmetric binary relation Pv on

A. (d) If B is any y-member subset of A and o is any ordered n-tuple of

linear orderings of B, then d/a = 21 for some o’ E PROF. According to the unrestricted domain condition, every ordered n-tuple of

weak orderings of A belongs to PROF, which is stronger than (d) and (I have argued) excessively demanding. Note that the present domain condition does not state that every vector in PROF comprises only linear orderings of A, or even that every vector in PROF comprises only weak orderings of A. The latter is almost always assumed but never used in proofs of impossibility theorems.

INDEPENDENCE

Assume D,W E PROF, V/{X,JJ} = w/{x,,v}, and xP”v. Then S’“‘J.

PARETO CONDITION

If xp;y, xlFp”y.

NONBLOCKER

There is no i E N such that every subset of N to which i belongs is nearly decisive for every element of A versus every other element.

WEAK POSITIVE RESPONSIVENESS

If g is nearly decisive for x versus y and h is a af-member subset of N - g, then g U h is decisive for x versus y.

WEAK ACYCLICITY

3 B y =G 1 Al, and there is no o E PROF and y-member subset B of A such that:

(1) for all distinct x,y E B, either xP”y or xP”r,P”y or xlP”z1iP”z2PDy or xP”zIP”z2P”z3P”y for some zi, z2, z3 E B;

(2) if y > 4, xP;y for some x, y E B; and (3) if y= IA), B=A. Note that weak acyclicity implies that 1 A 1 > 3.

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The Theorem Proved

Theorem: domain, independence, Pareto condition, nonblocker, weak positive responsiveness, and weak acyclicity are jointly inconsistent.

To prove this, I assume the six conditions, from which I deduce nine consequences, the last of which contradicts nonblocker.

CONSEQUENCE 1

(a) There is no 0 E PROF such that, for some distinct x,y,z EA, some ( y - 3)-member subset B of A - { x,y, z), and some linear ordering A, of B, B U {x, y, z} = A if y = IA 1, P,“/B = X, for all i E N, and xP”yP”zP”xP “tP”y for all f E B. That is, there is no o E PROF for which P” has the form:

(where IBI=y-3 and every Pi” linearly orders B in the same way as does b>.

(b) There is no v E PROF such that, for all distinct x, y EA, some (y - 2)-element subset B of A - {x, y}, and some linear ordering h, of B, BU(x,y}=Aify=~A~,P,“/B=~,foralliEN,andxP”yP”rP”xforall t E B. That is, there is no 0 E PROF for which P” has the form:

(where 1 B 1 = y - 2 and every Pi” linearly orders B in the same way as does

u. Proof of (a). Suppose, on the contrary, that there existed such D, x, y, z, B,

and X,. Then (1) I B U {x,y,z} 1 = y; (2) P”/B = X, by the Pareto condition, so that P” linearly orders B; (3) for all distinct w, u E B U {x, y, I}, either wP”z.4 or wP”t,~Ou or w~Dt,~0t2~0u or w(FD0t,P”t2PDt3P”u for some t,, t,, t, E B U {x, y, z}; and (4) if I B U {x, y, z} I > 4 then there are distinct w, u E B such that wP,$u. But this contradicts weak acyclicity.

The proof of (b) is similar.

CONSEQUENCE 2

Assume that g c N, xP,“y, yP,&,,x, and not yP”x. Then g is nearly decisive for x versus y.

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Proof. Suppose that xP,“*y and yP;l,x; to prove that not yP”*x. But 2)*/(x, y} = O/(X, y}. By the independence condition, then, yP”x if yP”*x. But not yP”x. So not yP”*x.

CONSEQUENCE 3

Suppose that g is nearly decisive for x versus y, h is a al-member subset of N - g, and h is not nearly decisive for anything. Then:

(a) g is nearly decisive for x versus every z E A - {x) ; (b) g is nearly decisive for every z E A - { y} versus y; (c) g is nearly decisive for every z E A versus every w E A - {z} . Proof. g U h is decisive for x versus y by hypothesis and weak positive

responsiveness. Since (by weak acyclicity) 1 Al 2 3, there is a ( y - 3)-member subset B of A-(x, y, z} such that B U {x, y, z} = A if y = 1 Al. Let A, be a linear ordering of B.

(a) By the domain condition, there is a o E PROF whose component relations linearly order {x, y, z> U B as follows:

g h N-(guh)

X b Y Y Z A,

b X Z

Z Y X

Then Py/B = A, for all i EN. By the decisiveness of g U h, x[Ip’y. By Consequence 2 and the assumption that h is not nearly decisive for anything, yP”z and yP”t for all t E B. But tP”z for all t E B by the Pareto condition. Thus PD has the form:

(where 1 B 1 = y - 3 and every Pi” linearly orders B in the same way as does A,). By Consequence l(a), then, not zIFD”x. Hence, by Consequence 2, g is nearly decisive for x versus z.

(6) By the domain condition, there is a o E PROF whose component relations linearly order {x, y, z} U B as follows:

g h N-(guh) Z X Y b Y X ill

Y L3 X

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Then P,“/B = A, for all i EN. By the decisiveness of g U h, xP’y. By Consequence 2 and the assumption that h is not nearly decisive for anything, zlFD”x and tP*x for all t E B. But rP”t for all t E B by the Pareto condition. Thus P” has the form:

/

Y

x----z

\J V’tEB

(where IBj=y-3 and every P,” linearly orders B in the same way as does A,). By Consequence l(a), then, not yP”z. Hence, by Consequence 2, g is nearly decisive for z versus y.

(c) Because ( A 12 3, there is a t E A - {x, z}. By hypothesis and Conse- quence 3(a), g is nearly decisive for x versus t, whence g is nearly decisive for z versus t by Consequence 3(b), so g is nearly decisive for z versus w by Consequence 3(a).

CONSEQUENCE 4

Suppose that g is nearly decisive for x versus y, and h is an f-member subset of N-g. Then:

(a) g U h is nearly decisive for x versus every z E A - {x} ; (b) g U h is nearly decisive for every z E A - { y} versus y. Proof. g U h is decisive for x versus y by weak positive responsiveness.

Since 1 A( 2 3, there is a ( y - 3)-member subset B of A{ x, y, z} such that B U {x, z} = A if y = 1 Al. Let A, be a linear ordering of B.

(a) By the domain condition, some u E PROF has the form:

guh N-(guh)

X Y

Y xl3

xl3 Z

Z X

Then P,“/B = A, for all i E N. By the decisiveness of g U h, XP “y. But by the Pareto condition, yP”tP”z for all t E B, and yP”z. So P” has the form:

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(where ]B]=y-3 and every P,” linearly orders B in the same way as does h,). By Consequence l(a), then, not z5)“x. Hence, by Consequence 2, g is nearly decisive for x versus z.

(b) By the domain condition, some u E PROF has the form:

19”h N-(g”h)

Then P,“/B = X, for all i E B. By the decisiveness of g U h, x[Fp”y. But by the Pareto condition, zP”tP”x for all t E B, and ~119”~. Thus P” has the form:

(where IBI=y-3 and every Pi” linearly orders B in the same way as does X,). By Consequence l(a), then, not yP”z. Hence, by Consequence 2, g is nearly decisive for z versus y.

CONSEQUENCE 5

Assume that g and h are disjoint, f-member subsets of N, and g is nearly decisive for x versus y. Then for every z E A, either g is nearly decisive for x versus z, or h is nearly decisive for z versus y.

Proof. The proof is trivial if z =y or z = x. So suppose that z E A - {x, y}. Let B be a (y - 3)-member subset of A{x, y, z} such that B U {x, y, z} =A if y=]A], and let X, be a linear ordering of B. By the domain condition, N - (g U h) has an f-member subset p, and N - (g U h Up) has more than f members. By Consequence 4, since g is nearly decisive for x versus y, g U h is nearly decisive for x versus every t E B and for every t E B versus y. Thus, by weak positive responsiveness, g U h is decisive for x versus y,gUhUpisdecisiveforxversuseverytEB,andgUhU[N-(gUhUp)] is decisive for every t E B versus y.

By the domain condition, some 2) E PROF has the form:

g

;I, Y Z

h

Z

t3 Y

P N-(g”h”p)

Y b I Y X Z

&I X

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Then P,“/B = X, for all i E N. But by the decisiveness of g U h, g U h Up, and g U h U [N - (g U h Up)], it is the case that XP “y, and xP”tP “y for all t E B. Thus P” has the form:

(where IBI =y-3 and every P,” linearly orders B in the same way as does X,). Hence, by Consequence l(a), not both yP”z and zP”x. Thus, by Consequence 2, either h is nearly decisive for z versus y, or g is nearly decisive for x versus z.

CONSEQUENCE 6

Assume that g c N, ) g ) 2 4f, and some f-members subset of g is nearly decisive for some pair. Then g is decisive for every z E A versus every wEA-(

Proof. By hypothesis, g may be partitioned into four subsets, g,, g,, g,, g,, such that (a( = lg,( = (g,l =fG lg& and g, is nearly decisive, say, for x versus y. Because 1 A 12 3, there is a t E A - {x, z} .

Case 1. t = w. So w # x. Then g, U g, is nearly decisive for x versus w by Consequence 4(a), whence g, U g, U g, is nearly decisive for z versus w by Consequence 4(b), and thus g, U g, U g, U g, = g is decisive for z versus w by weak positive responsiveness.

Case 2. t # W. So t @ {x, z, w}. By Consequence 5, either g, is nearly decisive for x versus t, or g, is nearly decisive for t versus y.

Subcase 2(a). g, is nearly decisive for x versus t. Then g, U g, is nearly decisive for z versus t by Consequence 4(b), whence g, U g2 U g, is nearly decisive for z versus w by Consequence 4(a), and thus g, U g, U g, U g, = g is decisive for z versus w by weak positive responsiveness.

Subcase 2(b). g, is nearly decisive for t versus y. Then g, U g, is nearly decisive for t versus w by Consequence 4(a), whence g, U g, U g, is nearly decisive for z versus w by Consequence 4(b), and thus g, U g, U g, U g, = g is decisive for z versus w by weak positive responsiveness.

CONSEQUENCE 7

There do not exist disjoint sets g, h such that each is nearly decisive for some pair and lgl Gf = Jhl.

Proof. Suppose that such sets exist; proof will be accomplished by deducing a contradiction. Suppose that g is nearly decisive for x versus y. But I N -g I B 4f. So by Consequence 6, since h c N - g and h is nearly decisive for some pair, N - g is decisive for y versus x. But by the domain

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condition, there is a v E PROF for which xP,“y and yPl_,x. Then by the decisiveness of N - g, yP “x, but by the near decisiveness of g, not yP Ox.

CONSEQUENCE 8

For some i E N, {i} is nearly decisive for every z E A versus every M&A-(Z).

Proof. Because N is finite and (by the Pareto condition) decisive for any given pair of distinct elements of A, some subset M of N is a smallest decisive set: M is decisive for some pair, say x versus y, and no smaller set is decisive for any pair. I first show that some G ( f+ I)-member subset of A4 is nearly decisive for some pair. This is trivial if 1 M 1 <f+ 1. Otherwise, let i E M, and let F be an f-member subset of M - {i}. Then let B be a (y - 2)-member subset of A - {x, y) such that B U {x, .v} = A if y = ) A], and let A, be a linear ordering of B.

By the domain condition, some v E PROF has the form

M I * \ i F M-((i) U F) N-M

&I h3 X Y X X Y XL3 Y Y XL3 X

By Consequence 2, for every t E A, if not tP “x then M - ({i} U F) is nearly decisive for x versus f, whence [M - ((i} U F)] U F = M - {i} is decisive for x versus t by weak positive responsiveness. But that is impossible, because no set smaller than M can be decisive for any pair. Hence, tP”x for all t E B. But by the decisiveness of M, xP”y, and P,“/B = A, for all i E N. Therefore, by Consequence I(b), y does not bear the relation P” to every t E B. For some t E B, then, not yP”t. So F U {i}, which has f+ 1 members, is nearly decisive for t versus y by Consequence 2.

Now let L be any smallest near4 decisive set: L is nearly decisive for some pair, and no smaller set is nearly decisive for any pair. Because, as just proved, some C ( f+ l)- member subset of M is nearly decisive for some pair, 1 L 1 <f+ 1. By the Pareto condition, L has at least one member, say i. I now prove that {i} is nearly decisive for every z E A versus every w E A - {z>.

By the domain condition, there is anf-member subset g of N - L. Because no set smaller than L is nearly decisive for any pair, if g were nearly decisive for some pair, it would be the case that I L I Gf = I g 1, contrary to Conse- quence 7. Therefore, g is not nearly decisive for any pair, and thus, by Consequence 3, L is nearly decisive for every element of A versus every other element.

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If (i} is nearly decisive for some pair, then {i} is no smaller than L, so {i} = L, and thus {i} is nearly decisive for every element of A versus every other element. Hence, it suffices to show that {i} is nearly decisive for some pair.

Suppose that {i} were not nearly decisive for any pair; proof will be accomplished by deducing a contradiction. By the domain condition, there are disjoint, f-member subsets h,, h, of N - L. By weak positive responsive- ness, since L is nearly decisive for every element of A versus every other element, L U h, is decisive for every element of A versus every other element. Let x, y be distinct elements of A. Let B be a (y - 2)-member subset of A - {x, y} such that B U {x, v} = A if y = (Al, and let yg be a linear ordering of B.

By the domain condition, some 0 E PROF has the form:

L , A {i} L-(i) h; N-(LUh,)

h3 x x Y x Y Y b Y AlI ALI x

Then P,“/B = A, for all i E B. But by the decisiveness of L U h,, xlFp “y. By Consequence 2, since {i} is not nearly decisive for any pair, yP”t for all t E B. Therefore, by Consequence l(b), there is a t E B for which not tP”x. Thus, by Consequence 2, (L - {i}) U h, is nearly decisive for x versus t. Hence, by weak positive responsiveness, ( L - {i}) U h, U h 2 is decisive for x versus t.

Now let C be a ( y - 2)-member subset of A - (x, t} such that C U {x, t} =A if y=lAl, and let Xc be a linear ordering of C. By the domain condition, 1 N - (L U h, U h2)1 af. So by weak positive responsiveness, since L is nearly decisive for every element of A versus every other element, L U [N - ( L U h, U h *)] is decisive for every element of A versus every other element.

By the domain condition, some 0’ E PROF has the form:

L t A \ {i} L--(i) h, h, N-(Luh,uh,)

xc i, x

x x t t t A,

x t xc AC x

Then P,“‘/C = A, for all i E N. But by the decisiveness of L U [N - (L U h,

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U h,)], uP”‘x for all u EC. By the decisiveness of (L- {i}) U h, U h,.

xP”‘t. Hence, by Consequence l(b), there is a u E C such that not tP”‘u. Thus, by Consequence 2, L - {i} is nearly decisive for u versus t. But that is impossible, because no set smaller than L can be nearly decisive for any pair.

CONSEQUENCE 9

For some i E N, every subset of N to which i belongs is nearly decisive for every x E A versus every y E A - {x} (contrary to nonblocker).

Proof. By Consequence 8, there is an i E N such that {i} is nearly decisive for every element of A versus every other element. Suppose that i E g c N; the aim is to deduce that g is nearly decisive for x versus y. Let B be a (y - 2)-member subset of A - {x, y} such that B U (x, y} = A if y = (Al, and let A, be a linear ordering of B. By the domain condition, N - {i} may be partitioned into >f-member subsets h, and h,. By weak positive re- sponsiveness, {i} U h, and {i} U h 2 are each decisive for every element of A versus every other element.

By the domain condition, there is a o E PROF such that xPRt)y and ylFpi_,x, xP~tP,“y for all t E B, xlF’t,t and tP,“zy for all t E B, P,“/B = A, for allj E N. Then by the decisiveness of {i} U h, and {i} U h,, xP”t[FD”y for all t E B. Therefore, by Consequence l(b), not yP”x. By Consequence 2, then, since xlP:u and yPg_,x, g is nearly decisive for x versus y.

Note

1 Actually, my 1970 result was more general than this. Independence was drastically weakened, and individual preferences were represented by real-valued functions, the latter being done to allow collective preferences to reflect interpersonal utility comparisons to some degree. I did not assume, however, that collective preferences did reflect anything other than ordinal utilities-I just did not rule out the possibility. Because these conditions are inconsistent, they remain so when strengthened to include ordinality plus full independence. I have here described this less general result (discovered in 1968) in order more easily to compare my theorem with those of Arrow (1963) and Mas-Cole11 and Sonnenschein (1972).

References

Arrow, Kenneth (1963). Social Choice and Individual Values. New Haven: Yale University Press.

Mas-Colell, A. and Sonnenschein, H. (1972). “General possibility theorems for group deci- sions”, Review of Economic Studies 39: 18% 192.

Schwartz, Thomas (1970). “On the possibility of rational policy evaluation”, Theory and Decision 1: l-25.