Ordered Value Restriction

Ordered Value Restriction Salvador Barberà Bernardo Moreno Univ. Autònoma de Barcelona and Univ. de Málaga and centra March 1, 2006 Abstract In this paper, we restrict the pro les of preferences to be either singlepeaked or single-dipped for every triple of alternatives when there is a natural ranking of the alternatives. This condition is analytically equivalent to impose the condition of weak single-peakedness for every triple of alternatives proposed in Barberà and Moreno (2005). We then show that if a preference pro le is weakly single-peaked for every triple of alternatives relative to a particular ranking of the alternatives, we have that such a preference pro le is weakly single-peaked on the set of alternatives relative to this particular ranking. The opposite does not hold. Keywords: Single-peakedness, single-dippedness, weak single-peakedness. Jel Classi cation: D72, D71 We would like to thank William Thomson for helpful comments and suggestions. Financial support from Junta de Andalucia under SEJ426 and the research project SEC2002-01926 are gratefully acknowledged by Bernardo Moreno. Financial support of the Barcelona Economics program (CREA), the Spanish Ministry of Science and Technology through grant BEC2002-002130, and the Generalitat of Catalonia through grant SGR2001-00162 are gratefully acknowledged by Salvador Barberà. 1

1 Introduction Is there any justi cation to consider domains that are restricted in the sense of not allowing particular individual preference relations to occur? A classical example is that in the political arena a left-right ordering of the parties appear reasonable, and single-peaked preferences are meaningful, or in locations problems when, people want to live as close as possible to the city centre. In other instances, individuals wish to be as far as possible from a refuse disposal site, or from the downtown of the city. In these cases, individuals have single-dipped preferences. However, there are situations in which makes sense that both types of preferences coexist. There are individuals that wish to live as close as possible to the city centre, and some others that prefer to live as far as possible from the city centre. All of the above situations also have in common that there is a natural order over the set of alternatives. In this paper, we investigate the consequences of imposing that for every triple of alternatives individuals preferences satisfy either the condition of single-peakedness, or the condition of single-dippedness or both under the assumption that there is a particular ranking of the alternatives. An analytically equivalent aspect of such a requirement is to impose that a preference pro le satis es the condition of weak single-peakedness for every triple of alternatives relative to a ranking of the alternatives. The condition of weak single-peakedness was proposed by Barberà and Moreno (2005). A preference pro le is weakly single-peaked relative to an order if and only if (a) Each of the voters preference has a unique maximal element and (b) For any alternative that is a preferred alternative for some agent, all agents whose peak is to the left of such an alternative, consider this value to be better than any alternative to the right of it and all agents with peak to the right of such an alternative, consider this value to be better than any alternative to the left of it. We then show that if a preference pro le is weakly single-peaked for every triple of alternatives relative to a particular ranking of the alternatives, we have that such a preference pro le is weakly single-peaked on the set of alternatives relative to this particular ranking. However, the opposite does not hold. A pro le of preferences can be weakly single-peaked on the set of alternatives relative to a particular ranking but it is not always the case that the pro le of preferences satis es the condition of weak single-peakedness for every triple of alternatives for this particular ranking (or for any other ranking). 2 The model Let X = fx; y; z; ::g denote a nite set of alternatives and let N = f1; ::; ng be a nite set of alternatives or voters (n > 2). Let R i stands for a binary relation on X. We interpret R i as the individual i 0 s preference relation and we assume that it is re exive, complete and transitive. The strict preference relation (the asymmetric part of R i ) will be denoted by P i. Let W stands for the social 2

preference relation. Let < be the set of preference orderings, < n denote the cartesian product < :: <, n-times, and R = (R 1 ; ::; R n ) 2 < n be a pro le of an n-member society consisting of preference orderings. We shall be interested in the existence of a best element. De nition 1 The existence of a best element for all subsets S of X means that for each S, there exists x 2 S such that for all y : y 2 S! xw y. An important result due to Sen (1970) states that if W is re exive and complete, then a necessary and su cient condition for the existence of a best element for all nite subsets S of X is that W be acyclical. De nition 2 W is said to be acyclical if for all nite sequences fx 1 ; ::; x k g from X it is not the case that x 1 W x 2 ^ x 2 W x 3 ^ :: ^ x k 1 W x k and x k W x 1. The rst collective choice rule that was examined in the context of domain conditions was the method of simple majority decision (Black (1948)). De nition 3 The method of majority decision is a collective choice rule M such that 8R 2 < n ; 8x; y 2 X : xmy $ [N(xR i y) > N(yR i x) where N(xR i y) denote the number of individuals for whom xr i y. The best known restriction on agent s preferences guaranteeing the existence of a best element is Black s (1958) condition of single-peaked preferences. Intuitively, preferences are single-peaked if for every triple of alternatives there exists a ranking of the alternatives such that "more" is strictly preferred to "less" up to a point, and "less" to "more" beyond that point. An analytically equivalent aspect of single-peakedness, due to Sen (1966), uses a perspective which is quite di erent from the view on which Black (1958) and Arrow (1951,1963) focused. Preferences are single-peaked if for every triple of alternatives there is an alternative that is not considered as "the worst" of the three alternatives. The property of value restriction, due to Sen (1966), generalizes various conditions proposed in the literature and in particular the condition of singlepeakedness and the mirror image of single-peakedness, the so-called singledipped preferences. This condition is stated in terms of triple of distinct alternatives (x; y; z); a; b; c are three distinct variables, each of which can assume one of the values x; y and z. De nition 4 9a; 9b; 9c such that for all R i : [8i : ap i b _ ap i c] _ [8i : bp i a _ cp i a] _ [8i : (ap i b ^ ap i c) _ (bp i a ^ cp i a)] In the triple (x; y; z) there is an option a such that all individuals agree that it is not worst, or agree that it is not best, or agree that it is not medium. 3

Theorem 1 (Sen and Pattanaik (1969)). I a pro le of preference orderings satis es one of the conditions Value Restriction, Extremal Restriction, and Limited Agreement 1 for every triple of alternatives, the method of majority decision is a social decision function. The concept of social decision function requires the existence of best elements. The existence of a best element guarantees that there is a Condorcet winner. De nition 5 An alternative x is a Condorcet winner on X for R 2 < n i, for all y 2 X, xm(r)y. It is often interesting to endow the set of alternatives with some structure. One way to do it is by assuming that alternatives are ranked according to some criterion which is independent of the agent s preferences. While these rankings also belong to the set <, we shall distinguish them through notation, and denote them as >, > 0, etc.. In what follows, we consider a xed ranking of alternatives >. We now present the de nitions of single-peakedness and single-dippedness based on the view on which Black (1958) and Arrow (1951,1963) focused. De nition 6 A pro le of preference orderings R 2 < n is single-peaked on Y = fx; y; zg X relative to >2 < i (1) For each R i there exists a unique maximal element on fx; y; zg, denoted p i (Y ) and called i s top alternative on Y, or i s peak on Y and (2) For all i 2 N, a; b 2 Y, [t i (Y ) < b < a or a < b < t i (Y )]! bp i a A pro le of preference orderings is single-peaked on X relative to >2 < i it is single-peaked for every triple of alternatives relative to >2 <. De nition 7 A pro le of preference orderings R 2 < n is single-dipped on Y = fx; y; zg X relative to >2 < i (1) For each R i there exists a unique minimal element on Y, denoted d i (Y ) and called i s bottom alternative on Y, or i s dip on Y and (2) For all i 2 N, a; b 2 Y, 1 Extremal Restriction says that 8a; 8b; 8c : [d i (Y )<b<a or a<b<d i (Y )]! ap i b [9i : ap i b ^ bp i c! 8j : (cp j a! cp j b ^ bp j a)] and Limited Agreement says that 9a; 9b : [8i : ar i b]. 4

A pro le of preference orderings is single-dipped on X relative to >2 < i it is single-dipped for every triple of alternatives relative to >2 <. We de ne a new restriction on individual preferences, proposed in Barberà and Moreno (2005). This condition, called weak single-peakedness, generalizes those of single-peakedness and single-dippedness, while still having the same two basic implications of the classical restriction. It guarantees that Condorcet winners always exist, and they are the medians of the distribution of voters peaks. De nition 8 A pro le of preference orderings R 2 < n is weakly single-peaked on Y X relative to >2 < i (1) For each R i there exists a unique maximal element on Y, denoted t i (Y ) and called i s top alternative on Y, or i s peak on Y and (2) For all i 2 N, and for all j 2 N, x 2 Y [t j (Y ) < t i (Y ) and x < t j (Y ) or t j (Y ) > t i (Y ) and x > t j (Y )]! t j (Y ) i x We now show that weak single-peakedness on a triple (x; y; z) is equivalent to say that there is an option a such that all individuals agree that it is not worst, or agree that it is not best. Lemma 1 A pro le of preference orderings R 2 < n is weakly single-peaked for fx; y; zg X relative to >2 < i it is either single-peaked or single-dipped on fa; b; cg for >2 <. Proof. Let Y = fx; y; zg X be a triple of alternatives and let R be a pro le preference orderings. It is clear that if R is either single-peaked or singledipped on Y for > then it is weakly single-peaked on Y for >. Suppose that R is weakly single-peaked on fx; y; zg relative to >=(x < y < z). We distinguish two cases: Case 1. t j (Y ) 6= y for all j 2 N. Agents preferences are single-dipped relative to >=(x < y < z). Case 2. t j (Y ) = y for some j 2 N. If t j (Y ) = y for all j 2 N then agents preferences are single-peaked. Suppose that for some i 2 N, t i (Y ) 6= y, by weak single-peakedness if y < t i (Y ) then yp i x and, if y > t i (Y ) then yp i z. Therefore, agent i has single-peaked preferences on Y. We now present a parallel condition to Value Restriction relative to the case in which the alternatives follow a particular ranking. We call this condition Ordered Value Restriction: De nition 9 A pro le of preference orderings R 2 < n satis es the Ordered Value Restriction condition on X relative to >2 < if for every triple of alternatives satis es weak single-peakedness relative to >2 <. The condition of Ordered Value Restriction is illustrated in the following example. 5

Example 1 Let N = f1; 2; 3g, X = fx; y; z; wg and the preference orderings of the individuals as given in Figure 1. The vertical axis just indicates the order of preference. Ordering of preference x y z w Ranking of alternatives Figure 1. Example of a pro le of preference orderings satisfying the Ordered Value Restriction. Note that this pro le of preference orderings is weakly single-peaked relative to >= (x < y < z < w), for every triple of alternatives relative to >. We illustrate in the following example that contrary to the case of singlepeakedness and single-dippedness, a pro le of preference orderings can be weakly single-peaked on X but not on every triple. Example 2 Let N = f1; 2; 3g, X = fx; y; z; wg and the preference orderings of the individuals as given in Figure 2. Ordering of preference x y z w Ranking of alternatives 6

Figure 2. Example of weakly single-peaked pro le of preference orderings on X that it is not weakly single-peaked on every triple. Note that this pro le of preference orderings is weakly single-peaked relative to >= (x < y < z < w) but it does not satisfy the Ordered Value Restriction since there is a cycle on fx; w; zg. Finally, we show that if pro le of preference orderings is weakly single-peaked for every triple, then it is weakly single-peaked on X. Theorem 2 If a pro le of preference orderings R 2 < n is weakly single-peaked on every triple relative to >2 < then it is weakly single-peaked on X relative to >2 <. Proof. Let R be a pro le preference orderings that is weakly single-peaked on every triple relative to >2 <. Suppose that it is not weakly single-peaked on X. Then, w.l.g., for some t i (X) and for some agent j 2 N for which that t j (X) < t i (X) there exists x > t i (X) such that xp j t i (X). Therefore, (R 1 ; ::; R n ) is not weakly single-peaked on ft j (X); t i (X); xg relative to (t j (X) < t i (X) < x), a contradiction. References Arrow, K. J. (1951,1963). Social Choice and Individual Values, 2nd edn., 1963, New York: Wiley. Barberà and Moreno (2005). "Essential and Weak Single-Peakedness", Mimeo, Universidad de Málaga. Black, D. (1948). "On the Rationale of Group Decision Making", The Journal of Political Economy, 56: 23-34. Black, D. (1958). The Theory of Committees and Elections, Cambridge University Press. Sen, A. K. (1966). "A Possibility Theorem on Majority Decisions", Econometrica, 34: 491-499. Sen, A. K. (1970). Collective Choice and Social Welfare, San Francisco: Holden-Day. Sen, A. K. and Pattanaik, P. K. (1969). "Necessary and Su cient Conditions for Rational Choice under Majority Decision", Journal of Economic Theory, 1: 178-202. 7