Vol. 37 (1977), No. 1-2, pp. 61--66 Zeitschrift for N a t i o n a l 6 k o n o m i e Journal of Economics

�9 by Springer-Verlag 1977

On Two Sufficient Conditions for Transitivity of the Social Preference Relation*

By

Wulf Gaertner, Bielefeld, and Achim Heinecke, Miinster

(Received May 9, 1977)

1. Introduction

Several years ago I n a d a [5, 6] demonstrated that if each voter classifies all social alternatives into two groups in such a way that he is indifferent among alternatives in each group, the simple ma- jority decision rule always gives a transitive social preference rela- tion. This case where the classification of alternatives may differ from voter to voter but where for any three alternatives x, y, and z strict individual orderings like, for example, x Pc y P~ z are not al- lowed to occur, was called the case of "dichotomous preferences".

Very recently S a p o s n i k [7] has proved that under the method of majority decision the social preference relation is transitive if individual preferences satisfy what he calls "cyclical balance". This restriction on individual orderings is different from Inada's condi- tion as well as from the well-known conditions "value restriction", "limited agreement", and "extremal restriction ''1 in so far as all the latter restrictions rule out the joint occurrence of certain indi- vidual preference orderings whereas Saposnik's condition excludes no individual ranking.

Our purpose in this paper is twofold: First of all we wish to show that the two theorems by I n a d a and S a p o s n i k follow from each other if certain reductions are introduced that transform an originally given preference profile into a reduced set of preferences. Secondly, referring to a remark by S a p o s n i k that within the con-

* The authors are indebted to an anonymous referee of this Journal for his valuable comments and suggestions.

1 Cf. [8], [9] and [2] for an analysis and comparison of these con- ditions.

62 W. Gaertner and A. Heinecke:

text of his theorem "the number of individual rankings with at least one indifference has no bearing on the social transitivity issue" ([7], p. 6), we want to prove that cyclical balance is not only suffi- cient but also necessary for this result to hold.

2. Notation and Definitions

Let there be a finite number of individuals forming a society S and let A be a finite set of akernatives. Ri is individual i's pre- ference ordering on A, that is, a binary relation being reflexive, complete, and transitive. Strict preference Pi and indifference Ii are derived from Ri in the customary way. The collective preference relation R on A is a binary relation, reflexive and complete. From R strict collective preference P and collective indifference I can be defined in the same way as for individual relations.

In the following the individual preference ordering x P iy Ii z with respect to the given triple (x, y, z) of alternatives will often be denoted as a P - I ordering or a P - I sequence. Let N (a P b) be the number of individuals for whom a P~ b where a, b e A. Then the method of majority decision holds if and only if

V x, y e A : x R y ~ IN (xPy)>_N (yP x)].

According to Saposnik's terminology, for any ordered triple (x, y, z) of alternatives the "clockwise cycle" of individual orderings is de- fined to be

xP~ yPi z

y P~ z P~ x

z P~ x P~ y

and the "counterclockwise cycle" of individual rankings is similarly defined as

z Pi y P~ x

x P~ z P~ y

y P~ x P~ z

"Cyclical balance" of the preferences of society S is then given if and only if there is the same number of individual preference order- ings constituting the clockwise cycle and the counterclockwise cycle.

We say that two individual preference orderings are "inversely related" if they cancel each other out when the votes of the mem- bers of society are counted. As an example, y P~ z P~ x from the

Transitivity of the Social Preference Relation 63

clockwise cycle and x P~ z P~ y from the counterclockwise cycle are inversely related. Clearly two strict individual orderings that are inversely related never originate from the same cycle.

3. From Cyclical Balance to Dichotomy

The main tools for a proof of our propositions are the following two reductions which simplify the originally given preference pat- tern without changing the social outcome given by the method of majority decision.

These reductions are

(a) pairs of strict individual orderings that are inversely related are deleted;

(b) pairs of non-inverse strict individual orderings belonging to different cycles are replaced by replicas of that P - I or I - P sequence which lead to the same social outcome. As an example let individual 1 possess the ordering x P y P z and let individual 2 have the ranking x P z P y. Obviously, the social outcome in this voting situation is identical with one emanating from a situation where there are two individuals with identical pre- ferences, each of them having the P - I ordering x P y I z.

Let us assume now that both types of reductions have been performed in the way described. Obviously, the reduced set of orderings which we eventually obtain either contains no strict indi- vidual preference orderings or all strict individual orderings that still occur belong to the same cycle (cf. [4]).

Proposition 1: A given preference pattern is cyclically balanced if and only if the reduced set of orderings either comprises only dichotomous preferences or the reduced set is the empty set.

Proof. Remember the definition of cyclical balance and simply observe that each of the two procedures reduces the number of strict individual preference orderings in each cycle in such a way that the difference between the number of strict orderings in the clockwise cycle and in the counterclockwise cycle is left unchanged.

Corollary: By means of the above reduction procedure Saposnik's theorem follows from Inada's theorem.

Proof. Let there be given a preference profile that is cyclically balanced. From proposition 1 the reduced set either is the empty

64 W. Gaertner and A. Heinecke:

set or the original pattern can be reduced to a set containing dichotomous preferences only. From Inada's theorem we know that the social preference relation is transitive in this case. As the reduc- tion procedure is such that the social outcome is left unchanged under the method of majority decision, the social preference relation with respect to the original preference profile is transitive. This is Saposnik's theorem.

S a p o s n i k himself observed ([7], p. 6) that Inada's theorem is an obvious consequence of his. Therefore, we obtain the result that both theorems follow from each other, which was to be shown.

Proposition 1 and its corollary were discussing the case where the reduced set of preferences does not contain any strict individual orderings. Investigating the other possible cases leads to

Proposi t ion 2: If and only if the preference profile of society S is cyclically balanced, individual orderings with at least one in- difference have no bearing on the social transitivity issue.

Proof. From the discussion above it is clear that if a given pre- ference pattern satisfies cyclical balance the number of P - I and I - P orderings in this profile is irrelevant for the question of tran- sitivity of the collective relation. This was already observed by S a p o s n i k .

What remains to be shown is that in those cases where the reduced set of preferences comprises one or several strict individual orderings, the existence or non-existence of social transitivity depends on the profile of the individual orderings with at least one indif- ference. We consider these cases consecutively and complete the proof by constructing for each case a preference pattern that is not transitive.

Case I: One strict individual ordering, say x P y P z, is left which occurs, say, n times. One can easily find a distribution of individual orderings with one indifference such that no social preference order- ing exists for society S.

Consider the following distribution:

z P y I x, occurring (n + 1) times

y P z I x, occurring once.

For society S the majority decision rule leads to z P x, x P y, z I y for n > 1, x I y, y I z, z P x for n = 1 so that transitivity of the social relation does not hold.

Transitivity of the Social Preference Relation 65

Case Ih There are two strict individual orderings left belonging to the same cycle. Let these t w o orderings be

x P y P z, occurring nl times

y P z P x, occurring no. times,

wkh nl _> n2 wi thout loss of generality. Let us n o w add the fol low- ing distribution of individual orderings with one indifference:

z I y P x, occurring (nl - n~) + 1 times

z I x P y, occurring once.

The social outcome is y P z, z P x, y I x which is intransitive.

Case III: Three strict individual orderings are left belonging to the same cycle. In this case Condorcet 's wel l -known "paradox of voting" - - situation is readily constructed (cf. [1], pp. 2--3) , and no further explanation is needed.

Therewith the necessity of Saposnik's condition of cyclical balance is proved. Individual orderings with at least one indifference can arbitrarily be added to or subtracted from the whole set of indi- vidual preferences wi thout affecting the social transitivity issue if and only if the property of cyclically balanced preferences is satisfied. In cases I and II transitivity of the social relation is secured only under additional assumptions on the occurrence of P - I and I - P sequences (cf. [3, 4]). In case III no general statement seems to be possible but clearly transitivity holds if the number of people having one particular strict ordering is larger than the total number of individuals having one of the other occurring preference relations.

Refe rences

[1] K. J. A r r o w : Social Choice and Individual Values, 2nd ed., New York 1963.

[2] V.J. B o w m a n and C. S. Co lan ton i : The extended Condorcet condition: A necessary and sufficient condition for the transitivity of majority decision, Journal of Mathematical Sociology 2 (1972), pp. 267--283.

[3] W. Gae r tne r : Zum Problem der Existenz von Sozialen Wohlfahrts- funktionen im Sinne von Ar row, Zeitschrift ftir die gesamte Staatswissen- schaft I33 (1977), pp. 61--74.

[4] W. Gae r tne r and A. H e i n e c k e : Cyclically mixed preferences - - A necessary and sufficient condition for transitivity of the social preference relation. Presented at the European Meeting of the Econometric Society, Helsinki, August I976.

Zeitschr. f. National6konomie, 37. Bd., Heft 1-2 5

66 W. Gaertner et al.: Transitivity of the Social Preference Relation

[5] K. Inada : A note on the simple majority decision rule, Econo- metrica 32 (1964), pp. 525--531.

[6] K. Inada : The simple majority decision rule, Econometrica 37 (1969), pp. 490---506.

[7] R. Saposnik : On transitivity of the social preference relation under simple majority rule, Journal of Economic Theory 10 (1975), pp. 1--7.

[8] A. K. Sen: Collective Choice and Social Welfare, San Francisco, Edinburgh 1970.

[9] A. K. Sen and P. K. P a t t a n a i k : Necessary and sufficient condi- tions for rational choice under majority decision, Journal of Economic Theory 1 (1969), pp. 178--202.

Addresses of authors: Univ.-Ass. Dr. Wulf Gaer tner , Fakult~it fiir Wirtschaftswissenschaften, Universit~it Bielefeld, D-4800 Bielefeld; Univ.- Ass. Dr. Achim Heinecke, Institut f~ir Medizinische Informatik und Bio- mathematik, Universit~it Miinster, D-4400 Miinster, Federal Republic of Germany.