Majority Voting on Restricted Domains

Philosophy

Majority voting on restricted domains Franz Dietrich London School of Economics & University of Maastricht Christian List London School of Economics 21 November 2007, nal version 14 February 2010 Abstract In judgment aggregation, unlike preference aggregation, not much is known about domain restrictions that guarantee consistent majority outcomes. We introduce several conditions on individual judgments su¢ cient for consistent majority judgments. Some are based on global orders of propositions or individuals, others on local orders, still others not on orders at all. Some generalize classic socialchoice-theoretic domain conditions, others have no counterpart. Our most general condition generalizes Sen's triplewise value-restriction, itself the most general classic condition. We also prove a new characterization theorem: for a large class of domains, if there exists any aggregation function satisfying some democratic conditions, then majority voting is the unique such function. Taken together, our results support the robustness of majority rule. 1 Introduction In the theory of preference aggregation, it is well known that majority voting on pairs of alternatives may generate inconsistent (i.e., cyclical) majority preferences even when all individuals'preferences are consistent (i.e., acyclical). The most famous example is Condorcet's paradox. Here one individual prefers x to y to z, a second y to z to x, and a third z to x to y, and thus there are majorities for x against y, for y against z, and for z against x, a 'cycle'. But it is equally well known that if individual preferences fall into a suitably restricted domain, majority cycles can be avoided (see Gaertner [14] for an overview). The most famous domain restriction with this e¤ect is Black's single-peakedness [1]. A pro…le of individual preferences is single-peaked if the alternatives can be ordered from 'left'to 'right'such that each individual has a most preferred alternative with decreasing preference for other alternatives as we move away from it in either direction. Inada [17] showed that another condition called singlecavedness and interpretable as the mirror image of single-peakedness also su¢ ces for avoiding majority cycles: a pro…le is single-caved if, for some left-right order of the This paper draws on an unpublished draft circulated in July 2006. We are grateful to Lars Ehlers, Ashley Piggins, Ben Polak, Clemens Puppe, Alejandro Saporiti and two anonymous reviewers for helpful comments and discussions. Our work was supported by a Nu¢ eld Foundation New Career Development Fellowship. Christian List's work was also supported by a Philip Leverhulme Prize. 1 Preprint of an article in Journal of Economic Theory 145(2): 512-543, 2010 Official Publication: https://www.sciencedirect.com/science/article/pii/S0022053110000141 alternatives, each individual has a least preferred alternative with increasing preference for other alternatives as we move away from it in either direction. Sen [38] introduced a very general domain condition, called triplewise value-restriction, that guarantees acyclical majority preferences and is implied by Black's, Inada's and other conditions; it therefore uni…es several domain-restriction conditions, yet has a technical ‡avour without straightforward interpretation. The wealth of domain-restriction conditions for avoiding majority cycles was supplemented by another family of conditions based not on left-right orders of the alternatives, but on left-right orders of the individuals. Important conditions in this family are Grandmont's intermediateness [16] and Rothstein's order restriction ([34], [35]) with its special case of single-crossingness (e.g., Roberts [32], Saporiti and Tohmé [36], Saporiti [37]). To illustrate, a pro…le of individual preferences is order-restricted if the individuals – rather than the alternatives – can be ordered from left to right such that, for each pair of alternatives x and y, the individuals preferring x to y are either all to the left, or all the right, of those preferring y to x. In the theory of judgment aggregation, by contrast, domain restrictions have received much less attention (the only exception is the work on unidimensional alignment, e.g., List [22]). This is an important gap in the literature since, here too, majority voting with unrestricted but consistent individual inputs may generate inconsistent collective outputs, while on a suitably restricted domain such inconsistencies can be avoided. As illustrated by the much-discussed discursive paradox (e.g., Pettit [31]), if one individual judges that a, a! b and b, a second that a, but not a! b and not b, and a third that a! b, but not a and not b, there are majorities for a, for a! b and yet for not b, an inconsistency. But if no individual rejects a ! b, for example, this problem can never arise. Surprisingly, however, despite the abundance of impossibility results generalizing the discursive paradox as reviewed below, very little is known about the domains of individual judgments on which discursive paradoxes can occur (as opposed to agendas of propositions susceptible to such problems, which have been extensively characterized in the literature). If we can nd compelling domain restrictions to ensure majority consistency, this allows us to re…ne and possibly ameliorate the lessons of the discursive paradox. Going beyond the standard impossibility results, which all assume an unrestricted domain, we can then ask: in what political and economic contexts do the identi…ed domain restrictions hold, so that majority voting becomes safe, and in what contexts are they violated, so that majority voting becomes problematic? This paper has two goals. The rst is to introduce several conditions on pro…les of individual judgments that guarantee consistent majority judgments. These can be distinguished in at least two respects: rst, in terms of whether they are based on orders of propositions, on orders of individuals, or not on orders at all; and second, if they are based on orders, in terms of whether these are 'global'or 'local'. We further draw a distinction between product and non-product domains, which is relevant to game-theoretic applications. The second goal of the paper is to present a characterization result demonstrating the robustness of majority voting. In analogy with May's classic characterization of majority voting in binary choices [25] and Dasgupta and Maskin's theorem on the robustness of majority voting in preference aggregation [2], we show that, for a very 2 large class of domains, if there exists any aggregation function that satis…es some minimal democratic conditions including consistency of its outcomes, then majority voting is the unique such function. In combination with our domain-restriction conditions, this theorem provides a powerful argument for majority voting in a broad range of circumstances. We pursue our two goals in reverse order, beginning with the characterization of majority voting, followed by the discussion of domain restrictions. We state our results for the general case in which individual and collective judgments are only required to be consistent; they need not be complete (i.e., they need not take a view on every proposition-negation pair). But we also consider the important special case of full rationality, i.e., the conjunction of consistency and completeness. Some of our proofs are given in the main text, others in the appendix. Let us brie‡y comment on how the two central distinctions underlying the domainrestriction conditions discussed in this paper relate to the literature on domain restrictions in preference aggregation. First, as noted, some of our conditions are based on orders of propositions, others on orders of individuals, and yet others not on orders at all. The conditions based on orders of individuals generalize some of the conditions on preferences reviewed above, notably Grandmont's intermediateness and Rothstein's order restriction, and reduce to them when applied to judgments on binary ranking propositions that can represent preferences (such as xPy, yPz, xPz and so on). By contrast, the conditions based on orders of propositions are not obviously analogous to any standard conditions on preferences. (An exception may be La¤ond and Lainé's [20] condition of single-switch preferences in the di¤erent context of Ostrogorski's paradox, where individuals'most preferred positions on multiple, albeit unconnected, issues are restricted relative to some order of issues.) While an order of individuals can be interpreted similarly in judgment and preference aggregation –namely in terms of the individuals'positions on a normative or cognitive dimension –an order of propositions in judgment aggregation is conceptually distinct from an order of alternatives in preference aggregation. Propositions, unlike alternatives, are not mutually exclusive. It is therefore surprising that su¢ cient conditions for consistent majority judgments can be given even based on orders of propositions. We also introduce a very general domain-restriction condition not based on orders at all, which generalizes Sen's condition of triplewise value-restriction, and characterize the maximal domain on which majority voting yields consistent collective judgments. Secondly, as we have also pointed out, our domain-restriction conditions based on orders admit global and local variants. In the global case, the individuals'judgments on all propositions on the agenda are constrained by the same left-right order of propositions or individuals, whereas in the local case, that order may di¤er across subsets of the agenda. To relate this to the more familiar context of preference aggregation, single-peakedness and single-cavedness are global conditions, whereas the restriction of these conditions to triples of alternatives yields local ones. But while in preference aggregation local conditions result from the restriction of global conditions to triples of alternatives, the picture is more general in judgment aggregation. Here di¤erent left-right orders may apply to di¤erent subagendas, which correspond to different semantic elds. We give precise criteria for selecting appropriate subagendas. An individual can be left-wing on a 'social'subagenda and right-wing on an 'economic' one, for example. 3 Finally, a few remarks about the literature on judgment aggregation are due. The recent eld of judgment aggregation emerged from the areas of law and political philosophy (e.g., Kornhauser and Sager [19] and Pettit [31]) and was formalized social-choice-theoretically by List and Pettit [23]. The literature contains several impossibility results generalizing the observation that on an unrestricted domain majority judgments can be logically inconsistent (e.g., List and Pettit [23] and [24], Pauly and van Hees [30], Dietrich [3], Gärdenfors [15], Nehring and Puppe [29], van Hees [39], Mongin [26], Dietrich and List [7], and Dokow and Holzman [12]). Some of these impossibility results build on Nehring and Puppe's [27] results on strategy-proof social choice in the framework of property spaces. Earlier precursors include works on abstract aggregation (Wilson [40], Rubinstein and Fishburn [33]). But so far the only domain-restriction condition known to guarantee consistent majority judgments is List's unidimensional alignment ([21], [22]), a global non-product domain condition based on orders of individuals. 2 The model We consider a group of individuals N = f1; 2;...; ng (n  2) making judgments on some propositions. To represent propositions, we use Dietrich's [4] model of general logics, which generalizes the approach in List and Pettit [23] and [24]. Logic. A logic is given by a language and a notion of consistency. The language is a non-empty set L of sentences (called propositions) closed under negation (i.e., p 2 L implies :p 2 L, where : is the negation symbol). For example, in standard propositional logic, L contains propositions such as a, b, â b, a_ b, :(a! b), where ^, _,! denote 'and', 'or', 'if-then', respectively. In other logics, the language may involve additional connectives, such as modal operators ('it is necessary/possible that'), deontic operators ('it is obligatory/permissible that'), subjunctive conditionals ('if p were the case, then q would be the case'), or quanti…ers ('for all/some'). The notion of consistency captures the logical connections between propositions by stipulating that some sets of propositions S  L are consistent (and the others inconsistent), subject to some regularity axioms.1 A proposition p 2 L is a contradiction if fpg is inconsistent and a tautology if f:pg is inconsistent. We further say that a set S  L entails a proposition p 2 L if S[f:pg is inconsistent. For example, in standard logics, fa; a ! b; bg and fa ^ bg are consistent and fa;:ag and fa; a ! b;:bg inconsistent; a ^ :a is a contradiction and a _ :a a tautology; and fa; a! bg entails b. Agenda. The agenda is the set of propositions on which judgments are to be made. It is a non-empty set X  L expressible as X = fp;:p : p 2 X+g for some set X+ of unnegated propositions (this avoids double-negations inX). In our introductory example, the agenda is X = fa;:a; a ! b;:(a ! b); b;:bg. For convenience, we assume that X is nite.2 As a notational convention, we cancel double-negations in 1Self-entailment: Any pair fp;:pg  L is inconsistent. Monotonicity: Subsets of consistent sets S  L are consistent. Completability: ; is consistent, and each consistent set S  L has a consistent superset T  L containing a member of each pair p;:p 2 L. See Dietrich [4]. 2For in…nite X, our results hold either as stated or under compactness of the logic. 4 front of propositions in X.3 Further, for any Y  X, we write Y  = fp;:p : p 2 Y g to denote the (single-)negation closure of Y . Judgment sets. An individual's judgment set is the set A  X of propositions in the agenda that he or she accepts (e.g., 'believes'). A pro…le is an n-tuple (A1; : : : ; An) of judgment sets across individuals. A judgment set is consistent if it is consistent in L; it is complete if it contains at least one member of each proposition-negation pair p;:p 2 X; it is opinionated if it contains precisely one member of each propositionnegation pair p;:p 2 X. (Clearly, consistency and completeness jointly imply opinionation.) Our results mostly do not require completeness, in line with several works on the aggregation of incomplete judgments (Gärdenfors [15]; Dietrich and List [9], [10], [11]; Dokow and Holzman [13]; List and Pettit [23]). This strengthens our possibility results as the identi…ed possibilities hold on larger domains of pro…les. But we also consider the complete case. Aggregation functions. A domain is a set D of pro…les, interpreted as admissible inputs to the aggregation. An aggregation function is a function F that maps each pro…le (A1; : : : ; An) in a given domain D to a collective judgment set F (A1; : : : ; An) = A  X. While the literature focuses on the universal domain, which consists of all pro…les of consistent and complete judgment sets, we here focus mainly on domains that are less restrictive in that they allow for incomplete judgments, but more restrictive in that we impose some structural conditions. We call an aggregation function consistent or complete, respectively, if it generates a consistent or complete judgment set for each pro…le in its domain. The majority outcome on a pro…le (A1; :::; An) is the judgment set fp 2 X : there are more individuals i 2 N with p 2 Ai than with p =2 Aig. The aggregation function that generates the majority outcome on each pro…le in its domain D is called majority voting on D.4 Preference aggregation as a special case. To relate our results to existing results on preference aggregation, we must explain how preference aggregation can be represented in our model (following Dietrich and List [7] and List and Pettit [24]). Since preference relations are binary relations on a set of alternatives K = fx; y; :::g, they can be represented as judgments on an agenda of binary ranking propositions of the form xPy ('x is preferable to y'), where x; y 2 K. Formally, the preference agenda is XK = fxPy 2 L : x; y 2 Kg  L, where L is a simple predicate language with the set of constants K (representing alternatives) and the two-place predicate P (representing strict preference), and any set S  L is consistent if it is consistent with the rationality conditions on strict 3More precisely, if p 2 X is already of the form p = :q, we write :p to mean q rather than ::q. This ensures that, whenever p 2 X, then :p 2 X. 4Other widely discussed aggregation functions include dictatorships, supermajority functions, and premise-based or conclusion-based functions. 5 preferences.5 Now preference relations and opinionated judgment sets stand in a oneto-one correspondence:  To any preference relation (arbitrary binary relation) on K corresponds the opinionated judgment set A  XK such that A = fxPy : x; y 2 K and x yg [ f:xPy : x; y 2 K and x 6 yg.  Conversely, to any opinionated judgment set A  XK corresponds the preference relation A on K such that, for all x; y 2 K, x A y , xPy 2 A. Since we have built the rationality conditions on preferences into the notion of consistency governing the logic, a preference relation is fully rational (i.e., asymmetric, transitive and connected) if and only if A is consistent. Moreover, a judgment aggregation function (for opinionated judgment sets) represents a preference aggregation function, and majority voting as de…ned above corresponds to pairwise majority voting in the standard Condorcetian sense. 3 Why majority voting? To motivate our focus on majority voting, we begin by presenting a new characterization of it on a large class of domains. We use two democratic conditions in addition to the requirement of consistent collective judgment sets. Anonymity. For any pro…les (A1; : : : ; An), (A1; : : : ; A  n) in the domain of F that are permutations of each other, F (A1; : : : ; An) = F (A1; : : : ; A  n): Acceptance/rejection neutrality. For any pro…les (A1; : : : ; An), (A1; : : : ; A  n) in the domain of F and any proposition p 2 X, [for all i 2 N , p 2 Ai , p 62 Ai ] ) [p 2 F (A1; : : : ; An), p 62 F (A1; : : : ; An)]. Both conditions are familiar from May's classic characterization of majority voting in a single binary choice [25].6 Anonymity requires equal treatment of all individuals, and acceptance/rejection neutrality prevents the aggregation function from favouring the acceptance of a proposition over its rejection or vice versa; i.e., if the individuals accepting a given proposition in one pro…le are the same as those rejecting it in another, then the proposition must be collectively accepted in the rst pro…le if and only if it is collectively rejected in the second.7 5Formally, this requires S [ Z to be consistent in the standard sense of predicate logic, where Z consists of (8v1)(8v2)(v1Pv2 ! :v2Pv1) (asymmetry), (8v1)(8v2)(8v3)((v1Pv2 ^ v2Pv3) ! v1Pv3) (transitivity), (8v1)(8v2)(: v1 = v2 ! (v1Pv2 _ v2Pv1)) (connectedness) and, for each pair of distinct constants x; y 2 K, :x = y (exclusiveness of alternatives). 6Further, if we require consistency and completeness of individual and collective judgment sets, acceptance/rejection neutrality becomes equivalent to 'unbiasedness'(Dietrich and List [6]) and, suitably translated, 'neutrality-within-issues'(Nehring and Puppe [28]). 7Majority voting satis…es acceptance/rejection neutrality as stated here only if n is odd, since rejection by exactly n=2 individuals leads to rejection but acceptance by the same n=2 individuals does not lead to acceptance. This problem can be bypassed by subtly weakening acceptance/rejection neutrality, but for simplicity we set these technicalities aside. 6 Suppose, for the purposes of our rst theorem, that the agenda X contains no tautologies or contradictions.8 Call a domain D minimally rich if it includes all bipolar pro…les, where a pro…le (A1; :::; An) is bipolar if there exists a proposition p 2 X such that every non-empty Ai is either fpg or f:pg. Theorem 1 If an aggregation function F on a minimally rich domain D is consistent, anonymous and acceptance/rejection neutral, then it is majority voting on D. This result is surprising in at least two respects. First, unlike May's theorem, it requires no monotonicity condition on the aggregation function; monotonicity follows from the other conditions. Second, unlike almost all results in the eld of judgment aggregation, it requires no assumptions about the agenda, apart from the exclusion of tautologies and contradictions. Existing theorems usually need some agenda complexity assumptions, for example to derive monotonicity if it is not explicitly imposed; so the validity of a theorem for essentially all agendas is rather atypical. How can we interpret Theorem 1? As noted in the introduction, its lesson is somewhat similar to that of Dasgupta and Maskin's much-discussed result on the robustness of majority voting in preference aggregation [2]. Theorem 1 shows that, for all minimally rich domains, if there is any consistent aggregation function that satis…es anonymity and acceptance/rejection neutrality, then majority voting is the unique such function. Practically all interesting and non-degenerate domains, such as those introduced below, fall into this class of domains. To prove Theorem 1, we rst state a lemma, proved in the appendix. Using standard terminology, call aggregation function F independent if, for any pro…les (A1; : : : ; An), (A1; : : : ; A  n) in the domain of F and any proposition p 2 X, [for all i 2 N , p 2 Ai , p 2 Ai ] ) [p 2 F (A1; : : : ; An), p 2 F (A1; : : : ; An)]. Lemma 1 Every consistent and acceptance/rejection neutral aggregation function F on a minimally rich domain D is independent. Proof of Theorem 1. Consider any agenda X without tautologies, and let F and D be as speci…ed. By Lemma 1, F is independent. For every p 2 X, let Kp be the set of numbers k 2 f0; :::; ng such that p 2 F (A1; :::; An) for some (and hence, by independence and anonymity, every) pro…le (A1; :::; An) 2 D with jfi : p 2 Aigj = k. We prove three claims, the second one being the key step. Claim 1: For all p 2 X and all k 2 f0; :::; ng, k 2 Kp , n k =2 Kp. 8Our other results do not require this restriction. Whether Theorem 1 continues to hold when tautologies and contradictions are permitted in X depends on how the de…nition of bipolarity (and by implication minimal richness) is extended to this case. If, in the de…nition of bipolarity, we quantify over all propositions p 2 X, including tautologies and contradictions, minimally rich domains will be forced to include pro…les containing inconsistent judgment sets, which renders minimal richness less interpretationally plausible. But then the theorem continues to hold. If, on the other hand, we extend bipolarity by quantifying only over non-tautological and non-contradictory propositions p 2 X, there are counterexamples to the theorem: Let X contain a tautology t, and let X have no minimal inconsistent subset of size three or more (so that majority voting is consistent). Let n be odd and de…ne F on the smallest minimally rich agenda D as follows: (i) t 2 F (A1; :::; An) , jfi : t 2 Aigj < n=2; (ii) :t =2 F (A1; :::; An); (iii) for all p 2 Xnft;:tg, p 2 F (A1; :::; An), jfi : p 2 Aigj > n=2. 7 Consider any p 2 X and any k 2 f0; ::; ng. Let C  N be a coalition of size k. As D is minimally rich, it contains a pro…le (A1; :::; An) for which fi 2 N : p 2 Aig = C (take the bipolar pro…le given by Ai = fpg for i 2 C, and Ai = ? for i =2 C). Analogously, there exists a pro…le (A1; :::; A  n) 2 D such that fi 2 N : p 2 Ai g = NnC. By acceptance/rejection neutrality, p 2 F (A1; :::; An) , p =2 F (A1; :::; An). In this equivalence, the left-hand-side is equivalent to k 2 Kp, and the right-hand-side to n k =2 Kp. So k 2 Kp , n k =2 Kp, as required. Claim 2: For all p 2 X and all k 2 f0; :::; ng, k 2 Kp ) k > n=2. Let p 2 X, and assume for a contradiction that Kp contains k  n=2. By Claim 1, K:p contains exactly one of k; n  k. De…ne k as k if k 2 K:p and as n  k if n  k 2 K:p. As k  n=2, we have k + k  n. So, there is a pro…le (A1; :::; An) in which exactly k of the sets Ai are fpg, exactly k of them are f:pg, while the rest (if any) of them are empty. As D is minimally rich, it contains this pro…le. As k 2 Kp and k 2 K:p, we have p;:p 2 F (A1; :::; An), contradicting consistency. Claim 3: For all p 2 X and all k 2 f0; :::; ng, k 2 Kp , k > n=2 (which completes the proof that F is majority voting on D). Let p 2 X and k 2 f0; :::; ng. By Claim 2, k 2 Kp ) k > n=2. Conversely, let k =2 Kp. Then n  k 2 Kp by Claim 1. So, by Claim 2, n  k > n=2, i.e. k < n=2. Hence k 6> n=2, as required.  4 Conditions for majority consistency based on global orders We have seen that, on every minimally rich domain, if there is any consistent aggregation function at all that satis…es anonymity and acceptance/rejection neutrality, then majority voting is the unique such function. But we already know from the discursive paradox that without any domain restriction majority voting can be inconsistent. Majority inconsistencies can arise on the universal domain whenever the agenda has a minimal inconsistent subset of three or more propositions (i.e., an inconsistent subset of that size all of whose proper subsets are in turn consistent), such as the set fa; a ! b;:bg in the example from the introduction.9 However, we now show that there exist many compelling domains on which majority voting is consistent. On these domains, then, majority voting not only follows from the conditions of Theorem 1 but also satis…es them.10 4.1 Conditions based on orders of propositions We begin with two conditions based on global orders of the propositions. An order of the propositions (in X) is a linear order  on X.11 9For a proof of this fact under consistency alone, see Dietrich and List [9]; under full rationality, see Nehring and Puppe [28]. 10For odd n; recall the earlier note about even n. 11Thus  is re‡exive (x  x 8x), transitive ([x  y and y  z] ) x  z 8x; y; z), connected (x 6= y ) [x  y or y  x] 8x; y) and antisymmetric ([x  y and y  x]) x = y 8x; y). 8 Single-plateauedness. A judgment set A is single-plateaued relative to  if A = fp 2 X : pleft  p  prightg for some pleft; pright 2 X, and a pro…le is (A1; :::; An) is single-plateaued relative to  if every Ai is singleplateaued relative to . Single-canyonedness. A judgment set A is single-canyoned relative to  if A = Xnfp 2 X : pleft  p  prightg for some pleft; pright 2 X, and a pro…le is (A1; :::; An) is single-canyoned relative to  if every Ai is singlecanyoned relative to .12 An order  that renders a pro…le single-plateaued or single-canyoned is called a structuring order ; it need not be unique. If a pro…le is single-plateaued or singlecanyoned relative to some , we also call it single-plateaued or single-canyoned simpliciter. The order  may represent a normative or cognitive dimension on which propositions are located. Informally, single-plateauedness requires that every individual's judgment set constitute an interval (a 'plateau') relative to a particular left-right order of the propositions; single-canyonedness that every individual's set of rejected propositions (i.e., the complement of his or her judgment set) form such an interval (a 'canyon'). As an illustration, consider the agenda X = fa; b; a! bg, with the following interpretation: a : 'CO2 emissions will increase dramatically by 2020.' b : 'The frequency of hurricanes will double by 2030.' a! b : 'If CO2 emissions increase dramatically by 2020, then the frequency of hurricanes will double by 2030.' Now it is conceivable that individuals hold single-plateaued judgment sets on this agenda relative to an order of the propositions from 'most pessimistic'to 'most optimistic'. Proposition a can plausibly be described as more pessimistic than b, because of its consequences over and above the occurrence of hurricanes; b as more pessimistic than a ! b, since the latter entails b only under the pessimistic circumstances of a; and the three unnegated propositions as more pessimistic than the three negated ones. Table 1 shows a pro…le of single-plateaued judgment sets relative to this order. The location of each individual's plateau re‡ects his or her viewpoint on the issue of global warming. Propositions (in the order) a b a! b :a :b :(a! b) Individual 1 (a 'pessimist') X X X Individual 2 (a 'moderate') X X X Individual 3 (an 'optimist') X X Table 1: A single-plateaued pro…le 12 In the de…nitions of single-plateauedness and single-canyonedness, we do not require pleft  pright, i.e., fp : pleft  p  prightg may be empty. 9 By contrast, an individual who accepts only very pessimistic propositions and very optimistic ones, but nothing in between, holds a single-canyoned judgment set relative to the given order, as illustrated in Table 2. (Important special cases of singlecanyoned judgment sets are also those in which only 'extreme'propositions of one kind –i.e., only pessimistic ones or only optimistic ones –are accepted.) Propositions (in the order) a b a! b :a :b :(a! b) An individual (an 'extremist') X X X Table 2: A single-canyoned judgment set relative to the given order We can easily think of other cases in which single-plateauedness is plausible. If the agenda contains propositions about various tax or budget policies, for instance, the propositions may be ordered on a classical socio-economic dimension from 'socialist' to 'libertarian', with the individuals'plateaus representing their political positions. If the agenda contains propositions about science education in public schools, the order may range from 'closest to endorsing evolutionary theory' to 'closest to endorsing creationism', with individual plateaus representing di¤erent educational viewpoints. Before we state our main result about the implications of single-plateauedness and single-canyonedness, we observe that every single-canyoned pro…le is single-plateaued, as proved in the appendix. Our proof reorders the propositions so as to 'glue together' any individual's two extreme sets of propositions into a single plateau. Remark 1 Every single-canyoned pro…le (A1; :::; An) of consistent judgment sets is single-plateaued. As anticipated, majority voting preserves consistency on single-plateaued pro…les. On single-canyoned pro…les, it does even more: it also preserves single-canyonedness. Proposition 1 For any pro…le (A1; :::; An) of consistent judgment sets, (a) if (A1; :::; An) is single-plateaued, the majority outcome is consistent; (b) if (A1; :::; An) is single-canyoned, the majority outcome is consistent and singlecanyoned (relative to the same structuring order). Proof. Consider a pro…le (A1; :::; An). The following notation is used in this and other proofs. Let A be the majority outcome. For each p 2 X, de…ne Np = fi 2 N : p 2 Aig. Whenever we consider an order  of X, let [p; q] = fr 2 X : p  r  qg, for each p; q 2 X. An order  is sometimes identi…ed with the corresponding ascending list of propositions p1:::p2k (from left to right), where 2k is the size of X (which is even as X is a union of pairs fp;:pg). Now let each Ai be consistent. (a) Assume single-plateauedness, say relative to . Among all propositions in A, let p and q be, respectively, the smallest and largest proposition with respect to . So A  [p; q]. As Np and Nq each contain a majority of the individuals, we have Np \Nq 6= ?, and so there is an i 2 Np \Nq. As Ai is single-plateaued and p; q 2 Ai, we have [p; q]  Ai and thus A  Ai. Therefore A is consistent. (b) Let (A1; :::; An) be single-canyoned, say relative to . By part (a) and Remark 1, A is consistent. As one easily checks, A is single-canyoned relative to  if and only 10 if, for all p 2 A, we have fq 2 X : q  pg  A or fq 2 X : q  pg  A. So it su¢ ces to establish the right-hand side of this equivalence. Consider any p 2 A. Check that either (i) jfq 2 X : q  pgj  k < jfq 2 X : p  qgj or (ii) jfq 2 X : p  qgj  k < jfq 2 X : q  pgj. We assume (i) and show that fq 2 X : q  pg  A (analogously, (ii) implies fq 2 X : p  qg  A). For each i 2 Np, single-canyonedness implies that fq 2 X : q  pg  Ai or fq 2 X : p  qg  Ai. But the latter is impossible: otherwise jAij > k by (i), so that Ai would contain a pair p;:p, contradicting consistency. So we have fq 2 X : q  pg  Ai for all i 2 Np and thus for a majority of the individuals. It follows that fq 2 X : q  pg  A, as required.  4.2 Conditions based on orders of individuals Let us now turn to two conditions based on global orders of the individuals. An order of the individuals (in N) is linear order on N . For any sets of individuals N1; N2  N; we write N1 N2 if i j for all i 2 N1 and j 2 N2. Unidimensional orderedness.13 A pro…le (A1; :::; An) is unidimensionally ordered relative to if, for all p 2 X, fi 2 N : p 2 Aig = fi 2 N : ileft i irightg for some ileft; iright 2 N . Unidimensional alignment. (List [22]) A pro…le (A1; :::; An) is unidimensionally aligned relative to if, for all p 2 X, fi 2 N : p 2 Aig fi 2 N : p =2 Aig or fi 2 N : p =2 Aig fi 2 N : p 2 Aig. In analogy to the earlier de…nition, an order that renders a pro…le unidimensionally ordered or unidimensionally aligned is called a structuring order ; again, it need not be unique. If a pro…le is unidimensionally ordered or unidimensionally aligned relative to some , we also call it unidimensionally ordered or unidimensionally aligned simpliciter. Unidimensional alignment is the special case of unidimensional orderedness in which, for every p 2 X, at least one of ileft; iright is the left-most or right-most individual in the structuring order . Remark 2 Every unidimensionally aligned pro…le (A1; :::; An) is unidimensionally ordered. Informally, a pro…le is unidimensionally ordered if the individuals can be ordered from left to right such that, for each proposition, the individuals accepting it are all adjacent to each other. A pro…le is unidimensionally aligned if, in addition, the individuals accepting each proposition are either all to the left or all to right of those rejecting it. The order of the individuals can be interpreted as re‡ecting their location on some underlying normative or cognitive dimension. To illustrate unidimensional orderedness and unidimensional alignment, consider again the agenda X = fa; b; a! bg, this time with the following interpretation: 13 In this de…nition, we do not require ileft iright, i.e., fi : ileft i irightg may be empty. 11 a : 'A growth in government expenditure is acceptable.' b : 'Defence spending should be increased.' a! b : 'If a growth in government expenditure is acceptable, then defence spending should be increased.' We can imagine a political left-right order of individuals such that those on the left tend to nd a growth in government expenditure acceptable (proposition a), while those further to the right accept its negation (:a); moreover, those on the right tend to favour an increase in defence spending (proposition b), while those far enough to the left accept its negation (:b); and nally, those in the middle tend to accept a connection between the two (proposition a! b), while others are either uncommitted on this matter or accept its negation (:(a ! b)). The resulting pro…le satis…es unidimensional orderedness, as shown in Table 3. Individuals (in the order) 1 2 3 4 5 a X X X b X X X a! b X X :a X X :b X :(a! b) X X Table 3: A unidimensionally ordered pro…le In this example, the pro…le would become unidimensionally aligned if individual 5 accepted rather than rejected a ! b, thereby making it the case that the individuals accepting each proposition are opposite those rejecting it on the given left-right order. Table 4 shows the required modi…cation of the pro…le. Individuals (in the order) 1 2 3 4 5 a! b X X X Table 4: A unidimensionally aligned combination of judgments relative to the given order Below we o¤er a uni…ed interpretation of all four domain-restriction conditions introduced so far. Let us now turn to the implications of unidimensional orderedness and unidimensional alignment. On unidimensionally ordered pro…les, majority voting preserves consistency, and its outcome is always a subset of the middle individual's judgment set (or, for even n, a subset of the intersection of the two middle individuals' judgment sets). If the pro…le is unidimensionally aligned, the majority outcome is not just included in that set but coincides with it. Proposition 2 For any pro…le (A1; :::; An) of consistent judgment sets, (a) if (A1; :::; An) is unidimensionally ordered, the majority outcome A is consistent and A   Am if n is odd, Am1 \Am2 if n is even, where m is the middle individual (if n is odd) and m1;m2 the middle pair of individuals (if n is even) in any structuring order ; 12 (b) (List [22]) if (A1; :::; An) is unidimensionally aligned, the majority outcome is as stated in part (a) with  replaced by =. Proof. Let each Ai be consistent. We use earlier proof notation. (a) Suppose unidimensional orderedness, say relative to . For all p 2 A, Np is some interval [ileft; iright]. By jNpj > n=2, [ileft; iright] is long enough to contain the middle individual m (if n is odd) or the middle pair of individuals m1;m2 (if n is even); so p 2 Am (if n is odd) or p 2 Am1 \Am2 (if n is even). Therefore A  Am (if n is odd) or A  Am1 \Am2 (if n is even), as required. By implication, A is consistent. (b) See List [22], or check that, under unidimensional alignment, the converse inclusions Am  A (if n is odd) or Am1 \Am2  A (if n is even) also hold in the proof of (a).  4.3 A uni…ed interpretation of all four conditions Although the four domain-restriction conditions introduced up to this point are quite distinct from each other, they can all be interpreted in terms of a common spatial framework. To introduce this framework, suppose that there exists a single left-right axis on which both propositions and individuals are located, as illustrated in Figure 1 for three individuals and six propositions. Each of the four conditions can now be Figure 1: The positions of three individuals and six propositions interpreted in terms of the particular way in which the locations of the individuals and the propositions constrain judgments. Single-plateauedness and single-canyonedness are de…ned in terms of acceptance regions assigned to individuals, relative to such a spatial representation. Speci…cally, a pro…le is single-plateaued if each individual accepts all propositions away from his or her location by at most a certain distance, where the 'cut-o¤'distance may di¤er from individual to individual (see Case 1 in Figure 2). If each individual's acceptance interval is left-justi…ed or right-justi…ed (a property met only by individual 1 in the example of Case 1), we obtain a special case of single-canyonedness. A general interpretation of single-canyonedness is obtained by reinterpreting someone's 'location'as the position he or she deems least acceptable, rather than most acceptable, and by assuming that each individual rejects, rather than accepts, all propositions away from his or her location by at most a certain distance (see Case 2 in Figure 2). Unidimensional orderedness and unidimensional alignment, on the other hand, are de…ned in terms of intervals associated with propositions, rather than individuals. A pro…le is unidimensionally ordered if each proposition is accepted by those individuals who are away from it by at most a certain distance, where the 'cut-o¤' distance depends on the proposition, rather than the individual (Case 3 in Figure 2). If each individual's acceptance interval is left-justi…ed or right-justi…ed, the pro…le becomes unidimensionally aligned. An alternative interpretation of unidimensional alignment 13 Figure 2: A uni…ed spatial interpretation of the four conditions 14 is obtained by reinterpreting a proposition's 'location'as constituting an acceptance threshold (which may vary across propositions) and assuming that the individuals accepting the proposition are either all to the left or all to the right of this threshold (Case 4 in Figure 2). Thus the extreme positions on the left-right axis correspond either to clear acceptance or to clear rejection of each proposition, and the relevant threshold divides the 'acceptance interval'from the 'rejection interval'. Given the present spatial representation, each four domain-restriction conditions, like single-peakedness in preference aggregation, can be interpreted as indicating a form of 'meta-agreement'among the individuals on a single normative or cognitive dimension in terms of which their di¤erent judgment sets can be rationalized, as distinct from a 'substantive agreement'on which judgment set to hold (List [21]). 4.4 The logical relationships between the four conditions We have already seen that single-canyonedness implies single-plateauedness, and that unidimensional alignment implies unidimensional orderedness. A natural question is how the rst two conditions, which are based on orders of the propositions, are related to the second two, which are based on orders of the individuals. The following result answers this question.14 Proposition 3 (a) Restricted to pro…les of consistent judgment sets,  unidimensional alignment implies any of the other three conditions;  single-canyonedness implies single-plateauedness;  there are no other pairwise implications between the four conditions. (b) Restricted to pro…les of consistent and complete (or just of opinionated) judgment sets, the four conditions are equivalent. Proof. (a) We already know that single-canyonedness implies single-plateauedness, and that unidimensional alignment implies unidimensional orderedness. To show that unidimensional alignment implies the other conditions too, it su¢ ces to establish that it implies single-canyonedness. We do this in the appendix, where we also show by counterexamples that there are no other implications. (b) Let (A1; :::; An) be a pro…le of consistent and complete (or just opinionated) judgment sets. Then each Ai contains exactly k = jXj=2 propositions. Since, by part (a), unidimensional alignment implies single-canyonedness, and single-canyonedness implies single-plateauedness, the equivalence of all four conditions follows from the following additional implications, which we now prove using the fact that jAij = k for all i. We use the notation from an earlier proof. Single-plateauedness ) unidimensional orderedness. Suppose single-plateauedness, say relative to the order p1:::p2k. Then, for all i, there is (using jAij = k) an index j(i) 2 f1; :::; 2kg such that Ai = [pj(i); pj(i)+k1]. Consider an order of the individuals i1:::in such that j(i1)  j(i2)  :::  j(in). To check unidimensional orderedness 14The non-implication claims in (a) do not refer to a xed agenda X and group size n. Rather, for some (in fact, most) X and n, there are pro…les satisfying one condition but not the other. For special X or n, e.g., for X = fp;:pg or n = 2, all conditions hold trivially. 15 relative to i1:::in, note that, for all p = pl 2 X, we have fi : pl 2 Aig = fi : pl 2 [pj(i); pj(i)+k1]g = fi : j(i)  l < j(i) + kg = fi : l  j(i) < k  lg = fi : l  k < j(i)  lg, which is an interval of the order i1:::in, as required. Unidimensional orderedness ) unidimensional alignment. Let (A1; :::; An) be unidimensionally ordered, say relative to the order . To see that it is also unidimensionally aligned relative to the same order , consider any p 2 X. As each Ai contains exactly one member of each pair p;:p 2 X, N:p = NnNp. Further, by unidimensional orderedness, Np and N:p are ( -)intervals. So Np and NnNp are intervals. Hence Np NnNp or NnNp Np, as required.  4.5 Applications to preference aggregation: order restriction and intermediateness What do our present domain-restriction conditions amount to when translated into the classical framework of preference aggregation? As we have already noted, the conditions based on orders of propositions, although formally applicable to the preference agenda, have no obvious standard counterparts when applied to it. They do, however, resemble the condition of single-switch preferences (La¤ond and Lainé [20]), which de…nes a domain restriction in a framework in which individuals have preferences over combinations of positions on multiple logically unconnected issues, such as multiple referendum items. La¤ond and Lainé show that this condition, which is de…ned in terms of an order of issues, is su¢ cient for ensuring that two distinct majoritarian voting procedures yield the same outcome, and thus for avoiding 'Ostrogorski's paradox'.15 Our conditions based on orders of individuals are more closely related to standard conditions on preferences. We now relate unidimensional orderedness to Grandmont's intermediateness [16] and unidimensional alignment to Rothstein's order restriction ([34], [35]). To introduce intermediateness and order restriction, de…ne a (strict) preference relation be a binary relation on K (so far, we do not impose any rationality conditions on preferences), and de…ne a preference pro…le to be an n-tuple (1; :::;n) of such relations.16 15Ostrogorski's paradox identi…es a con‡ict between issue-by-issue majority voting and pairwise majority voting over combinations of issues. It shows that, if each individual's preferences over combinations of positions on those issues are determined by their symmetrical distance from the individual's most preferred combination, issue-by-issue majority voting may lead to a combination of positions that would lose in pairwise majority voting over combinations of issues. La¤ond and Lainé [20] show that when individuals'most preferred combinations satisfy a particular restriction based on an ordering of issues –every ideal combination is characterized by a single switch from accepted issues to rejected issues or vice versa –then Ostrogorski's paradox cannot occur. The order of issues used in La¤ond and Lainé's condition is analogous to the order of propositions used in the conditions of single-plateauedness and single-canyonedness discussed here. We thank an anonymous reviewer for raising this point. 16Rothstein and Grandmont formulate their de…nitions more generally for weak preference relations i. 16 Intermediateness. (Grandmont [16]) A preference pro…le (1; :::;n) is intermediate relative to if, for all x; y 2 K for all i; j; k 2 N with i j k, [x i y and x k y]) x j y: Order restriction. (Rothstein [34], [35]) A preference pro…le (1; :::;n) is order restricted relative to if, for all x; y 2 X, fi 2 N : x i yg fi 2 N : y i xg or fi 2 N : y i xg fi 2 N : x i yg. The following is easy to check: Remark 3 (a) A preference pro…le (1; :::;n) is order restricted (relative to some

) if and only if the corresponding judgment pro…le (A1 ; :::; An) is unidimensionally aligned (relative to the same ). (b) An opinionated preference pro…le (1; :::;n) is intermediate (relative to some

) if and only if the corresponding judgment pro…le (A1 ; :::; An) is unidimensionally ordered (relative to the same ), where opinionation means that, for each i 2 N and all distinct x; y 2 K, precisely one of x i y or y i x holds. The restriction to opinionated preference pro…les in part (b) can be dropped under an alternative correspondence between preference relations and judgment sets.17 5 Conditions for majority consistency based on local orders For many agendas, the four domain-restriction conditions discussed so far are stronger than necessary for achieving majority consistency. Our goal in this section is to weaken them by applying them not to judgments on all propositions in X, but rather 17Without opinionation of each i, intermediateness of (1; :::;n) is not equivalent to unidimensional orderedness of (A1 ; :::; An). For all x; y 2 K, the former requires that fi 2 N : xPy 2 Aig be an interval, the latter that fi 2 N : :xPy 2 Aig be an interval too. But under another correspondence between preference relations 2 KK and judgment sets A  XK , intermediateness becomes equivalent to unidimensional orderedness (and order restriction remains equivalent to unidimensional alignment). On our earlier de…nition, the judgment set A corresponding to a preference relation is always opinionated. But a judgment set A  XK need not be opinionated. In particular, if x 6 y, this can have two distinct interpretations: either 'not considering x preferable to y'or 'considering x not preferable to y', corresponding to not accepting p and accepting :p, where p is 'x is preferable to y'. Our earlier de…nition of A assumes the second (stronger) interpretation of x 6 y, because A contains :xPy if x 6 y. While a preference relation  KK is ambiguous between the two interpretations, a judgment set A  XK is not. For any distinct x; y 2 K, a preference relation can display four di¤erent patterns: x y&y 6 x, x 6 y&y x, x 6 y&y 6 x, or x y&y x; a judgment set A  XK can display 24 = 16 di¤erent patterns, depending on which of xPy;:xPy; yPx;:yPx are contained in A. Under the weaker interpretation of x 6 y, we de…ne A = fxPy : x; y 2 K&x yg (an incomplete judgment set, unless is the total relation). Now a preference relation is fully rational (i.e., asymmetric, transitive and connected) if and only if A is consistent and contains a member of each pair xPy; yPx 2 X with x 6= y. Intermediateness of (1; :::;n) then translates into unidimensional orderedness of (A1 ; :::; An). 17 to judgments on various subagendas of X, thereby allowing the relevant structuring order of individuals or propositions to vary across di¤erent subagendas. Consider, for instance, the agenda X = fa; b; c; a! b; a! cg, where a and b are the propositions 'CO2 emissions will increase dramatically by 2020'and 'The frequency of hurricanes will double by 2030'as in one of our earlier examples, and c is the proposition 'We should introduce a scheme of carbon taxes in which taxes on CO2 emissions increase over time'. Here the agenda has two semantically very di¤erent non-trivial subagendas, namely fa; b; a! bg and fa; c; a! cg, one concerning environmental aspects of global warming, the other concerning policy responses. Although some of our domain-restriction conditions may well be plausible when applied to each subagenda separately, it seems unduly demanding to require the same structuring order for both subagendas. Instead, di¤erent 'local' structuring orders corresponding to di¤erent subagendas may be warranted. The move from global to local structuring orders parallels the move in preference aggregation from single-peakedness to single-peakedness restricted to triples of alternatives. We begin by introducing the general form of our local domain restriction conditions; then we discuss two approaches to specifying the relevant subagendas. 5.1 The general form of the local conditions A subagenda (of X) is a subset Y  X that is itself an agenda (i.e., non-empty and closed under single negation). For each of our four global domain-restriction conditions, we say that a pro…le (A1; :::; An) satis…es the given condition on a subagenda Y  X if the restricted pro…le (A1 \ Y; :::; An \ Y ), viewed as a pro…le of judgment sets on the agenda Y , satis…es it. The relevant structuring order is then called a structuring order on Y and denoted Y (if it is an order of propositions) or Y (if it is an order of individuals). Whenever one of the conditions is satis…ed globally, then it is also satis…ed on every Y  X. But we now de…ne a local counterpart of each global condition. Let Y be some set of subagendas. Local single-plateauedness / single-canyonedness / unidimensional orderedness / unidimensional alignment. A pro…le (A1; :::; An) satis…es the local counterpart of each global condition (with respect to a given set of subagendas Y) if it satis…es the global condition on every Y 2 Y. This allows di¤erent structuring orders Y or Y for di¤erent subpro…les (A1 \ Y; :::; An \ Y ) (with Y 2 Y). Any implications and equivalences between our four global conditions, as stated in Proposition 3, carry over to their local counterparts (each de…ned with respect to the same Y).18 Corollary 1 (a) Restricted to pro…les of consistent judgment sets,  local unidimensional alignment implies any of the other three local conditions; 18Analogously to proposition 3, the non-implication claims in (a) do not refer to a xed agenda X, set of subagendas Y, and group size n. Rather, for some (in fact, most) X, Y and n, there are pro…les satisfying one condition but not the other. In special cases, e.g., for Y = ;, all conditions hold trivially. 18  local single-canyonedness implies local single-plateauedness;  there are no other pairwise implications between the four local conditions. (b) Restricted to pro…les of consistent and complete (or just of opinionated) judgment sets, the four local conditions are equivalent. Our choice of subagendas in Y, with respect to which our local conditions are de…ned, is guided by two goals. The rst is to ensure that a consistent majority outcome for every subagenda implies a consistent majority outcome overall (just as acyclicity on triples of alternatives in preference aggregation implies acyclicity overall). The second is to minimize the total number and size of subagendas, so as to make our local domain-restriction conditions as unrestrictive as possible. Accordingly, the subagendas in Y must be carefully chosen. Choosing them according to their size (e.g., by including in Y all subagendas of size less than some k) or according to the syntactic form of propositions in them (e.g., by including in Y all subagendas whose propositions contain only a certain type or number of logical connectives) does not generally work. 5.2 Selecting subagendas I: minimal inconsistent sets What set of subagendas Y should be chosen? In this subsection, we take the following approach. Note that a judgment set A  X is inconsistent if and only if it has a minimal inconsistent subset Y  X, i.e., a subset that is inconsistent but all of whose proper subsets are consistent. So a consistent majority outcome can be achieved by each of our local domain-restriction conditions where Y is de…ned as Y = fY  : Y is a minimal inconsistent subset of Xg. (1) Proposition 4 For any pro…le (A1; :::; An) of consistent judgment sets, (a) if (A1; :::; An) satis…es any of the four local conditions with respect to Y as de…ned in (1), the majority outcome A is consistent; (b) in the case of local unidimensional orderedness, A   [Y 2Y(AmY \ Y ) if n is odd, [Y 2Y(AmY;1 \AmY;2 \ Y ) if n is even, where, for each Y 2 Y, mY is the middle individual (if n is odd) and mY;1;mY;2 the middle pair of individuals (if n is even) in any structuring order Y on Y ;19 (c) in the case of local unidimensional alignment, A is as stated in part (b) with  replaced by =. Proof. Let Y and (A1; :::; An) be as speci…ed, with majority outcome A. (a) To prove A's consistency, it su¢ ces to prove that A has no minimal inconsistent subset, hence to prove that A\Y is consistent for all Y 2 Y. So consider any subagenda 19The result continues to hold if every occurrence of the quanti…cation Y 2 Y in part (b) is weakened to the quanti…cation Y 2 Y, where Y  Y is any subset of subagendas covering X, i.e., with [Y 2YY = X. There are many ways to cover X; trivial ones are Y = ffp;:pg : p 2 X+g and Y = Y. The representation of A becomes slim if Y minimally covers X, i.e., covers X but no Z ( Y does so too. 19 Y 2 Y. As (A1; :::; An) is, for example, single-plateaued on Y (the proof is similar for single-canyonedness or unidimensinoal orderedness/alignment), (A1 \Y; :::; An \Y ) is single-plateaued for the agenda Y and hence has a consistent majority outcome by Proposition 1. But this majority outcome is A\Y . So A\Y is consistent, as required. (b) Assume unidimensional orderedness and let the individuals (mY )Y 2Y (if n is odd) or (mY;1;mY;2)Y 2Y (if n is even) be as speci…ed. To show that A  [Y 2Y(AmY \ Y ) (if n is even) or A  [Y 2Y(AmY;1\AmY;2\Y ) (if n is odd), it is by A = [Y 2Y(A\Y ) su¢ cient to show that, for all Y 2 Y, A \ Y  AmY \ Y (if n is even) or A \ Y  AmY;1\AmY;2\Y (if n is odd). This follows from part (a) of Proposition 2 because A\Y is the majority outcome on the unidimensionally ordered pro…le (A1 \ Y; :::; An \ Y ). (c) The proof is analogous to that of part (b), with each "" replaced by "=" and where we now make use of part (b) (not (a)) of Proposition 2.  5.3 Selecting subagendas II: irreducible sets The set of subagendas generated from all minimal inconsistent subsets of the agenda can be large, but using this rich set has been necessary in order to guarantee majority consistency on domains that allow even for incomplete individual judgment sets. However, in the important special case of individual completeness, it is enough for majority consistency to impose any of our four local domain-restriction conditions with a much slimmer de…nition of the relevant set of subagendas. We generate these subagendas not from all minimal inconsistent subsets of the agenda, but only from those that are irreducible in the following sense.20 For any inconsistent set Y  X, we call another inconsistent set Z  X a reduction of Y if jZj < jY j and each p 2 ZnY is entailed by some V  Y with jY nV j > 1, and we call Y irreducible if it has no reduction.21 For instance, the inconsistent set fa; a ! b; b ! c;:cg (where a; b; c are distinct atomic propositions) is reducible to Z = fb; b ! c;:cg, since b is entailed by fa; a ! bg, whereas Z is irreducible. Now de…ne Y = fY  : Y is an irreducible subset of Xg. (2) The set of subagendas de…ned in (2) is a subset of the one de…ned in (1) above,22 since every irreducible set is minimal inconsistent (a non-minimal inconsistent set is reducible to any of its inconsistent proper subsets). The local domain-restriction conditions resulting from (2) are therefore less restrictive than those resulting from (1) above. The following lemma is crucial; a proof is given in the appendix. Lemma 2 Every complete and inconsistent judgment set A  X has an irreducible subset. 20Dietrich [5] has subsequently generalized this concept. 21 In the de…nition of reduction, the clause jY nV j > 1 is essential. Dropping it would render all inconsistent sets Y  X of size three or more reducible, namely to any pair fp;:pg with p 2 Y ; :p is entailed by Y nfpg. 22 It is usually a proper subset since many minimal inconsistent subsets of the agenda, such as fa; a! b; b! c;:cg, are reducible. 20 Using Lemma 2, we can prove our central claim: if individuals hold not only consistent but also complete judgment sets, our local domain-restriction conditions de…ned in terms of irreducible sets are enough to guarantee majority consistency. The assumption of individual completeness ensures an (apart from ties) complete majority outcome, so as to make Lemma 2 applicable in the proof. Proposition 5 For any pro…le (A1; :::; An) of consistent and complete judgment sets, if (A1; :::; An) satis…es any (hence by corollary 1 all) of the four local conditions with respect to Y as de…ned in (2), the majority outcome is consistent. Proof. We consider a pro…le (A1; :::; An) of the speci…ed kind and use the earlier notation. Case 1: n is odd. Then A is complete. So, by Lemma 2, to prove A's consistency, it su¢ ces to prove that A has no irreducible subset, hence to prove that A \ Y is consistent for all Y 2 Y. The latter follows by an argument analogous to the one in the proof of part (a) of Proposition 4. Case 2: n is even. Let An+1 be any complete and consistent judgment set such that (A1; :::; An+1) still satis…es the local condition, e.g. single-plateauedness on Y, now for group size n + 1 (one might take An+1 = A1). By Case 1 the majority outcome on (A1; :::; An+1) is a consistent judgment set eA. Check that A  eA. So A is consistent, as required.  5.4 Applications to preference aggregation: order restriction and intermediateness on k-tuples of alternatives What do our local conditions look like when applied to the preference agenda? To answer this question, we must identify the set of subagendas Y under each of our two criteria for selecting subagendas. A few de…nitions are needed. By our de…nition of the logic of preferences, for any distinct x; y 2 K, :xPy and yPx are equivalent. Call two judgment sets essentially identical if one arises from the other by (zero, one or more) replacements of propositions by equivalent propositions. For any distinct x1; :::; xk 2 K (k  1), the cyclical preferences x1 x2 ::: xk x1 can be represented by the set fx1Px2; x2Px3; :::; xk1Pxk; xkPx1g. We call such a set, and any set essentially identical to it, a cycle (of length k). We are now in a position to identify the minimal inconsistent subsets of the preference agenda. Remark 4 The minimal inconsistent sets Y  XK are the cycles. Proof. This follows from the de…nition of the logic L for representing preferences. First, any cycle is obviously minimal inconsistent in L. Second, suppose Y  XK is minimal inconsistent. One can check that, by Y 's inconsistency, some subset Y   Y is a cycle. By minimal inconsistency, then, Y = Y .  Next let us identify the irreducible subsets of the preference agenda. Not all cycles fall into this category. To illustrate, observe that any cycle of length k, Y = fx1Px2; x2Px3; :::; xk1Pxk; xkPx1g, 21 with k  4 is reducible, e.g., to the 3-cycle fx1Px2; x2Px3; x3Px1g, as x3Px1 is entailed by fx3Px4; x4Px5; :::; xkPx1g. Remark 5 The irreducible sets Y  XK are the cycles of length 1, 2 or 3. Proof. First, consider any cycle Y of length at most three. If Y is a 1-cycle, i.e., Y = fxPxg for some x 2 K, or a 2-cycle, i.e., Y = fxPy; yPxg with distinct x; y 2 K, then Y is obviously irreducible. Now let Y be a 3-cycle, i.e., Y = fxPy; yPz; zPxg for distinct x; y; z 2 K. Suppose, for a contradiction, that Y is reducible, say to Z  X. Then jZj  2. Moreover each p 2 Z is entailed by a single member of Y , i.e. by one of xPy; yPz; zPx. But the only proposition in X entailed by xPy is xPy (and the logically equivalent :yPx), and similarly for yPz and zPx. So each p 2 Z is one of xPy; yPz; zPx (or one of :yPx;:zPy;:xPz). Hence Z is (essentially identical to) a proper subset of Y = fxPy; yPz; xPxg. So Z is consistent, a contradiction. Second, suppose Y  XK is irreducible. Hence Y is minimal inconsistent. So, by Remark 4, Y is a cycle, hence (essentially identical to) a set of type fx1Px2; x2Px3; ::; xk1Pxk; xkPx1g (k  1). Now k  3, as otherwise Y would be reducible to Z := fx1Px2; x2Px3; x3Px1g. So Y is a 1or 2or 3-cycle.  By Remark 4, the set of subagendas generated from minimal inconsistent sets is Y = fY  : Y  XK is a cycleg, and by Remark 5, the set of subagendas generated from irreducible sets is the smaller set Y = fY  : Y  XK is a cycle of length 1, 2 or 3g. Just as in the global case, we are thus able to relate local unidimensional orderedness and local unidimensional alignment to local versions of intermediateness and order restriction. Consider the following two local conditions on preference pro…les: Intermediateness on triples. (Grandmont [16]) A preference pro…le (1; :::;n) is intermediate on triples if, for every subset K 0  K with jK 0j = 3, the preference pro…le restricted to K 0, i.e., (1 jK0 ; :::;n jK0), is intermediate (as de…ned above). Order restriction on triples. (Rothstein [34], [35]) A preference pro…le (1; :::;n ) is order restricted on triples if, for every subset K 0  K with jK 0j = 3, the preference pro…le restricted to K 0, i.e., (1 jK0 ; :::;n jK0), is order restricted (as de…ned above). It is easy to see that, when Y is de…ned as the set of subagendas of XK generated from all cycles, unidimensional orderedness and unidimensional alignment with respect to Y are more demanding than intermediateness and order restriction on triples, respectively. Unlike the two triplewise conditions on preference pro…les, our conditions require a structuring order of the individuals for every k-tuple of alternatives, not just for every triple. As already noted, our stronger requirement is warranted when we want to guarantee majority consistency even in the absence of individual completeness; order restriction or intermediateness on triples do not guarantee acyclic majority preferences when individual incompleteness is allowed. 22 But in the case of individual completeness, it su¢ ces for majority consistency to de…ne our local conditions in terms of irreducible sets, i.e., by de…ning Y as the set of subagendas of XK generated from all cycles of length up to three. Local unidimensional orderedness and alignment then become equivalent to the triplewise variants of Grandmont's and Rothstein's conditions, as shown in the appendix: Proposition 6 A pro…le (1; :::;n) of strict linear orders23 on K is intermediate (equivalently, order restricted) on triples if and only if the associated judgment pro…le (A1 ; :::; An) is locally unidimensionally ordered (equivalently, aligned) with respect to Y as de…ned by (2). 6 Conditions for majority consistency not based on orders Although our domain-restriction conditions based on local orders are already much less restrictive than those based on global orders, it is possible to weaken them further. Just as the various conditions based on orders in preference aggregation – singlepeakedness, single-cavedness etc. – can be generalized to a weaker, but less easily interpretable, condition –namely Sen's triplewise value-restriction [38] –so in judgment aggregation the conditions based on orders can be weakened to a more abstract condition, to be called value-restriction. When applied to the preference agenda, this condition becomes non-trivially equivalent to Sen's condition. But despite generalizing Sen's condition, our condition is simpler to state; we thus also hope to o¤er a new perspective on Sen's condition. 6.1 Value-restriction We state two variants of our condition, one based on minimal inconsistent sets, the other based on irreducible sets. Value-restriction. A pro…le (A1; :::; An) is value-restricted if every (non-singleton24) minimal inconsistent set Y  X has a two-element subset Z  Y that is not a subset of any Ai. Weak value-restriction. A pro…le (A1; :::; An) is weakly value-restricted if every (non-singleton) irreducible set Y  X has a two-element subset Z  Y that is not a subset of any Ai. Informally, value-restriction re‡ects a particular kind of agreement: for every minimal inconsistent (or irreducible in the weak case) subset of the agenda, there exists a particular conjunction of two propositions in this subset that no individual endorses. 23A strict linear order is an irre‡exive, transitive and connected binary relation. 24The quali…cation 'non-singleton'in this de…nition and the next is unnecessary if X contains only contingent propositions, since this rules out singleton inconsistent sets. 23 Like our previous domain-restriction conditions, the two new conditions are each su¢ cient for consistent majority outcomes (the weaker condition in the important special case of individual completeness). Proposition 7 For any pro…le (A1; :::; An) of consistent judgment sets, (a) if (A1; :::; An) is value-restricted, the majority outcome is consistent; (b) if (A1; :::; An) is weakly value-restricted and each Ai is complete, the majority outcome is consistent. Proof. Let (A1; :::; An) consist of consistent judgment sets. (a) Suppose (A1; :::; An) is value-restricted, but the majority outcome, A, is inconsistent. Then A has a minimal inconsistent subset Y . Obviously, Y is non-singleton (otherwise a majority would support a contradiction). So, by value-restriction, Y has a two-element subset Z  Y that is not a subset of any Ai. However, since Z  A, there is a majority for each of the two elements of Z. Since two majorities must overlap, some Ai contains both of these elements, whence Z  Ai for some i 2 N , a contradiction. (b) Suppose (A1; :::; An) is weakly value-restricted and each Ai is complete. There are two cases. Case 1: n is odd. Then A is also complete (because there cannot be majority ties). Suppose for a contradiction that the majority outcome, A, is inconsistent. Then A has an irreducible subset Y by Proposition 2, and one can derive a contradiction analogously to part (a). Case 2: n is even. Let An+1 be any complete and consistent judgment set such that (A1; :::; An+1) is still weakly value-restricted, now for group size n+1 (of course, there is such an An+1: e.g., take An+1 = A1). Let A0 be the majority outcome on (A1; :::; An+1). By Case 1, A0 is consistent. Check that the majority outcome on (A1; :::; An) is a subset of A0; hence it is consistent too, as required.  How general are our two value-restriction conditions? The following proposition, proved in the appendix, answers this question. Proposition 8 Restricted to pro…les of consistent judgment sets, (a) each of our four conditions based on global orders implies value-restriction; (b) each of our four conditions based on local orders, with respect to Y de…ned in terms of minimal inconsistent sets, implies value-restriction; (c) each of our four conditions based on local orders, with respect to Y de…ned in terms of irreducible sets, implies weak value-restriction. 6.2 Applications to preference aggregation: triplewise value-restriction We now show that, when applied to the preference agenda, our two value-restriction conditions surprisingly both collapse into Sen's triplewise value-restriction. Let us recapitulate Sen's condition: 24 Triplewise value-restriction. (Sen [38]) A preference pro…le (1; :::;n) is triplewise value-restricted if, for every triple of distinct alternatives x; y; z 2 K, there is one alternative, say x, that is either not ranked top by any individual (no i has x i y and x i z), or not ranked middle by any individual (no i has y i x i z or z i x i y) or not ranked bottom by any individual (no i has y i x and z i x). An alternative, but equivalent de…nition of triplewise value-restriction requires that, for each triple of alternatives, the individuals'preferences be either single-peaked or single-caved or separable in a sense de…ned by Inada [17]. (See also Elsholtz and List [18].) The following is the central result of this subsection, proved in the appendix. Proposition 9 For any pro…le (A1; :::; An) of consistent and complete judgment sets on the preference agenda, the following are equivalent: (a) (A1; :::; An) is value-restricted, (b) (A1; :::; An) is weakly value-restricted, (c) the associated preference pro…le (A1 ; :::;An) is triplewise value-restricted. 7 Conclusion We have introduced several domain-restriction conditions on pro…les of individual judgment sets that are su¢ cient for consistent majority outcomes. Some of our conditions are based on global orders of either the propositions or the individuals, others on local orders of them, and yet others not on orders at all. We have justi…ed our focus on majority voting by providing a new characterization result showing that, for all minimally rich domains, if there is any consistent aggregation function at all that satis…es certain democratic conditions, then majority voting is the unique such function. While all domain-restriction conditions discussed in this paper are su¢ cient for consistent majority outcomes, it is useful to compare them with a necessary and su¢ cient condition. Majority-consistency. A pro…le (A1; :::; An) is majority-consistent if every minimal inconsistent set Y  X contains a proposition not contained in a majority of the Ais.25 If (and only if) this condition is met, no minimal inconsistent set of propositions can be accepted under majority voting, and thus the majority outcome is consistent. But there are some important di¤erences between majority-consistency and the various conditions introduced earlier. First, unlike majority-consistency, the various earlier conditions are easily interpretable: they embody particular types of agreement within the group, for instance agreements on normative or cognitive dimensions underlying individual judgments. Secondly, the earlier conditions are structural (as opposed to numerical) in the sense of depending only on whether or not certain patterns occur 25 It is easy to see that, when the majority outcome is complete, it is enough to quantify over all irreducible (as opposed to all minimal inconsistent) sets Y  X. 25 in each judgment set in a given pro…le, but not on how often those patterns occur (Elsholtz and List [18]). By contrast, majority-consistency is a numerical condition. Thirdly, as we show in a moment, each of our earlier conditions can be used to de…ne product domains, whereas majority-consistency cannot. A domain D of admissible pro…les of an aggregation function (say, majority voting) is called a product domain if it can be expressed as D = D1 D2  :::Dn, where, for each i 2 N , Di is the set of admissible judgment sets of individual i (typically, Di is the same for all i). A domain is called a non-product domain if it does not admit such an expression, i.e., if the judgment set an individual can submit may depend on the judgment sets submitted by others. For example, in preference aggregation, single-peakedness and single-cavedness relativized to an antecedently xed order of alternatives specify product domains, while single-peakedness and single-cavedness simpliciter do not. The distinction between product and non-product domains is important both theoretically and practically. It is theoretically important in game-theoretic analyses of aggregation problems. If we want to interpret an aggregation problem as a game, where the individuals'possible inputs –i.e., their preferences or judgments –are their strategies, then the domain of admissible pro…les must be a Cartesian product of the strategy sets across individuals. Standard de…nitions of strategy-proofness following Gibbard and Satterthwaite employ precisely this representation, although they can be modi…ed so as to accommodate non-product domains (Saporiti and Tohmé [36]; see also Dietrich and List [8]). Practically, product domains matter when an aggregation function represents a voting procedure in the ordinary sense. Here each voter must be given a list of admissible choices –i.e., a set Di of admissible judgment sets (typically the same across voters) – and cannot be told that certain choices are inadmissible depending on the choices made by others. The product domains induced by our various conditions are as follows:  The product domain of single-plateaued/canyoned pro…les relative  (a xed order on X): each Di is the set of consistent judgment sets A  X that are single-plateaued/canyoned relative to . In the case of unidimensional orderedness/alignment, the construction is slightly more elaborate.26  The product domain of locally single-plateaued/canyoned pro…les relative to (Y )Y 2Y (a family of xed orders Y on the subagendas Y 2 Y): each Di is the set of consistent judgment sets A  X that are single-plateaued/canyoned on each Y 2 Y relative to Y . Again, a more elaborate construction is possible for local unidimensional orderedness/alignment.  The product domain of value-restricted pro…les relative to (ZY )Y 2MI (a family of xed two-element subsets ZY  Y , with Y ranging over the setMI of minimal inconsistent subsets of X): each Di is the set of consistent judgment sets A  X that are not supersets of any ZY .27 26Let (A1; :::; An) be any pro…le of consistent judgment sets satisfying unidimensional orderedness or alignment (relative to some ), where n  1 is any arbitrary group size (not necessarily identical to n). If we de…ne each Di to be the set of all Ajs occurring in (A1; :::; An), then D = D1D2 :::Dn is a product domain of unidimensionally ordered or aligned pro…les of consistent judgment sets. 27 In the case of weak value restriction, Di can be de…ned analogously, with MI replaced by the 26 It can be shown that any maximal product domain D with identical Dis that guarantees consistent and complete majority judgments in D must be value-restricted relative to some family (ZY )Y 2MI , as de…ned in the last bullet point.28 In this speci…c sense, in the case of product domains, value-restriction constitutes the most general domain restriction one can give to achieve consistent majority judgments. Unlike value-restriction, the non-product domain condition of majority-consistency does not induce a product domain, since it is a numerical condition, not a structural one. In conclusion, Figures 3 and 4 summarize the logical relationships between all the domain-restriction conditions discussed in this paper, in Figure 3 applied to pro…les majority-consistency value-restriction single-plateauedness single-canyonedness unidimensional orderedness unidimensional alignment local unidimensional alignment local unidimensional orderedness local single-canyonedness local single-plateauedness Figure 3: The logical relationships between the domain-restriction conditions for proles of consistent judgment sets majority consistency value-restriction local unidimensional orderedness local single-plateauedness local single-canyonedness local unidimensional alignment unidimensional orderedness single-plateauedness single-canyonedness unidimensional alignment Figure 4: The logical relationships between the domain-restriction conditions for proles of consistent and complete judgment sets (smaller) set of irreducible subsets of X. 28We show this result in follow-up work on domain restrictions that guarantee majority and supermajority consistency. 27 of consistent individual judgment sets and in Figure 4 applied to pro…les of consistent and complete individual judgment sets. 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Journal of Economic Theory 10: 89-99 A Appendix: Additional proofs Proof of Lemma 1. Consider any agenda (possibly containing tautologies), and let F and D be as speci…ed. Consider any p 2 X and any (A1; :::; An); (A1; :::; An) 2 D in which the same set of individuals C  N accepts p. We must show that p 2 F (A1; :::; An) , p 2 F (A1; :::; An). By consistency of F , if p is a contradiction, it belongs to neither F (A1; :::; An) nor F (A1; :::; A  n), hence p 2 F (A1; :::; An) , p 2 F (A1; :::; A  n). Now suppose p is not a contradiction (but perhaps a tautology). The pro…le (A01; :::; A 0 n) given by A 0 i = ? for all i 2 C and A0i = fpg for all i =2 C is inD (it is bipolar). By acceptance/rejection neutrality, p 2 F (A1; :::; An) , p =2 F (A01; :::; A0n). Further, by acceptance/rejection neutrality, p 2 F (A1; :::; An) , p =2 F (A01; :::; A0n). So p 2 F (A1; :::; An), p 2 F (A1; :::; An), as required.  Proof of Remark 1. We use the notation introduced in the proof of Proposition 1. Consider a pro…le (A1; :::; An) of consistent individual judgment sets, and let (A1; :::; An) be single-canyoned, say relative to the order p1:::p2k. We consider any Ai and show that Ai is single-plateaued relative to the new order pk+1:::p2kp1:::pk. By assumption, (i) Ai = fp1; :::pjg [ fpj0 ; :::; p2kg for some 0  j  j0  2k + 1. As Ai is consistent, Ai contains no pair p;:p 2 X; so jAij  jXj=2 = k, whence (ii) j  k and j0  k+1. Using both (i) and (ii), one can check that Ai is an interval relative to the new order pk+1:::p2kp1:::pk, as required. More precisely, Ai = 8>><>>: [pj0 ; pj ] if j 6= 0 and j0 6= 2k + 1, [p1; pj ] if j 6= 0 and j0 = 2k + 1, [pj0 ; p2k] if j = 0 and j0 6= 2k + 1, ? if j = 0 and j0 = 2k + 1.  30 Supplementary parts of the proof of Proposition 3. We use the notation from the earlier proofs as well as the abbreviations SP (single-plateauedness), SC (singlecanyonedness), UO (unidimensional orderedness) and UA (unidimensional alignment). UA =) SC. Let (A1; :::; An) be a pro…le of consistent judgment sets, and suppose UA, for simplicity relative to the order ( ) 1; 2; :::; nWe show SC relative to the order () p1p2:::p2k that begins with the propositions p 2 X with Np = f1; :::; ng, followed by the propositions p 2 X with Np = f1; :::; n 1g, ... followed by the propositions p 2 X with Np = f1g, followed by the propositions p 2 X with Np = ?, followed by the propositions p 2 X with Np = fng, followed by the propositions p 2 X with Np = fn 1; ng, ... ending with the propositions p 2 X with Np = f2; :::; ng. 1 2 3 4 5 p1 X X X X X p2 X X X X p3 X p4 p5 X p6 X X X X Table 5: Example of the order p1; :::; p2k for n = 5 individuals and 2k = 6 propositions; a 'Y'indicates acceptance of the row proposition by the column individual This procedure to construct p1:::p2k is well-de…ned, since, by UA, each p 2 X is of one of the forms considered in the procedure. In the example pro…le of Table 5, it is obvious that (A1; :::; An) is SC relative to p1:::p2k: A1 = Xn[p4; p6], A2 = A3 = A4 = Xn[p3; p5] and A4 = A5 = Xn[p2; p4]. For the general proof of SC, consider any Ah (1  h  n) and let us show that Ah is SC relative to . It su¢ ces to prove that, for all p 2 X, either [p1; p]  Ah or [p; p2k]  Ah. Consider any p 2 X. By UA, either Np = f1; :::; kg for some k, or Np = fk; :::; ng for some k  2. By construction of the order p1:::p2k, in the rst case [p1; p]  Ah and in the second case [p; p2k]  Ah, as required. SP 6) SC. Consider an agenda X and a pro…le (A1; :::; An) consisting of pairwise disjoint consistent judgment sets, at least three of which are non-empty. The pro…le is SP, namely relative to an order starting with the propositions in A1; followed by those in A2; ..., and ends with those in An. But the pro…le is not SC: if it were SC, say relative to an order , then each non-empty Ai would contain an extreme (i.e., leftor right-most) proposition; so that, as at least three Ais are non-empty but there are only two extreme propositions, the Ais would not be pairwise disjoint, a contradiction. SC 6) UO. Consider an agenda X, group N and pro…le (A1; :::; An) such that n = 4, A1 = fp; p0; q; q0g; A2 = fp; p0g, A3 = fq; q0g, A4 = fp; qg, where p; p0; q; q0 2 X 31 are pairwise distinct. This pro…le is SC: consider an order  such that p < p0 < ::: < q0 < q (where ':::'contains all remaining propositions). Suppose for a contradiction UO holds, say relative to an order i1:::in. As Np0 = f1; 2g, individuals 1 and 2 are neighbours (in i1:::in). As Nq0 = f1; 3g, 1 and 3 are neighbours. So 1 is 'surrounded' by 2 and 3, i.e., i1:::in contains the sublist 213 or 312; suppose it contains the sublist 213 (the proof continues analogously for the sublist 312). Also, as Np = f1; 2; 4g, 4 is a neighbour of 1 or of 2; since 4 cannot be a neighbour of 1 (which is surrounded by 2 and 3), it is a neighbour of 2. So i1:::in contains the sublist 4213. Finally, as Nq = f1; 3; 4g, 4 is a neighbour of 1 or 3, which is not the case since i1:::in contains the sublist 4213. SC 6) UA. This follows from SC 6) UO by UO ) UA. SP 6) UO. This follows from SC 6) UO by SC ) SP. SP 6) UA. This follows from SC 6) UA by SC ) SP. UO 6) UA. Consider an agenda X, group N and pro…le (A1; :::; An) such that n  3 and the Ais are pairwise disjoint and singleton. As every Np is empty or singleton, the pro…le is UO (relative to any order of N). It is not UA: if it were, say relative to the order of N , then each i 2 N would have to be extreme, i.e., smallest or largest in (as i is the only individual accepting the proposition in Ai), which is not possible as there are n  3 individuals but only two extreme positions. UO 6) SP. Consider a group, agenda X and pro…le (A1; :::; An) with n = 3 and A1 = fp; p1g, A2 = fp; p2g and A3 = fp; p3g, where p; p1; p2; p3 2 X are pairwise distinct. This pro…le is UO, relative to any order of N . But it is not SP: if it were SP, say relative to an order p1:::p2k of X, then in this order p would have to be a neighbour of p1 (by A1 = fp; p1g), and one of p2 (by A2 = fp; p2g), and also one of p3 (by A3 = fp; p3g), a contradiction. UO 6) SC. This follows from UO 6) SP by SC ) SP.  Proof of Lemma 2. Let A  X be complete and inconsistent. Among all inconsistent subsets of A, let B be one of smallest size jBj. We show that B is irreducible. Suppose for a contradiction that B is reducible to C  X. We will de…ne an inconsistent subset of A smaller than B, in contradiction to the choice of B. By jCj < jBj and the choice of B, we have C 6 A. So there is a p 2 CnA. Since A is complete, we have :p 2 A. As C is a reduction of B, there is a subset B  B with jBnBj  2 and B ` p. Now B [ f:pg is an inconsistent subset of A smaller than B: B [ f:pg is a subset of A by B  B  A and :p 2 A; B [ f:pg is inconsistent by B ` p; jB [ f:pgj  jBj+ 1 = jBj  jBnBj+ 1  jBj  2 + 1 < jBj.  Proof of Proposition 6. Let (1; :::;n) be as speci…ed, and denote by (A1; :::; An) the corresponding judgment pro…le, whose judgment sets Ai (= Ai) are complete and consistent as each i is also fully rational. For all i and all distinct x; y 2 K, x i y , y 6i x; so that for (1; :::;n) intermediateness on triples is indeed equivalent to order restriction on triples. Moreover, as each Ai is complete and consistent, for (A1; :::; An) local unidimensional orderedness with respect to Y is indeed equivalent to local unidimensional alignment with respect to Y (see corollary 1). So it remains 32 to show that (1; :::;n) is intermediate on triples if and only if (A1; :::; An) is locally unidimensionally ordered (with respect to Y). To prove the latter, recall that the irreducible sets are, by Remark 5, the cycles of length 1 or 2 or 3, i.e. the subagendas essentially identical to a subagenda of type fxPxg or fxPy; yPxg (x 6= y) or fxPy; yPz; zPxg (x; y; z distinct): (3) So, using that unidimensional orderedness on a subagenda is equivalent to unidimensional orderedness on any essentially identical subagenda, (A1; :::; An) is locally unidimensionally ordered if and only if it is unidimensionally ordered on any subagenda of one of the three types in (3). Unidimensional orderedness holds trivially on subagendas of the rst type fxPxg; and similarly for subagendas of type fxPy; yPxg (x 6= y): consider an order of N beginning with the individuals i with x i y, and followed by the individuals i with y i x. So local unidimensional orderedness is equivalent to unidimensional orderedness on each of the subagendas fxPy; yPz; zPxg (x; y; z 2 K distinct). But this is equivalent to intermediateness of (1; :::;n), as one easily checks (using that, for distinct x; y 2 K, :xPy 2 Ai , yPx 2 Ai for all Ai).  The proof of Proposition 8 requires a lemma: Lemma 3 Let S 6= ? be a set of subsets I  N that are each intervals relative to some xed linear order on N . If the elements of S are pairwise non-disjoint (i.e., I \ J 6= ? for all I; J 2 S), they are all non-disjoint (i.e., \I2SI 6= ?). Proof. Let S be as de…ned in the lemma. Note that S must be nite. So a proof by induction on the size of S is possible. More precisely, we prove by induction that \i2SI = [maxi2I min I;min I2S max I] 6= ?. First let S have size 1, say S = fIg. The claim then holds, since \J2SJ = I = [min I;max I], which is non-empty because it can be written as I \ I, a non-empty set by pairwise non-disjointness. Now suppose the claim holds for sets of a size k ( 1), and consider a set S of size k+1, say S = S0 [fJg where S0 has size k. We have \I2SI = J \ (\I2S0I), where by induction hypothesis, \I2S0I = [maxI2S0 min I;minI2S0 max I] 6= ?. So \I2SI = J \ [max I2S0 min I;min I2S0 max I]: This set obviously equals [maxI2S min I;minI2S max I]. To complete the proof, suppose for a contradiction that this interval is empty. The intersection of two intervals (here, of J and [maxI2S0 min I;minI2S0 max I]) can only be empty if the largest element of one of the intervals is smaller than the smallest element of the other interval. So either minI2S0 max I < minJ or max J < maxI2S0 min I. In the rst case, there is an I 2 S0 such that max I < minJ , so that I \ J = ?. In the second case, there is an I 2 S0 with max J < min I, so that again I \ J = ?. So in any case pairwise non-disjointness is violated, a contradiction.  33 Proof of Proposition 8. We prove part (a). Parts (b) and (c) follow analogously. Consider a pro…le (A1; :::; An) of consistent judgment sets. By Remarks 1 and 2, it su¢ ces to show that (i) single-plateauedness implies value-restriction and that (ii) unidimensional orderedness implies value-restriction. (i) Suppose (A1; :::; An) is single-plateaued, say relative to the order . To show value-restriction, consider any non-singleton minimal inconsistent set Y . We must specify a two-element subset of Y not contained in any Ai. De…ne it as consisting of the smallest element p and the largest element q of Y (relative to the order ). As required, no Ai can contain both p and q: otherwise it would include the entire interval from p to q (by single-plateauedness), hence include the inconsistent set Y , a contradiction. (ii) Suppose, for a contradiction, that (A1; :::; An) is unidimensionally ordered but not value-restricted. Let Y be a non-singleton minimal inconsistent set for which value-restriction is violated. Let S be the set ffi 2 N : p 2 Aig : p 2 Y g. By unidimensional orderedness, S consists of intervals (relative to a structuring order ). Further, these intervals are pairwise non-disjoint: otherwise there would be p; q 2 Y such that fi 2 N : p 2 Aig \ fi 2 N : q 2 Aig = ?, so that no Ai contains both p and q, whence value-restriction would not be violated for Y . So, by Lemma 3, S has a non-empty intersection. In other words, some Ai contains all p 2 Y . But then Ai is inconsistent, a contradiction.  Proof of Proposition 9. Let (A1; :::; An) be as speci…ed, and denote by (1; :::;n) the corresponding preference pro…le. We rst show that (b) is equivalent to (c), and then that (a) is equivalent to (b). (c) =) (b). First suppose (1; :::;n) is triplewise value-restricted. Consider any non-singleton irreducible Y  XK . By Remark 5, Y is a cycle of length 2 or 3. If Y has length 2, hence is a binary inconsistent set, we can take Z = Y , and by individual consistency no Ai includes Z. Now let Y be a 3-cycle, hence essentially identical to a set of the form fxPy; yPz; zPxg for distinct x; y; y 2 K. By triplewise value-restriction, some of x; y; z is in (1; :::;n) either never ranked between, or never above, or never below, the two other alternatives. We go through all nine cases: if x is never ranked between y and z, no Ai is a superset of Z = fzPx; xPyg; if y is never ranked between x and z, no Ai is a superset of Z = fxPy; yPzg; if z is never ranked between x and y, no Ai is a superset of Z = fyPz; zPxg; if x is never ranked above y and z, no Ai is a superset of Z = fxPy; yPzg; if y is never ranked above x and z, no Ai is a superset of Z = fyPz; zPxg; if z is never ranked above x and y, no Ai is a superset of Z = fzPx; xPyg; if x is never ranked below y and z, no Ai is a superset of Z = fyPz; zPxg; if y is never ranked below x and z, no Ai is a superset of Z = fzPx; xPyg; if z is never ranked below x and y, no Ai is a superset of Z = fxPy; yPzg. (b) =) (c). Now let (A1; :::; An) be weakly value-restricted. To show that (1 ; :::;n) is triplewise value-restricted, consider any distinct alternatives x; y; z 2 K. By Remark 5, the sets Y = fxPy; yPz; zPxg and Y 0 = fzPy; yPx; xPzg are irreducible and non-singleton. So, by weak value-restriction, Y has a two-element subset Z not included in any Ai, and similarly Y 0 has a two-element subset Z 0 not included in any 34 Ai. Assume Z = fxPy; yPzg (the proof is analogous for other binary subsets of Y ). Since each Ai is neither a superset of Z nor one of Z 0, we can conclude the following: if Z 0 = fzPy; yPxg, then no Ai ranks y between x and z; if Z 0 = fyPx; xPzg, then no Ai ranks z below x and y; if Z 0 = fxPz; zPyg, then no Ai ranks x above y and z. So, whatever Z 0 is, we have triplewise value-restriction. (a) =) (b). Trivial, since irreducible sets are minimal inconsistent. (b) =) (a). Suppose (A1; :::; An) is weakly value-restricted. To show valuerestriction, consider any non-singleton minimal inconsistent set Y  XK . By Remark 4, Y is a cycle of some length k, hence is essentially identical to –we may assume identical to –a set of type Y = fx1Px2; x2Px3; :::; xk1Pxk; xkPx1g, with distinct x1; :::; xk 2 K for some k  2. We show by induction on the size k of Y that Y has a two-element subset Z that is not included in any Ai. First let k = 2 or k = 3. Then Y is by Remark 5 irreducible, hence has by weak value restriction a two-element subset Z not included in any Ai. Now suppose k  4, and let the claim hold for sets of size less than k. Consider the non-singleton irreducible sets Y 0 = fx1Px2; x2Px3; x3Px1g and Y 00 = fx1Px3; x3Px4; :::; xk1Pxk; xkPx1g. By induction hypothesis, (*) Y 0 has a binary subset Z 0 not included in any Ai; and (**) Y 00 has a binary subset Z 00 not included in any Ai. We distinguish three cases. Case 1: x3Px1 =2 Z 0. Then Z 0  Y , and we may put Z = Z 0. Case 2: x1Px3 =2 Z 00. Then Z 00  Y , and we may put Z = Z 00. Case 3: x3Px1 2 Z 0 and x1Px3 2 Z 00. Then Z 0 = fp; x3Px1g for some p 2 fx1Px2; x2Px3g, and Z 00 = fq; x1Px3g for some q 2 fx3Px4; :::; xk1Pxk; xkPx1g. De…ne Z = fp; qg. Obviously, Z is a two-element subset of Y . Further, no Ai includes Z: if p 2 Ai, then x3Px1 =2 Ai (as Ai does not include Z 0), so x1Px3 2 Ai (as Ai is complete and consistent), and so q =2 Ai (as Ai does not include Z 00); if q 2 Ai, then x1Px3 =2 Ai (as Ai does not include Z 00), so x3Px1 2 Ai (as Ai is complete and consistent), and so p =2 Ai (as Ai does not include Z 0).