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Impossibility results for choice correspondences
mathematical social sciences ELSEVIER
Mathematical Social Sciences 33 (1997) 129-148
Impossibility results for choice correspondences Anton Stefanescu Bucharest University, Faculty of Mathematics, 14, Academiei st, Bucharest 70109, Romania Received April 1995; revised November 1995
Abstract One of the most important results in social choice theory is the Impossibility Theorem. Since 1951 when its first version was stated by K. Arrow, many other versions and extensions have been done. Although conditions differ, most results are concerned with relational aggregation rules which transform the individual preference relations into a social binary relation. The mixed case in which the social choice is defined in terms of choice functions while the individual opinions are represented by preference relations, also allows some interesting extensions of the original result. The study of the choice aggregation rules, when both individual and social choices are expressed in terms of choice functions, is well represented by the works of Aizerman and Aleskerov (Mathematical Social Sciences, 1986, 11, 201-242). Our approach concerns this latter framework. We introduce new normative conditions under which impossibility results are derived. These results are not dependent upon rationality conditions and extend the Impossibility Theorem to general sets of voters and alternatives. © 1997 Elsevier Science B.V. Keywords: Choice correspondence; Aggregation rule; Impossibility theorem
1. Introduction Choice processes are widely studied owing to their incidence in all fields of social life, and a systematic study based on mathematical modelling can be observed especially in the second half of this century. The most commonly used models for choice processes are choice correspondences (set-to-set mappings) and ordinal preferences (binary relations, also called preference relations). Considering a universal set of alternatives, a choice correspondence specifies for each given admissible set of alternatives a non-empty subset of this set - the result of the choice process. A n ordinal preference can, theoretically, be any 0165-4896/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PII S0165-4896(96)00829-3
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binary relation on the set of alternatives, but usually some specific properties like reflexivity, completeness or transitivity are required. The main problems studied in social choice theory are related to the aggregation of the individual choices in a society. If we represent a society as the set of individuals (voters) endowed with preferences over the set of alternatives, then it is commonly assumed that social choice is obtained from these preferences by an aggregation rule. Therefore, it is expected that the properties of the individual preferences will be reflected by the choice of the whole society, and it would be interesting to inquire if this choice can satisfy certain basic democratic and ethical principles. The celebrated Arrow's Impossibility theorem emphasizes the incompatibility of a set of fair properties if the social choice determined by an aggregation rule is subjected to some rationality conditions. Obviously, the social choice, as well as any individual one, can be defined in the same way, i.e. by a choice correspondence or by an ordinal preference relation, so that different types of aggregation rules could, theoretically, be considered. Arrow's theorem deals with the case in which the individuals are endowed with preference orderings on the set of alternatives. In one version (Arrow, 1963) this result stated that if society consists of finitely many individuals, then any aggregation rule satisfying the weak Pareto principle and the independence of irrelevant alternatives must be dictatorial if the social choice is guided by a weak order too. (Such an aggregation rule is called the 'social welfare function'.) The same conclusion holds if, equivalently, the social choice is expressed in terms of a choice function (choice correspondence in the present paper) defined on all subsets of alternatives and satisfying some 'rationality conditions' (Arrow's choice axiom.) Several extensions of Arrow's results have been sought in different ways. While in the original Impossibility Theorem the set of voters was assumed to be finite, Fishburn (1970) has shown that its conclusion fails if the number of voters is infinite. An elegant proof both of Arrow's and of Fishburn's results has been done by Kirman and Sondermann (1972). Moreover, they have pointed out that, even in the infinite case, a special form of dictatorship still exists. (The term invisible dictator has been proposed in this case.) Extensions of Arrow's results have been obtained by weakening the condition of collective rationality. In its social welfare function form, Arrow's theorem is also dependent on the social preference relation being transitive. It was shown (Sen, 1970) that the impossibility result does not hold if this is required to be only quasi-transitive. However, Gibbard (1969) showed that in this case society must admit an 'oligarchy', so that a special form of group dictatorship still exists. Another way Arrow's results have been extended is to restrict the collection of admissible sets when the social choice is defined in terms of choice functions. In the choice function version of Arrow's theorem all non-empty subsets of alternatives are assumed to be admissible and its usual proof relies on the
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assumption that every pair of alternatives is an admissible set. A variant of Arrow's theorem involving individual ordinal preferences and social choices correspondences is due to Grether and Plott (1982). They considered that the domain of the admissible sets consists of all subsets containing at least k alternatives, where k must be strictly less than the number of alternatives in the universal set. Donaldson and Weymark (1988) have proved an impossibility theorem in an economic environment. The domain of the admissible sets is restricted to the family of compact and comprehensive subsets of the positive orthant of a Euclidean space. However, the cited authors asked for the social choice correspondences to satisfy a strong rationality condition (Arrow's choice axiom.) Since in both cases the family of admissible sets is closed under the finite union, the social choice can be represented by an ordering (see Hansson, 1968). A particular case when the universal set is the only admissible set (the fixed agenda case) has been considered by Hansson (1969) and Denicolo (1987). They obtained impossibility results without rationality condition but by strengthening the usual independence condition, In an extensive study, Aizerman and Aleskerov (1986) reformulated the social choice problem only in terms of choice functions. In their terminology a 'voting operator' (or functional operator) transforms the individual opinions represented by choice functions into a collective choice function. They introduced normative properties analogous to Arrow's axioms in the classical framework and used them in conjunction with standard rationality properties to isolate voting operators that respond to each set of imposed conditions. Particularly, by isolating dictatorship operators we obtain some impossibility results. A more recent paper by Aleskerov and Duggan (1993) has extended these investigations for larger classes of such aggregation rules. Note that in the aggregation framework of Aizerman and Aleskerov, the sets of voters and alternatives are finite and all subsets of alternatives are admissible. However, the emptiness of the individual and social choices was allowed, which is rather unusual. (Consequently, the term 'choice correspondence' is not appropriate in this case.) In the same framework, Popov and El'kin (1990) ~ studied a class of voting operators characterized by the property called 'partialness' (see Section 4). They maintained the finiteness assumptions on the sets of voters and alternatives, but unlike Aizerman and Aleskerov the collection of admissible sets does not necessarily coincide with the family of all subsets of alternatives. (They have assumed only that each alternative is included in at least one admissible set.) Note also that the non-emptiness of the choice function was generally assumed. Within
1 W h e n the first version of the present paper was submitted, the author did not know about the works of Popov and El'kin. He is grateful to the referees for signalling and sending him a copy of their papers.
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this modified framework Popov and El'kin obtained impossibility results without explicit rationality conditions. Our study deals basically with the Aizerman-Aleskerov framework, but considers arbitrary sets of voters and alternatives and more general families of admissible sets of alternatives. The non-emptiness of the choice functions is always assumed (so that these are actually choice correspondences) and the rationality is not explicitly required, as in Popov and El'kin. Our impossibility results follow from two new normative conditions on social aggregation rules. The main result of the paper shows that the set of all decisive coalitions of voters is an ultrafilter if the social aggregation rule satisfies these conditions; and, conversely, if all admissible sets are finite, then every ultrafilter of coalitions can be characterized as the collection of decisive sets of a social aggregation rule. Section 2 formalizes the framework of our study and introduces the two normative properties called strong monotonicity and decisiveness-neutrality, respectively. The main results is proved in Section 3. Its corollaries extend, within the present framework, Arrow's theorem for finitely many voters, Fishburn's possibility result for infinitely many voters, as well as Kirman and Sondermann's result concerning 'invisible dictators'. Everywhere Section 3 the social aggregation rules are defined for all possible choice correspondences which characterize the individual opinions. Section 4 explains the two conditions introduced in Section 2 in the context of the normative properties of the Aizerman-Aleskerov framework. The most significant result shows that our conditions are implied by the normative conditions used in Popov and El'kin (1990). From here and from the main theorem we obtain impossibility results which extend those of Popov and El'kin. In the final section we point out that the results of Section 3 hold even if the individual choices are required to satisfy rationality conditions. Arrow's choice axiom and Chernoff's condition are used to argue these conclusions. It is also shown that the social choice determined by a social aggregation rule satisfying strong monotonicity and decisiveness-neutrality inherits the above rationality conditions from individual choices.
2. Choice correspondences and social aggregation rules Everywhere in this paper X will denote the non-empty set of alternatives and £~ a (finite or infinite) set whose elements are called voters or consumers. Each voter has his own opinion with respect to the set of alternatives and, as we have already mentioned, this opinion can be represented in several ways. We will focus on the case when it is expressed in terms of a choice correspondence defined below.
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Let us denote by ~ a non-void subfamily of ~0(X) = {S C_X I S ~ 0} the family of all admissible sets of alternatives. Definition 1. A choice correspondence on X is a mapping C: ~ 0 ( X ) that C(S) C S for all S E ~.
such
Denote by F ( ~ ) the set of all choice correspondences on X and by q~ the set of all choice profiles, i.e. = {c I c : a
A choice profile associates with each voter i E 12 its own choice correspondence c(i), conventionally denoted by c r The set qg contains all logically possible choice profiles. Some of them could seem to be rather irrational and their use by voters is not justified. In Section 5 we will discuss the possibility of restricting the set of choice profiles by imposing rationality conditions on the elements of F ( ~ ) . Beginning with the pioneering works in social choice theory, several notions with a similar meaning are used for a social aggregation rule according to the considered situations. Since our approach concerns only one type of choice process, we will use the generic notion of social aggregation rule. The formal definition follows. Definition 2. A social aggregation rule (s.a.r.) is any mapping F : ~ ~-~F(~). Here F(c) (also denoted by F~) represents the social choice correspondence associated with the choice profile c. Let ~ be the set of all s.a.r, satisfying the following conditions: (A1) VS E ~, x, y E S, c, c' E q¢ such that {i I x E
ci(S)} C {i I Y E c'i(S)}
and x E
Fc(S ) ~ y E F¢,(S). (A2) V c E ~ , S , S ' E ~ , xES, yES' then x E Fc(S)C:> y , ~ Fc(S' ).
such that { i l x E c i ( S ) } = { i l y , ~ c i ( S ' ) }
Condition (AI) could be called strong monotonicity. It states that if a group of voters (coalition) once imposes an alternative from a given set S, then any other larger coalition can impose on society any alternative from S if the opinions of its members are convergent. According to the second condition, (A2), any 'accepting coalition' with respect to a given profile (i.e. a coalition which can impose the alternatives agreed by all its members) is also a 'blocking coalition' (i.e. it can block any alternative which is
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rejected by all its members even if all other voters have a different opinion). It has been suggested that condition (A2) could be called decisiveness-neutrality. For an intuitive justification of this condition let us first consider the case of binary comparison. Let S be any two-element admissible set, say S = {x, y}, and assume that two complementary coalitions, A and B, have different opinions towards the alternatives x and y. A naive 'democratic' principle requires that society should choose that alternative which has obtained the largest number of votes. If, in addition, a special rule is adopted when tml = IBI to break ties, then we arrive at the conclusion required by (A2) for S' = S. Furthermore, if S = {x, z} and S' = {y, w} are two distinct admissible sets and x is chosen by all members of A while y is chosen only by all members of B, then it is obvious that either x and w have similar positions or w has a better situation than x and therefore, as in the above, y should be rejected by society if it chooses x. Of course, (A2) does not say exactly this, but it extends the naive ideas involved in the above to more general collections of admissible sets. In a formal framework we can show that both conditions (ml) and (A2) can be derived from other normative conditions. For instance, Proposition 5 of Section 4 proves that any 'partial' and 'neutral' s.a.r, must satisfy (ml) and (Az). Roughly speaking, partialness says that society cannot be totally indifferent if none of the individuals is. Specifically if all members of a coalition chose one alternative from a two-element admissible set while all other individuals choose the second one, then society must choose one and only one of these two alternatives, so that we arrive to the conclusion required by (A2). Neutrality requires that if a coalition imposes an alternative, then the votes of its members suffice for any alternative of any other admissible set to be chosen by society. In conjunction with neutrality, the partialness extends the conclusions emphasized in the particular situation of the two-element set-up to the full condition (A2). Moreover, condition (A~) (which concerns only one fixed admissible set) is also satisfied if the s.a.r, satisfies both neutrality and partialness. Note that (A1) is satisfied by some plurality-type rules, as is shown in the following example. For any c E ~ and x E S E ~ denote by vc(x, S) = 1(i ~ 12 Ix ~ ci(S)}l. Example 1. Plurality rule (see Aizerman and Aleskerov, 1986, for the k-plurality rule). Assume 12 is finite and let k be a positive integer k ~< 1121. Define the aggregation rule F p by F~ (S) = {x E S I Vc(x, S ) >l k } .
Since in the present framework non-emptiness is assumed for all choice correspondences, a necessary condition for consistency of the above definition is that
(k - 1)1SI < 1121 for all S E ~ .
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Obviously, this condition is satisfied for non-trivial O and ~ and then F p satisfies (A1). U n d e r some special assumptions, the plurality rule can also satisfy (A2). Example 2. For the plurality rule F p defined in Example 1, assume n = 1121 is odd and take k = (n + 1)/2. Also consider that ISl ~<2 for all S E ~ . In this case both (A1) and (A2) are satisfied. Moreover, we note that this is the only non-trivial case when the plurality rule, defined as in the previous example, satisfies (A2) (we must have 1~21~ < 2 k - 1 which, together with the necessary condition imposed in Example 1, is fulfilled only when ~f consists of one-element sets, or in the case of the present example). Otherwise, the plurality rule satisfies (A1) but not (A2). In the next example, condition (Az) will be satisfied by introducing a special tie-breaking rule. However, the s.a.r, may not satisfy (A1), so that it will be shown that the two conditions are mutually independent. Example 3. For I~1 = n, where n is even and for ~ consisting of subsets with at most two elements, define:
F~(S) = ~ Fp~(s)' Lcl(S) ,
if F~(S) ~ fJ, otherwise,
where F p is the s.a.r, of Example 1 for k = (n/2) + 1. Clearly, F satisfies (A2). Let S E ~ be a two-element admissible set, S = {x, y}. Define c, c'@ q~ such that ci(S ) = {x} and c'i(S ) = S, for i = 1 . . . . . n/2, and ci(S ) = c'i(S ) = {y}, otherwise. Then, {i Ix E ci(S) } = {i Ix E c'i(S) } but x ~ Fc(S ) while x ~_ Fc,(S) = {y}, so that F fails (A1). Now, let us note two interesting propositions. Proposition 1. I f F satisfies (A1) and (A2), then it also satisfies the Pareto
principle:
(A3) Fc(S ) C LJi~,c1 ci(S ) for all S E ~ and c E q¢. Proof. An immediate consequence of (A1) is that for every S and c then either F¢(S) = S or F~(S)C LJiea ci(S). All that we must do is to prove that the first situation is impossible if LJiea ci(S) ~ S. T o the contrary, pick an x E S \ LJiea ci(S) and an y ~ U~ea ci(S) and suppose that x E F~(S). Construct c ' E cg such that x~c'i(S) and y E c'~(S) for all i E g2. T h e n it follows by (A1) that x ~ F~,(S) and y E Fc,(S) contradicting (A2). []
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Remark. Condition (A3) is called negative unanimity in Aizerman and Aleskerov (1986). Under assumption (A1) it implies that ' n ~ E a q(s)c__ F~(S)c Uiea ci(S) for all c and S E ~ ' .
Proposition 2. If F satisfies
(A1) and
(A2), then it also satisfies
(A~) c,c' E ~ , S , S ' E ~ , x E S , y E S ' such that
(ilx e
ci(S)} =
{il yg~c;(S')} then x E fc(S)C~y,~Fc,(S' ).
Proof. Note first that, as a consequence of Proposition 1, if x E Fc(S), then the set c;(S)} must be non-empty. Hence, the non-trivial situation is when both S and S' have at least two elements. Pick an z E S'\{x} and construct c"E ~¢ such that
A = {ilx E
{i Ix E c'~(S)} = {i I x E ci(S)} , {il z e c"(S')} = {i I y E c'~(S')}. Then it follows by (A1) that xEF~(S)CC, xEFc.,(S ) and by (A2) that x E Fc,,(S)C=>z~.F~,,(S'). Again, (A1) implies that z~_Fc,,(S')C=>y~_Fc,(S' ). []
3. Impossibility results Since the proofs of the main results of this section are based on the theory of ultrafilters, let us first recall some fundamental notions and results coming from this theory. Let E be a non-empty set and ~ C 2 e, ~ ~ l~.
Definition 3. ~ is a filter on E if it satisfies the following conditions: (a) 0 ~ ; (b) A E $ , A C B ~ B E $ ; (c) A, B E $ ~ A O B E $ . Obviously, the intersection of an arbitrary collection of filters is again a filter. If, for two filters $1 and $2 we have $1C $2 (i.e. A E $1 ~ A E $2), then ~2 is said to be finer than $1. Since the filters on E form.a partially ordered set under C, then the following definition makes sense:
Definition 4. Any maximal filter on E is called an ultrafilter. The existence of ultrafilters follows from Zorn's lemma since for any totally ordered family of filters the least upper bound exists (the union of all filters of the
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family). Moreover, it follows that for any filter ~ there exists an ultrafilter ~ ' finer than ~. Finally, note that a filter ~ is an ultrafilter if and only if for every subset A C_E either A or its complementary .,~ belongs to ~. Denote now by 12 the set of all ultrafilters on O. Let us associate with each F E ~ the family D e of 'decisive coalitions':
D F = { A C_12 I Fc(S) C_ i ~
ci(S)
for all c E c¢ and S E ~ )
and define the mapping h on ~ by h(F)= D r.
Theorem 1. Assume that Ixl/> 3 and ~ c_ 2x\{~} such that Isl 3 for at least one S E ~. Then: (a) h(o%) C g2. (b) If, additionally, IsI < for all S ~ ~g, then h sudectively maps ~ onto ~. The following auxiliary lemmas will be helpful for our proof. Fix an s.a.r. F and denote for each S E ~ :
D(S) = {A C_12 I Fc(S) c_ U
iEA
ci(S), for all c ~ c~}.
Every coalition of D(S) can block any alternative of S if all its members reject that alternative. Obviously, D e = n s s ~ ~ ( s ) . Note also that D(S) = 2a\{l~} if Isl : 1 L e m m a 1. If IS I I>2 and F satisfies (A1), then the following statements are
equivalent: (i) A ~ D(S). (ii) c E c~, x E S, x ~ UieA ci(S), x E ni~a ci(S) ~ x , ~ F c ( S ) . (iii) There exist c E c~ and x E S such that x , ~ U~A c~(S), x E ni~a c~(S) and
Proof. The implication ( i ) f f (ii) is trivial and the equivalence of (ii) and (iii) immediately follows from (A~). To show that ( i i ) ~ (i), let A satisfy (ii) and show that A E D(S). To the contrary, assume that there exist y E S and c E ~ such that y EF~(S) and Y,~ UieA ci(S). Choose c' E ~ such that y ~ Ui~_Aeli(S) and y E f-]ie.~ c'i(S). Then y,~F~,(S) and it follows by (A1) that y~[F~(S), a contradiction. []
Lemma 2. I f F E ~ a n a
S,S'G~,
IsI,
Is'l >~2, then D(S) = D(S').
Proof. Assume that A ~ D ( S ) and show that A ~ D ( S ' ) . To the contrary, if
A,q~D(S'), then it follows by Lemma 1 that there exist c ~ ~ and z ~ S' such that z ~ F~(S'), z~. UicA ci(S'), and z ~ ni~a ci(S'). Pick an x ~ S, x ~ z and con-
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struct c' E ~ such that x ~ UiEA c~(S), x E ('~iE,~ cti(S) , z E ( i E a cti(at), and Zfi~ U/E~ cti(S'). Since A E @(S) we have x ~ F c , ( S ). Then it follows by (A2) that zEF~,(S'). Now, it results from Proposition 2 that z , ~ F c(S'), a contradiction. [] L e m m a 3. If F ~
and
Is1~>3, then
~(S) E~.
Proof. O E @(S) by Proposition 1. If A E ~ ( S ) and A £ B, then obviously B E ~ ( S ) . Let us prove that if A ~ I~ and A , ~ ( S ) , then A E ~ ( S ) . To the contrary, suppose that A, .4 ,~ ~ ( S ) for some A ¢ / 2 . Then there exist x, y E S and c, c' E such that xfi~ Ui~A ci(S), x ~ niE ~ ci(S ) and x E Fc(S ), and y , ~ Uie~. c'i(S), y E Mic a cti(S) and y E Fc,(S), contradicting (A2). Pick now A, B E 9 ( S ) and show that A M B E 9 ( S ) . Since IsI >I 3, it is always possible to consider a non-trivial partition {$1, $2, $3} of S. Construct c such that:
ci(S ) = S 1,
for a l l i ~ A f q B ,
ci(S ) = S 2,
for a l l i E A k B ,
ci(S ) = S 3 , for all i E BLA, ci(S ) = S 2 U S 3,
for a l l i E A f q / ~ .
T h e n since A ~ 9 ( S ) it follows that Fc(S ) C_S 1 U S 2 and since B E 9 ( S ) it follows that F¢(S) C S 1 U S 3. Consequently, F~(S) C S 1 = UiEAA B ci(S ). Particularly, A n B # ~. It follows from the above that either A fq B ~ 9 ( S ) or A fq B E ~ ( S ) . O f course, the second alternative is impossible since F~(S) and UiE~-~ ci(S ) are disjoint. Therefore A M B E 9 ( S ) . [] Proof of Theorem I. (a) As was already noted, ~ ( S ) = 2a\{~} if ISI = 1. By the assumptions of the theorem, there exists S Owith Is01 I> 3. By L e m m a 2 it follows that 9F = 9(S0) and then, by L e m m a 3, it results that 9F E I}. (b) Assume ~ is an ultrafilter on O and define an s.a.r. F by its values on q¢, F~ = F(c), where, for each S E ~ :
Fc(S ) = { x E S I {i I x ~ ci(S)} ~ ~ } . We show first that F¢(S)~ ~ for all S ~ ~. To the contrary, assume that for each x ~ S there exists Ax E ~ such that x , ~ U;e,a ~ ci(S ) and x E n~ea~ ci(S ). Set A = f ' l ~ s A~. Since S is finite, then A ~ ~, particularly A # ~. Therefore, there exists i such that i ~ A~ for all x ~ S, i.e. x,~ci(S) for all x ~ S. Hence, c~(S) = ~), contradicting the definition of c. Let us show now that F satisfies (A~). Pick S ~ ~, c, c' U ~¢ and x, y ~ S such
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that a = {i I x E ci(S)} C_ {i I Y E c'i(S)} = B and assume that x E F~(S). Then A E and since ~ is a filter it follows that B E ~. Hence, y E Fc,(S ). Finally, we verify (A2). Consider S, S' E ~, c E ~ and x E S, y E S' such that { i l x E c i ( S ) } = { i l y , ~ c ~ ( S ' ) } . Denote by A the coalition defined by the previous equality. We have x E Fc(S ) <:~A E ~ <:>A ~ ~ <:>{ i I Y E ci(S')},~ ~ <:>y~F~(S'). Hence, (A2) is also satisfied by F. All that we have to do now is to show that ~ = ~V, i.e.
~= [a~121Fc(S)C_ I.
k_J ci(S) for all cECg and S E m i , .
iEA
J
Pick an A E ~ and suppose that there exist c E ~g and S E ~ such that Fc(S)k~_ UiE a ci(S ). Thus there exists x E F~(S) such that A C {i I x ~ c i ( S ) } . Since ~ is an ultrafilter it follows that {i [ x,~ci(S)} ~ ~ or, equivalently, {i I x E c i ( S ) } ~ ~. H e n c e , x ~ F ~ ( S ) , a contradiction. Therefore ~ C_~F" Conversely, let A belong to ~F" Then, for any fixed S E ~ it results that A E ~ ( S ) . It follows from lemma 1 that there exist x ~ S and c E cg such that A = { i l x ~ c i ( S ) } and x~F~(S). Hence A E ~, since otherwise A = { i l x ci(S)} E ~ and, by the definition of F it follows that x E F~(S). Consequently, D r C ~ and therefore, the identity ~ = ~F has been proved. [] The impossibility results can now be derived from our main theorem. The dictatorship property will be emphasized in the following by the next two conditions:
(A)4 There exists i E O such that F~(S) C_ci(S ) for all c E cg and S E ~. (A)5 There exists i E 12 such that Fc(S ) = ci(S ) for all c E cg and S E ~. Although the sense of the property (A4) seems to be very close to the classical Arrowian definition of the dictatorship, a voter i satisfying this property has been called a 'rejecting voter' in Popov and El'kin (1990). Following Aizerman and Aleskerov (1982) a voter satisfying (As) is said to be a dictator, but note that this stronger form of dictatorship property, when the social choice is identified with the personal choice of the dictator, is not standard in the classical framework.
Corollary 1. Suppose
lal<~
and ISl>~3 for a least one S ESE. Then ( m l ) a n d
(A2) imply (As). Proof. The existence of a dictator i in the sense of (A4) is a consequence of T h e o r e m 1. Since lal < ~ every ultrafilter contains one (and only one) singleton coalition. Particularly, if F E ~ , then {i} E ~p for some i C g2. Obviously, i is a dictator. We show now that (As) is also verified by i. T o the contrary, assume that
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there exists an x E ci(S ) such that x ~ F ~ ( S ) for some c E q¢ and S E ~. Construct c ' E q¢ such that { i l x ~ c i ( S ) } = {i[iy~.c'i(S)}. By Proposition 2, it follows that x E F~,(S). However, x ~ c ~ ( S ) , contradicting (A4). [] Corollary 2. Assume lal = oo and ~ consist of finite subsets such that IsI 3 for at least one S E ~. Then there exist s.a.r, without rejecting voters satisfying (A1) and (A2). Proof. Following Kirman and Sondermann (1972, Corollary 4 of Theorem 1), consider the collection of coalitions ~ = {A C_~ [ [.4[ < co}. Obviously, ~ is a filter on O, and then there is an ultrafilter ~ ' E ~ finer than ~. By Theorem l(b) there exists an s.a.r. F such that ~e = ~'. There is no one-element decisive coalition. Otherwise, if {i} E @~ = ~ ' for some i, then {i-} = I2\{i} should belong to ~ C ~', contradicting the properties of the ultrafilter. [] Corollary 3. Under all assumptions of Corollary 2 let (O, ~f, A) be any atomless measure space on O. Then for every F E ~; and for any ~ > 0 there exists A E and A ¢ l~ such that A(A) < E and A E ~ . Proof. We closely follow the proof of Proposition 5 in Kirman and Sondermann (1972). By a standard result in probability theory, for any e > 0 there exists a finite partition M - - { A 1 , . . . , A k } of ~ such that A i ~ i~, A,. ~ ~ and h ( A i ) ~ e for all i = 1 , . . , k (see, for instance, Renyi, 1970, ex. 49, II-12). Since ~ f is an ultrafilter, then at least one A~ belongs to ~F since otherwise, -4i E ~F for all i, n~kl .4~ = ~, contradicting the definition of the ultrafilter. (In fact this A i is unique in M since the elements of M are mutually disjoint.) Thus, the conclusion of the corollary holds for A = Ag.
4. Related conditions and results The assumptions employed in our results are all of consistency (normative) type. Such properties were introduced in Aizerman and Aleskerov (1986) and in subsequent papers by Aleskerov and Duggan (1993), and Popov and El'kin (1990) respectively. For the sake of completeness, let us recall some of these and define them in the present framework. As we have already mentioned in Section 2, in the original Aizerman-Aleskerov version the choice functions are allowed to have empty values, while our choice correspondences are all non-empty valued (our s.a.r, can be identified with the restrictions to the set of all non-empty valued choice profiles of the non-empty voting operators of Popov and El'kin). Hence, most voter operators introduced in the above cited papers are not well-defined in the present context.
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Having in mind this remark, the main normative properties of the voting operators can be defined in terms of an s.a.r, as follows: (NI) Non-imposedness (sovereignty): For every S E ~ and x E S there exist c, c' E c¢ such that x E Fc(S), x,~F~,(S). (L) Locality: If S E ~, x E S, c, c' E q¢ satisfy {i Ix E ci(S)} ~- {i I x E c'i(S)}, then x E F~(S)<=~x E F~,(S). (M) Monotonicity: If S E ~, x E S, c, c' E qg satisfy {i Ix E ci(S)} C {i I x E c'i(S)}, then x E Fc(S ) ~ x E F~,(S).
(IC) Independence of context: If S, S' E ~, x E S fq S', c, c' E ~ satisfy {i [ x E ci(S)} = {i Ix E c'i(S')}, then x E Fc(S)C:>x E F~,(S'). (IV) Independence of variants: If S E ~, x, y E S, c, c' E c¢ satisfy {i I x E c i ( S ) } = {i I Y E c'i(S)}, then x E Fc(S)C=>y E Fc,(S). (N) Neutrality to variants: If S, S'E~g, x E S , y E S ' , c, c ' E ~ ci(S) } = {i [y E c'i(S') }, then x E F~(S)<~ y E F~,(S').
satisfy {i [ x E
(P) Partiality: F~(S)~ S whenever c E ~ is such that ci(S ) ~ S for all i E/2. Obviously, (N) implies both (IC) and (IV), but the converse is not always true, as will be shown in the following example.
Example 4. Take ;~ = { S 1 , 5 2 } , where 51 ['~ 52 = • and F~(SI) = cl(S~), Fc(S2) = c2($2) for all c E ~.
Is, I ~
2, Is2l/> 2. Define
Clearly, F satisfies (IC) and (IV) trivially. If Xk, Yk E S k for k = 1, 2, then choose c such that cl(Sk) ----{Xk}, cz(Sk)---- {Yk} for k = 1, 2. Then {i Ix1 E ci(S1) } = {i I x2 E ci(S2) ) but x 1 E Fc(S1) while XE~ Fc(S2). [] However, if ~ is closed under the finite union (i.e. $1, S 2 E ~ ~ S 1 U S 2 E ~ ) , then we can easily verify that (IC) and (IV) altogether imply (N). The discussion in Popov and El'kin (1990) focuses on 'partialness' which plays an important role in their impossibility results. Theorem 3 of their paper gives a full characterization of the local non-empty and partial operators, while its first two corollaries establish some relationships between the partialness and other normative properties listed above. In the framework of the present paper the cited corollaries say that if 1121< ~, IXI < ~, then any s.a.r. F satisfying (NI), (L) and (P) also satisfies (IV) and (M) provided that each admissible set has at least three (resp. two) alternatives. Since the original proof was dependent on the finiteness assumptions, a direct proof in a more general framework will be done in
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the next proposition. (See also the proof of our Theorem 3 for further developments.)
Proposition 3. Let ~, X, ~ be arbitrary and F a non-imposed local and partial s.a.r. Then F satisfies (A3) and (M). Moreover, if, additionally, IS I >13 for all S E ~, then (IV) also holds. Proof. To prove the first assertion, assume that x E Fc(S ), x,~ U~caci(S) for some c, S and x E S. Obviously, this implies that IsI 1>2. Construct c' such that c'i(S) = S\{x} for all i. By (L) it follows that x •F¢,(S), and by (P) there exists y • S\{x} such that y • ci(S ) for every i • / 2 and y~F¢,(S). By (NI), for each z • S there exists c z E cg such that z~F~z(S). Define now ? • qg such that ?~(S) ~ {z • S I z E c~(S)} U {y}. Clearly, ~ is well-defined and by (L) we obtain F~(S)= f), which is impossible. To prove monotonicity, let us assume that there exist x E S • ~ and c, c' • ~¢ such that A C B , x • F ~ ( S ) but x~F~,(S), where, A = { i l x • c ~ ( S ) } and B = {ilx • c'~(S)). Obviously this situation cannot occur if IS[ = 1. If B = 12 it also contradicts (A3). Assume that B ~ / 2 and construct c I such that c~(S) = {x} iff i E A and c~(S) = S\{x} iff i • A . Since (L) implies x • Fcl(S ), then it follows by (P) that there exists y • S k { x } such that y •c~(S) for all i • , 4 and y ~ . F c l ( S ). Define c 2 such that c~(S) = {x} iff i • A, c~(S) = {y} iff i • / ~ and c2(S) = {x, y} otherwise. Then, by (L) and (A3) it results that F¢2(S)= f~, a contradiction. For the final statement of the proposition, let S be an admissible set with IsI I> 3 and assume that {i Ix ~ ci(S)} = {i I Y ~ c'i(S)) (denoted by D), for two different alternatives x, y ~ S and two choice profiles c, c'. To prove (IV) suppose, by way of contradiction, that x ~ Fc(S) but y)~ Fc,(S ). As a consequence of (m3), I ~ D C O . For each z E S\{y} pick c z • qg such that c~(S) = {y} if i E D and cZ(S) = {z} if i ~ / 5 . Then it follows by (A3) and (L) that Fc,(S ) = {z}. Define ? E q g by: ?i(S)= {x, y} if i ~ D and ~ ( S ) = S \ { x } if i ~ / 5 . Then it follows by (L) that S\{x}CF~(S) and by (L) and (A3) that xEF~(S) (as a consequence of (L) and (A3), Ale a ci(S)C_F~(S) for every c and S.) Thus, F~(S) = S contradicting (P). [] It is obvious that (A1) is equivalent with (M) and (IV). Also, (A1) and (A2) together imply (N), as immediately follows from Proposition 2. However, the converse of the latter implication is not true (any plurality rule of Example 1 is neutral.) Another connection between the main normative conditions of the present paper is pointed out in the next proposition.
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Proposition 4. Let 0 and X be arbitrary and assume that ISI >>-3 for every S E ~. Then an s.a.r. F satisfies (A1) and (A2) if and only if it satisfies, simultaneously, (L), (A2) and (A3). Proof. Since the 'only if' implication immediately follows from Proposition 1, let us prove the converse. By way of contradiction, suppose that for some S ~ ~, x, y E S and c, c' E c~ we have: x E Fc(S), y,~Fc,(S ) and A C B where, A --- {i I x E ci(S)}, B = {i I Y E c'i(S)}. Choose c" E c¢ such that c'~(S) = {x} if i U A and c'[(S) = {z}, if i E ,4, for some z E S \ { x , y } . By (L) it follows that xEFc,,(S) and then (Az) implies z , ~ F~,,(S). Define ? ~ ~ by ?i(S) = {y} if i ~ A, d~(S) = {y, z} if i ~ B \ A , and (~(S) = {z} if i ~ / ~ . Clearly, (L) and (A3) imply Fe(S ) = I~, which is impossible. [] As a consequence of Proposition 4, ff in Theorem 1 can be characterized by the properties (L), (A2) and (A3) if IsI >1 3 for all S E ~. Consequently, Theorem 1 could be restated and in the new context the Pareto principle (condition (A3)) is explicitly involved in the impossibility results. Condition (A:) plays a crucial role for our results. Theorem 1 fails in the balance of this condition. The plurality rule of Example 1 (except the case point out in Example 2) is non-dictatorial and satisfies (NI) (for 1 < k < IoI), (M), (N) and, consequently, (IC), (IV), (L) and (A1). However, this condition is not stronger than other normative properties involved in the impossibility theorems, as will be shown below. Two impossibility results without an explicit rationality condition have been obtained in Popov and El'kin (1990). The first is contained in the Corollary 3 of Theorem 3. In the present framework it claims that:
If lal < ~, Ixl <
and ISI 3 for at least one admissible set S E ~, then any non-imposed neutral and partial s.a.r, must satisfy (A 5). The second impossibility result which the authors refer to as an analog of Arrow's theorem is Corollary 4 of Theorem 3. Accordingly,
if lal <
3
Ixl <
and all non-empty subsets of X are admissible, then every s.a.r, satisfying (NI), (IC) and (P) also satisfies (As). We show now that by Theorem 1 these results can be extended to more general situations. The extensions will be done by the next two theorems and we make use of the following basic result.
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Proposition 5. For arbitrary O, X, and ~ any non-imposed neutral and partial s.a.r, satisfies (A1) and (A2). Proof. As was shown in Proposition 3, every non-imposed local and partial s.a.r. is monotonic too. Hence, (A1) follows. Now let us show that (A2) is also satisfied. We will proceed in two steps. In the first step it will be proved that Ifx, yES~,x~y
and cEC~ are such that {i I x ~ c i ( S ) } =
{ilyy~ci(S)},
then x E Fc(S ) ¢~ y , ~ Fc(S) . Denoting by A the coalition defined in the above equality, let us define c ' E c~ such that c ' i ( S ) = { x } for i ~ A and c ' i ( S ) = S \ { x } for i ~ , 4 . If xy~Fc(S ) and yyi~Fc(S), then (N) implies that Fc,(S ) = ~, which is impossible by definition. If x E F~(S) and y E Fc(S), then it follows by (N) that Fc,(S ) = S, contradicting (P). The second step concerns the general case in (A2). Let x ~ S and y ~ S' be such that x E ci(S ) iff i ~ A and y E ci(S') iff i E A , for some coalition A and for some cEC¢. Obviously, if S = {x}, then A = / ) and x ~ F~(S) by the definition of the choice correspondences. Then, if y ~ Fc(S'), the monotonicity and neutrality of F should imply F~(S')= S', contradicting (P). In the non-trivial case, consider ISI ~>2 and pick an z E S \ { x } . Define c ' ~ c~ such that x E c'~(S) iff i E A and z E c'i(S ) iff i E,4. As in the first step, it results that x~F~,(S)C=>z,~F~,(S) and by (N) it will follow that x E Fc(S)C~y,~F~(S'). [] As a corollary of the above proposition and of Theorem 1, we obtain.
Theorem 2. Let FI, X, and ~ be as in Theorem 1. I f ~;' denotes the family of all s.a.r, satisfying (NI), (N) and (P), then ~F E ~ for every F E 3~'. Particularly, for [/21
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145
T h e o r e m 3. Assume IX[ >I 3 and ~ are closed under a finite union and satisfy (*).
I f J;' denotes the family o f all s.a.r, satisfying (NI), (IC) and (P), then ~ r E 12, for every F E ~ ' . Proof. If follows from Proposition 5 that we must show that any F ~ o~' satisfies (IV). In this case neutrality is verified as a consequence of the definition of ~f. As it was stated in Proposition 3, non-imposedness, locality and partiality also imply the independence of variants if all admissible sets have at least three alternatives. In fact, since all conditions implied refer to a single admissible set, we can say that if ISI I> 3 and (NI), (L) and (P) are satisfied, then (IV) also holds. All we must do is to verify (IV) for any two-element set. Pick an S E ~ f , ISI = 2 , say S = {x, y}. Suppose that {i I x E c , ( S ) } = {i I Y E c'i(S)}, x E Fc(S), y ~ F c , ( S ) for some c, c' E ~. Choose S' E ~f such that {x, y} C s', Is'l >13 (it exists by our assumption of X and ~ ) . Define c " E c¢ such that c'[(S') = {x, y} i f f x E ci(S ) and c'[(S') = {z} for some z E S ' \ S otherwise. By (IC) we have x E Fc,,(S'), y,~Fc°(S' ) contradicting (IV) for admissible sets with at least three alternatives. [] The counterpart of Corollary 1 for this theorem extends Corollary 4 of Theorem 3 of Popov and El'kin (1990) to more general :~.
5. Restricted domain - rationality of choice profiles
As can be seen, in the previous sections explicit rationality conditions are not used. The absence of these conditions does not mean that we cannot restrict the domain of the s.a.r, to a subset of rationalizable choice profiles. In fact, most usual rationality conditions are compatible with the properties introduced in the present paper. We can easily verify that the proof of Theorem 1 works even if sufficient conditions for the rationalization of the individual choices are imposed. To exemplify this assertion let us consider two well-known rationality conditions, namely: (Chernoff's axiom): S, S' E ~, S C_S' f f S n C(S') c C(S). (Arrow's axiom): S, S' E ~f, S C_S' ~ S n C(S') ~ {0, C(S)}. All we must do is to check the compatibility of these axioms with the arguments used in the proofs. Obviously, if it sufficient to refer to those situations when two
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admissible sets of alternatives, in an inclusion relation, are simultaneously involved. Such situations appear only in the proofs of Proposition 2 and Lemma 2. Denote by ¢gl and qg2 the sets of all choice profiles satisfying Chernoff's and Arrow's axiom, respectively. In Proposition 2, denote A = (ilx~ci(S)). It is easy to verify that the conclusion holds if ~g is replaced by (~1 or by ¢g2 if the choice profile c" is defined as follows: ,, f {x}, ci(S)~s\{x},
ifiEA, if x E , 4 ,
and
fS'\{z}, c'~(S') = ] { x } , [S'\{x},
ifiEA andxES', if i E a and x , ~ S ' , ifiC,4,
respectively. Clearly, c" E cg2 C qgl and it also satisfies the condition required in the proof of Proposition 2. Similarly, the proof of Lemma 2 agrees with the restricted domain of F. Take, for instance, c' defined as follows:
fS\{x}, c~(S) = ~ { z } , I.{x},
ifiEA if i E Z if i ~ A
andxES', and x ~ . S ' , ,
and fS'X{x}, j{z}, c'i(S') = / { x } ,
I.S'\{z},
ifiCA andxES' if i E A and x , ~ S ' , if i ~ A and x E S ' , ifiEA
andx~S',
respectively, and easily verify that c' E ¢g2. [] Moreover, we can show that if an s.a.r, satisfies (A1) and ( A 2 ) , then the rationality properties of the choice profiles are transferred to the social choice.
Proposition 6. Let 12, X and ~g be arbitrary and F satisfies (A1) and (Az). Then: (a) Fc satisfies Chernoff's axiom for every c @ ~gl. (b) Fc satisfies Arrow's axiom for every c E ~2.
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Proof. Let S, S' E ~ such that S C S'. Obviously, if IS[ = 1, then the relation F~(S') n S E {0, Fc(S)} follows from the definition of an s.a.r. Therefore, let us assume that IS I ~>2. (a) Pick an c E (~1 and let x ~ Fc(S' ) n S. Denoting by A = {i I x ~ ci(S') } it will follow by Chernoff's axiom that A C_B, where B = {i I x E ci(S)}. Pick an c' E qgl such that c'i(S)= {y} iff i E / i and c'i(S)={x } iff i ~ A , for some y E S \ { x } . Then, by Proposition 2, it results that yg;Fc,(S ) and by (A2) it follows that x ~ Fc,(S ). Hence, x E Fc(S ) by (A1). (b) Suppose that c satisfies Arrow's axiom. Since the inclusion Fc(S')n S c F~(S) follows from (a), let us show that F~(S) C_Fc(S' ) n S if Fc(S' ) n S # O. Suppose, to the contrary, that there exists an x EF~(S) such that x~.Fc(S' ). Pick an y E S \ { x } (it exists, since F ~ ( S ' ) A S # O ) . Denote A = {i I x E c i ( S ) } , B = { i l y E ci(S')}. Two situations may occur: (k)A n B = 0 and (kk)A n B # 0 . In the case (k), define c ' E c¢ such that: c;(S')= {x} iff i E A ; c~(S')= {y} if i E B; c'i(S' ) = {x, y} otherwise. It follows from Proposition 2 that ygf. F~,(S') (since x E F~(S)), and x gf. Fc,(S') (since y E F~(S')). Then F~,(S') = 0 (Proposition 1), contradicting the non-emptiness assumption. In the case (kk), pick and z E S ' \ S and define c' E ~ such that: c'i(S' ) = {x} if i~A\a; c ; ( S ' ) = { y } if i ~ B X A ; c ; ( S ' ) = { z } if i E A A B and c ; ( S ' ) = { x , y } , otherwise. By proposition 2 and (A1) it follows that x~.Fc,(S' ), y ~ F c , ( S ' ). Obviously, A n B = { i l x ~ c'(S')}. Hence, z ~_ Fc, (S ' ) , otherwise it would follow by (A1) that x E F~(S). Again we have obtained F~,(S')= 0, a contradiction. [] Finally, let us note that the condition 'lsI ~ 3 for at least one S ~ ~ ' cannot be removed from the assumptions of our results.
Example 5. For the s.a.r, considered in example 1, take 1•1 : 3, [xI ~ 3, k = 2 and as the family of all two-element sets. Then (A1) and (A2) are simultaneously verified. Obviously, ~ * is not a filter because each two-element coalition is decisive. Consequently, there are no rejecting voters.
Acknowledgements I thank Michel Le Breton for the very useful discussions during his visit to the California Institute of Technology in 1994 and R.D. McKelvey for his excellent lectures in Social Choice Theory (McKelvey, 1993). I am also grateful to the referees for their helpful suggestions and remarks.
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References M.A. Aizerman and F.T. Aleskerov, Voting operators in the space of choice functions, Math. Soc. Sci. 11 (1986) 201-242. F. Aleskerov and J. Duggan, Functional voting operators: The non-monotonic case, Math. Soc. Sci. 26 (1993) 175-201. K.J. Arrow, Social Choice and Individual Values, 2nd edn. (Wiley, New York, 1963). V. Denicolo, Some further results on non-binary social choice, Soc. Choice Welfare 4 (1987) 277-285. D. Donaldson and J.A. Weymark, Social choice in economic environments, J. Econ. Theory 46 (1988) 291-308. P.C. Fishburn, Arrow's Impossibility Theorem: Concise proof and infinite voters, J. Econ. Theory 21 (1970) 103-106. A Gibbard, Intransitive social indifference and the Arrow dilemma (unpublished), 1969. D.M. Grether and C.R. Plott, Nonbinary social choice: An impossibility theorem, Rev. Econ. Stud. 49 (1982) 143-149. B. Hansson, Choice structures and preference relations, Synthese 18 (1968) 443-458. B. Hansson, Voting and group decision function, Synthese 20 (1969) 526-537. A.E Kirman, and D. Sondermann, Arrow's theorem, many agents and invisible dictator, J. Econ. Theory 5 (1972) 267-278. R.D. McKelvey, Social choice, Lecture Notes, California Institute of Technology, 1993. B.V. Popov and L.N. El'kin, Local functional operators of partial and nonempty choice of alternatives, Autom. Remote Control 7 (1990) 987-992 (translated from Avtomatika i Telemekhanika 7 (1989) 168-175). A. Renyi, Probability Theory (Akademiai Kiado, Budapest, 1970). A.K. Sen, Collective Choice and Social Welfare (Holden-Day, San Francisco, 1970).
Mathematical Social Sciences 33 (1997) 129-148
Impossibility results for choice correspondences Anton Stefanescu Bucharest University, Faculty of Mathematics, 14, Academiei st, Bucharest 70109, Romania Received April 1995; revised November 1995
Abstract One of the most important results in social choice theory is the Impossibility Theorem. Since 1951 when its first version was stated by K. Arrow, many other versions and extensions have been done. Although conditions differ, most results are concerned with relational aggregation rules which transform the individual preference relations into a social binary relation. The mixed case in which the social choice is defined in terms of choice functions while the individual opinions are represented by preference relations, also allows some interesting extensions of the original result. The study of the choice aggregation rules, when both individual and social choices are expressed in terms of choice functions, is well represented by the works of Aizerman and Aleskerov (Mathematical Social Sciences, 1986, 11, 201-242). Our approach concerns this latter framework. We introduce new normative conditions under which impossibility results are derived. These results are not dependent upon rationality conditions and extend the Impossibility Theorem to general sets of voters and alternatives. © 1997 Elsevier Science B.V. Keywords: Choice correspondence; Aggregation rule; Impossibility theorem
1. Introduction Choice processes are widely studied owing to their incidence in all fields of social life, and a systematic study based on mathematical modelling can be observed especially in the second half of this century. The most commonly used models for choice processes are choice correspondences (set-to-set mappings) and ordinal preferences (binary relations, also called preference relations). Considering a universal set of alternatives, a choice correspondence specifies for each given admissible set of alternatives a non-empty subset of this set - the result of the choice process. A n ordinal preference can, theoretically, be any 0165-4896/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PII S0165-4896(96)00829-3
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binary relation on the set of alternatives, but usually some specific properties like reflexivity, completeness or transitivity are required. The main problems studied in social choice theory are related to the aggregation of the individual choices in a society. If we represent a society as the set of individuals (voters) endowed with preferences over the set of alternatives, then it is commonly assumed that social choice is obtained from these preferences by an aggregation rule. Therefore, it is expected that the properties of the individual preferences will be reflected by the choice of the whole society, and it would be interesting to inquire if this choice can satisfy certain basic democratic and ethical principles. The celebrated Arrow's Impossibility theorem emphasizes the incompatibility of a set of fair properties if the social choice determined by an aggregation rule is subjected to some rationality conditions. Obviously, the social choice, as well as any individual one, can be defined in the same way, i.e. by a choice correspondence or by an ordinal preference relation, so that different types of aggregation rules could, theoretically, be considered. Arrow's theorem deals with the case in which the individuals are endowed with preference orderings on the set of alternatives. In one version (Arrow, 1963) this result stated that if society consists of finitely many individuals, then any aggregation rule satisfying the weak Pareto principle and the independence of irrelevant alternatives must be dictatorial if the social choice is guided by a weak order too. (Such an aggregation rule is called the 'social welfare function'.) The same conclusion holds if, equivalently, the social choice is expressed in terms of a choice function (choice correspondence in the present paper) defined on all subsets of alternatives and satisfying some 'rationality conditions' (Arrow's choice axiom.) Several extensions of Arrow's results have been sought in different ways. While in the original Impossibility Theorem the set of voters was assumed to be finite, Fishburn (1970) has shown that its conclusion fails if the number of voters is infinite. An elegant proof both of Arrow's and of Fishburn's results has been done by Kirman and Sondermann (1972). Moreover, they have pointed out that, even in the infinite case, a special form of dictatorship still exists. (The term invisible dictator has been proposed in this case.) Extensions of Arrow's results have been obtained by weakening the condition of collective rationality. In its social welfare function form, Arrow's theorem is also dependent on the social preference relation being transitive. It was shown (Sen, 1970) that the impossibility result does not hold if this is required to be only quasi-transitive. However, Gibbard (1969) showed that in this case society must admit an 'oligarchy', so that a special form of group dictatorship still exists. Another way Arrow's results have been extended is to restrict the collection of admissible sets when the social choice is defined in terms of choice functions. In the choice function version of Arrow's theorem all non-empty subsets of alternatives are assumed to be admissible and its usual proof relies on the
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assumption that every pair of alternatives is an admissible set. A variant of Arrow's theorem involving individual ordinal preferences and social choices correspondences is due to Grether and Plott (1982). They considered that the domain of the admissible sets consists of all subsets containing at least k alternatives, where k must be strictly less than the number of alternatives in the universal set. Donaldson and Weymark (1988) have proved an impossibility theorem in an economic environment. The domain of the admissible sets is restricted to the family of compact and comprehensive subsets of the positive orthant of a Euclidean space. However, the cited authors asked for the social choice correspondences to satisfy a strong rationality condition (Arrow's choice axiom.) Since in both cases the family of admissible sets is closed under the finite union, the social choice can be represented by an ordering (see Hansson, 1968). A particular case when the universal set is the only admissible set (the fixed agenda case) has been considered by Hansson (1969) and Denicolo (1987). They obtained impossibility results without rationality condition but by strengthening the usual independence condition, In an extensive study, Aizerman and Aleskerov (1986) reformulated the social choice problem only in terms of choice functions. In their terminology a 'voting operator' (or functional operator) transforms the individual opinions represented by choice functions into a collective choice function. They introduced normative properties analogous to Arrow's axioms in the classical framework and used them in conjunction with standard rationality properties to isolate voting operators that respond to each set of imposed conditions. Particularly, by isolating dictatorship operators we obtain some impossibility results. A more recent paper by Aleskerov and Duggan (1993) has extended these investigations for larger classes of such aggregation rules. Note that in the aggregation framework of Aizerman and Aleskerov, the sets of voters and alternatives are finite and all subsets of alternatives are admissible. However, the emptiness of the individual and social choices was allowed, which is rather unusual. (Consequently, the term 'choice correspondence' is not appropriate in this case.) In the same framework, Popov and El'kin (1990) ~ studied a class of voting operators characterized by the property called 'partialness' (see Section 4). They maintained the finiteness assumptions on the sets of voters and alternatives, but unlike Aizerman and Aleskerov the collection of admissible sets does not necessarily coincide with the family of all subsets of alternatives. (They have assumed only that each alternative is included in at least one admissible set.) Note also that the non-emptiness of the choice function was generally assumed. Within
1 W h e n the first version of the present paper was submitted, the author did not know about the works of Popov and El'kin. He is grateful to the referees for signalling and sending him a copy of their papers.
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this modified framework Popov and El'kin obtained impossibility results without explicit rationality conditions. Our study deals basically with the Aizerman-Aleskerov framework, but considers arbitrary sets of voters and alternatives and more general families of admissible sets of alternatives. The non-emptiness of the choice functions is always assumed (so that these are actually choice correspondences) and the rationality is not explicitly required, as in Popov and El'kin. Our impossibility results follow from two new normative conditions on social aggregation rules. The main result of the paper shows that the set of all decisive coalitions of voters is an ultrafilter if the social aggregation rule satisfies these conditions; and, conversely, if all admissible sets are finite, then every ultrafilter of coalitions can be characterized as the collection of decisive sets of a social aggregation rule. Section 2 formalizes the framework of our study and introduces the two normative properties called strong monotonicity and decisiveness-neutrality, respectively. The main results is proved in Section 3. Its corollaries extend, within the present framework, Arrow's theorem for finitely many voters, Fishburn's possibility result for infinitely many voters, as well as Kirman and Sondermann's result concerning 'invisible dictators'. Everywhere Section 3 the social aggregation rules are defined for all possible choice correspondences which characterize the individual opinions. Section 4 explains the two conditions introduced in Section 2 in the context of the normative properties of the Aizerman-Aleskerov framework. The most significant result shows that our conditions are implied by the normative conditions used in Popov and El'kin (1990). From here and from the main theorem we obtain impossibility results which extend those of Popov and El'kin. In the final section we point out that the results of Section 3 hold even if the individual choices are required to satisfy rationality conditions. Arrow's choice axiom and Chernoff's condition are used to argue these conclusions. It is also shown that the social choice determined by a social aggregation rule satisfying strong monotonicity and decisiveness-neutrality inherits the above rationality conditions from individual choices.
2. Choice correspondences and social aggregation rules Everywhere in this paper X will denote the non-empty set of alternatives and £~ a (finite or infinite) set whose elements are called voters or consumers. Each voter has his own opinion with respect to the set of alternatives and, as we have already mentioned, this opinion can be represented in several ways. We will focus on the case when it is expressed in terms of a choice correspondence defined below.
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Let us denote by ~ a non-void subfamily of ~0(X) = {S C_X I S ~ 0} the family of all admissible sets of alternatives. Definition 1. A choice correspondence on X is a mapping C: ~ 0 ( X ) that C(S) C S for all S E ~.
such
Denote by F ( ~ ) the set of all choice correspondences on X and by q~ the set of all choice profiles, i.e. = {c I c : a
A choice profile associates with each voter i E 12 its own choice correspondence c(i), conventionally denoted by c r The set qg contains all logically possible choice profiles. Some of them could seem to be rather irrational and their use by voters is not justified. In Section 5 we will discuss the possibility of restricting the set of choice profiles by imposing rationality conditions on the elements of F ( ~ ) . Beginning with the pioneering works in social choice theory, several notions with a similar meaning are used for a social aggregation rule according to the considered situations. Since our approach concerns only one type of choice process, we will use the generic notion of social aggregation rule. The formal definition follows. Definition 2. A social aggregation rule (s.a.r.) is any mapping F : ~ ~-~F(~). Here F(c) (also denoted by F~) represents the social choice correspondence associated with the choice profile c. Let ~ be the set of all s.a.r, satisfying the following conditions: (A1) VS E ~, x, y E S, c, c' E q¢ such that {i I x E
ci(S)} C {i I Y E c'i(S)}
and x E
Fc(S ) ~ y E F¢,(S). (A2) V c E ~ , S , S ' E ~ , xES, yES' then x E Fc(S)C:> y , ~ Fc(S' ).
such that { i l x E c i ( S ) } = { i l y , ~ c i ( S ' ) }
Condition (AI) could be called strong monotonicity. It states that if a group of voters (coalition) once imposes an alternative from a given set S, then any other larger coalition can impose on society any alternative from S if the opinions of its members are convergent. According to the second condition, (A2), any 'accepting coalition' with respect to a given profile (i.e. a coalition which can impose the alternatives agreed by all its members) is also a 'blocking coalition' (i.e. it can block any alternative which is
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rejected by all its members even if all other voters have a different opinion). It has been suggested that condition (A2) could be called decisiveness-neutrality. For an intuitive justification of this condition let us first consider the case of binary comparison. Let S be any two-element admissible set, say S = {x, y}, and assume that two complementary coalitions, A and B, have different opinions towards the alternatives x and y. A naive 'democratic' principle requires that society should choose that alternative which has obtained the largest number of votes. If, in addition, a special rule is adopted when tml = IBI to break ties, then we arrive at the conclusion required by (A2) for S' = S. Furthermore, if S = {x, z} and S' = {y, w} are two distinct admissible sets and x is chosen by all members of A while y is chosen only by all members of B, then it is obvious that either x and w have similar positions or w has a better situation than x and therefore, as in the above, y should be rejected by society if it chooses x. Of course, (A2) does not say exactly this, but it extends the naive ideas involved in the above to more general collections of admissible sets. In a formal framework we can show that both conditions (ml) and (A2) can be derived from other normative conditions. For instance, Proposition 5 of Section 4 proves that any 'partial' and 'neutral' s.a.r, must satisfy (ml) and (Az). Roughly speaking, partialness says that society cannot be totally indifferent if none of the individuals is. Specifically if all members of a coalition chose one alternative from a two-element admissible set while all other individuals choose the second one, then society must choose one and only one of these two alternatives, so that we arrive to the conclusion required by (A2). Neutrality requires that if a coalition imposes an alternative, then the votes of its members suffice for any alternative of any other admissible set to be chosen by society. In conjunction with neutrality, the partialness extends the conclusions emphasized in the particular situation of the two-element set-up to the full condition (A2). Moreover, condition (A~) (which concerns only one fixed admissible set) is also satisfied if the s.a.r, satisfies both neutrality and partialness. Note that (A1) is satisfied by some plurality-type rules, as is shown in the following example. For any c E ~ and x E S E ~ denote by vc(x, S) = 1(i ~ 12 Ix ~ ci(S)}l. Example 1. Plurality rule (see Aizerman and Aleskerov, 1986, for the k-plurality rule). Assume 12 is finite and let k be a positive integer k ~< 1121. Define the aggregation rule F p by F~ (S) = {x E S I Vc(x, S ) >l k } .
Since in the present framework non-emptiness is assumed for all choice correspondences, a necessary condition for consistency of the above definition is that
(k - 1)1SI < 1121 for all S E ~ .
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Obviously, this condition is satisfied for non-trivial O and ~ and then F p satisfies (A1). U n d e r some special assumptions, the plurality rule can also satisfy (A2). Example 2. For the plurality rule F p defined in Example 1, assume n = 1121 is odd and take k = (n + 1)/2. Also consider that ISl ~<2 for all S E ~ . In this case both (A1) and (A2) are satisfied. Moreover, we note that this is the only non-trivial case when the plurality rule, defined as in the previous example, satisfies (A2) (we must have 1~21~ < 2 k - 1 which, together with the necessary condition imposed in Example 1, is fulfilled only when ~f consists of one-element sets, or in the case of the present example). Otherwise, the plurality rule satisfies (A1) but not (A2). In the next example, condition (Az) will be satisfied by introducing a special tie-breaking rule. However, the s.a.r, may not satisfy (A1), so that it will be shown that the two conditions are mutually independent. Example 3. For I~1 = n, where n is even and for ~ consisting of subsets with at most two elements, define:
F~(S) = ~ Fp~(s)' Lcl(S) ,
if F~(S) ~ fJ, otherwise,
where F p is the s.a.r, of Example 1 for k = (n/2) + 1. Clearly, F satisfies (A2). Let S E ~ be a two-element admissible set, S = {x, y}. Define c, c'@ q~ such that ci(S ) = {x} and c'i(S ) = S, for i = 1 . . . . . n/2, and ci(S ) = c'i(S ) = {y}, otherwise. Then, {i Ix E ci(S) } = {i Ix E c'i(S) } but x ~ Fc(S ) while x ~_ Fc,(S) = {y}, so that F fails (A1). Now, let us note two interesting propositions. Proposition 1. I f F satisfies (A1) and (A2), then it also satisfies the Pareto
principle:
(A3) Fc(S ) C LJi~,c1 ci(S ) for all S E ~ and c E q¢. Proof. An immediate consequence of (A1) is that for every S and c then either F¢(S) = S or F~(S)C LJiea ci(S). All that we must do is to prove that the first situation is impossible if LJiea ci(S) ~ S. T o the contrary, pick an x E S \ LJiea ci(S) and an y ~ U~ea ci(S) and suppose that x E F~(S). Construct c ' E cg such that x~c'i(S) and y E c'~(S) for all i E g2. T h e n it follows by (A1) that x ~ F~,(S) and y E Fc,(S) contradicting (A2). []
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Remark. Condition (A3) is called negative unanimity in Aizerman and Aleskerov (1986). Under assumption (A1) it implies that ' n ~ E a q(s)c__ F~(S)c Uiea ci(S) for all c and S E ~ ' .
Proposition 2. If F satisfies
(A1) and
(A2), then it also satisfies
(A~) c,c' E ~ , S , S ' E ~ , x E S , y E S ' such that
(ilx e
ci(S)} =
{il yg~c;(S')} then x E fc(S)C~y,~Fc,(S' ).
Proof. Note first that, as a consequence of Proposition 1, if x E Fc(S), then the set c;(S)} must be non-empty. Hence, the non-trivial situation is when both S and S' have at least two elements. Pick an z E S'\{x} and construct c"E ~¢ such that
A = {ilx E
{i Ix E c'~(S)} = {i I x E ci(S)} , {il z e c"(S')} = {i I y E c'~(S')}. Then it follows by (A1) that xEF~(S)CC, xEFc.,(S ) and by (A2) that x E Fc,,(S)C=>z~.F~,,(S'). Again, (A1) implies that z~_Fc,,(S')C=>y~_Fc,(S' ). []
3. Impossibility results Since the proofs of the main results of this section are based on the theory of ultrafilters, let us first recall some fundamental notions and results coming from this theory. Let E be a non-empty set and ~ C 2 e, ~ ~ l~.
Definition 3. ~ is a filter on E if it satisfies the following conditions: (a) 0 ~ ; (b) A E $ , A C B ~ B E $ ; (c) A, B E $ ~ A O B E $ . Obviously, the intersection of an arbitrary collection of filters is again a filter. If, for two filters $1 and $2 we have $1C $2 (i.e. A E $1 ~ A E $2), then ~2 is said to be finer than $1. Since the filters on E form.a partially ordered set under C, then the following definition makes sense:
Definition 4. Any maximal filter on E is called an ultrafilter. The existence of ultrafilters follows from Zorn's lemma since for any totally ordered family of filters the least upper bound exists (the union of all filters of the
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family). Moreover, it follows that for any filter ~ there exists an ultrafilter ~ ' finer than ~. Finally, note that a filter ~ is an ultrafilter if and only if for every subset A C_E either A or its complementary .,~ belongs to ~. Denote now by 12 the set of all ultrafilters on O. Let us associate with each F E ~ the family D e of 'decisive coalitions':
D F = { A C_12 I Fc(S) C_ i ~
ci(S)
for all c E c¢ and S E ~ )
and define the mapping h on ~ by h(F)= D r.
Theorem 1. Assume that Ixl/> 3 and ~ c_ 2x\{~} such that Isl 3 for at least one S E ~. Then: (a) h(o%) C g2. (b) If, additionally, IsI < for all S ~ ~g, then h sudectively maps ~ onto ~. The following auxiliary lemmas will be helpful for our proof. Fix an s.a.r. F and denote for each S E ~ :
D(S) = {A C_12 I Fc(S) c_ U
iEA
ci(S), for all c ~ c~}.
Every coalition of D(S) can block any alternative of S if all its members reject that alternative. Obviously, D e = n s s ~ ~ ( s ) . Note also that D(S) = 2a\{l~} if Isl : 1 L e m m a 1. If IS I I>2 and F satisfies (A1), then the following statements are
equivalent: (i) A ~ D(S). (ii) c E c~, x E S, x ~ UieA ci(S), x E ni~a ci(S) ~ x , ~ F c ( S ) . (iii) There exist c E c~ and x E S such that x , ~ U~A c~(S), x E ni~a c~(S) and
Proof. The implication ( i ) f f (ii) is trivial and the equivalence of (ii) and (iii) immediately follows from (A~). To show that ( i i ) ~ (i), let A satisfy (ii) and show that A E D(S). To the contrary, assume that there exist y E S and c E ~ such that y EF~(S) and Y,~ UieA ci(S). Choose c' E ~ such that y ~ Ui~_Aeli(S) and y E f-]ie.~ c'i(S). Then y,~F~,(S) and it follows by (A1) that y~[F~(S), a contradiction. []
Lemma 2. I f F E ~ a n a
S,S'G~,
IsI,
Is'l >~2, then D(S) = D(S').
Proof. Assume that A ~ D ( S ) and show that A ~ D ( S ' ) . To the contrary, if
A,q~D(S'), then it follows by Lemma 1 that there exist c ~ ~ and z ~ S' such that z ~ F~(S'), z~. UicA ci(S'), and z ~ ni~a ci(S'). Pick an x ~ S, x ~ z and con-
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struct c' E ~ such that x ~ UiEA c~(S), x E ('~iE,~ cti(S) , z E ( i E a cti(at), and Zfi~ U/E~ cti(S'). Since A E @(S) we have x ~ F c , ( S ). Then it follows by (A2) that zEF~,(S'). Now, it results from Proposition 2 that z , ~ F c(S'), a contradiction. [] L e m m a 3. If F ~
and
Is1~>3, then
~(S) E~.
Proof. O E @(S) by Proposition 1. If A E ~ ( S ) and A £ B, then obviously B E ~ ( S ) . Let us prove that if A ~ I~ and A , ~ ( S ) , then A E ~ ( S ) . To the contrary, suppose that A, .4 ,~ ~ ( S ) for some A ¢ / 2 . Then there exist x, y E S and c, c' E such that xfi~ Ui~A ci(S), x ~ niE ~ ci(S ) and x E Fc(S ), and y , ~ Uie~. c'i(S), y E Mic a cti(S) and y E Fc,(S), contradicting (A2). Pick now A, B E 9 ( S ) and show that A M B E 9 ( S ) . Since IsI >I 3, it is always possible to consider a non-trivial partition {$1, $2, $3} of S. Construct c such that:
ci(S ) = S 1,
for a l l i ~ A f q B ,
ci(S ) = S 2,
for a l l i E A k B ,
ci(S ) = S 3 , for all i E BLA, ci(S ) = S 2 U S 3,
for a l l i E A f q / ~ .
T h e n since A ~ 9 ( S ) it follows that Fc(S ) C_S 1 U S 2 and since B E 9 ( S ) it follows that F¢(S) C S 1 U S 3. Consequently, F~(S) C S 1 = UiEAA B ci(S ). Particularly, A n B # ~. It follows from the above that either A fq B ~ 9 ( S ) or A fq B E ~ ( S ) . O f course, the second alternative is impossible since F~(S) and UiE~-~ ci(S ) are disjoint. Therefore A M B E 9 ( S ) . [] Proof of Theorem I. (a) As was already noted, ~ ( S ) = 2a\{~} if ISI = 1. By the assumptions of the theorem, there exists S Owith Is01 I> 3. By L e m m a 2 it follows that 9F = 9(S0) and then, by L e m m a 3, it results that 9F E I}. (b) Assume ~ is an ultrafilter on O and define an s.a.r. F by its values on q¢, F~ = F(c), where, for each S E ~ :
Fc(S ) = { x E S I {i I x ~ ci(S)} ~ ~ } . We show first that F¢(S)~ ~ for all S ~ ~. To the contrary, assume that for each x ~ S there exists Ax E ~ such that x , ~ U;e,a ~ ci(S ) and x E n~ea~ ci(S ). Set A = f ' l ~ s A~. Since S is finite, then A ~ ~, particularly A # ~. Therefore, there exists i such that i ~ A~ for all x ~ S, i.e. x,~ci(S) for all x ~ S. Hence, c~(S) = ~), contradicting the definition of c. Let us show now that F satisfies (A~). Pick S ~ ~, c, c' U ~¢ and x, y ~ S such
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that a = {i I x E ci(S)} C_ {i I Y E c'i(S)} = B and assume that x E F~(S). Then A E and since ~ is a filter it follows that B E ~. Hence, y E Fc,(S ). Finally, we verify (A2). Consider S, S' E ~, c E ~ and x E S, y E S' such that { i l x E c i ( S ) } = { i l y , ~ c ~ ( S ' ) } . Denote by A the coalition defined by the previous equality. We have x E Fc(S ) <:~A E ~ <:>A ~ ~ <:>{ i I Y E ci(S')},~ ~ <:>y~F~(S'). Hence, (A2) is also satisfied by F. All that we have to do now is to show that ~ = ~V, i.e.
~= [a~121Fc(S)C_ I.
k_J ci(S) for all cECg and S E m i , .
iEA
J
Pick an A E ~ and suppose that there exist c E ~g and S E ~ such that Fc(S)k~_ UiE a ci(S ). Thus there exists x E F~(S) such that A C {i I x ~ c i ( S ) } . Since ~ is an ultrafilter it follows that {i [ x,~ci(S)} ~ ~ or, equivalently, {i I x E c i ( S ) } ~ ~. H e n c e , x ~ F ~ ( S ) , a contradiction. Therefore ~ C_~F" Conversely, let A belong to ~F" Then, for any fixed S E ~ it results that A E ~ ( S ) . It follows from lemma 1 that there exist x ~ S and c E cg such that A = { i l x ~ c i ( S ) } and x~F~(S). Hence A E ~, since otherwise A = { i l x ci(S)} E ~ and, by the definition of F it follows that x E F~(S). Consequently, D r C ~ and therefore, the identity ~ = ~F has been proved. [] The impossibility results can now be derived from our main theorem. The dictatorship property will be emphasized in the following by the next two conditions:
(A)4 There exists i E O such that F~(S) C_ci(S ) for all c E cg and S E ~. (A)5 There exists i E 12 such that Fc(S ) = ci(S ) for all c E cg and S E ~. Although the sense of the property (A4) seems to be very close to the classical Arrowian definition of the dictatorship, a voter i satisfying this property has been called a 'rejecting voter' in Popov and El'kin (1990). Following Aizerman and Aleskerov (1982) a voter satisfying (As) is said to be a dictator, but note that this stronger form of dictatorship property, when the social choice is identified with the personal choice of the dictator, is not standard in the classical framework.
Corollary 1. Suppose
lal<~
and ISl>~3 for a least one S ESE. Then ( m l ) a n d
(A2) imply (As). Proof. The existence of a dictator i in the sense of (A4) is a consequence of T h e o r e m 1. Since lal < ~ every ultrafilter contains one (and only one) singleton coalition. Particularly, if F E ~ , then {i} E ~p for some i C g2. Obviously, i is a dictator. We show now that (As) is also verified by i. T o the contrary, assume that
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there exists an x E ci(S ) such that x ~ F ~ ( S ) for some c E q¢ and S E ~. Construct c ' E q¢ such that { i l x ~ c i ( S ) } = {i[iy~.c'i(S)}. By Proposition 2, it follows that x E F~,(S). However, x ~ c ~ ( S ) , contradicting (A4). [] Corollary 2. Assume lal = oo and ~ consist of finite subsets such that IsI 3 for at least one S E ~. Then there exist s.a.r, without rejecting voters satisfying (A1) and (A2). Proof. Following Kirman and Sondermann (1972, Corollary 4 of Theorem 1), consider the collection of coalitions ~ = {A C_~ [ [.4[ < co}. Obviously, ~ is a filter on O, and then there is an ultrafilter ~ ' E ~ finer than ~. By Theorem l(b) there exists an s.a.r. F such that ~e = ~'. There is no one-element decisive coalition. Otherwise, if {i} E @~ = ~ ' for some i, then {i-} = I2\{i} should belong to ~ C ~', contradicting the properties of the ultrafilter. [] Corollary 3. Under all assumptions of Corollary 2 let (O, ~f, A) be any atomless measure space on O. Then for every F E ~; and for any ~ > 0 there exists A E and A ¢ l~ such that A(A) < E and A E ~ . Proof. We closely follow the proof of Proposition 5 in Kirman and Sondermann (1972). By a standard result in probability theory, for any e > 0 there exists a finite partition M - - { A 1 , . . . , A k } of ~ such that A i ~ i~, A,. ~ ~ and h ( A i ) ~ e for all i = 1 , . . , k (see, for instance, Renyi, 1970, ex. 49, II-12). Since ~ f is an ultrafilter, then at least one A~ belongs to ~F since otherwise, -4i E ~F for all i, n~kl .4~ = ~, contradicting the definition of the ultrafilter. (In fact this A i is unique in M since the elements of M are mutually disjoint.) Thus, the conclusion of the corollary holds for A = Ag.
4. Related conditions and results The assumptions employed in our results are all of consistency (normative) type. Such properties were introduced in Aizerman and Aleskerov (1986) and in subsequent papers by Aleskerov and Duggan (1993), and Popov and El'kin (1990) respectively. For the sake of completeness, let us recall some of these and define them in the present framework. As we have already mentioned in Section 2, in the original Aizerman-Aleskerov version the choice functions are allowed to have empty values, while our choice correspondences are all non-empty valued (our s.a.r, can be identified with the restrictions to the set of all non-empty valued choice profiles of the non-empty voting operators of Popov and El'kin). Hence, most voter operators introduced in the above cited papers are not well-defined in the present context.
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Having in mind this remark, the main normative properties of the voting operators can be defined in terms of an s.a.r, as follows: (NI) Non-imposedness (sovereignty): For every S E ~ and x E S there exist c, c' E c¢ such that x E Fc(S), x,~F~,(S). (L) Locality: If S E ~, x E S, c, c' E q¢ satisfy {i Ix E ci(S)} ~- {i I x E c'i(S)}, then x E F~(S)<=~x E F~,(S). (M) Monotonicity: If S E ~, x E S, c, c' E qg satisfy {i Ix E ci(S)} C {i I x E c'i(S)}, then x E Fc(S ) ~ x E F~,(S).
(IC) Independence of context: If S, S' E ~, x E S fq S', c, c' E ~ satisfy {i [ x E ci(S)} = {i Ix E c'i(S')}, then x E Fc(S)C:>x E F~,(S'). (IV) Independence of variants: If S E ~, x, y E S, c, c' E c¢ satisfy {i I x E c i ( S ) } = {i I Y E c'i(S)}, then x E Fc(S)C=>y E Fc,(S). (N) Neutrality to variants: If S, S'E~g, x E S , y E S ' , c, c ' E ~ ci(S) } = {i [y E c'i(S') }, then x E F~(S)<~ y E F~,(S').
satisfy {i [ x E
(P) Partiality: F~(S)~ S whenever c E ~ is such that ci(S ) ~ S for all i E/2. Obviously, (N) implies both (IC) and (IV), but the converse is not always true, as will be shown in the following example.
Example 4. Take ;~ = { S 1 , 5 2 } , where 51 ['~ 52 = • and F~(SI) = cl(S~), Fc(S2) = c2($2) for all c E ~.
Is, I ~
2, Is2l/> 2. Define
Clearly, F satisfies (IC) and (IV) trivially. If Xk, Yk E S k for k = 1, 2, then choose c such that cl(Sk) ----{Xk}, cz(Sk)---- {Yk} for k = 1, 2. Then {i Ix1 E ci(S1) } = {i I x2 E ci(S2) ) but x 1 E Fc(S1) while XE~ Fc(S2). [] However, if ~ is closed under the finite union (i.e. $1, S 2 E ~ ~ S 1 U S 2 E ~ ) , then we can easily verify that (IC) and (IV) altogether imply (N). The discussion in Popov and El'kin (1990) focuses on 'partialness' which plays an important role in their impossibility results. Theorem 3 of their paper gives a full characterization of the local non-empty and partial operators, while its first two corollaries establish some relationships between the partialness and other normative properties listed above. In the framework of the present paper the cited corollaries say that if 1121< ~, IXI < ~, then any s.a.r. F satisfying (NI), (L) and (P) also satisfies (IV) and (M) provided that each admissible set has at least three (resp. two) alternatives. Since the original proof was dependent on the finiteness assumptions, a direct proof in a more general framework will be done in
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the next proposition. (See also the proof of our Theorem 3 for further developments.)
Proposition 3. Let ~, X, ~ be arbitrary and F a non-imposed local and partial s.a.r. Then F satisfies (A3) and (M). Moreover, if, additionally, IS I >13 for all S E ~, then (IV) also holds. Proof. To prove the first assertion, assume that x E Fc(S ), x,~ U~caci(S) for some c, S and x E S. Obviously, this implies that IsI 1>2. Construct c' such that c'i(S) = S\{x} for all i. By (L) it follows that x •F¢,(S), and by (P) there exists y • S\{x} such that y • ci(S ) for every i • / 2 and y~F¢,(S). By (NI), for each z • S there exists c z E cg such that z~F~z(S). Define now ? • qg such that ?~(S) ~ {z • S I z E c~(S)} U {y}. Clearly, ~ is well-defined and by (L) we obtain F~(S)= f), which is impossible. To prove monotonicity, let us assume that there exist x E S • ~ and c, c' • ~¢ such that A C B , x • F ~ ( S ) but x~F~,(S), where, A = { i l x • c ~ ( S ) } and B = {ilx • c'~(S)). Obviously this situation cannot occur if IS[ = 1. If B = 12 it also contradicts (A3). Assume that B ~ / 2 and construct c I such that c~(S) = {x} iff i E A and c~(S) = S\{x} iff i • A . Since (L) implies x • Fcl(S ), then it follows by (P) that there exists y • S k { x } such that y •c~(S) for all i • , 4 and y ~ . F c l ( S ). Define c 2 such that c~(S) = {x} iff i • A, c~(S) = {y} iff i • / ~ and c2(S) = {x, y} otherwise. Then, by (L) and (A3) it results that F¢2(S)= f~, a contradiction. For the final statement of the proposition, let S be an admissible set with IsI I> 3 and assume that {i Ix ~ ci(S)} = {i I Y ~ c'i(S)) (denoted by D), for two different alternatives x, y ~ S and two choice profiles c, c'. To prove (IV) suppose, by way of contradiction, that x ~ Fc(S) but y)~ Fc,(S ). As a consequence of (m3), I ~ D C O . For each z E S\{y} pick c z • qg such that c~(S) = {y} if i E D and cZ(S) = {z} if i ~ / 5 . Then it follows by (A3) and (L) that Fc,(S ) = {z}. Define ? E q g by: ?i(S)= {x, y} if i ~ D and ~ ( S ) = S \ { x } if i ~ / 5 . Then it follows by (L) that S\{x}CF~(S) and by (L) and (A3) that xEF~(S) (as a consequence of (L) and (A3), Ale a ci(S)C_F~(S) for every c and S.) Thus, F~(S) = S contradicting (P). [] It is obvious that (A1) is equivalent with (M) and (IV). Also, (A1) and (A2) together imply (N), as immediately follows from Proposition 2. However, the converse of the latter implication is not true (any plurality rule of Example 1 is neutral.) Another connection between the main normative conditions of the present paper is pointed out in the next proposition.
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Proposition 4. Let 0 and X be arbitrary and assume that ISI >>-3 for every S E ~. Then an s.a.r. F satisfies (A1) and (A2) if and only if it satisfies, simultaneously, (L), (A2) and (A3). Proof. Since the 'only if' implication immediately follows from Proposition 1, let us prove the converse. By way of contradiction, suppose that for some S ~ ~, x, y E S and c, c' E c~ we have: x E Fc(S), y,~Fc,(S ) and A C B where, A --- {i I x E ci(S)}, B = {i I Y E c'i(S)}. Choose c" E c¢ such that c'~(S) = {x} if i U A and c'[(S) = {z}, if i E ,4, for some z E S \ { x , y } . By (L) it follows that xEFc,,(S) and then (Az) implies z , ~ F~,,(S). Define ? ~ ~ by ?i(S) = {y} if i ~ A, d~(S) = {y, z} if i ~ B \ A , and (~(S) = {z} if i ~ / ~ . Clearly, (L) and (A3) imply Fe(S ) = I~, which is impossible. [] As a consequence of Proposition 4, ff in Theorem 1 can be characterized by the properties (L), (A2) and (A3) if IsI >1 3 for all S E ~. Consequently, Theorem 1 could be restated and in the new context the Pareto principle (condition (A3)) is explicitly involved in the impossibility results. Condition (A:) plays a crucial role for our results. Theorem 1 fails in the balance of this condition. The plurality rule of Example 1 (except the case point out in Example 2) is non-dictatorial and satisfies (NI) (for 1 < k < IoI), (M), (N) and, consequently, (IC), (IV), (L) and (A1). However, this condition is not stronger than other normative properties involved in the impossibility theorems, as will be shown below. Two impossibility results without an explicit rationality condition have been obtained in Popov and El'kin (1990). The first is contained in the Corollary 3 of Theorem 3. In the present framework it claims that:
If lal < ~, Ixl <
and ISI 3 for at least one admissible set S E ~, then any non-imposed neutral and partial s.a.r, must satisfy (A 5). The second impossibility result which the authors refer to as an analog of Arrow's theorem is Corollary 4 of Theorem 3. Accordingly,
if lal <
3
Ixl <
and all non-empty subsets of X are admissible, then every s.a.r, satisfying (NI), (IC) and (P) also satisfies (As). We show now that by Theorem 1 these results can be extended to more general situations. The extensions will be done by the next two theorems and we make use of the following basic result.
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Proposition 5. For arbitrary O, X, and ~ any non-imposed neutral and partial s.a.r, satisfies (A1) and (A2). Proof. As was shown in Proposition 3, every non-imposed local and partial s.a.r. is monotonic too. Hence, (A1) follows. Now let us show that (A2) is also satisfied. We will proceed in two steps. In the first step it will be proved that Ifx, yES~,x~y
and cEC~ are such that {i I x ~ c i ( S ) } =
{ilyy~ci(S)},
then x E Fc(S ) ¢~ y , ~ Fc(S) . Denoting by A the coalition defined in the above equality, let us define c ' E c~ such that c ' i ( S ) = { x } for i ~ A and c ' i ( S ) = S \ { x } for i ~ , 4 . If xy~Fc(S ) and yyi~Fc(S), then (N) implies that Fc,(S ) = ~, which is impossible by definition. If x E F~(S) and y E Fc(S), then it follows by (N) that Fc,(S ) = S, contradicting (P). The second step concerns the general case in (A2). Let x ~ S and y ~ S' be such that x E ci(S ) iff i ~ A and y E ci(S') iff i E A , for some coalition A and for some cEC¢. Obviously, if S = {x}, then A = / ) and x ~ F~(S) by the definition of the choice correspondences. Then, if y ~ Fc(S'), the monotonicity and neutrality of F should imply F~(S')= S', contradicting (P). In the non-trivial case, consider ISI ~>2 and pick an z E S \ { x } . Define c ' ~ c~ such that x E c'~(S) iff i E A and z E c'i(S ) iff i E,4. As in the first step, it results that x~F~,(S)C=>z,~F~,(S) and by (N) it will follow that x E Fc(S)C~y,~F~(S'). [] As a corollary of the above proposition and of Theorem 1, we obtain.
Theorem 2. Let FI, X, and ~ be as in Theorem 1. I f ~;' denotes the family of all s.a.r, satisfying (NI), (N) and (P), then ~F E ~ for every F E 3~'. Particularly, for [/21
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T h e o r e m 3. Assume IX[ >I 3 and ~ are closed under a finite union and satisfy (*).
I f J;' denotes the family o f all s.a.r, satisfying (NI), (IC) and (P), then ~ r E 12, for every F E ~ ' . Proof. If follows from Proposition 5 that we must show that any F ~ o~' satisfies (IV). In this case neutrality is verified as a consequence of the definition of ~f. As it was stated in Proposition 3, non-imposedness, locality and partiality also imply the independence of variants if all admissible sets have at least three alternatives. In fact, since all conditions implied refer to a single admissible set, we can say that if ISI I> 3 and (NI), (L) and (P) are satisfied, then (IV) also holds. All we must do is to verify (IV) for any two-element set. Pick an S E ~ f , ISI = 2 , say S = {x, y}. Suppose that {i I x E c , ( S ) } = {i I Y E c'i(S)}, x E Fc(S), y ~ F c , ( S ) for some c, c' E ~. Choose S' E ~f such that {x, y} C s', Is'l >13 (it exists by our assumption of X and ~ ) . Define c " E c¢ such that c'[(S') = {x, y} i f f x E ci(S ) and c'[(S') = {z} for some z E S ' \ S otherwise. By (IC) we have x E Fc,,(S'), y,~Fc°(S' ) contradicting (IV) for admissible sets with at least three alternatives. [] The counterpart of Corollary 1 for this theorem extends Corollary 4 of Theorem 3 of Popov and El'kin (1990) to more general :~.
5. Restricted domain - rationality of choice profiles
As can be seen, in the previous sections explicit rationality conditions are not used. The absence of these conditions does not mean that we cannot restrict the domain of the s.a.r, to a subset of rationalizable choice profiles. In fact, most usual rationality conditions are compatible with the properties introduced in the present paper. We can easily verify that the proof of Theorem 1 works even if sufficient conditions for the rationalization of the individual choices are imposed. To exemplify this assertion let us consider two well-known rationality conditions, namely: (Chernoff's axiom): S, S' E ~, S C_S' f f S n C(S') c C(S). (Arrow's axiom): S, S' E ~f, S C_S' ~ S n C(S') ~ {0, C(S)}. All we must do is to check the compatibility of these axioms with the arguments used in the proofs. Obviously, if it sufficient to refer to those situations when two
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admissible sets of alternatives, in an inclusion relation, are simultaneously involved. Such situations appear only in the proofs of Proposition 2 and Lemma 2. Denote by ¢gl and qg2 the sets of all choice profiles satisfying Chernoff's and Arrow's axiom, respectively. In Proposition 2, denote A = (ilx~ci(S)). It is easy to verify that the conclusion holds if ~g is replaced by (~1 or by ¢g2 if the choice profile c" is defined as follows: ,, f {x}, ci(S)~s\{x},
ifiEA, if x E , 4 ,
and
fS'\{z}, c'~(S') = ] { x } , [S'\{x},
ifiEA andxES', if i E a and x , ~ S ' , ifiC,4,
respectively. Clearly, c" E cg2 C qgl and it also satisfies the condition required in the proof of Proposition 2. Similarly, the proof of Lemma 2 agrees with the restricted domain of F. Take, for instance, c' defined as follows:
fS\{x}, c~(S) = ~ { z } , I.{x},
ifiEA if i E Z if i ~ A
andxES', and x ~ . S ' , ,
and fS'X{x}, j{z}, c'i(S') = / { x } ,
I.S'\{z},
ifiCA andxES' if i E A and x , ~ S ' , if i ~ A and x E S ' , ifiEA
andx~S',
respectively, and easily verify that c' E ¢g2. [] Moreover, we can show that if an s.a.r, satisfies (A1) and ( A 2 ) , then the rationality properties of the choice profiles are transferred to the social choice.
Proposition 6. Let 12, X and ~g be arbitrary and F satisfies (A1) and (Az). Then: (a) Fc satisfies Chernoff's axiom for every c @ ~gl. (b) Fc satisfies Arrow's axiom for every c E ~2.
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Proof. Let S, S' E ~ such that S C S'. Obviously, if IS[ = 1, then the relation F~(S') n S E {0, Fc(S)} follows from the definition of an s.a.r. Therefore, let us assume that IS I ~>2. (a) Pick an c E (~1 and let x ~ Fc(S' ) n S. Denoting by A = {i I x ~ ci(S') } it will follow by Chernoff's axiom that A C_B, where B = {i I x E ci(S)}. Pick an c' E qgl such that c'i(S)= {y} iff i E / i and c'i(S)={x } iff i ~ A , for some y E S \ { x } . Then, by Proposition 2, it results that yg;Fc,(S ) and by (A2) it follows that x ~ Fc,(S ). Hence, x E Fc(S ) by (A1). (b) Suppose that c satisfies Arrow's axiom. Since the inclusion Fc(S')n S c F~(S) follows from (a), let us show that F~(S) C_Fc(S' ) n S if Fc(S' ) n S # O. Suppose, to the contrary, that there exists an x EF~(S) such that x~.Fc(S' ). Pick an y E S \ { x } (it exists, since F ~ ( S ' ) A S # O ) . Denote A = {i I x E c i ( S ) } , B = { i l y E ci(S')}. Two situations may occur: (k)A n B = 0 and (kk)A n B # 0 . In the case (k), define c ' E c¢ such that: c;(S')= {x} iff i E A ; c~(S')= {y} if i E B; c'i(S' ) = {x, y} otherwise. It follows from Proposition 2 that ygf. F~,(S') (since x E F~(S)), and x gf. Fc,(S') (since y E F~(S')). Then F~,(S') = 0 (Proposition 1), contradicting the non-emptiness assumption. In the case (kk), pick and z E S ' \ S and define c' E ~ such that: c'i(S' ) = {x} if i~A\a; c ; ( S ' ) = { y } if i ~ B X A ; c ; ( S ' ) = { z } if i E A A B and c ; ( S ' ) = { x , y } , otherwise. By proposition 2 and (A1) it follows that x~.Fc,(S' ), y ~ F c , ( S ' ). Obviously, A n B = { i l x ~ c'(S')}. Hence, z ~_ Fc, (S ' ) , otherwise it would follow by (A1) that x E F~(S). Again we have obtained F~,(S')= 0, a contradiction. [] Finally, let us note that the condition 'lsI ~ 3 for at least one S ~ ~ ' cannot be removed from the assumptions of our results.
Example 5. For the s.a.r, considered in example 1, take 1•1 : 3, [xI ~ 3, k = 2 and as the family of all two-element sets. Then (A1) and (A2) are simultaneously verified. Obviously, ~ * is not a filter because each two-element coalition is decisive. Consequently, there are no rejecting voters.
Acknowledgements I thank Michel Le Breton for the very useful discussions during his visit to the California Institute of Technology in 1994 and R.D. McKelvey for his excellent lectures in Social Choice Theory (McKelvey, 1993). I am also grateful to the referees for their helpful suggestions and remarks.
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