May’s theorem in one dimension

Abstract

This paper provides three versions of May’s theorem on majority rule, adapted to the one-dimensional model common in formal political modeling applications. The key contribution is that single peakedness of voter preferences allows us to drop May’s restrictive positive responsiveness axiom. The simplest statement of the result holds when voter preferences are single peaked and linear (no indifferences), in which case a voting rule satisfies anonymity, neutrality, Pareto, and transitivity of weak social preference if and only if the number of individuals is odd and the rule is majority rule.

Keywords

Majority rule May’s theorem single peaked transitivity

1. Introduction

Majority rule occupies a central role in democratic decision making, and it has accordingly received close attention in formal political theory. A well-known axiomatization due to May (1952) provides some theoretical justification for the status of majority rule: when individual preferences are unrestricted, it is the only voting rule that is unbiased toward individuals, unbiased toward alternatives, and positively responsive to changes individuals preferences. In the terminology of this paper, majority rule is uniquely characterized by the axioms of anonymity, neutrality, and strong tie break. The latter axiom is indeed strong: it requires that if social indifference holds between two alternatives and if even just one individual increases her support for one of these alternatives (e.g. indifference is replaced by strict preference for one alternative), then the social ‘tie’ between the alternatives is broken in favor of the one with increased support. Although desirable in a world where each vote counts and social ties represent an exact balancing of individual preferences, the strong tie break axiom is violated by most voting rules —for example, it is violated by all quota rules other than majority rule—and its normative importance is less compelling than the other axioms.

In this paper, I provide an alternative to May’s axiomatization that is closer to the subject matter of political science and eschews his strong tie break axiom. The key structure added to the problem is the assumption of single peakedness: whereas May assumes individual preferences are unrestricted, consistent with the social choice literature, formal modeling in political science often assumes alternatives are one-dimensional and individual preferences are single peaked. This preference restriction is prevalent in work on electoral modeling in the tradition of Calvert (1983); it was used in the analysis of agenda control by Romer and Rosenthal (1978), introduced to legislative bargaining by Banks and Duggan (2000), and Penn et al. (2011) recently explore the implications of single peakedness in an axiomatic analysis of strategy-proof voting mechanisms. A compelling property of single-peaked preferences, established by Arrow (1951) and Black (1948, 1958), is that it precludes the Condorcet paradox and confers desirable transitivity properties on majority rule: majority voting generates transitive strict social preferences, and when the number of individuals is odd, majority indifference relation is also transitive. This full transitivity of majority rule with an odd number of individuals is a highly specialized property—for example, it is violated by all quota rules other than majority rule—and in this sense, transitivity with single-peaked individual preferences seems to impose restrictions similar to strong tie break with unrestricted preferences.

I maintain May’s other axioms and investigate the question, ‘Assuming single-peaked preferences, does May’s characterization carry over if we replace the tie break axiom with transitivity?’ A positive answer to this question would be of interest because it would provide further justification for majority rule in many applications considered in political science, and because, compared to the tie break axiom, transitivity arguably has greater normative relevance: when a voting rule generates an ordering of alternatives, we can view social choices as deriving from a representative agent; this simplifies the process of choosing from a finite set of feasible options, as we simply choose the top-ranked of the available options; and it precludes inconsistencies when options are added or deleted. The answer to the above question depends on the details—on the possibility of individual indifferences between alternatives, on the form of majority rule considered, and on the nature of the transitivity condition imposed. Regarding the definition of majority rule, a majority preference for x over y may hold if more than half of all individuals strictly prefer x to y (simple majority rule), or it may hold if more individuals strictly prefer x than strictly prefer y (relative majority rule). One transitivity condition is that strict majority preferences are transitive, and a stronger one is that weak majority preferences are transitive (i.e. both strict majority preference and indifference are transitive). The sharpest result holds under the stronger transitivity condition in the absence of individual indifference, where an exact characterization of majority rule is obtained, and May’s theorem carries over in the conjectured manner.

The conclusion distills the findings of this paper into three corollaries, each of which provides a version of May’s theorem in one dimension. First, assuming that there are at least three alternatives and all single-peaked preferences are possible, if we consider a voting rule that satisfies anonymity, neutrality, and the Pareto axiom, then the weak social preference relation is transitive if and only if the number of individuals is odd and the voting rule is nested between strict and weak simple majority rule. Second, assuming there are at least three alternatives and only the single-peaked preferences without indifferences are possible (so simple and relative majority rule coincide), if we consider a voting rule that satisfies anonymity, neutrality, and the Pareto axiom, then strict social preferences are transitive if and only if strict social preference implies majority preference and an additional non-negative responsiveness axiom (used implicitly by May) is satisfied. The third corollary continues to preclude individual indifference and provides an exact characterization of majority rule: a voting rule satisfies anonymity, neutrality, the Pareto axiom, and always generates transitive weak social preference if and only if the number of individuals is odd and it is exactly equal to majority rule. This result provides a close analogue of May’s theorem that exploits the unidimensionality prevalent in formal models of politics and fits naturally within the analytical framework of political science.

Section 1 sets the formal framework of the analysis, and Section 2 states May’s axioms and provides two statements of May’s theorem from the social choice literature, depending on the domain of individual preferences. Section 3 deduces implications of the two transitivity conditions mentioned above assuming single-peaked preferences, depending on whether individual preferences may admit indifferences. Section 4 presents sufficient conditions for the transitivity conditions, and Section 5 contains a number of technical examples to confirm that the results of the paper are tight. The conclusion states three versions of May’s theorem for the one-dimensional model in the form of three corollaries, and it compares the results to those of Dasgupta and Maskin (2008) on the robustness of majority rule.

2. Formal preliminaries

A collective choice environment is defined by four things: a set Θ of social states, which represent information that we want to treat as variable and which capture the uncertainty of the analyst; a set N of individual decision makers; a set X of alternatives from which they must make a collective choice; and finally individual preferences (P_i (θ), R_i (θ)) over alternatives in each state θ, where P_i (θ) represents strict preference and R_i (θ) represents weak preference. We assume that there is a finite number n of individuals, who are indexed 1,…,n, that the strict preference is asymmetric and the weak preference is complete,¹ and that these relations are dual, in the sense that for all x, y ∈ X, xP_i (θ)y if and only if not yR_i (θ)x. We define the indifference relation I_i (θ) by the standard convention, i.e. for all x, y ∈ X, xI_i (θ)y if and only if neither xP_i (θ)y nor yP_i (θ)x. We always assume that individual preferences form an ordering of alternatives. Formally, we say a dual pairing (P, R) of an asymmetric relation P with a complete relation R is an order if P is negatively transitive and R is transitive.² A stronger restriction is that (P, R) is a linear order, which adds the requirement that indifference cannot hold between any two distinct alternatives.

To denote the groups for which strict preference, weak preference, and indifference holds between any two alternatives, we use

\begin{matrix} P (x, y | θ) = {i \in N | {xP}_{i} (θ) y} \\ R (x, y | θ) = {i \in N | {xR}_{i} (θ) y} \\ I (x, y | θ) = {i \in N | {xI}_{i} (θ) y}, \end{matrix}

and we write p(x, y|θ) = |P(x, y|θ)| and r(x, y|θ) = |R(x, y|θ)| for the numbers of individuals strictly and weakly preferring one alternative to another. The profile of individual preferences at θ is denoted by

PR (θ) = ((P_{1} (θ), R_{1} (θ)), \dots, (P_{n} (θ), R_{n} (θ))),

and the set PR(Θ) = {PR(θ) |θ ∈ Θ} is the domain of possible preference profiles. I impose the following minimal richness condition on the domain of preferences: there exists a free pair {s, t} ⊆X such that for all x, y ∈ X and all θ ∈ Θ, there exists θ′ ∈ Θ such that P(s, t|θ′) = P(x, y|θ) and P(t, s|θ′) = P(y, x|θ). This assumption is satisfied in most of the environments of interest in formal political theory, including the domains defined below.³

A classical example of a collective choice environment is that of a group N of voters who must choose from a finite set X of candidates to fill a political office. Each voter has preferences over the candidates, and states simply index the set of possible preference profiles. Alternatively, a voting body (e.g. an electorate, a city council, a board of directors) must choose from a finite set of projects to undertake, or a department faculty must choose from a finite set of job candidates to fill a position. In these kinds of environments, it is often difficult to impose a priori restrictions on individual preferences, and it is natural to allow for every profile of orderings on X. Let

U = {((P_{1}, R_{1}), \dots, (P_{n}, R_{n})) | \forall i \in N : (P_{i}, R_{i}) is an order of X}

denote the set of all profiles of orders of X. We refer to the assumption that PR(Θ) = U as unrestricted domain. Sometimes, especially when the set of alternatives is small, we may wish to consider the possibility that individuals can always discern a strict preference between distinct alternatives. Let

L = {((P_{1}, R_{1}), \dots, (P_{n}, R_{n})) | \forall i \in N : (P_{i}, R_{i}) is a linear order of X} .

We refer to the assumption that PR(Θ) = L as linear domain.

Other classical examples of a collective choice environment are that of a society that must choose a single proportional income tax rate between zero and one, or that of a society that must choose the level of provision of a single public good. More generally, suppose that public policies can be ordered according to a single summary statistic that contains all the information relevant for voter preferences, with the standard interpretation that policies corresponding to smaller values of the statistic are more ‘liberal,’ while those corresponding to greater values of the statistic are more ‘conservative.’ Given this structure, it is natural to assume individual preferences are single peaked, in the sense defined formally as follows. Letting $(≺, ⪯)$ be a linear order of X, a dual pair (P, R) is single peaked with respect to $(≺, ⪯)$ if there exists $\hat{x} \in X$ satisfying the following conditions:

for all y ∈ X with $y \neq \hat{x}$ , $\hat{x} Py$ ;

for all y, z ∈ X, if $z ≺ y ≺ \hat{x}$ , then yPz;

for all y, z ∈ X, if $\hat{x} ≺ y ≺ z$ , then yPz.

We say a profile ((P ₁, R ₁),…,(P_n , R_n )) of individual preferences is single peaked with respect to $(≺, ⪯)$ if for each i ∈ N, (P_i , R_i ) is single peaked with respect to $(≺, ⪯)$ . In this case, we denote the unique maximal element of (P_i , R_i ) by ${\hat{x}}^{i}$ , and we refer to ${\hat{x}}^{i} \in X$ as the individual’s ideal point. We say ((P ₁, R ₁), …, (P_n , R_n )) is single peaked if it is single peaked with respect to some $(≺, ⪯)$ . Let

S = {((P_{1}, R_{1}), \dots, (P_{n}, R_{n})) | ((P_{1}, R_{1}), \dots, (P_{n}, R_{n})) is single peaked} .

We refer to the assumption that PR(Θ) = S as single-peaked domain. An implication of single peakedness is that given any three distinct alternatives {x, y, z}, there is one alternative among the three, namely the alternative that is ordered in the middle, that is not bottom ranked among the three for any individual, e.g. if x≺y≺z, then for every individual i, we have either yP_i (θ)x or yP_i (θ)z.

To analyze collective decisions, we model social preferences in each state as determined by a mapping F, where we denote the values of F by F(θ) = (P_F (θ), R_F (θ)). Thus, we have

θ \overset{F}{\mapsto} (P_{F} (θ), R_{F} (θ))

and we call F a social preference rule (SPR). Here, P_F (θ) is the strict social preference relation at θ, and we interpret a preference xP_F (θ)y to indicate that the alternative x is normatively superior to y. Also, R_F (θ) is the weak social preference relation at θ, and xR_F (θ)y indicates that x is normatively at least as desirable as y. We assume that the latter relation is complete, the former is asymmetric, and the two relations are dual. We denote by I_F (θ) the social indifference relation defined from strict or weak social preference in the usual way.

The following examples give two versions of well-known SPRs based on majority voting and Pareto dominance.

Simple majority F_SM .

{xP}_{SM} (θ) y if and only if p (x, y | θ) > n / 2

Relative majority F_RM .

{xP}_{RM} (θ) y if and only if p (x, y | θ) > p (y, x | θ)

Simple Pareto F_SP .

{xP}_{SP} (θ) y if and only if p (x, y | θ) = n

Relative Pareto F_RP .

{xP}_{RP} (θ) y if and only if r (x, y | θ) = n and p (x, y | θ) > 0

Weak social preferences are obtained via duality, e.g. xR_SM (θ)y if and only if not yP_SM (θ)x, which holds if and only if not p(y, x|θ) > n/2, which is equivalent to r(x, y) ≥ n/2. Of course, under linear domain, the distinction between simple and relative majority rule disappears, so that F_SM = F_RM , and similarly for simple and relative Pareto.

3. Axioms for majority rule

This section states the axioms that are standard for the characterization of majority rule and in the broader analysis of social choice, and it presents two axiomatizations for majority rule that are essentially due to May (1952). We begin with an axiom that formalizes the idea that social preferences are not biased for or against any particular alternative. Note that it strengthens Arrow’s independence of irrelevant alternatives (IIA).⁴

Neutrality. For all x, y, w, z ∈ X and all θ, θ′ ∈ Θ,

\begin{matrix} P (x, y | θ) = P (w, z | θ'), \\ R (x, y | θ) = R (w, z | θ'), \\ and {xP}_{F} (θ) y \end{matrix}} \Rightarrow {wP}_{F} (θ') z .

In particular, the above axiom captures the idea that if individual preferences between x and y in one state are the same as individual preferences between w and z in another, then social preferences should be the same too (with w playing the same role as x and z playing the same role as y).

The next axiom formalizes the notion that social preferences are not biased for or against any individual: the social preference between x and y depends only on the numbers of individuals who prefer (strictly and weakly) x over y, and not on their identities. Note that it also strengthens IIA.

Anonymity. For all x, y ∈ X and all θ, θ′ ∈ Θ,

\begin{matrix} p (x, y | θ) = p (x, y | θ'), \\ r (x, y | θ) = r (x, y | θ'), \\ and {xP}_{F} (θ) y \end{matrix}} \Rightarrow {xP}_{F} (θ') y .

The next axiom says, essentially, that social indifferences are sensitive to changes in individual preferences: if x and y are socially indifferent and if some individual increases their support for x then, after the change, x is strictly socially preferred to y. Note that the axiom only applies when two alternatives are initially socially indifferent, so it does not generally imply that a strict social preference is preserved by an increase in support for the preferred alternative. Nevertheless, this initial formulation of the axiom does impose restrictive conditions on the way individual indifferences are handled.

Strong tie break. For all x, y ∈ X and all θθ′ ∈ Θ,

\begin{matrix} P (x, y | θ) \subseteq P (x, y | θ'), \\ R (x, y | θ) \subseteq R (x, y | θ'), \\ at least one inclusion strict, \\ and {xI}_{F} (θ) y \end{matrix}} \Rightarrow {xP}_{F} (θ') y .

In particular, if x and y are socially indifferent, if one individual initially prefers y to x strictly, and if we consider another state where the individual is just indifferent between the two alternatives (all other individuals’ preferences between x and y held constant), then this breaks the social indifference in favor of x. The same is true if the individual is initially indifferent and then moves x above y. Of the above SPRs, only relative majority rule generally satisfies this condition.

The first proposition provides a formal statement of the restrictiveness of strong tie break among the anonymous and neutral SPRs. It is the closer of two results presented in this section to May’s original theorem. Note, however, that the axiom of strong tie break used here is weaker than May’s Condition IV (positive responsiveness), which incorporates the monotonicity condition of non-negative responsiveness, defined below, as well.

Proposition 1 (May (1952)). Assume PR(Θ) = U. A SPR F satisfies anonymity, neutrality, and strong tie break if and only if F = F_RM .

Proof. It is clear that F_RM satisfies these conditions. Now assume F does, and consider any a, b ∈ X and any θ ∈ Θ. I claim that p(a, b|θ) = p(b, y|θ) implies aI_F (θ)b. Consider a state θ′∈Θ such that P(a, b|θ′) = P(b, a|θ) and P(b, a|θ′) = P(a, b|θ). By anonymity, aP_F (θ′)b, but by neutrality, bP_F (θ′)a, a contradiction. It is now sufficient to show that p(a, b|θ) > p(b, a|θ) implies aP_F (θ)b. Assuming the inequality holds, let θ′ be a state such that P(b, a|θ′) = P(b, a|θ) and P(a, b|θ′) ⊊P(a, b|θ) with p(a, b|θ′) = p(b, a|θ′). From the above claim, we have aI_F (θ′)b, and strong tie break implies aP_F (θ)b, as required.□

As we have mentioned, Proposition 1 uses a strong tie break condition, one that exploits the possibility of individual indifferences in a perhaps undesirable way. When individual indifference is precluded, we can prove a similar result using a weaker version of the tie break condition that requires a strict preference reversal in order to break a social indifference.

Tie break. For all x, y ∈ X and all θ, θ′ ∈ Θ,

\begin{matrix} P (x, y | θ) \subseteq P (x, y | θ'), \\ R (x, y | θ) \subseteq R (x, y | θ'), \\ P (y, x | θ) \cap P (x, y | θ') \neq \emptyset, \\ and {xI}_{F} (θ) y \end{matrix}} \Rightarrow {xP}_{F} (θ') y .

The next axiom, which strengthens IIA, imposes a monotonicity condition to the effect that increased support for any x over any y should preserve a social preference for x over y. Note that in comparing individual preferences across states θ and θ′, the antecedent assumes that no individuals are indifferent between x and y in θ′, so in particular every individual who is indifferent in θ has a strict preference for x in θ′. This leads to a weaker non-negative responsiveness (NNR) condition than normally used.

NNR. For all distinct x, y ∈ X and all θ, θ′ ∈ Θ,

\begin{matrix} R (x, y | θ) \subseteq P (x, y | θ') \\ and {xP}_{F} (θ) y \end{matrix}} \Rightarrow {xP}_{F} (θ') y .

The following proposition assumes that only profiles of linear orders are possible and uses the weaker tie break axiom, above. Now, however, we must add the NNR axiom separately; see Example 1 in Section 6 for a counterexample to the statement of the proposition without NNR.

Proposition 2. Assume PR(Θ) = L. A SPR F satisfies anonymity, neutrality, NNR, and tie break if and only if F = F_SM = F_RM .

Proof. It is clear that F_SM satisfies these conditions. Now assume F does, and consider any a, b ∈ X and any θ ∈ Θ. We need to show aP_F (θ)b if and only if p(a, b|θ) > p(b, a|θ). First, assume aP_F (θ)b, and suppose p(a, b|θ) ≤ p(b, a|θ). Letting θ′ ∈ Θ be such that P(b, a|θ′) = P(a, b|θ) and P(a, b|θ′) = P(b, a|θ), neutrality implies bP_F (θ′)a. Since p(a, b|θ′) ≥ p(b, a|θ′), there is a state θ″ in which p(a, b|θ′) − p(b, a|θ′) ≥ 0 members of P(a, b|θ′) have their ab-preferences reversed (so now they prefer b to a), while all others’ab-preferences are unchanged. By NNR, bP_F (θ″)a; but

\begin{matrix} p (a, b | θ ″) = p (a, b | θ') - (p (a, b | θ') - p (b, a | θ')) \\ = p (b, a | θ') \\ = p (a, b | θ) \end{matrix}

and

\begin{matrix} p (b, a | θ ″) = p (b, a | θ') + (p (a, b | θ') - p (b, a | θ')) \\ = p (a, b | θ') \\ = p (b, a | θ) \end{matrix}

so aP_F (θ)b and anonymity imply aP_F (θ″)b, a contradiction. Second, assume p(a, b|θ) > p(b, a|θ), and suppose bR_F (θ)a. Let θ′ ∈ Θ be such that p(a, b|θ) − p(b, a|θ) > 0 members of P(a, b|θ) have their ab-preferences reversed (so now they prefer b to a), while all others’ab-preferences are unchanged. In the case bP_F (θ)a, NNR implies bP_F (θ′)a, and in the case bI_F (θ)a, tie break implies bP_F (θ′)a again. Letting θ″ ∈ Θ be such that P(a, b|θ″) = P(b, a|θ′) and P(b, a|θ″) = P(a, b|θ′), neutrality implies aP_F (θ″)b, but

\begin{matrix} p (a, b | θ ″) = p (b, a | θ') \\ = p (b, a | θ) + (p (a, b | θ) - p (b, a | θ)) \\ = p (a, b | θ) \end{matrix}

and

\begin{matrix} p (b, a | θ ″) = p (a, b | θ') \\ = p (a, b | θ) - (p (a, b | θ) - p (b, a | θ)) \\ = p (b, a | θ) \end{matrix}

so aP_F (θ″)b and anonymity implies aP_F (θ)b, a contradiction.□

Thus, the ostensibly intuitive conditions of neutrality and anonymity, together with the monotonicity conditions of NNR and tie break, lead us to majority rule. It is arguably for this reason, at least in part, that majority rule has occupied a central role in democratic collective choice. Before proceeding, we define a well-known axiom that plays a central role in normative social choice analysis.

Pareto. For all x, y ∈ X and all θ ∈ Θ, if p(x, y|θ) = n, then xP_F (θ)y.

That is, if every individual strictly prefers one alternative to another, then social preferences must agree with the unanimous assessment of the members of society. In what follows, we examine the implications of dropping the tie break axioms, while maintaining the Pareto axiom, when individual preferences are restricted to being single peaked.

4. Necessary conditions for transitivity

We begin with the observation that if all profiles of single-peaked linear orders are possible, then neutrality and transitivity of strict social preferences imply NNR. This elementary result is well known under unrestricted domain or linear domain; thus, Proposition 3 confirms that it carries over to the one-dimensional model, a fact that will be useful in drawing together results from this section and the next.

Proposition 3. Assume|X| ≥ 3 and PR(Θ) = S∩L. Let F be an SPR satisfying neutrality and Pareto. If P_F (θ) is transitive for all θ ∈ Θ, then F satisfies NNR.

Proof. Consider a, b ∈ X and θ, θ′∈Θ such that R(a, b|θ) ⊆P(a, b|θ′) and aP_F (θ)b. Let {G ₁, G ₂, G ₃} be the partition of N such that G ₁ = P(a, b|θ), G ₂ = P(b, a|θ) ∩P(a, b|θ′), and G ₃ = P(b, a|θ) ∩P(b, a|θ′). Now let c ∈ X \{a, b} be a distinct alternative, and consider a state θ″ ∈ Θ such that preferences restricted to {a, b, c} are as follows. In particular, individual preferences between a and b are unchanged compared to θ.

\begin{matrix} G_{1} G_{2} G_{3} \\ c c b \\ a b c \\ b a a \end{matrix}

Note that these preferences are single peaked with respect to the ordering a≺c≺b, so this specification is permissible. Since P(a, b|θ″) = G ₁ = P(a, b|θ) and P(b, a|θ″) = G ₂∪G ₃ = P(b, a|θ), neutrality and aP_F (θ)b imply aP_F (θ″)b. By Pareto, cP_F (θ″)a. Then transitivity and cP_F (θ″)aP_F (θ″)b imply cP_F (θ″)b. Since P(a, b|θ′) = G ₁∪G ₂ = P(c,b|θ″) and P(b, a|θ′) = G ₃ = P(b, c|θ″), neutrality and cP_F (θ″)b imply aP_F (θ′)b, as required.□

For completeness, the next result shows that the implication of Proposition 3 carries over to single-peaked domain with indifferences, as long as there are at least four alternatives. The assumption that there are at least four alternatives is needed for the result, as Section 6 contains a three-alternative example (Example 2) of an SPR satisfying anonymity, neutrality, Pareto, and transitivity of strict social preferences, yet violating NNR.⁵

Proposition 4. Assume|X| ≥ 4 and PR(Θ) = S. Let F be an SPR satisfying neutrality and Pareto. If P_F (θ) is transitive for all θ ∈ Θ, then F satisfies NNR.

Proof. Consider a, b ∈ X and θ, θ′ ∈ Θ such that R(a, b|θ) ⊆P(a, b|θ′) and aP_F (θ)b. Let {G ₁, G ₂, G ₃, G ₄, G ₅} be the partition of N such that G ₁ = P(a, b|θ), G ₂ = I(a, b|θ), G ₃ = P(b, a|θ) ∩P(a, b|θ′), G ₄ = P(b, a|θ) ∩I(a, b|θ′), and G ₅ = P(b, a|θ) ∩P(b, a|θ′). Now let c, d ∈ X\{a, b} be distinct alternatives, and consider a state θ″ ∈ Θ such that preferences restricted to {a, b, c, d} are as follows; in particular, individual preferences between a and b are unchanged compared to θ.

\begin{matrix} G_{1} G_{2} G_{3} G_{4} G_{5} \\ d d d d d \\ c c c bc b \\ a ab b a c \\ b a a \end{matrix}

Note that these preferences are single peaked with respect to the ordering a≺c≺d≺b, so this specification is permissible. Since P(a, b|θ″) = G ₁ = P(a, b|θ) and P(b, a|θ″) = G ₃∪G ₄∪G ₅ = P(b, a|θ), neutrality and aP_F (θ)b imply aP_F (θ″)b. By Pareto, cP_F (θ″)a, then transitivity and cP_F (θ″)aP_F (θ″)b imply cP_F (θ″)b. Since P(a, b|θ′) = G ₁∪G ₂∪G ₃ = P(c, b|θ″) and P(b, a|θ′) = G ₅ = P(b, c|θ″), neutrality and cP_F (θ″)b imply aP_F (θ′)b, as required.□

Next, we investigate the majoritarian structure implied by transitive strict social preference under linear, single-peaked domain: a strict social preference for one alternative over another can hold only if the first is majority preferred to the second. In this result and others, we view a relation P on X as a subset of ordered pairs, i.e. P⊆X × X. In line with this, the set inclusion P⊆P′ indicates that if xPy holds for a pair of alternatives, then xP′y also holds, so that preferences under P carry over to P′. Thus, Proposition 5 instructs us that under anonymity, neutrality, and Pareto, if an SPR always generates transitive strict social preferences, then xP_F (θ)y holds only if a strict majority preference also holds in the same direction.

Proposition 5. Assume|X| ≥ 3 and PR(Θ) = S∩L. Let F be an SPR satisfying anonymity, neutrality, and Pareto. If P_F (θ) is transitive for all θ ∈ Θ, then P_F (θ) ⊆P_SM (θ) = P_RM (θ) for all θ ∈ Θ.

Proof. Consider a, b ∈ X and θ ∈ Θ such that aP_F (θ)b, and suppose in order to deduce a contradiction that bR_SM (θ)a, so that p(a, b|θ) ≤ p(b, a|θ). Let {G ₁, G ₂, G ₃} be the partition of N such that G ₁ = P(a, b|θ) and G ₂∪G ₃ = P(b, a|θ), with |G ₂| = p(a, b|θ), and |G ₃| = p(b, a|θ) − p(a, b|θ). Consider a state θ′ such that preferences restricted to {a, b} are as follows.

\begin{matrix} G_{1} G_{2} G_{3} \\ a b a \\ b a b \end{matrix}

By Proposition 3, the SPR F satisfies NNR. Since R(a, b|θ) ⊆P(a, b|θ′), NNR and aP_F (θ)b imply aP_F (θ′)b. Now consider state θ″ such that preferences over a and b are as follows.

\begin{matrix} G_{1} G_{2} G_{3} \\ b a a \\ a b b \end{matrix}

Section 6 provides two examples to illustrate the boundaries of Proposition 5. First, Example 3 shows that the result does not hold for single-peaked domain when individual indifferences are allowed, even if the transitivity condition is strengthened to transitivity of weak social preference: there are SPRs satisfying anonymity, neutrality, and Pareto and that always generate transitive weak social preferences, yet admit the possibility that P_F (θ) ⊈P_RM (θ). Thus, the implications of transitive strict social preference are dulled by the presence of individual indifferences. Second, Example 4 shows that the result cannot be strengthened to the equality P_SM (θ) = P_RM (θ) = P_F (θ). Thus, transitivity of strict social preference is consistent with SPRs based on criteria that are more demanding than majority rule, and our axioms must be strengthened to obtain an exact characterization of majority rule.

In accordance with these observations, to deduce necessary conditions for transitivity under single-peaked domain while permitting individual indifferences, we consider transitivity of weak social preferences. With this stronger transitivity condition, we show that the number of individuals is odd, and that strict social preferences are nested between strict and weak simple majority preferences. Note that Example 3, in Section 6, shows that the conclusion of Proposition 6 cannot be strengthened to the equality P_F (θ) = P_SM (θ), or even to the nesting P_SM (θ) ⊆P_F (θ) ⊆P_RM (θ).⁶

Proposition 6. Assume|X| ≥ 3 and PR(Θ) = S. Let F be an SPR satisfying anonymity, neutrality, and Pareto. If R_F (θ) is transitive for all θ ∈ Θ, then n is odd and P_SM (θ) ⊆P_F (θ) ⊆R_SM (θ) for all θ ∈ Θ.

Proof. We first show that under the conditions of the proposition, for all θ ∈ Θ and all x, y ∈ X, xR_F (θ)y implies r(x, y|θ) > n/2. To this end, consider a, b ∈ X and θ ∈ Θ such that aR_F (θ)b, and suppose in order to deduce a contradiction that r(a, b|θ) ≤ n/2. Let {G ₁, G ₂, G ₃} be the partition of N such that G ₁ = P(a, b|θ), G ₂ = I(b, a|θ), and G ₃ = p(b, a|θ), and note that |G ₃| ≥ n/2. Let c ∈ X \{a, b} be a distinct alternative, and consider a state θ′ such that preferences restricted to {a, b, c} are as follows; in particular, individual preferences between a and b are unchanged compared to θ.

\begin{matrix} G_{1} G_{2} G_{3} \\ c c b \\ a ab c \\ b a \end{matrix}

Note that these preferences are single peaked with respect to the ordering a≺c≺b, so this specification is permissible. Since p(a, b|θ′) = p(a, b|θ) and p(b, a|θ′) = p(b, a|θ), anonymity and aR_F (θ)b implies aR_F (θ′)b. By Pareto, cP_F (θ′)a, then transitivity and cP_F (θ′)aR_F (θ′)b imply cP_F (θ′)b. Now let {H ₁, H ₂, H ₃} be a partition of N such that |H ₁| = |G ₁| + |G ₂|, |H ₂| = |G ₁| + |G ₂|, and |H ₃| = |G ₃| − |G ₁| − |G ₂|, and consider a state θ″ with preferences restricted {b, c} as follows.

\begin{matrix} H_{1} H_{2} H_{3} \\ c b b \\ b c c \end{matrix}

Since p(c, b|θ″) = p(c, b|θ′) and p(b, c|θ″) = p(b, c|θ′), anonymity and cP_F (θ′)b imply cP_F (θ″)b. Now consider a state θ‴ with preferences restricted to {a, b, c} as follows.

\begin{matrix} H_{1} H_{2} H_{3} \\ a b b \\ b c a \\ c a c \end{matrix}

These preferences are single peaked with respect to the ordering a≺b≺c, so this specification is admissible. Since P(a, b|θ‴) = P(c, b|θ″) and P(b, a|θ‴) = P(b, c|θ″), neutrality and cP_F (θ″)b imply aP_F (θ‴)b. By Pareto, bP_F (θ‴)c, and then transitivity and aP_F (θ‴)bP_F (θ‴)c imply aP_F (θ‴)c. Finally, consider a state θ″″ with preferences restricted to {a, c} as follows.

\begin{matrix} G_{1} G_{2} G_{3} \\ c c a \\ a a c \end{matrix}

Since p(a, c|θ″″) = |G ₃| = |H ₁| + |H ₃| = p(a,c|θ‴) and p(c, a|θ″″) = |G ₁| + |G ₂| = |H ₂| = p(c, a|θ‴), anonymity and aP_F (θ‴)c imply aP_F (θ″″)c. But since P(c, a|θ″″) = P(c, b|θ′) and P(a, c|θ″″) = P(b, c|θ′), neutrality and cP_F (θ′)b imply cP_F (θ″″)a, contradicting asymmetry of P_F (θ″″). We conclude that for all x, y ∈ X and all θ, xR_F (θ)y implies r(x, y|θ) > n/2, or equivalently, p(y, x|θ) ≥ n/2 implies yP_F (θ)x. An implication is that P_SM (θ) ⊆P_F (θ) ⊆R_SM (θ). Another implication is that n is odd, otherwise we can choose state θ and alternatives x and y such that p(x, y|θ) = p(y, x|θ) = n/2, and then xP_F (θ)y and yP_F (θ)x, contradicting asymmetry.□

An implication of Proposition 6 is an extension of Proposition 4 to the case of three alternatives. While Example 2 shows that the result does not carry over directly to the three-alternative case while maintaining only transitivity of strict social preference, it actually does extend if we strengthen the transitivity assumption to transitivity of weak social preference. In this case, retaining the background assumptions of anonymity, neutrality, and Pareto, Proposition 6 implies that n is odd and for all states θ, we have P_SM (θ) ⊆P_F (θ) ⊆R_SM (θ). Now, suppose R(x, y|θ) ⊆P(x, y|θ′) and xP_F (θ)y, as in the antecedent of NNR. We then have n/2 < r(x, y|θ) ≤ p(x, y|θ′), which implies xP_SM (θ)y and xP_F (θ′)y, fulfilling NNR. Because the case of exactly three alternatives is quite special, the formal statement of this corollary is omitted.

Finally, we return to single peakedness in the absence of individual indifference, and we continue to maintain transitivity of weak social preference. In case only the single-peaked linear orders are possible, indifference cannot arise in the proof of Proposition 6, and the proof goes through (with G ₂ = ∅). With Proposition 5, moreover, we can state the next proposition as an equivalence result: the only SPR that satisfies anonymity, neutrality, and Pareto and generates transitive weak social preferences in every state is majority rule.

Proposition 7. Assume|X| ≥ 3 and PR(Θ) = S∩L. Let F be an SPR satisfying anonymity, neutrality, and Pareto. If R_F (θ) is transitive for all θ ∈ Θ, then n is odd and F = F_SM = F_RM .

5. Sufficient conditions for transitivity

We now focus on the opposite logical direction and provide sufficient conditions under which single peakedness implies desirable transitivity properties for social preferences. First, we consider transitivity of strict preference and establish that for every SPR satisfying neutrality and NNR, if strict social preference implies strict relative majority preference, then the SPR always generates transitive strict social preferences. Note that anonymity and Pareto are not used. Example 5, in Section 6, defines a class of SPRs that satisfy neutrality, NNR, anonymity, and Pareto, and are intermediate between simple and relative majority rule; an implication of Proposition 8 is that the strict social preferences generated by this class are transitive.

Proposition 8. Assume PR(Θ) ⊆S. Let F be an SPR satisfying neutrality and NNR. For all θ ∈ Θ, if P_F (θ) ⊆P_RM (θ), then P_F (θ) is transitive.

Proof. Consider any θ ∈ Θ such that PR(θ) is single peaked with respect to $(≺, ⪯)$ , and consider any distinct a, b, c ∈ X. Assume aP_F (θ)bP_F (θ)c. Then one of the following cases must obtain

\begin{matrix} 1 . a ≺ b ≺ c 3 . b ≺ c ≺ a 5 . c ≺ a ≺ b \\ 2 . b ≺ a ≺ c 4 . c ≺ b ≺ a 6 . a ≺ c ≺ b \end{matrix}

Case 1. Consider any i ∈ R(a, b|θ). By single peakedness, we have ${\hat{x}}^{i} ≺ b$ , and then single peakedness implies bP_i (θ)c. Since aR_i (θ)bP_i (θ)c, transitivity implies aP_i (θ)c and therefore, R(a, b|θ) ⊆P(a, c|θ). Let {G ₁, G ₂, G ₃, G ₄, G ₅} be the partition of N such that G ₁ = P(a, b|θ), G ₂ = I(a, b|θ), G ₃ = P(b, a|θ) ∩P(a, c|θ), G ₄ = P(b, a|θ) ∩I(a, c|θ), and G ₅ = P(b, a|θ) ∩P(c, a|θ). Let {s, t} be a free pair, and consider states θ′, θ″ ∈ Θ such that individual preferences restricted to {s, t} are

\begin{matrix} G_{1} G_{2} G_{3} G_{4} G_{5} \\ s st t t t \\ t s s s \end{matrix}

(1)

in state θ′, and they are

\begin{matrix} G_{1} G_{2} G_{3} G_{4} G_{5} \\ s s s st t \\ t t t s \end{matrix}

(2)

Case 2. Consider any i ∈ R(b, c|θ), by single peakedness, ${\hat{x}}^{i} ≺ c$ . If ${\hat{x}}^{i} ⪯ a$ , then single peakedness implies aP_i (θ)c. If $a ≺ {\hat{x}}^{i} ≺ c$ , then single peakedness implies aP_i (θ)b. Since aP_i (θ)bR_i (θ)c, transitivity implies aP_i (θ)c and therefore, R(b, c|θ) ⊆P(a, c|θ). Let {G ₁, G ₂, G ₃, G ₄, G ₅} be the partition of N such that G ₁ = P(b, c|θ), G ₂ = I(b, c|θ), G ₃ = P(c, b|θ) ∩P(a, c|θ), G ₄ = P(c, b|θ) ∩I(a, c|θ), and G ₅ = P(c, b|θ) ∩P(c, a|θ). Let {s, t} be a free pair, and consider states θ′, θ″ ∈ Θ such that individual preferences restricted to {s, t} are as in (1) in state θ′ and as in (2) in state θ″. Since P(s, t|θ′) = G ₁ = P(b, c|θ) and P(t, s|θ′) = G ₃∪G ₄∪G ₅ = P(c, b|θ), neutrality and bP_F (θ)c imply sP_F (θ′)t. Since R(s, t|θ′) ⊆P(s, t|θ″), NNR then implies sP_F (θ″)t. Since P(a, c|θ) = G ₁∪G ₂∪G ₃ = P(s, t|θ″), neutrality then implies aP_F (θ)c, as required.

Case 3. Since aP_F (θ)b, we have aP_RM (θ)b, implying p(a, b|θ) > p(b, a|θ) and therefore r(a, b|θ) > n/2. Since bP_F (θ)c, we similarly have r(b, c|θ) > n/2. Then there is some individual i ∈ R(a, b|θ) ∩R(b, c|θ), but then c is bottom ranked among {a, b, c} for individual i, contradicting single peakedness with respect to $(≺, ⪯)$ . Thus, this case cannot occur. Case 4 is symmetric to Case 1, Case 5 is symmetric to Case 2, and Case 6 is symmetric to Case 3.□

The next result shows that the converse direction of Proposition 6 holds, even without the ancillary assumptions of anonymity, neutrality, and Pareto, and without the NNR axiom used in the preceding result. That is, if the number of individual is odd, if individual preferences are single peaked, and if strict social preferences respect simple majority rule, then the weak social preference is transitive. Compared to Proposition 8, we add the assumption that the number of individuals is odd and deduce the stronger conclusion that weak social preferences are transitive. Obviously, the result implies the well-known result that weak social preference for simple and relative majority rule are transitive.

Proposition 9. Assume PR(Θ) ⊆S and n is odd. Let F be an SPR. For all θ ∈ Θ, if P_SM (θ) ⊆P_F (θ), then R_F (θ) is transitive.

Proof. Consider any θ ∈ Θ such that PR(θ) is single peaked with respect to $(≺, ⪯)$ , and consider any distinct a, b, c ∈ X. Assume aR_F (θ)bR_F (θ)c. Then one of the six cases in the proof of Proposition 8 must be obtained.

Case 1. Since aR_F (θ)b, we have aR_SM (θ)b, which means r(a, b|θ) ≥ n/2. Since n is odd, the latter inequality must in fact hold strictly. Consider any i ∈ R(a, b|θ). By single peakedness, we have ${\hat{x}}^{i} ≺ b$ , and then single peakedness implies bP_i (θ)c. By transitivity, aP_i (θ)c, and therefore, R(a, b|θ) ⊆P(a, c|θ). Thus, p(a, c|θ) ≥ r(a, b|θ) > n/2, so aP_SM (θ)c, which implies aP_F (θ)c.

Case 2. Since bR_F (θ)c, we have bR_SM (θ)c, so r(b, c|θ) > n/2. Consider any i ∈ R(b, c|θ). By single peakedness, ${\hat{x}}^{i} ≺ c$ . If ${\hat{x}}^{i} ⪯ a$ , then single peakedness implies aP_i (θ)c. If $a ≺ {\hat{x}}^{i} ≺ c$ , then single peakedness implies aP_i (θ)b. By transitivity, aP_i (θ)c. Therefore, R(b, c|θ) ⊆P(a, c|θ), and it follows that p(a, c|θ) > n/2, so aP_SM (θ)c, which implies aP_F (θ)c.

Case 3. Since aR_F (θ)b, we have r(a, b|θ) > n/2. Since bR_F (θ)c, we similarly have r(b, c|θ) > n/2. Then there is some individual i ∈ R(a, b|θ) ∩R(b, c|θ), but then c is bottom ranked among {a, b, c} for individual i, contradicting single peakedness with respect to $(≺, ⪯)$ . Thus, this case cannot occur. Case 4 is symmetric to Case 1, Case 5 is symmetric to Case 2, and Case 6 is symmetric to Case 3.□

6. Examples

Example 1. To see that Proposition 2 does not hold without NNR, even if we impose Pareto, consider the non-majoritarian SPR specifying that one alternative is strictly socially preferred to another if either every individual prefers it or, failing that, a minority of individuals prefer it. Formally, this SPR is defined as follows: for all x, y ∈ X and all θ ∈ Θ,

{xP}_{F} (θ) y \Leftrightarrow {\begin{matrix} p (x, y | θ) = n or \\ 0 < p (x, y | θ) < \frac{n}{2} \end{matrix} .

Assuming n is odd, tie break is vacuously satisfied by this SPR, since social indifference never obtains. Note also that replacing tie break with transitivity of strict social preference does not preserve Proposition 2, because simple Pareto F_SP satisfies anonymity, neutrality, NNR, and transitivity of strict social preferences.

Example 2. Proposition 4 does not hold when there are just three alternatives. For example, assume n ≥ 5 is odd, and define the SPR F so that for all x, y ∈ X and all θ ∈ Θ, xP_F (θ)y if and only if both xP_RM (θ)y and p(x, y|θ) + p(y, x|θ) < n. This SPR satisfies anonymity, neutrality, and Pareto. I argue that strict social preferences are vacuously transitive. Consider any a, b, c ∈ X and any state θ, and suppose in order to deduce a contradiction that aP_F (θ)bP_F (θ)c. Then there is at least one individual i such that aI_i (θ)b, and for each such individual, it must be that cP_i (θ)aI_i (θ)b; otherwise, aI_i (θ)bR_i (θ)c, and the individual does not have an ideal point. Moreover, for every individual j with aP_j (θ)b, we must have cP_j (θ)b; otherwise, each alternative is bottom ranked by someone among {a, b, c}, contradicting single peakedness. Thus, we have R(a, b|θ) ⊆P(c, b|θ). Note that n odd and aP_F (θ)b imply r(a, b|θ) > n/2, which implies p(c, b|θ) > n/2, and thus not bP_F (θ)c, a contradiction. Nevertheless, F violates NNR. To see this for the n = 5 case, consider states θ, θ′ ∈ Θ such that preferences restricted to {a, b} are

\begin{matrix} 1 2 3 4 5 \\ a a a ab b \\ b b b a \end{matrix}

in θ and

\begin{matrix} 1 2 3 4 5 \\ a a a a b \\ b b b b a \end{matrix}

in θ′. Then we have aP_F (θ)b, and since R(a, b|θ) ⊆P(a, b|θ′), NNR would imply aP_F (θ′)b, but this does not hold.

Example 3. Proposition 5 does not extend to single-peaked domain when individual indifferences are allowed, even if transitivity of P_F (θ) is increased to transitivity of R_F (θ). For example, assume n ≥ 7 is odd, and define the SPR F so that for all x, y ∈ X and all θ ∈ Θ, xP_F (θ)y if and only if either (i) xP_SM (θ)y or (ii) both p(x, y|θ) = 1 and p(y, x|θ) = 2. Note that in case (ii), we have |I(x, y|θ)| = n − 3 > n/2, so P_F (θ) is asymmetric; in particular, we cannot have xP_SM (θ)y and |I(x, y|θ)| = n − 3. This SPR is anonymous and neutral, and it satisfies Pareto. I claim that R_F (θ) is transitive, or equivalently that P_F (θ) is negatively transitive, for all θ. Indeed, consider distinct alternatives a, b, c ∈ X and a state θ such that aP_F (θ)b. There are two cases. First, if aP_SM (θ)b, then by negative transitivity of P_SM (θ), from Proposition 9, it follows that either aP_SM (θ)c or cP_SM (θ)b, and therefore either aP_F (θ)c or cP_F (θ)b. Second, if p(a, b|θ) = 1 and p(b, a|θ) = 2, then |I(a, b|θ)| > n/2. For every individual i such that aI_i (θ)b, single peakedness implies that we cannot have aI_i (θ)c. Thus, there are two possibilities: (i) aP_i (θ)c and (ii) cP_i (θ)a. Furthermore, observe that by single peakedness, it cannot be that (i) holds for one individual i with aI_i (θ)b but (ii) holds for another individual j with aI_j (θ)b. Indeed, if (i) holds for i, then c is bottom ranked among {a, b, c} for i, and if (ii) holds for j, then a and b are bottom ranked among {a, b, c} for j, contradicting single peakedness. This observation implies that if aP_i (θ)c holds for one individual with aI_i (θ)b, then it holds for |I(a, b|θ)| > n/2 individuals, and thus aP_SM (θ)c, which implies aP_F (θ)c. Similarly, if cP_i (θ)a holds for one individual with aI_i (θ)b, then it hold for a majority of individuals, and we have cP_F (θ)a. This establishes negative transitivity of P_F (θ), as required.

Example 4. The conclusion of Proposition 5 cannot be strengthened by adding the inclusion P_SM (θ) ⊆P_F (θ), because strict simple Pareto preference, P_SP (θ), is transitive and we can have P_SM (θ) ⊈P_SP (θ).

Example 5. To construct a class of SPRs with social preferences nested between simple and relative majority rule, and given any k ∈ {1,…,(n + 1)/2}, define F_k as follows: for all x, y ∈ X and all θ ∈ Θ

{xP}_{F_{k}} (θ) y \Leftrightarrow p (x, y | θ) \geq k > p (y, x | θ)

This SPR clearly satisfies anonymity, neutrality, and Pareto, and if 1 < k < (n + 1)/2, then F_k is not equal to either version of majority rule. Since we have P_SM (θ) ⊆P_Fk (θ) ⊆P_RM (θ) for all θ ∈ Θ, Proposition 8 implies P_Fk (θ) is transitive in every state.

7. Conclusion

Combining the results of the previous sections, we obtain three versions of May’s theorem that exploit the structure of single peakedness, which plays a central role in formal political theory. Importantly, this structure allows us to replace the strong tie-break axiom of May with normatively compelling transitivity properties. When there are three or more alternatives and all profiles of single-peaked preferences are possible, Propositions 6 and 9 immediately yield the following corollary: among SPRs satisfying the background conditions of anonymity, neutrality, and Pareto, transitivity of weak social preference holds if and only if the number of individuals is odd and social preferences are nested between strict and weak simple majority rule. In particular, under anonymity, neutrality, and Pareto, Proposition 6 tells us that transitivity of weak majority preference implies that n is odd and that the inclusion P_SM (θ) ⊆P_F (θ) ⊆R_SM (θ) holds; and Proposition 9 provides the converse direction.

Corollary 10. Assume|X| ≥ 3 and PR(Θ) = S. Let F be an SPR satisfying anonymity, neutrality, and Pareto. Then R_F (θ) is transitive for all θ ∈ Θif and only if n is odd and P_SM (θ) ⊆P_F (θ) ⊆R_SM (θ) for all θ ∈ Θ.

When only the single-peaked linear orders are possible, using Propositions 3, 5, and 8 we obtain the following corollary: among SPRs satisfying anonymity, neutrality, and Pareto transitivity of strict social preferences holds if and only if an SPR satisfies NNR and strict social preference implies majority preference. For the result, we no longer need the full force of transitivity of weak social preference, and we obtain the result for an arbitrary (possibly even) number of individuals. Here, with anonymity, neutrality, and Pareto, Proposition 5 tells us that transitivity of strict social preference implies the inclusion P_F (θ) ⊆P_SM (θ) = P_RM (θ), and Proposition 3 tells us that if strict social preferences are transitive, then F satisfies NNR; and Proposition 8 provides the converse direction.

Corollary 11. Assume|X| ≥ 3 and PR(Θ) = S∩L. Let F be an SPR satisfying anonymity, neutrality, and Pareto. Then P_F (θ) is transitive for all θ ∈ Θif and only if F satisfies NNR and P_F (θ) ⊆P_SM (θ) = P_RM (θ).

Finally, we return to transitivity of weak social preference and obtain an exact characterization of majority rule under single-peaked linear domain that sharpens Corollary 11 and provides a close analogue of May’s theorem that fits naturally within the rubric of formal political theory: an SPR satisfies anonymity, neutrality, Pareto and always generates transitive weak social preference if and only if the number of individuals is odd and it is exactly equal to majority rule. Indeed, Proposition 7 shows that transitivity of weak social preference implies n is odd and F is equal to majority rule; and Proposition 9 yields the converse direction.

Corollary 12. Assume|X| ≥ 3 and PR(Θ) = S∩L. Then the SPR F satisfies anonymity, neutrality, and Pareto and is such that R_F (θ) is transitive for all θ ∈ Θif and only if n is odd and F = F_SM = F_RM .

The preceding corollary is related to Proposition 2 of Dasgupta and Maskin (2008), who show that majority rule satisfies anonymity, neutrality, Pareto, and ‘generic decisiveness’ on more domains than any other SPR. (Note that our definitions of anonymity and neutrality imply IIA, whereas those of Dasgupta and Maskin do not.) The framework of the latter authors is, however, rather different from that of this paper: (i) they assume a finite set of alternatives and a continuum of voters, (ii) they restrict the possible preferences of each individual to be a linear ordering of alternatives, and (iii) their generic decisiveness axiom has the effect of excluding preference profiles for which an SPR generates social indifference. Outside this set of excluded profiles, the decisiveness axiom is equivalent to transitivity of weak (which is the same as strict) social preference. In light of (iii), it is most appropriate to compare their result with Corollary 12, where n is assumed odd. The second part of Dasgupta and Maskin’s Theorem 2 states that for every SPR F distinct from majority rule, there is some domain on which majority rule satisfies their axioms, whereas F does not. Corollary 12 is more specific: it states that majority rules satisfies the axioms on the particular domain of profiles of single-peaked linear orders, whereas F does not.

Footnotes

Acknowledgements

I am grateful for discussions with Mark Fey. All errors are my own.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Notes

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