the possibility of democratic pluralism

8/3/2019 The Possibility of Democratic Pluralism

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Economica 45 , 401-406

The Possibility of Democratic PluralismBy D A N A N . STEVENS and JAMES E . FOSTER

New College, University of South Florida and Cornell University

One method of extending Amartya Sen's (1970) theorems on the impos-sibility of a Paretian liberal is to allow groups, rather than individuals, to bethe decisive agents in social choice. Such a decisive group approach has beenexplored by Batra and Pattanaik (1972), Kelly (1976) and Sen (1976). W hileimpossibility results have been generated, the results are of limited interestsince they rely upon two unrealistic restrictions: (a) groups are disjoint; i.e.,no individual belongs to more than one group; and (b) groups are decisive insocial choice only where there is a unanimous preference by group mem-bers. But when groups are disjoint and member preferences are unanimous,decisive groups become indistinguishable from decisive individuals. It is notsurprising, therefore, that the group impossibility theorems have mirroredSen's original individualistic formulation. Indeed, Sen (1976) generatesgroup theorems by simply replacing "two persons" with "two disjointsubsets of individuals" while otherwise leaving his proofs intact.

In modern pluralistic societies, individuals commonly partic ipate in severalinterest groups. Moreover, as Olson (1965) and Hirschman (1970) haveargued, interest groups may still exert political pressure despite internaldissent among members over stated group goals and preferences. Conse-quently, this paper allows for groups that are not disjoint and for groups thatremain politically decisive despite a lack of member unanimity. Any finitenumber of distinct, decisive groups is permitted, where distinctness meansonly that "different" groups do not have precisely the same members anddecisiveness requires only that a majority of group members have a (weak)preference over the assigned pair of choice alternatives. Given this majorityrule characteristic of group decisiveness and potential existence of manyinterconnected, politically powerful interest groups, this paper examines thefeasibility of what will be termed "democratic pluralism".

1. NOTATION AND DEFINITIONS


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402 ECONOMICA [NOVEMBERrelation R for every n-tuple of individual orderings (R^), one ordering foeach individual in society, R = f{{RJ}.It should be noted that, implicit in the use of a social decision functionor SDF, is the assumption that choice functions should be defined over everynon-empty subset of X. If instead, choice functions are defined over a familyof subsets of X, what Plott (1971) terms an "admissible agenda", then sociachoice can be cyclic over any non-admissible subset of X without causingsocial consternation. The Sen dilemma can then be easily avoided (as well asthe theorems of this paper) by simply making non-admissible those subsetsof X containing all the alternatives over which individuals (or groups) aredecisive. Such a manipulation of the admissible agenda underlies the "solu-tions" of Sen's dilemma by Ramachandra (1972) and Farrell (1976).

Let S denote a collection of r ^ 2 many non-empty groups of individualsS = ( G \ . . . , G'), subject to the condition that each group G' of S is distinctfrom every other group G' of S; i.e., for all i, j=l,...,n if iV) thenG" # G'. N ote that distinctness permits G' to be a proper subset (orsuperset) of G'. The collection S is termed a group structure. Let R' denothe group preference of G' as determined by the group voting rule f, wherR' = fiiRa'. ae G')}. No restrictions are placed upon group preference R'\e.g., R' may be intransitive, incomplete, etc.

For X and y in X, the condition xRy is termed a weak (social)preference for x over y. A strict (social) preference for x over y, denotedxPy, is defined byxPy N ' ( y P x ) } . x P ' y ]

and


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1 9 7 8 ] THE POSSIBILITY OF DEMOCRATIC PLURALISM 40 3Condition DP (democratic pluralism). For each group G' of groupstructure S, there exists a pair of distinct alternatives (x', y') over which G'is decisive and over which /' is democratic.To simplify the proof of the next section, several assumptions will bemade. First, without loss of generality, it is assumed that groups are indexedaccording to size, such tha t NiG') ^ N{G^) g . . . N{G') where N(H) is thenumber of individuals in a subset H of society. Next, let the number ofindividuals common to all groups be denoted by m = N(n G"). Finally,assume that the pairs assigned to the decisive groups do not overlap at anypoint; i.e., for i,j = l,... ,t, such that i# /, xV y', xV x' and y V y'. Sincedecisiveness is defined in both directions, this last assumption can bepartially relaxed without affecting the validity of the theorems. A particulargroup structure can now be defined.

Definition. A group structure S is termed a two-set if and only if m = 1,N ( G ' ) ^ 2 , and for i = 1 , . . . , t - 1 , N (G ') = 2.IL THE POSSIBILITY OF DEMOCRATIC PLURALISM

For most group structures, it is clear that the Sen dilemma can arise forsome preference configurations. When any two groups are disjoint, forexample, the Batra and Pattanaik theorems give an immediate impossibilityresult. It follows that one restriction on group structures must be that theintersection of all decisive groups is non-empty; at least one individual mustbe a member of every decisive group. While this is a necessary condition forpossibility, it is not sufficient. Those group structures where the commongroup members can be outvoted will also yield cyclic social preferences.Indeed, it can be shown that when there are three or more decisive groups,then there is only one type of group structure that eliminates the possibilityof a cyclic social preference over the set of decisive pairs.Theorem 1. Where group structure S is made up of three or more groups, anecessary and sufiicient condition for a SDF to exist that satisfies conditionsU, P and DP is for S to be a two-set.

Proof. For necessity, assume that such a SDF exists. If so, then it will beshown that S must be a two-set. By condition U, let every individual a havea preference chain of the typey'^P^X-'^'P^ . . . Pay'-'PaX'Pay'PaX'Pay'Pa Py'^PaX"for some k = 1 , . . . , t, abbreviated to y^Pa . . . P^x\ For each individual a


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404 ECONOMICA [NOVEMBERy'Pa... PaX'. Clearly, if m = 0, and fl G ' is empty, then for each j = 1,...,(x" Pa y' for all a e G\ so that x ' P ' y'; and by DP, x'Py' for all i = 1 , . . . , f.This means that social preference is cyclic. Therefore, if a SDF is to exist,n G' must be non-empty and m > 0.Note that when f] G ' is non-empty, then by distinctness and the assumedsize indexing, if fl G ' is a group in S, then fl G ' = G'. Partition fl G' intothree sets, S\ S^ and S If m is even, let N{S^) = N{S^) = m/2 and letN{S^) = 0. If m is odd, let JV(S ) = N(S^) = (m - 1 )/ 2 and let N (S') = 1. Forall members in f] G', assign the preferences

for allfor all c6S^for all d e S\

By distinctness, N ( G ^ - n G ' ) ^ l and JV (G 2 -n G ') S 1. Therefore ,x^ P^ y^ and x^P^ y^ whether m is even or odd, since votes in S^ and S^cancel, and votes in S^ are not needed to assure a majority. As before,x' P ' y' for all i = 3 , . . . , ( - 1. If m is even, then clearly x ' P' y', since there isno opposition from S^ But if so, then by P and DP, social preference iscyclic. Therefore, if a SDF exists, m must be odd.To prevent cyclicity, a SDF must assign y'Rx', which byDP implies notx 'P 'y ' . With the preferences above, only the single individual d in S^prefers y' to x'; all other members of G' prefer x' to y'. To prevent x' P ' y',the group structure must be such that N'(y'Px')^N'{x'Py'). With'N ' ( y ' P x ' ) = l, then N ' ( x ' P y ' ) g l , which implies that m = l, since m isodd. If n G ' = G', then N{G') = m = 1; if p GV G', then N(G' - fl G ') S 1,whereas N'(x' P y') S l implies that NiG' - fl G') S 1 , so that N (G ') = 2 andm = 1.Having established that if a SDF exists, then m = 1, and N ( G ' ) S 2 , by U

let individual d have preference y'Pa . . . Pdx\ For i = 2 , . . . , r, x' P ' y'; butfor j = l, there is one vote for y^ against x\ Since all individuals in G'except for d prefer x' to y\ d will be outvoted if N(G^)^3. Thus,N{G^) = 2, which by the assumed index ordering on S implies that N{G')^2, for all j = 1 , . . . , (; by the distinctness of groups, N(G') = 1 only if i = t.Therefore, if a SDF exists, the group structure must be such that m = 1,N(G ') = 2 for i = 1 , . . . , f- 1, and NiG')g2. But this is precisely a two-set.Sufficiency is obtained by example. In fact, let(1) (xP^y for all a ) ^


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1 9 7 8 ] T H E POSSIBILITY OF DEMOCRATIC PLURALISM 405

for all i = l,...,t, and definexi?y ^not yPx.

The above rule clearly satisfies P and DP, and yields a reflexive, completeR. Moreover, if S is a two-set, R is acyclic for any n-tuple of individualorderings. To see this, let social preference be cyclic, with the cyclez^ Pz^P ... Pz' Pz'^^

where z' = z'''*^\ for some configuration of individual preferences.Consider individual d, the member common to all groups. For each/ = 1 , . . . , s, either (1) or (2) must apply in order for z'Pz'^^ to be true. If(1) applies, then in particular z'P^ z'*^; if z'Pz''*'^ is due to (2), then by theconstruction of the two-set z'Raz'^\ Clearly, if the Pareto component (1)of the rule is invoked even once, then the preference of d must beintransitive. The cycle must be due to the democratic pluralistic component(2) alone. However, since the decisive pairs are assumed to be non-overlapping, this too is impossible. Thus, social preference cannot be cyclic,and the given rule is a SDF with the desired properties. Q .E .D .As might be expected, when there are only two decisive groups, moregroup structures can be found which eliminate the chance of social cyclicity.However, these additional group structures are similar to a two-set in thateach group can have at most one member who is not also a member of theother group. The number of common group members no longer needs to bea single individual; but the number of common group members must be oddto prevent their votes from cancelling each other. The proof of Theorem 2 isclear from the proof of Theorem 1, and will not be given.Theorem 2. Where the group structure S is made up of exactly two groups,a necessary and sufficient condition for a SDF to exist that satisfies condi-tions U, P and DP is for S to satisfy N(G*) = m + 1 where m is odd.

The "possible" group structures identified in the above theorems aresensitive to the voting process f within the decisive groups. If majority ruleis replaced by some other group preference function, then more groupstructures become possible. For example, ifx P ' y - { N ' ( x P y ) > 2 / 3 }

is the relevant group voting rule, then two-sets can be expanded to "three-sets", i.e., m = 1, N ( G ' ) ^ 3 . In general, when ( g 3 , group structures can befound that avoid the Sen dilemma only when those group structures forcesome single individual d to be (at least) semi-decisive over every group


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406 ECONOMICA [NOVEMBER

If condition DP is accepted as a reasonable restriction on social choice,then in a Paretian society with unrestricted preferences, democratic plural-ism is "possible" only when the group structure frustrates the spirit of bothdemocracy and pluralism. Democratic pluralism is possible only when thegroup structure makes some individual a mini-dictator over every groupchoice. The notion of a mini-dictator is not very attractive, and it comes asno surprise that examples of two-sets in the free world are difficult to find.Apparently, most pluralistic societies are willing to risk the possibility ofinconsistency in social choice.

Of course, condition DP can always be rejected for the same argumentsused against Sen's conditions of liberalism; and alternative formulations ofpluralism may be generated that eliminate the need for dictatorial two-sets.But just as Sen's theorems are valid whenever individuals do possess socialdecision-making power, irrespective of the liberal or illiberal nature of thepower relationships, so too condition DP may be valid as a positivedescription of those societies where special interest groups do exert sufficientpressure to influence social policies.

ACKNOWLEDGMENTSWe would like to thank Amartya Sen for helpful comments and encouragementon an earlier draft of this paper.

REFERENCESBATRA, R . N. and PATTANAIK, P. K. (1972). On some suggestions for having non-binary socialchoice functions. Theory and Decision, 3, 1-lLFARRELL, M. J. (1976). Liberalism in the theory of social choice. Review of Economic Studies,43, 3-10.HIRSCHMAN, A. O. (1970). Exit, Voice and Loyalty, Cambridge, Mass.: H arvard UniversityPress.KELLY, J. S. (1976). The impossibility of a just liberal. Economica, 43, 67-76.OLSON, M . Jr (1965). The Logic of Collective A ction. Cam bridge, Mass.: H arvard UniversityPress.PLOTT, C . R . (1971). Recent results in the theory of voting. In Frontiers of QuantitativeEconomics (M. D. Intriligator, ed.), pp. 109-129 . Amsterdam: North-H olland.RAMACHANDRA, V. S. (1972). Liberalism, non-binary choice and Pareto principle. Theory andDecision, 3, 49-54.SEN, A . K . (1970). The impossibility of a Paretian liberal. Journal of Political Economy 78.152-157. ^'

, (1976). Liberty, unanimity and rights. Economica, 43, 217-245.


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the possibility of democratic pluralism

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