The Single-Peaked Domain Revisited: A Simple Global Characterization *

Transcription

1 The Single-Peaked Domain Revisited: A Simple Global Characterization * Clemens Puppe Department of Economics and Management Karlsruhe Institute of Technology (KIT) D Karlsruhe, Germany clemens.puppe@kit.edu Preliminary and incomplete version: February 14, 2016 First version: October 2015 * I am indebted to Tobias Dittrich and Michael Müller who provided excellent research assistance. The computer program for classifying majority consistent preference domains (on a set with four alternatives) that Tobias Dittrich developed was crucial in formulating the conjectures that led to the results reported here. He also created the figures based on a graphic tool developed by David McCooey. This work was presented in seminars at Higher School of Economics in Moscow (Oct. 2015), Université de Cergy- Pontoise and Université Paris Dauphine (Nov. 2015), at the Workshop on Game Theory and Social Choice at Corvinus University Budapest (Dec. 2015). I am grateful to the participants for valuable discussion and comments. Special thanks go to Ulle Endriss, Gleb Koshevoy, Jérôme Lang, Jean-Francois Laslier, Bernard Monjardet, Matias Nunez, Hans Peters, Marcus Pivato, Ernesto Savaglio, Attila Tasnádi and Bill Zwicker for particularly helpful comments.

2 Abstract It is proved that, among all restricted preference domains that guarantee consistency (i.e. transitivity) of pairwise majority voting, the single-peaked domain is the only minimally rich and connected domain that contains two completely reversed strict preference orders. It is argued that this result explains the predominant role of single-peakedness as a domain restriction in models of political economy and elsewhere. The main result has a number of corollaries, among them a dual characterization of the single-dipped domain; it also implies that a single-crossing ( order-restricted ) domain can be minimally rich only if it is a subdomain of a single-peaked domain. The main conclusions are robust as they apply both to strict and weak preference orders. JEL Classification D71, C72 Keywords: Social choice, single-peakedness, single-dippedness, pairwise majority voting, median voter theorem, single-crossing property, distributive lattice.

3 1 Introduction A subset of preference orders on a finite set of alternatives is called single-peaked if there exists a left-to-right arrangement of alternatives such that all upper contour sets are connected ( convex ) with respect to the given left-to-right arrangement of alternatives. The celebrated median voter theorem of Black [1948] and Arrow [1951] states that the domain of all single-peaked linear orders with respect to a fixed underlying spectrum of alternatives form a Condorcet domain, i.e. pairwise majority voting with an odd number of individuals induces a transitive relation. Moreover, the domain of all single-peaked preferences is minimally rich in the sense that every alternative is on top of at least one preference ordering; it is connected in the sense that every two single-peaked orders can be obtained from each other by a sequence of transpositions of neighboring alternatives such that the resulting order remains single-peaked at each step; and it contains two completely reversed orders (namely, the order that has the left-most alternative at the top and the order that has the right-most alternative at the top). This paper s main result shows that, conversely, every minimally rich and connected Condorcet domain which contains at least two completely reversed orders must be singlepeaked. As is easily seen, any single-peaked domain contains at most one pair of completely reversed orders. As a corollary to our main result we obtain that, for any given pair of completely reversed orders, there is a unique maximal Condorcet domain that contains them and is minimally rich as well as connected: the domain of all orders that are single-peaked with respect to either of the given pair of completely reversed orders. This result is remarkable in particular in view of the fact that quite a number of nonsingle-peaked Condorcet domains have been identified in the literature, among others the domains satisfying Sen s value restriction condition (Sen [1966]) with the single-dipped domain (Inada [1964]) as a special case, the domains satisfying the so-called intermediateness property (Grandmont [1978], Demange [2012]), and the order-restricted domains identified by Rothstein [1990]; the latter domains are sometimes also referred to as the domains with the single-crossing property (Roberts [1977], Saporiti [2009], Puppe and Slinko [2015]). Our analysis shows that none of these domains can jointly satisfy the three simple conditions of minimal richness, connectedness and the inclusion of a pair of completely reversed orders unless it is also single-peaked. In particular, a single-crossing domain can be minimally rich only if it is at the same time also single-peaked. Our main result also implies that Condorcet domains with the maximal number of elements on a given set of alternatives, the so-called maximum Condorcet domains, are never minimally rich. This may seem particularly surprising, as it is known that that the cardinality of a maximum Condorcet domain may exceed the cardinality the domain of all single-peaked domain considerably; for instance, with n alternatives a single-peaked domain has at most 2 n 1 elements, while the cardinality of the maximum Condorcet domains are 9, 20, 45, for n = 4, 5, 6, respectively. 1 The main result is robust as it generalizes with some additional work, but without 1 The exact cardinality of the maximum Condorcet domains for larger n is unknown, but it is known that it always exceeds 2 n 1 (see Monjardet [2009]). 1

4 major novel difficulties to the case of weak preference orders, i.e. to the case in which individual preferences may display indifferences. In this case, the domain of all singlepeaked weak orders does not form a Condorcet domain in our sense since, even with an odd number of voters, the indifference relation corresponding to pairwise majority voting may be intransitive. Related to this observation, the notion of connectedness has to be suitably adapted since two neighboring weak orders may differ in the ranking of more than one pair of alternatives if one of these orders displays an indifference class with more than two elements. Nevertheless, we still obtain that any connected, minimally rich Condorcet domain that contains (at least) two completely reversed strict orders must be single-peaked with respect to either one of the pair of completely reversed orders. However, if one allows for indifferences in particular also on the top of individual preferences there are different possibilities to formulate the condition of minimal richness. If one only requires that each alternative be among the top elements of at least one order in the domain, the stated conditions characterize a class of different non-isomorphic maximal Condorcet domains, all of which are proper subdomains of the domain of all single-peaked orders. If, on the other hand, one imposes the stronger requirement that each alternative be the unique top element of at least one order in the domain, the stated conditions characterize a unique maximal Condorcet domain of single-peaked orders (with respect to the left-to-right arrangement of alternatives given by either one of the completely reversed strict orders). The above characterization results of the single-peaked domain imply dual characterization results for the single-dipped domain in a straightforward way. Specifically, any connected Condorcet domain with maximal width such that every alternative is (among) the least preferred alternative(s) for some order in the domain must be single-dipped. 2 Characterizing the single-peaked domain: The case of strict preference orders 2.1 Statement of main result Consider a finite set of alternatives X and the set P(X) of all linear (strict) orders (i.e., complete, transitive and antisymmetric binary relations) on X. A subset D P(X) will be called a domain of preferences or simply a domain. A profile π = (P 1,..., P n ) on D is an element of the Cartesian product D n for some number n N of voters, where the linear order P i represents the preferences of the ith voter over the alternatives from X. A profile with an odd number of voters will simply be referred to as an odd profile. Frequently, we will denote linear orders simply by listing the alternatives in descending order, e.g. the linear order that ranks a first, b second, c third, etc., is denoted by abc.... The majority relation associated with a profile π is the binary relation Pπ maj on X y if and only if more than half of the voters rank x above y. Note that, such that xpπ maj according to this definition, the majority relation is asymmetric and for any odd profile π and any two distinct alternatives x, y X, we have either xpπ maj y or xpπ maj y. The class of domains D P(X) such that, for all odd n, the majority relation associated with any 2

5 profile π D n is transitive has received significant attention in the literature, see the excellent survey of Monjardet [2009] and the references therein. In the following, we will refer to any such domain as a Condorcet domain. A domain D is called a maximal Condorcet domain if every Condorcet domain (on the same set of alternatives) that contains D as a subset must in fact coincide with D. It is well-known that any maximal Condorcet domain D is closed in the sense that the majority relation of any odd profile from D is again an element of D (and not only of P(X)), cf. [Puppe and Slinko, 2015, Lemma 2.1]. A domain D is single-peaked with respect to the linear order > on X if, for all P D and all w X, the upper contour sets U P (w) := {y X : yp w} are connected ( convex ) in the order >, i.e. {x, z} U P (w) and x < y < z jointly imply y U P (w). A domain D is simply called single-peaked if there exists some linear order > such that D is single-peaked with respect to >. The domain of all orders that are single-peaked with respect to the fixed order > is denoted by SP >. If a domain is single-peaked with respect to >, we will often call the linear order > the spectrum underlying the single-peaked domain. A path in P(X) is subset {P 1,..., P m } P(X) with m 2 such that for all j = 1,..., m 1, the two consecutive orders P j and P j+1 differ in the ranking of exactly one pair x, y of (distinct) alternatives; note in that case x and y must be adjacent alternatives in both orders P j and P j+1. A domain D will be called connected if, for every pair P, P D of distinct orders in D, there exists a path {P 1,..., P m } that connects P and P (i.e. P 1 = P and P m = P ) and that lies entirely in D (i.e. P j D for all j = 1,..., m). Two orders P and P inv are called completely reversed if P and P inv rank the alternatives in X in exactly the opposite order, i.e. for all distinct x and y, xp y (xp inv y). Note that by the completeness assumption, two orders P, P P(X) are completely reversed if and only if xp y yp inv x. A domain is said to have maximal width if it contains at least one pair of completely reversed orders. Finally, a domain D will be called minimally rich if, for every alternative x X, there exists an order P D such that P has x as the top alternative. The following characterization of the single-peaked domain is this paper s main result. Theorem 1. a) For every linear order > on X, the domain SP > of all single-peaked orders with respect to > is a connected and minimally rich Condorcet domain with maximal width. b) Conversely, let D P(X) be a connected and minimally rich Condorcet domain with maximal width. Then, D is single-peaked. Except perhaps for the connectedness, the properties of the domain of all singlepeaked orders stated in part a) are straightforward to verify. Clearly, the two completely reversed orders are > itself and its reverse. Note that a single-peaked domain can contain at most one pair of completely reversed orders, 2 therefore such pair uniquely determines a corresponding maximal single-peaked domain, and we have the following corollary. Corollary 1. Let D P(X) be a maximal Condorcet domain that is connected, minimally rich and contains the pair P, P inv of completely reversed orders, then D = SP > where the spectrum > is given by either P or P inv. 2 This follows, e.g., immediately from Fact 2.2 below. 3

6 The following figure depicts the (unique) maximal single-peaked domain containing the pair abcd and dcba of completely reversed orders on the set X = {a, b, c, d} (the domain consists of the orders marked in red color). Fig. 1: A maximal single-peaked domain on X = {a, b, c, d}. The proof of Theorem 1 is provided in Subsection 2.3 below. The most interesting part of the theorem is of course the sufficiency for single-peakedness of the three simple conditions stated in part b). Note that it is not asserted that D must contain all orders that are single-peaked with respect to some given linear order (and this does in fact not follow); on the other hand, we will show that due to the connectedness and the inclusion of a pair of completely reversed orders, any domain satisfying the conditions of Theorem 1b) must contain at least #X (#X 1)/2 + 1 elements (because every ordered pair of alternatives has to be switched at least once on any path connecting the two completely reversed orders). Theorem 1b) is proved via a series of intermediate results which are interesting in their own right. In particular, we shall see that in fact a somewhat stronger result holds, since the connectedness of D can be replaced in part b) by the weaker condition that D contains a path connecting the two completely reversed orders (see Subsection 2.3 where this condition is referred to as quasi-connectedness ). Before providing the proof, we discuss the meaning and significance of the characterizing conditions in Theorem 1 in the next subsection. We will also show there that the characterization of the single-peaked domain provided by Theorem 1 is tight in the sense that each condition in part b) is indeed necessary to obtain the conclusion. 2.2 Discussion The conditions imposed on preference domains in Theorem 1 will now be discussed. In particular, we demonstrate by means of examples that each condition in part b) is neces- 4

7 sary to obtain the single-peakedness of the domain. A secondary purpose of this subsection is to illustrate the great diversity of the class of (maximal) Condorcet domains Consistency of majority voting The condition of consistency of pairwise majority voting lies of course at the heart of the present analysis and defines the problem. Clearly, the condition that D be a Condorcet domain is necessary for the conclusion that D is single-peaked. For instance, the universal domain P(X) evidently satisfies all other conditions in Theorem 1b), but the universal domain is clearly not single-peaked Maximal width: The existence of two completely reversed orders One may think of a domain as the description of a society. Under this interpretation, the existence of two orders in the domain that are completely reverses of each other is a condition of maximal width of opinions. The underlying society is required to admit the most extreme opinions with respect to at least one dimension. As with the other conditions in Theorem 1, the maximal width condition describes a substantial requirement. 3 Figure 2 illustrates the necessity of the maximal width condition in Theorem 1b) by displaying a subdomain of P(X) on X = {a, b, c, d} that is not single-peaked and that satisfies all conditions of Theorem 1b) except for the existence of two completely reversed orders. Fig. 2: A connected and minimally rich Condorcet domain without a pair of two completely reversed orders. 3 Also mathematically, it has significant consequences. Indeed, as we shall see in Section 2.3 below, together with the consistency of majority voting it implies that the domain can be embedded in a distributive lattice. 5

8 The connectedness and minimal richness of the depicted domain (the red marked orders) is evident. To verify that it is not single-peaked, note first that a and d are the only two alternatives that occur at the bottom of each marked order. Since abcd is a member, this implies that if the domain is to be single-peaked with respect to >, we must have either a > b > c > d or d > c > b > a. However, in either case the contained order bdca, for instance, would not qualify as single-peaked. The fact that the domain depicted in Fig. 2 is indeed a Condorcet domain can be easily inferred from the following well-known result. 4 Fact 2.1. Let D P(X) with X finite. The following statements are equivalent. i) D is a Condorcet domain, i.e. the majority relation corresponding to any odd profile on D is an element of P(X). ii) The majority relation corresponding to every profile on D is acyclic. 5 iii) In any triple x, y, z X of pairwise distinct alternatives, there exists one element that either never has rank 1, or never has rank 2, or never has rank 3 in the restrictions of the orders in D to the set {x, y, z}. iv) For no triple P 1, P 2, P 3 D, and no triple x, y, z X of pairwise distinct alternatives one has xp 1 yp 1 z, yp 2 zp 2 x and zp 3 xp 3 y. We finally note that the domain depicted in Fig. 2 is in fact a maximal Condorcet domain Connectedness Continuing with the metaphor of a domain representing a society, connectedness has a clear meaning as well: it must be possible to reach any admissible opinion from any other admissible opinion by a series of minimal changes in the corresponding rankings while staying in the domain at each step. This may be viewed as a homogeneity condition which forbids that opinions are clustered around a few representative opinions. Figure 3 illustrates this; it depicts a maximal Condorcet domain on X = {a, b, c, d} that satisfies all conditions of Theorem 1b) except connectedness. The society corresponding to this domain is clustered around the opinion that the pair of alternatives {a, b} dominates the pair {c, d} (the 4-cycle in the front) and the opposite opinion that the pair {c, d} dominates the pair {a, b} (the 4-cycle in the back). That the depicted domain is a Condorcet domain follows again easily using Fact 2.1; that it is not single-peaked follows at once from the fact that it violates the following simple (and well-known) necessary condition for singlepeakedness. 4 See, e.g. [Monjardet, 2009, p. 142]). Condition iii) is Sen s [1966] value restriction and condition iv) has been introduced by Ward [1965] as the absence of a Latin square (in other terminology, it requires the absence of a Condorcet cycle ; cf. Condorcet, 1785). In light of this condition, Condorcet domains of (linear) orders are sometimes referred to as acyclic sets of linear orders (e.g. by Fishburn [1996]). 5 An asymmetric binary relation P is acyclic if there does not exist a subset {x 1,..., x m } X such that x 1 P x 2, x 2 P x 3,..., x m 1 P x m and x m P x 1. 6 The verification of this statement is straightforward if somewhat tedious. 6

9 Fact 2.2. Suppose that D P(X) is single-peaked. Then there are at most two alternatives in X which can occur at the bottom of any order in D. 7 Fig. 3: A non-connected, minimally rich Condorcet domain containing pairs of completely reversed orders Minimal richness Minimal richness has a straightforward interpretation as well: no alternative should a priori be ruled as the individually most desired choice. The condition is termed minimal here because in the literature much stronger richness conditions have been imposed. 8 Note that the domain of all single-peaked preferences with respect to some fixed linear order > on X in fact also satisfies a stronger richness condition, namely that each alternative occurs not only sometimes as the best but also as the second-best alternative. Despite its innocuous appearance, the minimal richness condition has quite some bite, as illustrated by the two domains depicted in Figure 4. Both domains are maximal, connected Condorcet domains and contain the pair abcd and dcba of completely reversed orders. Evidently, neither domain is minimally rich, and by Fact 2.2 above, neither domain is single-peaked. That the depicted domains are indeed Condorcet domains follows again from Fact 2.1, and their respective maximality can be verified in a straightforward way. Interestingly, among all maximal connected Condorcet domains on X = {a, b, c, d} with maximal width, the domain on the left hand side of Fig. 4 has the minimal number of elements (in this case, #X (#X 1)/2 + 1 = 4 3/2 + 1 = 7) and the domain on the 7 Note that this condition is clearly not sufficient for single-peakedness as the domain depicted in Fig. 2 shows. 8 Our terminology follows Aswal et al. [2003] and Chatterji and Sen [2011]; the latter work also discusses domains violating the minimal richness requirement. Stronger richness conditions have been used, e.g., in Chatterji et al. [2013], Nehring and Puppe [2007]). 7

10 right hand side has the maximal number of elements. In fact, the domain depicted on the r.h.s. of Fig. 4 has the maximal number of elements among all Condorcet domains on a set of four alternatives, namely 9 (Monjardet [2009]). A Condorcet domain with the maximal number of elements is sometimes referred to as a maximum Condorcet domain. It is known that for #X 6, the maximal number of elements of a Condorcet domain is attained by connected domains with maximal width, 9 and that the maximal number of elements of such domains always exceeds the number of elements of any single-peaked domain (Monjardet [2009]). Theorem 1b) thus implies that a maximum Condorcet domain is never minimally rich, and that the maximal cardinality of a connected and minimally rich Condorcet domain with maximal width is 2 #X 1, the number of all single-peaked orders with respect to some fixed linear order of X. Fig. 4: Two connected but not minimally rich Condorcet domains containing a pair of completely reversed orders. The strength of the minimal richness condition, at least when imposed jointly with connectedness and the maximal spread condition, can also be inferred from the following immediate corollary of Theorem 1b) and Fact 2.2. Corollary 2. Let #X 3. There does not exist a connected and minimally rich Condorcet domain on X containing two completely reversed orders such that every alternative in X is the at the bottom of some order in D. Note that the domain depicted in Fig. 3 above satisfies all conditions in Corollary 2 except for the connectedness. 9 For #X = 3, 4, 5, 6, the maximum Condorcet domains have 4, 9, 20, 45 elements, respectively. For #X = 7, the maximal number of elements of a connected Condorcet domain with maximal width is 100 (Galambos and Reiner [2008]), but it is not known whether this is also the maximal number of elements among all Condorcet domains on a set with 7 elements. 8

11 2.3 Proof of the main result The proof of the connectedness of the domain SP > of all single-peaked orders with respect to any spectrum > in Theorem 1a) is given through a straightforward induction argument in the appendix. The proof of Theorem 1b) is more involved. We start with some preparatory observations. The universal domain P(X) is naturally endowed with a betweenness structure as follows. An order Q is between the orders P and P if Q (P P ), i.e., if Q agrees with all binary comparisons in which P and P agree (Kemeny and Snell [1960]). 10 The set of all orders that are between P and P is called the interval spanned by P and P and is denoted by [P, P ]. The domain P(X) endowed with this betweenness relation is sometimes referred to as the permutahedron. A subset D P(X) is called median stable if, for any triple of elements P 1, P 2, P 3 D, there exists an element P med = P med (P 1, P 2, P 3 ) D, the median order corresponding to P 1, P 2, P 3, such that P med [P 1, P 2 ] [P 1, P 3 ] [P 2, P 3 ]. It is well-known that the median order of any triple P 1, P 2, P 3 P(X), if it exists, is unique. The following observation is well-known but fundamental, and will be useful in the following (cf. [Nehring and Puppe, 2007, Cor. 5]). Fact 2.3. A triple P 1, P 2, P 3 P(X) admits a median order if and only if the majority relation Pπ maj corresponding to the profile π = (P 1, P 2, P 3 ) is transitive, in which case P med (P 1, P 2, P 3 ) and Pπ maj coincide. In particular, a domain D is a closed Condorcet domain if and only if it is median stable Assuming maximality without loss of generality The general proof strategy for Theorem 1b) is as follows. In the literature, maximal connected Condorcet domains with maximal spread have been extensively studied and it is known that they display very regular behavior; in particular, any such domain has the structure of a distributive lattice (see Monjardet [2009]). Now, evidently every Condorcet domain D is contained in some maximal Condorcet domain D D. If one can prove Theorem 1b) for D using its regular structure, then one would obtain D = SP > for some linear order > on X, and since D D = SP > one could conclude the singlepeakedness of D as well. However, this proof strategy requires that D inherits the appropriate properties from D. Clearly, if D has maximal width, so has D ; moreover, if D is minimally rich, so is D. But prima facie, it is not at all clear if D inherits the connectedness from D. In the following, we will prove that it indeed does. Call a domain D P(X) semi-connected if it contains a pair of completely reversed orders and a path connecting these two orders (cf. Danilov et al. [2012]). Clearly, semiconnectedness is implied by the conjunction of connectedness and maximal width. Unlike connectedness, the property of semi-connectedness is inherited from a domain to any 10 Some authors such as, e.g., Grandmont [1978] and Demange [2012] refer to orders that are between two others in this sense as intermediate orders. 9

12 superdomain. In the following, we will refer to a pair P, P D of distinct orders satisfying [P, P ] D = {P, P } as D-neighbors, and to P(X)-neighbors often simply as neighbors. Note that P and P are P(X)-neighbors if and only if P and P differ in the ranking of exactly one pair of alternatives (which must be an adjacent pair in either order). Say that D P(X) is directly connected if all D-neighbors are neighbors. In the present case of linear orders, a domain is thus directly connected if and only if all D-neighbors differ in the ranking of exactly on pair of alternatives. For any two distinct orders P, P P(X), say that {P 1, P 2,..., P k } is a direct path connecting P and P if P 1 = P, P k = P, for all j {1,..., k 1}, the orders P j and P j+1 are neighbors, and for all 1 j < j k, {P j, P j+1,..., P j } [P j, P j ]. The following result follows from a more general result proved in the appendix (see Lemma??). Fact 2.4. A domain D P(X) is directly connected if and only if all pairs of distinct orders in D admit a direct path connecting them. Fact 2.4 shows in particular that direct connectedness of a domain implies its connectedness. The converse does not hold, as the following simple example demonstrates. Let D = {acb, bac, bca, cab, cba}, i.e. let D consist of all linear orders on the set {a, b, c} with the order abc removed. Evidently, D is connected, but not directly so, since the two orders acb and bac are D-neighbors but not neighbors. While the connectedness of a domain thus does not imply its direct connectedness in general, the following result shows that this implication does hold for closed Condorcet domains with maximal width. In fact, the implication holds even under the weaker assumption of semi-connectedness. The result represents the first important step towards our proof of Theorem 1b). Proposition 2.1. Let D P(X) be semi-connected. If D is a closed Condorcet domain, then D is directly connected. Proof. By contradiction, suppose that P, P D are distinct orders with [P, P ] D = {P, P } that differ in the ranking of more than one pair of alternatives, say ap b and cp d while bp a and dp c (note that it is not assumed that {a, b} {c, d} = ). Observe that since P and P are D-neighbors, one has P med (P, P, Q) {P, P } for all Q D. Now, by the semi-connectedness, let P 1, P m D be two completely reversed orders and let {P 1,..., P m } D be a path connecting them. Without loss of generality, let ap 1 b. Then, since P m is completely reversed, bp m a. Thus, there exist (at least) two neighbors P k, P k+1 {P 1,..., P m } such that ap k b and bp k+1 a. There are two cases, either (i) cp k d and cp k+1 d, or (ii) dp k c and dp k+1 c. In case (i), the triple P, P, P k+1 D does not admit a median order (since neither P [P, P k+1 ], nor P [P, P k+1 ]); in case (ii), the triple P, P, P k D does not admit a median order (since neither P [P, P k ], nor P [P, P k ]). Thus, in each case we obtain a contradiction to the assumption that D is a closed Condorcet domain. By Proposition 2.1, any Condorcet domain D satisfying the conditions of Theorem 1b) is contained in a maximal Condorcet domain satisfying the same conditions, and in fact the stronger condition of direct connectedness, as asserted by the following result. 10

13 Proposition 2.2. Let D P(X) be a connected and minimally rich Condorcet domain with maximal width, and suppose that D D is a maximal Condorcet domain. Then, D is directly connected, minimally rich and satisfies the maximal width condition. Proof. Clearly, the conditions on D guarantee its semi-connectedness. Since D contains D it is also semi-connected (containing in particular a pair of completely reversed orders) and minimally rich. As a maximal Condorcet domain D is closed, thus it is directly connected by Proposition 2.1. Since any subdomain of a single-peaked domain is single-peaked with respect to the same underlying spectrum, we may thus assume maximality of D without loss of generality in Theorem 1b). In fact, we may moreover assume that D is directly connected. And we could weaken the conjunction of the conditions of connectedness and maximal width in Theorem 1b) to the condition of semi-connectedness Maximal Condorcet domains as unions of maximal direct paths Henceforth, we will thus assume that D is a maximal, directly connected and minimally rich Condorcet domain with maximal width. In the following, we denote by P, P a fixed pair of completely reversed orders in D. In the following, we will make extensive use of the very regular structure of maximal directly connected Condorcet domains containing two completely reversed order. The following result gives a first hint at how to exploit this structure. Proposition 2.3. Let D P(X) be a maximal, directly connected Condorcet domain containing the pair P, P of completely reversed orders. Then, D is the union of all direct paths in D connecting P and P. Proposition 2.3 follows from a more general result proved in the appendix (see Lemma??), but the underlying idea is simple. Let Q D be an arbitrary order. By the direct connectedness there exist direct paths A D connecting Q and P, and B D connecting Q and P. Because P and P are completely reversed, one has A B = {Q} and one can show that the union A B is a direct path connecting P and P. It is easily verified that the direct paths in a domain D P(X) connecting two completely reversed orders are the direct paths of maximal length. In fact, along such a direct path all distinct pairs of alternatives have to be switched exactly once. In particular, all these paths have length #X (#X 1)/ Maximal chains in the Bruhat order and the single-crossing property The last remark of the previous paragraph suggests that additional insights can be obtained by studying domains of linear orders in terms of the set of pairs of alternatives 11 The property that all maximal direct paths connecting two ordered elements in a lattice have the same length is sometimes referred to as the Jordan-Dedekind chain condition, see, e.g., Grätzer and Schmidt [1957]. We note already here that this property does not generalize to the case of weak orders, in which maximal direct paths can have different length. 11

14 that their member orders switch with respect to some fixed reference order. This has indeed been done in the literature via the so-called weak Bruhat order, see Abello [1991], Chameni-Nembua [1989], Galambos and Reiner [2008], Danilov et al. [2012], among others. Specifically, fix any order P P(X); then, every order P P(X) is uniquely identified by the set β P := {(x, y) X X : yp x and xp y} of pairs of alternatives in the ranking of which P disagrees with P. The partial order on {β P : P P(X)} given by set inclusion is known as the weak Bruhat order. It is well-known that this partial order indeed has the structure of a lattice, and that every maximal and directly connected Condorcet domain with maximal width corresponds to a distributive sub-lattices of this order (see, e.g. Monjardet [2009]). 12 While the theory of the weak Bruhat order provides the abstract mathematical background of the present analysis, it is possible to give elementary proofs of all results needed for our purpose, and we will do so now. It is evident that the maximal direct paths in P(X) connecting a given pair of completely reversed orders P and P correspond to the maximal chains in the weak Bruhat order, with minimal element β P = and maximal element β P = {(x, y) X X : xp y}. Moreover, the maximal direct paths have the single-crossing property, as follows. Let C = {P 1,..., P k } with P 1 = P and P k = P be a maximal direct path. Then, for any pair a, b of distinct alternatives the sets {P C : ap b} and {P C : bp a} are connected in the sense that, if ap l b and ap m b for l < m, then ap j b for all j {l, l + 1,..., m}, and similarly if bp l a and bp m a for l < m, then bp j a for all j {l, l + 1,..., m}. In general, call a domain D a single-crossing domain if it can be written in the form D = {P 1,..., P k } such that, for all pairs a, b X, ap l b and ap m b with l < m implies ap j for all j {l, l+1,..., m}. The following observation is immediate. Fact 2.5. The maximal direct paths between two completely reversed orders in P(X) correspond exactly to the maximal single-crossing subdomains of P(X). It is well-known that all single-crossing domains are closed Condorcet domains (Rothstein [1990, 1991]). However, only some but by far not all maximal single-crossing domains are maximal Condorcet domains (Monjardet [2007], Puppe and Slinko [2015]) A restriction on the order of how pairs of alternatives are switched We now describe an important observation, first made by Galambos and Reiner [2008], on the order of how pairs of alternatives are switched along maximal direct paths within a maximal Condorcet domain. As above fix two completely reversed orders P and P, and consider any triple x, y, z X, say with xp yp z. By the fact that along any maximal direct path from P to P every pair of alternatives is switched exactly once, we obtain a unique order in which the three pairs (x, y), (x, z) and (y, z) are switched along any maximal direct path, respectively; call this the pairwise switching order of the triple x, y, z. Furthermore, call the switching order (x, y), (x, z), (y, z) lexicographic, and the reverse switching order (y, z), (x, z), (x, y) anti-lexicographic. 12 Indeed, the latter assertion holds without the assumption of direct connectedness (Danilov and Koshevoy [2013]), and even for all closed (not necessarily maximal) Condorcet domains that contain a pair of completely reversed orders (Puppe and Slinko [2015]). 12

15 Fact 2.6 (Galambos and Reiner, 2008). Let D be a maximal, directly connected Condorcet domain with maximal width. a) The switching order of any triple along any given maximal direct path in D is either lexicographic or anti-lexicographic. b) The switching order of any triple must be the same along all maximal direct paths. Proof. a) Let x, y, z X be such that xp yp z, and consider any direct path connecting P and P. Evidently, since every pair of {x, y, z} is switched one at a time when traveling the path from P to P, the pair (x, z) cannot be switched first. Assume that (x, y) is switched first. Then, since every pair is switched only once along the path, the next pair from {x, y, z} which is switched must be (x, z) and we obtain the lexicographic switching order. By the same argument, we obtain the anti-lexicographic switching order if (y, z) is switched first along the given direct path, see Fig. 5. b) Now consider two maximal direct paths connecting P and P, say C and C, and let x, y, z X be such that xp yp z. By contradiction, suppose that the switching orders of the triple x, y, z are different along the two paths, say lexicographic along C and antilexicographic along C. Then one can choose an order P C with yp zp x, and an order P C with zp xp y. But in this case, the triple P, P, P restricted to {x, y, z} forms a Condorcet cycle and thus violates the necessary and sufficient condition iv) in Fact 2.1 for a domain to be a Condorcet domain. Fig. 5: The lexicographic and anti-lexicographic switching orders. Remark. Let D be a union of maximal direct paths connecting two fixed orders P and P. As Galambos and Reiner [2008] observe, the condition that the switching order of any given triple is either lexicographic along every maximal direct path connecting P and P, or anti-lexicographic along every maximal direct path connecting P and P is not only necessary for D to be Condorcet domain but also sufficient. While Galambos and Reiner [2008] refer to the theory of hyperplane arrangements and (higher) Bruhat orders for a proof, in particular to the work of Ziegler [1993], a simple elementary proof follows again from Fact 2.1 above. Indeed, consider any triple {x, y, z}, say with xp yp z. If the switching order of this triple is lexicographic along all maximal direct paths, no order Q D can have zqxqy; if, on the other hand, the switching order of this triple is antilexicographic along all maximal direct paths, no order Q D can have yqzqx. Hence, in 13

16 either case, the necessary and sufficient condition iv) in Fact 2.1 for D to be a Condorcet domain is satisfied Minimal richness implies lexicographic switching Consider a Condorcet domain D and a maximal direct path C D connecting the completely reversed orders P and P in D. Evidently, if a triple {x, y, z} with xp yp z is switched anti-lexicographically along C, the alternative y is never best among the triple (cf. Fig. 5). But this implies that y can in fact never be the top alternative of any order in C. Combining Proposition 2.3 and Fact 2.6, we thus obtain the following result. Proposition 2.4. Let D P(X) be a maximal connected and minimally rich Condorcet domain containing the completely reversed orders P and P. Then, all triples {x, y, z} with xp yp z are switched in lexicographic order along every maximal direct path connecting P and P Completing the proof of Theorem 1 We are now ready to complete the proof of Theorem 1b). Consider a connected and minimally rich Condorcet domain D containing the pair P and P of completely reversed orders. By Proposition 2.2, D is contained in a maximal Condorcet domain D that is directly connected and minimally rich. Clearly, D also contains P and P. By Proposition 2.3, D is the union of all (maximal) direct paths connecting P and P. By Proposition 2.4, all triples are switched in lexicographic order along every maximal direct path connecting P and P. The proof is thus completed by the following observation. Fact 2.7. Let C be a maximal direct path connecting the completely reversed order P and P. If all triples {x, y, z} with xp yp z are switched in lexicographic order, then C is single-peaked with respect to > = P. Proof. We have to show that all upper contour sets of all orders P C are connected in the order > = P. Thus, consider any order P C and any triple x, y, z with x > y > z, i.e. xp yp z, and suppose that {x, z} U P (w) for some w X. Since the triple is switched lexicographically along C, the restriction of every order P in C to the set {x, y, z} has one of the following form: xp yp z, yp xp z, yp zp x, or zp yp x (cf. Fig. 5). Evidently, in each case {x, z} U P (w) implies y U P (w). 2.4 A dual characterization of the single-dipped domain Theorem 1 above entails a dual characterization of the single-dipped domain in a straightforward way, as follows. A domain D is called single-dipped with respect to the linear order > on X if, for all P D and all w X, the lower contour sets L P (w) := {y X : wp y} are connected ( convex ) in the order >, i.e. {x, z} U P (w) and x < y < z jointly imply y U p (w). A domain D is called single-dipped if there exists some linear order > such that D is 14

17 single-dipped with respect to >. 13 The set of all linear orders that are single-dipped with respect to > is denoted by SD >. Theorem 2. a) For every linear order > on X, the domain SD > is a connected Condorcet domain with maximal width such that every alternative in X is the worst alternative for some order. b) Conversely, let D P(X) be a connected Condorcet domain with maximal width such that every alternative in X is the worst alternative for some order in D. Then, D is single-dipped. Proof. Part a) is analogous to the proof of Theorem 1a). For part b) note that, as in the proof of Theorem 1b), we may assume that D is a maximal connected Condorcet domain, and in fact that it is the union of all direct paths connecting two completely reversed orders P, P D. By an argument completely symmetric to the one that led to Proposition 2.4, the condition that every alternative must occur at the bottom of some order in D implies that every triple x, y, z with xp yp z is switched anti-lexicographically along all direct paths connecting P and P (cf. Fig. 5). By an argument symmetric to the one in the proof of Fact 2.7, this implies that the lower contour sets of every order P D are connected in the order given by P. Hence, D is single-dipped. 2.5 Corollary: all minimally rich single-crossing domains are single-peaked Our analysis has an immediate implication for all single-crossing domains. Recall from above that D is a single-crossing domain if it can be written as D = {P 1,..., P k } such that, for all pairs a, b X, the sets {P D : ap b} are connected in the set {1,..., k}. We have the following corollary. Corollary 3. Let D P(X) be a single-crossing domain. If D is minimally rich, then it is single-peaked. If every alternative in X occurs at the bottom of at least one order in D, then D is single-dipped. Proof. First, any single-crossing domain can be extended to a maximal single-crossing domain C D (note that we are not asserting that C is a maximal Condorcet domain, only that C is maximal among all single-crossing domains). Indeed, if D = {P 1,..., P k } does not contain two completely reversed orders, we can add the inverse of P 1 as the last order, i.e. consider the domain {P 1,..., P k, P k+1 } where P k+1 = P1 inv. On the other hand, if D does contain two completely reversed orders, we must have P k = P1 inv. In any case, we can now fill possible gaps in the sequence P 1,..., P k, P k+1 as follows. If P 1 and P 2 differ in the ranking of more than one pair of alternatives, at least one of them must be an adjacent pair in P 1. Then, we can add the order P 1 that switches exactly this pair and agrees with 13 Inada [1964] introduced the condition of single-dippedness under the name of single-cavedness. We follow here the more recent and, by now, well established terminology, see e.g. Klaus et al. [1997]. 15

18 P 1 in the ranking of all other pairs, and consider the domain {P 1, P 1, P 2,...}. Continuing in this fashion, we finally arrive at a maximal single-crossing domain C, which by Fact 2.5 forms a maximal direct path connecting two completely reversed orders. Thus, if C is minimally rich, it satisfies all conditions of Theorem 1b); similarly, if every alternative in X occurs at the bottom of some order in C, it satisfies all conditions of Theorem 2b). The desired conclusion thus follows from these results. 2.6 A further result on the structure of connected Condorcet domains with maximal width In this subsection, we derive a further remarkable result about the structure of connected Condorcet domains with maximal width. It concerns the sequence in which different alternatives occupy a fixed rank in the orderings along a maximal direct path. Specifically, consider a Condorcet domain containing the pair of completely reversed orders P and P and any direct path C D. Denote by s k C the sequence in which the alternatives occupy rank k in the orders along the path C in direction from P to P. We will refer to this sequence as the rank-k-sequence associated with the maximal direct path C. 14 Importantly, we omit possible repetitions in this definition. Thus, if s k C = (x 1, x 2,..., x m ) this means that alternative x 1 occupies rank k in the ordering P and perhaps also in the orders that follow P along the path C in direction to P. If the k-th ranked alternatives changes for the first time, it changes to x 2, etc.; the final alternative occupying rank k along C is x m, in particular, x m is the k-th alternative in the order P. To illustrate, consider the single-crossing domain C to the left of Fig. 4 above with P = abcd and P = dcba. We have s 1 C = (a, d) since a is the top alternative in the first four orders (including P ) and d is the top alternative in the remaining three order (including P ). Similarly, we obtain s 2 C = (b, c, d, a, c) as the sequence of second ranked alternatives, s3 C = (c, b, d, c, a, b) as the sequence of third ranked alternatives, and finally s 4 C = (d, b, a) as the sequence of alternatives at the bottom of the orders along the path C. Note that since all direct paths C connecting two completely reversed orders in P(X) have length #X (#X 1)/2 + 1 this is also maximal length of any sequence s k C. Since we omit possible repetitions, the sequences s k C will be shorter in general. Proposition 2.5. Let D be a connected Condorcet domain containing the pair of completely reversed orders P and P. Then, for all k {1,..., #X} and all direct paths C, C D that connect P and P, s k C = s k C, (2.1) i.e. all rank-k-sequences along C and C coincide. Proof. (to be completed) 14 The concept is distantly related to the notion of pseudoline used in Galambos and Reiner [2008]. However, unlike the present concept of rank-k-sequences, pseudolines are defined in terms of the pairwise transpositions. 16

19 To illustrate Proposition 2.5, consider the maximal Condorcet domain to the right of Fig. 4 above (recall that this is the maximum Condorcet domain on a set of four alternatives). There are four different maximal direct paths connecting the completely reversed orders abcd and dcba, since there are two bifurcations when traveling from abcd in direction of dcba: at abcd one can either go to the left by switching the pair (c, d), or to the right by switching (a, b); and at bdac one can again either go to the left by switching (b, d) or to the right by switching (a, c). For every path C among the resulting four direct paths, we have s 1 C = (a, b, d), s2 C = (b, a, d, b, c), s3 C = (c, d, a, c, b) and s4 C = (d, c, a). Observe that, in accordance with our analysis, the corresponding domain is neither single-peaked nor single-dipped. Indeed, minimal richness would require that s 1 C contains all alternatives from X (and in fact that these occur in s 1 C in exactly the same order as in P ). Similarly, the condition that every alternative be worst in at least one order of the domain would require that all alternatives in X occur in s #X C (in exactly the same order as in P ). 3 Characterizing the single-peaked and single-dipped domains: The case of weak preference orders 3.1 Statement of main results Consider now the set R(X) of all weak orders (i.e., complete and transitive binary relations) on X, and subdomains D R(X). Individual weak preferences are denoted by R, R etc., and profiles by ρ = (R 1,..., R n ) R(X) n. Frequently, we will denote weak orders by listing the alternatives in descending order and putting indifferent alternatives in brackets, e.g. the weak order that ranks a first and b and c indifferently as second best is denoted by a(bc)...; similarly, the weak order that has a and b indifferently at the top and c at the following rank is denoted by (ab)c...; finally, the weak order that has a, b and c indifferently on top is denoted by (abc)..., etc. The majority relation associated with a profile ρ is the binary relation Rρ maj on X such y if and only if xr i y for more than half of the voters. Note that, according that xr maj ρ to this definition, the majority relation is complete for every odd profile ρ. As above, the domains D R(X) such that, for all odd n, the majority relation associated with any profile ρ D n is transitive are referred to as Condorcet domains. Condorcet domains of weak orders have been studied much less than their counterparts with linear orders, see the monograph of Gaertner [2001] for a state-of-the-art survey. As above, a domain D R(X) is called a maximal Condorcet domain if every Condorcet domain (on the same set of alternatives) that contains D as a subset must in fact coincide with D. The following fact is easily verified. Fact 3.1. Every maximal Condorcet domain D R(X) is closed in the sense that the majority relation of any odd profile from D is again an element of D (and not only of R(X)). While some results of the analysis of linear orders carry over with straighforward modifications to the case of weak orders, there are also important differences. First, 17

20 the equivalences in Fact 2.1 do not carry over to the domain of weak orders. As a simple example, consider the domain on X = {a, b, c} consisting of the three weak orders (ab)c, a(bc) and cba, say. If ρ is the profile in which three voters have each one of these preferences, respectively, the majority relation is acyclic but not transitive, since a majority strictly prefers a to c while [arρ maj b and brρ maj a] as well as [brρ maj c and crρ maj b], i.e. both pairs of alternatives (a, b) and (b, c) are deemed indifferent, respectively, according to the majority relation. Thus, by our definition, the domain is not a Condorcet domain. As above, a domain D is called single-peaked with respect to the linear order > on X if, for all R D and all w X, the upper contour sets U R (w) := {y X : yrw} are connected ( convex ) in the order >, i.e. {x, z} U R (w) and x < y < z jointly imply y U R (w). A domain D is called single-peaked if there exists some linear order > such that D is single-peaked with respect to >. The second important difference to the case of linear orders is that not all single-peaked domains are Condorcet domains, in particular, the domain of all weak orders that are single-peaked with respect to some order > is no longer a Condorcet domain. The same example as above can be used to demonstrate this. We already argued above that the domain D = {(ab)c, a(bc), cba} is not a Condorcet domain. On the other hand, it is clearly single-peaked with respect to the linear order a < b < c. 15 As in the case of linear orders, a domain D of weak orders is called singledipped if there exists a linear order > on X such that, for all R D, the lower contour sets L R (w) := {y X : wry} are connected with respect to >. We also have to generalize the notion of connectedness, as follows. As above, for any two orders R, R R(X), denote by [R, R ] R(x) the set of orders that agree with R and R in all binary comparisons in which R and R agree; formally, [R, R ] := {Q R(X) : Q R R }. As above, we will refer to [R, R ] as the interval spanned by R and R, and to its elements as the orders between R and R ; furthermore, two distinct orders R and R are called D- neighbors if [R, R ] D = {R, R }. In this section, R(X)-neighbors are sometimes simply referred to as neighbors. Note that two R(X)-neighbors may differ in the ranking of more than one pair of alternatives. For instance, the six neighbors of the weak order (abc) on the set X = {a, b, c} each differ from (abc) in the ranking of two pairs of alternatives, respectively; see Fig. 6 for the betweenness structure of R({a, b, c}). The following simple observation will be useful (and is straightforward to verify). Fact 3.2. Two distinct orders R, R are R(X)-neighbors if and only if either (i) R R and, for all Q R(X) with R Q R, we have Q = R or Q = R, or (ii) R R and, for all Q R(X) with R Q R, we have Q = R or Q = R. 15 One can certainly argue about the correct extensions of the concepts of Condorcet domain and singlepeakedness to the case of weak orders. For instance, one could require only acyclicity of the majority relation for a domain to be called Condorcet domain, or require uniqueness of the top alternative in the definition of single-peakedness. I do not want to argue that the adopted definitions are the only reasonable ones. But for the purpose of the present study, these definitions turn out to be precisely the appropriate notions in order to demonstrate the robustness of the main characterization result. 18

21 A path in R(X) is subset {R 1,..., R m } R(X) with m 2 such that for all j = 1,..., m 1, the two consecutive orders R j and R j+1 are R(X)-neighbors. A domain D R(X) will be called connected if, for every pair R, R D of distinct orders in D, there exists a path {R 1,..., R m } that connects R and R (i.e. R 1 = R and R m = R ) and that lies entirely in D (i.e. R j D for all j = 1,..., m). Fig. 6: The betweenness structure of R({a, b, c}). As above, two orders R and R inv are called completely reversed if, for all distinct x and y, xry (xr inv y). Note that by the completeness assumption, two completely reversed weak orders must both in fact be linear orders, i.e. neither of the two can contain any non-trivial indifference. A domain is said to have maximal width if it contains at least one pair of completely reversed orders. The following fact lists the maximal connected Condorcet domains D R({a, b, c}) containing the two completely reversed orders abc and cba (see Fig. 6 for illustration). Fact 3.3. The following are the maximal connected Condorcet subdomains of R({a, b, c}) that contain the completely reversed orders abc and cba: D 1 = {abc, (ab)c, bac, b(ac), bca, (bc)a, cba}, D 2 = {abc, (ab)c, (abc), (bc)a, cba}, D 3 = {abc, (ab)c, (abc), c(ab), cba}, D 4 = {abc, a(bc), (abc), (bc)a, cba}, D 5 = {abc, a(bc), (abc), c(ab), cba}, D 6 = {abc, a(bc), acb, (ac)b, cab, c(ab), cba}. The verification is straightforward if tedious. A helpful observation is the fact that no Condorcet domain on {a, b, c} can simultaneously contain the universal indifference relation (abc) and two consecutive orders that admit exactly one indifference, such as 19