WHAT DEGREE OF CONFLICT IN THE DOMAIN OF INDIVIDUAL PREFERENCES COULD RESOLVE THE PARADOX OF DEMOCRATIC MAJORITY DECISION?

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1 WHAT DEGREE OF CONFLICT IN THE DOMAIN OF INDIVIDUAL PREFERENCES COULD RESOLVE THE PARADOX OF DEMOCRATIC MAJORITY DECISION? Ahmet KARA Fatih University Economics Department Hadimköy 34900, Büyüçkmece, İstanbul, Turkey. Phone: (90) Fax: (90) ABSTRACT This paper identifies the degree of conflict in the domain of individual preferences that resolves the paradox of democratic majority decision. The type of conflict that is incompatible with majority decisions is neither moderate nor maximal but somewhere in between. Key words: The paradox of democratic majority decision, degree of conflict BİREYLER TERCİHLERİ ALANINDAKİ HANGİ ÇELİŞKİ DERECESİ DEMOKRATİK ÇOĞUNLUK KURALI PARADOKSUNU ORTADAN KALDIRIR? ÖZET Bu makale, birey tercihleri arasındaki belirli tür farklılıkların (çelişkilerin) demokratik çoğunluk kuralı paradoksunu ortadan kaldıracağını göstermektedir. Ilgili paradoksla uyumlu çelişki derecesiö ılımlı ya da maximal değil arasında bir değere sahiptir. Anahtar kelimeler: Demokratik çoğunluk kuralı paradoksu, çelişki derecesi I. INTRODUCTION The paradox of democratic majority decisions (the voting paradox) is one of the most widely studied phenomena in social choice theory. The paradox, which arises when majority rule produces an intransitive social preference, has provoked a line of extensive inquiry attempting to analyze the paradox and identify the conditions that preclude its occurrence. Among these works are Arrow (1951), Inada (1964, 1969), Sen and Pattanaik (1969), Sen (1970), Fishburn (1970a, b), Fishburn and Gehrlein (1980), Jain (1985) and Usher (1994). This paper contributes to this literature by identifying a pattern of conflict in the domain of individual preferences that resolves the paradox in question.

2 II. THE GENERAL FRAMEWORK Let E be the set of a finite number of individuals forming a society, and let Z be a set of mutually exclusive social alternatives. Assume that the cardinalities of E and Z, denoted by, respectively, E and Z, are finite, and E > 1, Z > 2. Each individual i in the society has a preference relation R i, which is a binary relation on Z such that R i {(x, y): x, y are in Z and x, y are distinct}, and i = 1,, n. For any x, y in Z, xr i y will be interpreted as x is preferred to y by individual i. Define strict preference (P i ) and indifference (l i ) relations on {x,y} as follows: xp i y if and only if xr i y and not yr i x, xi i y if and only if xr i y and yr i x. Given distinct x, y in Z, exactly one of the following four possibilities holds: xp i y, yp i x, xi i y, and none of these. As such a preference relation can be specified by specifying P i and l i, specifically, xr i y if and only if xp i y or xi i y. Thus, we will often employ a particular specification of R i such that R i over an m-set in Z is a set, the elements of which are preferences over the pairs in that m-set. For example, R i = {xp i y, yp i z} is a possible preference relation over a triple {x,y,z} X (i.e. an m-set, where m=3), which illustrates the particular specification proposed here. A preference relation R i on Z is said to be complete if and only if xp i y or xi i y for all x, y in Z such that x y. R i on Z is incomplete if it is not complete. R i on Z is transitive if and only if for all distinct x, y, z in Z, (xp i y and yp i z implies xp i z). R i on Z is intransitive if it is not transitive. We will assume that individual preferences are transitive and complete. Let, for all x, y in Z, N (xpy) be the number of people who strictly prefers x to y. the method of democratic majority decision (majority rule) is a collective choice rule such that For all x, y in Z, xpy iff [N(xPy)>N(yPx)] and xly iff [N(xPy)=N(yPx)]. A preference relation R i over a triple {x, y, z} Z is said to be strongly strict if it has strict preference over every pair in {x, y, z}. It is said to be weakly strict if it has indifference over at least one pair in {x, y, z}. The degree to which preference relations represent, across individuals, compatible or incompatible orderings of alternatives will turn out to be crucial in our analysis of social choice paradoxes. Hence, we will define a few concepts to capture the relations of compatibility among different preference relations: a pair of preference relations R i and R i for any i and j are said to be incompatible (or in conflict) over a pair of alternatives {x,y} iff xp i y and yp i x. For any i and j, R i are said to be compatible over {x, y} if they are not incompatible over {x, y}. Define a conflict index C ij (x, y) such that C ij (x, y) = 1 if R i and R j are incompatible over {x,y} = 0 otherwise.

3 With R i and R j defined over m alternatives of Z, there are m (m-1)/2 pairwise comparisons in an m-set. Let C m (R i, R j ) be the sum of C ij (x,y)s over all alternatives {x,y} for preference relations R i and R j over an m-set. The value of C m (R i, R j ) depends on how conflicted preference relations are and on the number of alternatives over which they are defined. For instance, over a triple, the maximum and minimum values of C 3 (R i, R j ) are respectively 3 and 0. III. THE PARADOX OF DEMOCRATIC MAJORITY DECISION The following example presents an illustration of how majority rule could generate intransitive social preferences. Example: Consider a society of two individuals with the following preference relations over {x,y,z}: R 1 = {xp 1 y, yp 1 z, xp 1 z} R 2 = {yp 2 z, zp 2 x, yp 2 x}. Majority rule produces the following social preference relation over the pairs in {x,y,z}: Over {x,y}, N(xPy) = 1 = N(yPx), hence xly Over {y,z}, N(yPz) = 2 > 0 = N(zPy), hence ypz Over {x,z}, N(xPz) = 1 = N(zPx), hence xlz. Thus, the social preference R over {x,y,z} is R = {xly, ypz, xlz}, Which is intransitive. Sen (1970) presents the following theorem that formulates a condition called Extremal Restriction under which majority rule generates transitive social choice from transitive individual preferences. Extremal Restriction (ER): A triple {x,y,z} in Z satisfies ER if and only if the following condition holds for every ordered triple obtainable from it: Let {x,y,z} be an ordered triple. If there is someone who prefers x to y and y to z, then anyone regards z to be uniquely best if and only if she regards x to be uniquely worst, i.e. if for some i in E, xp i y and yp i z, then for all j in E, if zp i x, then zp j y and yp j x. Theorem 1 (Sen, 1970): A necessary and sufficient condition for the transitivity of majority decisions is that every triple of alternatives must satisfy ER. ER ensures the transitivity of majority decision by ruling out the joint presence of those individual preference relations that could, though majority rule, lead to an

4 intransitive social preference. 1 It turns out that there is an identifiable pattern of conflict between the preference relations whose joint presence violates ER. That is to say, and as proved in the theorem bellow, ER, where it holds non-vacuously, 2 is equivalent to the absence of a certain pattern of conflict between individual preferences. Theorem 2: ER holds non-vacuously for an arbitrary triple {x,y,z Z if and only if C 3 (R i,r j ) 2 over {x,y,z}, where at least one of R i and R j is a strongly strict preference relation over {x,y,z} and R i and R j are transitive for i,j = 1,, n. Proof: We first show that if ER holds non-vacuously for {x,y,z}, then C 3 (R i, R j ) 2 over {x,y,z}. Suppose for {x,y,z} Z, there is an individual i in the E with a strongly strict relation R i = {xp i y, yp i z, xp i z}. Then, assuming that it holds, ER would allow the presence of an individual j in E with any of the following preference relations over {x,y,z}: {xp i y, yp i z, xp i z} {xp i z, zp i y, xp i y} {zp i y, yp i x, zp i x} {yp i x, xp i z, yp i z} {xp i y, yl i z, xp i z} {xl i y, yp i z, xp i z} {yp i y, zl i x, yp i x} {zl i x, xp i y, zp i y} {xl i y, yl i z, xl i z} In the presence of R i, R j can take the form of any of the preference relations in this list. 3 In other words, ER would allow the joint presence of R i and any of these preference relations which form an exhaustive list of preferences, each of which can coexist with R i without violating ER. It is evident that the pairwise degree of conflict between R i and any of the preference relations in this list either 0, 1, or 3 but never 2. Hence, if ER holds, C 3 (R i,r j ) 2 over {x,y,z}. We have started with a particularly strongly strict R i to prove the result, but we could start the same result holds true. It remains to show that if ER does not hold for {x,y,z}, then C 3 (R i,r j ) = 2 over {x,y,z}. Suppose that there is an individual I in E with a strongly strict preference relations R i = {xp i y, yp i z, xp i z}. If does not hold, then there must exist an individual j in E with any of the following preference relations over {x,y,z}: {yp i z, zp ix, yp i x} {zp i x, xp i y, zp i y} 1 For instance, in the presence of {xp i y, yp i z, xp i z}, ER oes not allow the presence of {yp i z, zp i x, yp i x}, for if society consisted of individuals holding these two preference relations in equal numbers, the social preference over {x,y,z} inducted by majority rule would be {xly, ypz, xlz}, which is intransitive. 2 There are cases where ER holds vacuously: if ER is to be violated for a triple of Z, there must exist at least one strongly strict preference relation over that triple. Thus, cases involving only weakly strict preference relations over the triples of Z vacuously satisfy ER. 3 Though ER allows the joint presence of Ri and any of the preference relations in the list above it does not allow the joint presence of R I and all of the presence relations in the list as a group, for the joint presence of some preference relations in the list, such as R5 and R6, would be incompatible with ER.

5 {yl i z, zp i x, yp i x} {yp i x, xl i y, zp i y}. It is evident that the pairwise degree of conflict between R i and any of these preference relations is 2, i.e. C 3 (R i,r j ) = 2. Though the same procedure, it can be shown that the same result holds true for other pairs of preferences that do not satisfy ER. IV. CONCLUSION QED. The theorem presented in this paper proves that Sen s sufficient conditions for the transitivity of democratic majority decision over {x,y,z} is equivalent to the absence of a certain pattern of conflict in the domain of individual preferences, which is characterized by C 3 (R i, R j ) 2 over {x,y,z}, where i, j = 1,, n. Thus transitivity of democratic majority decision over {x,y,z} is compatible with zero conflicts (C 3 (R i,r j ) = 0), moderate conflicts (C 3 (R i,r j ) = 1) and maximal conflicts (C 3 (R i,r j ) = 3) between individual preferences over {x,y,z}, i.e. the type of conflict that is incompatible with transitive majority decisions is neither moderate nor maximal but somewhere in between. REFERENCES: Arrow, K. J Social Choice and Individual Values. New York: John Wiley. Fishburn, P.C. 1970a. Intransitive Individual Indifference and Transitive Majorities. Econometrica 38: b. Conditions for Simple Majority Decision with Intransitive Individual Indifference. Journal of Economic Theory 2: Fishburn, P.C. and W.V. Gehrlein Paradox of Voting: Effects of Individual Indifference and Intrasitivity. Journal of Public Economics 14: Inada, K A Note on the Simple Majority Decision Rule. Econometrica 32: The Simple Majority Decision Rule. Econometrica 37: Jain, S.K A Direct Proof of Inada-Sen-Pattanaik Theorem of Majority Rule. The Economic Studies Quarterly 36: Sen, A Collective Choice and Social Welfare. San Francisco: Holden-Day. Sen, A. and P. K. Pattanaik Necessary and Sufficient Conditions for Rational Choice Under Majority Decision. Journal Economic Theory 1: Usher, D The Significance of the Probabilistic Voting Theorem. Canadian Journal of Economics 27 (2):