Condorcet domains satisfying Arrow’s single-peakedness

Journal of Mathematical Economics 84 (2019) 166–175

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Journal of Mathematical Economics journal homepage: www.elsevier.com/locate/jmateco

Condorcet domains satisfying Arrow’s single-peakedness✩ Arkadii Slinko Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand

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Article history: Received 11 February 2019 Received in revised form 29 July 2019 Accepted 2 August 2019 Available online 14 August 2019 Keywords: Majority voting Transitivity Condorcet domains Median graphs Single-peaked property

a b s t r a c t Condorcet domains are sets of linear orders with the property that, whenever the preferences of all voters belong to this set, the majority relation of any profile with an odd number of voters is transitive. Maximal Condorcet domains historically have attracted a special attention. We study maximal Condorcet domains that satisfy Arrow’s single-peakedness which is more general than Black’s single-peakedness. We show that all maximal Black’s single-peaked domains on the set of m alternatives are isomorphic but we found a rich variety of maximal Arrow’s single-peaked domains. We discover their recursive structure, prove that all of them have cardinality 2m−1 , and characterise them by two conditions: connectedness and minimal richness. We also classify Arrow’s single-peaked Condorcet domains for m ≤ 5 alternatives. © 2019 Elsevier B.V. All rights reserved.

1. Introduction The famous Condorcet Paradox1 shows that if voters’ preferences are unrestricted, the majority voting can lead to intransitive group preference in which case the Condorcet Majority Rule (Marquis de Condorcet, 1785) does not determine a winner. Condorcet domains2 are sets of linear orders that are immune from this paradox which means that, whenever the preferences of all voters are drawn from a Condorcet domain, the majority voting never yields cycles. The first maximal Condorcet domains – one for each number of alternatives – were produced by Black (1948, 1958), who called them the domains of the single-peaked preferences. Such a domain for m alternatives contains 2m−1 linear orders. And although larger maximal Condorcet domains have been discovered (Kim and Roush, 1980; Abello and Johnson, 1984; Fishburn, 1996, 2002), the domain of single-peaked linear orders remains by far the most popular among social choice theorists and political scientists. The reason is that it is often plausible to assume that voters’ preferences are ‘one-dimensional’ and determined by a single parameter, e.g., the location of this voter on the left–right political spectrum. ✩ Arkadii Slinko was supported by the Marsden Fund 3706352 of The Royal Society of New Zealand. E-mail address: [email protected]. 1 This term appeared for the first time in Arrow (1951). 2 This term was coined by Monjardet (2006). Before that Condorcet domains have been also called consistent profiles (Ward, 1965), valued restricted domains (Kim and Roush, 1980), transitive simple majority domains or consistent sets (Abello and Johnson, 1984), acyclic sets (Fishburn, 1996, 2002), majority consistent sets (Craven, 1996). https://doi.org/10.1016/j.jmateco.2019.08.001 0304-4068/© 2019 Elsevier B.V. All rights reserved.

Arrow (1963), however, used a condition weaker than singlepeakedness requiring only that single-peaked condition is satisfied on triples, i.e., he required that Black’s single-peaked condition is satisfied only locally and not globally. He noted that domains that satisfy this weaker condition are still Condorcet domains and Inada (1964) showed that this condition is indeed more general and does not imply Black’s single-peakedness. Monjardet (2009) calls it Arrow–Black’s single-peakedness but we prefer to follow Raynaud (1981) and call it Arrow’s singlepeakedness. Dasgupta and Maskin (2008) claim that in practical elections voters are rarely choosing their orders from the universal domain and that ideology is the primary reason for that. In particular, they note that in 2002 French presidential elections and in 2000 U.S. presidential election voters were selecting their orders from two different Condorcet domains. In this paper we investigate the size and the structure of maximal Condorcet domains that satisfy Arrow’s single-peakedness. Unlike Black’s single-peakedness, where all maximal domains (on the sets of alternatives of the same cardinality) are isomorphic, we find a rich class of maximal Condorcet domains satisfying Arrow’s single-peakedness. However, surprisingly, all of them – as the maximal Black’s single-peaked domain – contain 2m−1 orders despite the restrictions on Arrow’s single-peaked domains are local and not global as with Black’s single-peakedness. We show that maximal Arrow’s single-peaked domains have a nice recursive structure, namely, a maximal Arrow’s single-peaked domain on n alternatives is constructed from two maximal Arrow’s singlepeaked domains on sets of n − 1 alternatives with isomorphic subdomains which are maximal Arrow’s single-peaked domain on n − 2 alternatives. Our structural results allow to obtain an

A. Slinko / Journal of Mathematical Economics 84 (2019) 166–175

if and only if characterisation of Arrow’s single-peaked domains by two conditions: connectedness and minimal richness. Finally, we classify all maximal Arrow single-peaked domains on a set of m ≤ 5 alternatives. The paper is organised as follows. In Section 2 we discuss the relation of this paper to the closest papers in the literature. Section 3 introduces the main concepts. In Section 4 we formulate and prove our main structural results. In Section 5 we obtain an if and only if characterisation of Arrow’s single-peakedness and show how all the existing characterisations of Black’s singlepeakedness follow from it. In Section 6 we classify, up to an isomorphism, maximal Arrow’s single-peaked domains on sets of four and five alternatives, respectively. In Section 7 we provide a polynomial algorithm for checking if a set of never-bottom conditions is consistent and, hence, defines a Condorcet domain. Section 8 concludes.

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3. Preliminaries 3.1. Condorcet domains and their graphs Let A be a finite set of alternatives and L(A) be the set of all (strict) linear orders on A. Any subset D ⊆ L(A) will be called a domain of linear orders on A or simply a domain. The set L(A) itself is called a universal domain. A profile of linear orders from a particular domain D is a sequence P = (v1 , . . . , vn ) of linear orders vi ∈ D. We will denote |P | = n. We also write a ≻i b instead of a vi b. Note that, unlike a domain, a profile can have several identical linear orders. A linear order v on the set A will be written as v = ai1 ai2 . . . aim , where ai1 is the top preference and aim is the bottom preference. We will also denote posv (ai ) = i indicating position of alternative ai in v , i.e., the top preference has position 1 and the bottom one has position m.

2. Relation to the literature Previous research has concentrated on Condorcet domains with two completely reversed orders like tiling Condorcet domains (Danilov et al., 2012), symmetric Condorcet domains (Danilov and Koshevoy, 2013), Black’s single-peaked Condorcet domains (Puppe, 2018). Danilov and Koshevoy (2013) call such a domain normal and Puppe (2018) says that such a domain has maximal width. Puppe (2018) even states that ‘‘it is not evident whether there are natural classes of maximal Condorcet domains that violate the maximal width condition’’. The closest to our paper in the literature is the aforementioned paper by Puppe (2018). In a way we extend his analysis from Black’s single-peakedness to much more general Arrow’s singlepeakedness. Our results show that the normality (or the maximal width condition) is a rather arbitrary requirement and there are natural domains that do not satisfy it. Moreover, these domains provide as much individual freedom as the Black’s single-peaked one as they have the same number of linear orders, which is 2m−1 , where m is the number of alternatives.3 They do not have two completely reversed orders but they satisfy a weaker normality condition, namely, they contain two orders, which we call extremal, with their top and bottom preferences reversed. Puppe (2018) showed that any maximal Black’s single-peaked domain is characterised by three properties: connectedness, minimal richness and existence of two completely reversed orders. We show that, in fact, the properties of connectedness and minimal richness characterise exactly the class of Arrow single-peaked domains and that, within the class of maximal Arrow Condorcet domains, the Black’s single-peaked domain is characterised by the existence of two completely reversed orders. The characterisation of single-peakedness by Ballester and Haeringer (2011) also immediately follows from our structural result. This paper builds on Puppe and Slinko (2016) but takes slightly different approach. In that paper domains of linear orders were considered as subsets of the permutahedron, so did (Raynaud, 1982). In this paper we pay attention to graphs of domains as defined in Puppe and Slinko (2016) without any reference to the permutahedron. This greatly simplifies the matter and allows us to classify Arrow’s single-peaked domains for up to five alternatives. A number of not so recent results on Condorcet domains presented in an excellent survey of Monjardet (2009) are also relevant to this paper. 3 It has long been known that 2m−1 is an upper bound for the size of a maximal Arrow’s single-peaked Condorcet domain (Raynaud, 1981; Köhler, 1978) but we prove that all maximal Arrow Condorcet domains actually achieve this bound.

Definition 1. The majority relation ⪰P of any profile P is defined as a ⪰P b ⇐⇒ |{i | a ≻i b}| ≥ |{i | b ≻i a}|. For an odd number of linear orders in the profile P this relation is a tournament, i.e., complete and asymmetric binary relation. In this case we denote it ≻P . Now we can define the main object of this investigation. Definition 2. A domain D ⊆ L(A) is a Condorcet domain if the majority relation of any profile P built from an odd number of linear orders of D is transitive. A Condorcet domain D is maximal if for any Condorcet domain D′ the inclusion D ⊆ D′ ⊆ L(A) implies D = D′ . A Condorcet domain is closed if the majority relation of any profile with an odd number of voters built from linear orders of this domain again belongs to this domain. There are a number of alternative definitions of Condorcet domains, see e.g., Puppe and Slinko (2016), where it is also proved that any maximal Condorcet domain is closed. In this paper we will make an attempt at classification of a certain class of Condorcet domains and, thus, we need to be able to identify similar Condorcet domain which, for example, are different only due to relabelling of alternatives or/and permuting voters. Hence we need the concept of an isomorphism. We will set [n] = {1, 2, . . . , n} and denote the symmetric group on [n] as Sn . We note that given two equinumerous sets of alternatives A1 and A2 , any bijection ψ : A1 → A2 induces a ˆ: L(A1 ) → L(A2 ) of respective sets of linear orders by mapping ψ setting for any a, b ∈ A1 and v ∈ L(A1 )

ˆ(v ) ψ (b). a v b ⇐⇒ ψ (a) ψ Now we can give the following definition of an isomorphism of domains. Definition 3. Let D1 = {u1 , . . . , un } and D2 = {v1 , . . . , vn } be two domains on sets of alternatives A1 and A2 , respectively, of equal cardinality, i.e., |A1 | = |A2 |. We say that domains D1 and D2 are isomorphic if there are a bijection ψ : A1 → A2 and a permutation ˆ(ui ) = vσ (i) for all i ∈ [n]. σ ∈ Sn such that ψ Example 1. Up to an isomorphism, there are only three maximal Condorcet domains on the set of alternatives {a, b, c }, namely, D3,1 = {abc , acb, cab, cba}, D3,2 = {abc , acb, bca, cba}, D3,3 = {abc , bac , bca, cba}.

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Fig. 1. Graphs of D3,1 , D3,2 and D3,3 .

The universal domain L(A) is naturally endowed with the following betweenness structure. An order v is between orders u and w if v ⊇ u ∩ w , i.e., v agrees with all binary comparisons in which u and w agree (Kemeny and Snell, 1960).4 The set of all orders that are between u and w is called the interval spanned by u and w and is denoted by [u, w]. The domain L(A) endowed with this betweenness relation is referred to as the permutahedron (Monjardet, 2009). Given a domain of preferences D, for any u, w ∈ D we define the induced interval as [u, w]D = [u, w] ∩ D. Puppe and Slinko (2016) defined a graph GD associated with this domain. The set of linear orders from D are the set of vertices VD of GD , and for two orders u, w ∈ D we draw an edge between them if there is no other vertex between them, i.e., [u, w]D = {u, w}. The set of edges is denoted ED so the graph is GD = (VD , ED ). As established in Puppe and Slinko (2016), for any Condorcet domain D the graph GD is a median graph (Mulder, 1978) and any median graph can be obtained in this way. The graphs of Condorcet domains from Example 1 are shown in Fig. 1. A domain D is called connected if its graph GD is a subgraph of the permutahedron; in Fig. 1 domains D3,1 and D3,3 are connected but D3,2 is not.5 3.2. Known constructions of Arrow’s single-peaked domains Black (1948, 1958) was the first to come up with a sequence of maximal Condorcet domains – one for each set of alternatives – which he called the domains of the single-peaked preferences. For a domain D = {v1 , . . . , vn } to be single-peaked he required the existence of a societal axis on A, i.e., he assumed that all alternatives can be written into a sequence a1 > a2 > · · · > am so that every order vi of D has its peak a∗i ∈ A such that b < c ≤ a∗i ⇒ c ≻i b, a∗i ≥ b > c ⇒ b ≻i c . Proposition 3.1. For every positive integer m all maximal singlepeaked domains over the set of m alternatives are isomorphic. Proof. Let A and B be two sets of alternatives with |A| = |B| = m, and D1 and D2 are two single-peaked domains over A and B, respectively. Let a1 > a2 > · · · > am ,

b1 > b2 > · · · > bm

be the two societal axes of these single-peaked domains. Then the isomorphism ψ between D1 and D2 can be defined as ψ (ai ) = bi , 4 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. 5 This old but well-entrenched terminology is at odds with the fact that the graph GD is always connected in the graph theory sense.

i = 1, . . . , m. It is easy to check that it is indeed an isomorphism. □ Inada (1964) noticed that a much weaker requirement is still sufficient for obtaining a Condorcet domain. He imposed the requirement of single-peakedness only on triples, i.e., he required that the restriction D{a,b,c } of a domain D on any triple {a, b, c } must be single-peaked relative to some societal axis specific for this triple. This condition appeared briefly in Arrow (1963) which gave (Monjardet, 2009) the reason to call this condition Arrow– Black’s single-peakedness (so do Raynaud and Arrow, 2011). We, however, prefer to follow Raynaud (1981) and call it Arrow’s single-peakedness while we often call Black’s single-peakedness as simply single-peakedness to comply with the majority of the literature.6 In Example 1 only D3,3 is single-peaked. It is worth noting that D3,1 and D3,3 have the same line graph but not isomorphic. Apart from Black’s single-peaked domains, whose construction is very well-known, there is only one known construction of Arrow’s single-peaked domains which is due to Romero (1978) (also translated into English by Raynaud and Arrow (2011) and mentioned by Monjardet (2009)). Given two linear orders p: x1 >1 x2 >1 · · · >1 xm ,

q: y1 >2 y2 >2 · · · >2 ym

on the set A of alternatives, we construct a domain D(p, q) by generating its orders as follows. By generating a particular order v we start with choosing a bottom-ranked alternative which can be either xm or ym , i.e., posv (xm ) = m or posv (ym ) = m. Then we remove the chosen alternative from both orders and repeat the procedure to choose the alternative whose position in v will be m − 1, etc. Unfortunately, as the following theorem shows, this construction never gives us new maximal Condorcet domains. Proposition 3.2. D(p, q) is Arrow’s single-peaked domain. This domain is maximal only if p and q are completely reversed in which case it is a maximal Black’s single-peaked domain. Proof. If p and q are completely reversed we get Black’s singlepeaked domain which is maximal (Romero, 1978; Monjardet, 2009). Let us show that if p and q are not completely reversed, then D(p, q) is not copious and by Theorem 1 cannot be maximal. Suppose a >1 b and a >2 b. Then in any order of D(p, q) alternative b will be ranked lower than a so for any third alternative c, in any order of the domain we can have either abc or acb or cab but not cba so D(p, q) is not copious. □ 6 Raynaud and Arrow (2011) sometimes call it Blackian single-peakedness.

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Example 2. If p = abcd and q = dbca, then D(p, q) = {abcd, bacd, bcad, bcda, bdca, dbca},

which is a proper subset of a maximal Arrow’s single-peaked domain D from Example 4 that will be introduced later. Hence, unfortunately, this construction does not produce any new maximal Arrow’s single-peaked Condorcet domains. 3.3. Defining Condorcet domains. The never conditions The domain D3,1 of Example 1 contains all linear orders on a, b, c where b is never ranked first, D3,2 contains all linear orders on a, b, c where a is never ranked second and D3,3 contains all linear orders where b is never ranked last. Following Monjardet, we denote these conditions as bN{a,b,c } 1, aN{a,b,c } 2 and bN{a,b,c } 3, respectively. We call the conditions xN{a,b,c } 1, xN{a,b,c } 2 and xN{a,b,c } 3 as never-top, never-middle and never-bottom conditions, respectively. The terminology of never conditions was introduced by Fishburn (1996, 2002), however, it has to be noted that these conditions are equivalent to Sen’s value restrictions (Sen, 1966) or Ward’s ‘absence of a Latin square’ (Ward, 1965) and ‘no-Condorcet-cycle property’ of Dasgupta and Maskin (2008).7 Given a set of alternatives A, we say that N = {xN{a,b,c } i | {a, b, c } ⊆ A, x ∈ {a, b, c } and i ∈ {1, 2, 3}}

is a complete set of never conditions if it contains at least one never condition for every triple a, b, c of distinct elements of A. Proposition 3.3. A domain of linear orders D ⊆ L(A) is a Condorcet domain if and only if it is non-empty and satisfies a complete set of never conditions. This proposition, in particular, means that the collection D(N ) of all linear orders that satisfy a certain complete set of never conditions N , if non-empty, is a Condorcet domain. Let us also denote by N (D) the set of all never conditions that are satisfied by all linear orders from D. Proposition 3.4. For any complete set of never conditions N the domain D(N ), if non-empty, is a closed Condorcet domain. Every maximal Condorcet domain is of this form. Proof. Let us first prove the closedness of D(N ). Suppose N is a complete set of never conditions and suppose there exist u1 , . . . , un from D(N ), with n being odd, such that the profile P = (u1 , . . . , un ) has the majority relation ≻P which is transitive (i.e., a linear order) but not in D(N ). Suppose, for example, aN{a,b,c } 3 ∈ N , i.e., a is never-bottom among a, b, c and the majority relation ≻P of P does not satisfy this never condition which means that it ranks a last. Restricting the profile P to {a, b, c } we get the profile P{a,b,c } with a voting situation n1 a b c

n2 a c b

n3 b a c

n4 c a b

Fig. 2. Graph of the single-peaked domain D4,1 on four alternatives.

For a maximal Condorcet domain D let N = N (D). Then D(N ) ⊇ D, and, due to the maximality of D, we have D = D(N ). □ Let us reformulate now Arrow’s single-peakedness in terms of never-bottom conditions. Proposition 3.5. A Condorcet domain D ⊆ L(A) is an Arrow’s single-peaked domain if and only if for every triple {a, b, c } ⊆ A this domain satisfies a never-bottom condition for one of the alternatives. Proof. As we saw in Example 1 single-peakedness on a triple is equivalent to one of the never-bottom conditions for this triple. □ Obviously, any Black’s single-peaked domain is Arrow’s singlepeaked but we will see that there are many interesting Condorcet domains which are Arrow’s single-peaked but not Black’s single-peaked. 3.4. Properties and further examples of Condorcet domains Definition 4. We call a Condorcet domain D copious if for any triple of alternatives a, b, c ∈ A the restriction D{a,b,c } of this domain to this triple has four distinct orders, that is, |D{a,b,c } | = 4. This definition is new to the literature. For some time we believed that any maximal Condorcet domain must be copious. However, it appeared that the maximal Condorcet domain D consisting only of four linear orders abcde, bdaec , ceadb, edcba is not copious. Indeed, D{b,d,e} = {bde, edb}. It was discovered by Danilov and Koshevoy (2013) and is defined by a complete set of never-middle conditions. Hence Definition 4 defines a non-trivial class of Condorcet domains and we will further see that all Arrow’s single-peaked domains which are defined by never-bottom conditions are copious. Example 3. Let us consider a maximal single-peaked domain on four alternatives: D4,1 = {abcd, bacd, bcad, cbad, bcda, cbda, cdba, dcba},

whose graph is presented in Fig. 2. We have D4,1 = D(N ), where N = {bN{a,b,c } 3, bN{a,b,d} 3, cN{a,c ,d} 3, cN{b,c ,d} 3},

where n1 , n2 , n3 and n4 are the number of voters with preferences abc, acb, bac and cab, respectively. Of course, n1 + n2 + n3 + n4 = n. Without loss of generality suppose that the majority relation for this profile has c ≻P b ≻P a, so a is bottom-ranked. Since in the majority relation b ≻P a we have n3 > n1 + n2 + n4 . But then also a ≻P c, which is a contradiction. Similar arguments work for aN{a,b,c } 1 and aN{a,b,c } 2.

and D4,1 is copious. Its subdomain D4,1 \ {bcda} is closed and copious Condorcet domain, which is of course not maximal. We learn two conclusions:

7 Danilov et al. (2012) and Danilov and Koshevoy (2013) use yet another terminology.

Here is an example of a maximal Arrow’s single-peaked domain which is not single-peaked.

• It is not true that any closed Condorcet domain is of the form D(N ) for some set of never conditions N ;

• A copious Condorcet domain is not necessarily maximal.

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Lemma 4.1. An Arrow’s single-peaked domain has at most two terminal alternatives. A maximal Arrow’s single-peaked domain has exactly two of them.

Fig. 3. Graph of an Arrow’s single-peaked domain D4,2 which is not Black’s single-peaked.

Example 4. Let us consider a maximal Arrow’s single-peaked domain on four alternatives: D4,2 = {abcd, bacd, bcad, cbad, bcda, cbda, bdca, dbca},

whose graph is presented in Fig. 3. We have D4,2 = D(N ), where N = {bN{a,b,c } 3, bN{a,b,d} 3, cN{a,c ,d} 3, bN{b,c ,d} 3}

and D4,2 is copious. This domain is not single-peaked (for example, because it does not have two completely reversed orders).

Proof. If a, b, c ∈ A are distinct bottom-ranked alternatives, then the triple {a, b, c } does not satisfy a never-bottom condition. Hence |Term(D)| ≤ 2. If |Term(D)| = 1 and, say, Term(D) = {z }, then z is the bottom ranked alternative in any order of D. Consider any triple with participation of z, say {a, b, z }. Both a and b are never-bottom in this triple, hence both aN{a,b,z } 3 and bN{a,b,z } 3 will be satisfied. If we move z in any order one position up, then for the new order one of the two never-bottom conditions will still be satisfied, hence we can add this new order to D and still have an Arrow’s single-peaked domain. In this case D cannot be maximal. □ Example 6. Terminal alternatives of a maximal Black’s singlepeaked domain are the leftmost and rightmost ones on the societal axis. 4.2. Recursive structure

Here is an example of a different kind. Example 5. aN{a,b,c } 3,

It is easy to see that for m = 4 the set bN{a,b,d} 3,

cN{a,c ,d} 3,

dN{b,c ,d} 3

of never-bottom conditions is inconsistent, that is defines an empty domain. Indeed, if there exists a linear order satisfying these conditions, then no one of the four alternatives can be bottom-ranked. In the literature various richness concepts for domains of preferences were considered. All of them are quite strong. The weakest of them – minimal richness – was one of the main concepts of Puppe’s characterisation (Puppe, 2018) of maximal Black’s single-peaked domains. Definition 5. A domain D ⊆ L(A) of linear orders on A is minimally rich if every alternative from A is the top alternative of at least one linear order in D. We note that in Example 1 the domains D3,2 and D3,3 are minimally rich while D3,1 is not. Both domains in Examples 3 and 4 are minimally rich. 4. Structural properties of Arrow’s single-peaked Condorcet domains As Example 4 shows Black’s single-peaked domains are not the only Arrow’s single-peaked domains (and later we will see that there is a large family of the latter). However, not surprisingly, Arrow’s single-peaked domains share some common features with Black’s single-peaked domains. This is reflected in their structural properties. 4.1. Terminal alternatives Definition 6. If D ⊆ L(A) is any domain over the set of alternatives A with |A| = m, then the alternatives from the set Term(D) = {x ∈ A | ∃v∈D posv (x) = m} will be called terminal. In other words, terminal alternatives are those which are bottom-ranked in at least one linear order of the domain. The following lemma (at least the first part of it) is known (Raynaud, 1981; Köhler, 1978) but we give its proof for completeness of exposition.

Now we start looking into the internal structure of maximal Arrow’s single-peaked domains. Suppose now that D ⊆ A is such a domain with |A| = m. Then by Lemma 4.1 |Term(D)| = 2, and let Term(D) = {a1 , a2 }. For i ∈ {1, 2}, let us consider subdomain ˜i = {v ∈ D | posv (ai ) = m}, and domain Di which is the D ˜i onto Ai = A \ {ai }. An important role will be also restriction of D ˜1 and D ˜2 : player by the following subdomains of D

ˆ1 = {v ∈ D | posv (a1 ) = m and posv (a2 ) = m − 1} ⊂ D ˜1 , D ˆ2 = {v ∈ D | posv (a2 ) = m and posv (a1 ) = m − 1} ⊂ D ˜2 . D The next several statements will use these assumptions and this notation.

ˆ1 and D ˆ2 Lemma 4.2. There exists an isomorphism σ between D such that σ (a1 ) = a2 , σ (a2 ) = a1 and σ being the identity mapping of A \ {a1 , a2 } onto itself. Proof. Suppose that for some order u ∈ L(A \ {a1 , a2 }) we ˆ2 . Then we claim that ua2 a1 ∈ D ˆ1 . Since ua2 a1 have ua1 a2 ∈ D ends with a2 a1 , it is sufficient to notice that ua2 a1 satisfies all the never-bottom conditions that D satisfied (since a1 cannot be never-bottom alternative in any triple). Then due to maximality of D ua2 a1 will be in D and hence in D1 . □ Lemma 4.3. D1 and D2 are maximal Arrow’s single-peaked domains on the sets A1 and A2 , respectively. Proof. It is obvious that D1 and D2 are Arrow’s single-peaked domains. To show that they are maximal we will firstly prove that for any triple of distinct alternatives {b, c , d} ⊆ A \{a1 , a2 } (if such exists) we have D1 {b,c ,d} = D2 {b,c ,d} = D{b,c ,d} .

(4.1)

˜1 with Suppose we have a linear order u = . . . x...y...z ...a1 in D {x, y, z } = {b, c , d} and a2 ranked somewhere (but not at the bottom as a2 ̸ = a1 ). We then move a2 to the last mth position without changing the order of other alternatives to obtain order v = . . . x...y...z ...a1 a2 . Since a2 is always a bottom alternative in any triple so moving it down means that v still satisfies all the never-bottom conditions that u satisfied. Hence it satisfies all defining never-bottom conditions that D satisfied. Due to maximality of D the order v is in D, hence in D2 . Thus D1 {b,c ,d} = D2 {b,c ,d} . This proves (4.1).

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Suppose now D1 is not maximal and there is a linear order u ∈ L(A1 ) such that D1 ∪ {u} is a larger Arrow’s single-peaked domain on A1 . Let us consider then u˜ = ua1 . We claim that D ∪{˜u} is also a Condorcet domain on A. For this we show that u˜ satisfies all never-bottom conditions not involving terminal alternatives. Let bcd be such alternatives and u{b,c ,d} be the restriction of u onto {b, c , d}. Then D1 {b,c ,d} ∪ u{b,c ,d} is a Condorcet domain which, due to (4.1) implies that D{b,c ,d} ∪ u{b,c ,d} is a Condorcet domain.

˜1 satisfy some never-bottom We know also that D and hence D condition xN{a2 ,x,y} 3 for every triple {a2 , x, y} with x ̸ = a2 and ˜1 onto {a2 , x, y} is contained y ̸ = a2 . Hence the restriction of D in {xa2 y, a2 xy, xya2 , yxa2 } and the restriction of u˜ onto {a2 , x, y} ˜2 onto {a2 , x, y} must be within this set. But the restriction of D is contained only in {xya2 , yxa2 }. Hence, if the addition of u˜ does not violate xN{a2 ,x,y} 3 for D1 , it would not violate the same never-bottom condition for D. Finally, any triple {a1 , x, y} with x ̸ = a1 and y ̸ = a1 in D satisfies a never-bottom condition xN{a1 ,x,y} 3 with x ̸ = a1 so adding u˜ with a1 at the bottom will not violate any never-bottom conditions of D involving a1 . Thus u˜ satisfies all the never-bottom conditions that D satisfies. This contradicts maximality of D and proves the lemma. □ Definition 7. Suppose a domain D ⊆ L(A) has Term(D) = {a1 , a2 } for some a1 , a2 ∈ A. Then a linear order v ∈ D such that posv (a1 ) = 1 and posv (a2 ) = m or posv (a2 ) = 1 and posv (a1 ) = m will be called extremal for D. In Examples 3 and 4 the pairs of extremal orders are abcd, dcba and

abcd, dbca,

respectively. Note that in the second case they are not completely reversed. Theorem 1. Any maximal Arrow’s single-peaked domain D on the set A of m alternatives satisfies the following properties: (a) (b) (c) (d) (e)

D D D D D

contains 2m−1 orders; is copious; is connected; is minimally rich; contains exactly two extremal linear orders.

Proof. (a) For m = 3 the unique (up to an isomorphism) Arrow’s single-peaked domain (domain D3 in Fig. 1) contains 4 = 23−1 orders, this will be a basis for the induction. Let D be a maximal Arrow’s single-peaked domain on m alternatives. We have D = ˜1 ∪ D ˜2 , where |D ˜1 | = |D1 | and |D ˜2 | = |D2 |. As by Lemma 4.3 D1 D and D2 are maximal on the set of m − 1 alternatives, by induction hypothesis we now deduce that |D1 | = |D2 | = 2m−2 , whence |D| = 2m−1 . (b) We will prove this also by induction. We note that for m = 3 all maximal Condorcet domains are copious. By induction hypothesis both D1 and D2 are copious, so the only triples {x, y, z } ⊆ A in D for which it is not clear that |D|{x,y,z } | = 4 are those which contain both a1 and a2 . Let {x, a1 , a2 } ⊂ A be such a triple. Let us choose y ∈ A \ {x, a1 , a2 }. The triple {x, y, a1 }, in D and hence in D2 , satisfies either xN{x,y,a1 } 3 or yN{x,y,a1 } 3. In the first case D{x,y,a1 } contains orders xya1 , yxa1 , a1 xy, xa1 y and in the second xya1 , yxa1 , a1 yx, ya1 x, respectively, hence we see that yxa1 and either a1 xy or a1 yx belongs to D2 {a1 ,x,y} . From this we see that D{x,a1 ,a2 } contains xa1 a2 and a1 xa2 . Similarly, we can prove it contains xa2 a1 and a2 xa1 . Thus |D|{x,a1 ,a2 } | = 4 and D is copious. (c) By the induction hypothesis D1 and D2 are connected (i.e., subgraphs of their respective permutahedra), which implies ˜1 and D ˜2 are connected as well. Since the order of a1 that D

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˜1 and D ˜2 is different, the one edge between them and a2 in D that is also an edge of the permutahedron is the edge between ...x...y...z ...a2 a1 and ...x...y...z ...a1 a2 . Let us show that there are no other edges between these two sets of vertices in GD . Suppose ˜1 is connected by an edge in GD with y = v a2 ∈ D ˜2 . x = ua1 ∈ D Let us move a2 down in x to obtain x′ = u′ a2 a1 and also move a1 down in y to obtain y′ = v ′ a1 a2 . Both x′ and y′ are between x and y, hence x = x′ = u′ a2 a1 and y = y′ = v ′ a1 a2 . If only v ′ ̸= u′ , then u′ a1 a2 is between x and y and different from both of them. Thus v ′ = u′ and the edge between x and y exists in the permutahedron. (d) By Lemma 4.3 D1 and D2 are maximal and by induction hypothesis both are minimally rich. Hence D1 has all alternatives of A1 = A\{a1 } on top in some rankings and D2 has all alternatives of A2 = A \ {a2 } on top. As the union of A1 and A2 is A, we have proved (c). (e) Let us consider Term(D1 ) = {b1 , b2 } and Term(D2 ) = {c1 , c2 }. We claim that a1 ∈ {c1 , c2 } and a2 ∈ {b1 , b2 }. The reason is that, if among b1 , b2 , c1 , c2 there were three distinct elements that are also different from a1 and a2 , then for this triple no neverbottom condition is satisfied. If Term(D1 ) = Term(D1 ) = {b, c }, then no never-bottom condition is satisfied for triples {ai , b, c }, i = 1, 2. Thus we may consider that Term(D1 ) = {a2 , b} and Term(D2 ) = {a1 , c }. By Lemma 4.3 D1 and D2 are maximal Arrow’s single-peaked domains on sets of m − 1 alternatives so by induction hypothesis D1 contains a linear order u with a2 on the top and D2 contains linear order v with a1 on the top. These ua1 and v a2 are two extremal orders of D sought for. To show the uniqueness of those linear orders suppose we have two different orders w1 and w2 satisfying posw1 (a1 ) = posw2 (a1 ) = 1 and posw1 (a2 ) = posw2 (a2 ) = m. Then w1 = a1 . . . x . . . y . . . a2 and w2 = a1 . . . y . . . x . . . a2 for some pair of alternatives x, y ∈ A. Then for the triple {a1 , x, y} only a1 can be the never-bottom alternative but it cannot be. This contradiction proves the statement. □ We note that Theorem 1(a) is quite a surprising result. The local Arrow’s single-peakedness is much weaker than the global Black’s single-peakedness but the former does not allow more individual freedom than the latter. This was discovered already by Raynaud (1981) that the cardinality of Arrow’s single-peaked domain cannot exceed 2m−1 . The novelty of our result is that every maximal Arrow’s single-peaked domain has exactly 2m−1 linear orders and hence provides no less freedom of choice than the Black’s single-peakedness. As is well-known (see e.g., Puppe (2018)) a maximal singlepeaked domain contains two completely reversed orders. However Theorem 1(e) shows that in the case of Arrow’s singlepeakedness we can salvage at least something, namely, the two extremal linear orders. These two extremal linear orders in Black’s single-peaked domain become completely reversed. But, in general, we can guarantee only that their top and bottom preferences are reversed. Example 4 shows that we cannot do any better. Let us now consider the structure of the graph of a maximal Arrow’s single-peaked domain. It has a recursive structure. The graph of a maximal Arrow’s single-peaked Condorcet domain on m alternatives is constructed from graphs of two maximal Arrow’s single-peaked Condorcet domains on m − 1 alternatives (which may not be isomorphic) each of which has a subgraph isomorphic to a graph of a maximal Arrow’s single-peaked Condorcet domains on m − 2 alternatives which is the same for both domains.

˜1 , D ˜2 , D ˆ1 , D ˆ2 be as before. Then the structure Corollary 1. Let D, D of the graph GD = (VD , ED ) is as follows:

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Proof. It is enough to show that an Arrow’s single-peaked domain D with two completely reversed orders is Black’s singlepeaked relative the axis defined by any of these orders. Let us choose one of those orders and denote it >. Suppose a linear order v ∈ D has a peak at a∗ , i.e., posv (a∗ ) = 1, and suppose that we also have b < c < a∗ . Then in our completely reversed orders (restricted to {a, b, c }) we have bca∗ and a∗ cb. The neverbottom condition which can be satisfied for the triple {a∗ , b, c } is cN{a∗ ,b,c } 3, hence the only two other remaining orders on this triple that can happen in the domain are cba∗ and ca∗ b. Since a∗ is the peak of v , the only option for v is a∗ cb, i.e., c is preferred to b as required. So D is Black’s single-peaked. □ The criterion of Ballester and Haeringer (2011) also immediately follows from our structural results.

Fig. 4. The anatomy of the graph of an Arrow’s single-peaked domain.

Corollary 3 (Ballester and Haeringer, 2011). A domain D ⊆ A is Black’s single-peaked if and only if 1. D is an Arrow’s single-peaked domain,9 and 2. There do not exist two orders, ≻1 , ≻2 ∈ D, and four alternatives x, y, z , t ∈ A such that the following four conditions simultaneously hold:

• VD = VD˜1 ∪ VD˜2 ; ˆ1 , D ˆ2 ), where E(D ˆ1 , D ˆ2 ) contains edges • ED = ED˜1 ∪ ED˜2 ∪ E(D connecting the vertices that correspond to each other under the isomorphism σ from Lemma 4.2 (see Fig. 4).

ˆ1 and D ˆ2 are Proof. According to Lemma 4.2 subdomains D isomorphic and the isomorphism only swaps a1 and a2 which ˆ1 is linked by an edge of permutahedron means that any order in D ˆ2 . And we showed in the proof of Theorem 1(c) to its image in D ˆ1 and D ˆ2 . □ that there are no other edges between D For example, for the two already presented maximal Arrow’s single-peaked Condorcet domains on four alternatives presented in Figs. 2 and 3 the horizontal lines are edges in the two singlepeaked domains on three alternatives and the vertical lines show the isomorphism between isomorphic subdomains on two alternatives. 5. A characterisation of Arrow’s single-peaked domains There are several characterisations of Black’s single-peaked domains of which the most notable ones are Ballester and Haeringer (2011) and Puppe (2018). In the latter it was shown that a domain is maximal single-peaked if and only if it is a Condorcet domain that is connected, minimally rich and contains two completely reversed orders. Theorem 1 allows us to obtain a similar characterisation of Arrow’s single-peaked domains. Theorem 2. A Condorcet domain is Arrow’s single-peaked domain if and only if it is connected and minimally rich. Proof. It is easy to observe that, if a Condorcet domain D ⊆ L(A) is connected and minimally rich, then so is its restriction to any triple {a, b, c } ⊆ A. In particular, D{a,b,c } must be isomorphic to D3,3 of Example 1 (neither D3,1 nor D3,2 nor any subdomain of D3,3 is connected and minimally rich) which means D{a,b,c } is single-peaked.8 Hence D is Arrow’s single-peaked. (We do not need maximality of D in this direction.) The converse follows from Theorem 1(c) and (d). □ Due to Theorem 2, the following is equivalent to Puppe’s characterisation. Corollary 2. A domain is Black’s single-peaked if and only if it is an Arrow’s single-peaked and contains two completely reversed orders. 8 These are observations Fact 2.3 (b) and (d) in Puppe (2018).

x ≻1 y ≻1 z ,

z ≻2 y ≻2 x,

t ≻1 y,

t ≻2 y.

Proof. Consider the two extremal orders of an Arrow’s singlepeaked domain D. If the second condition is satisfied, then they are completely reversed and D is Black’s single-peaked. □ 6. Classification of small maximal Arrow’s single-peaked condorcet domains 6.1. Four alternatives Theorem 3. Up to an isomorphism there are two maximal Arrow’s single-peaked domains on the set of 4 alternatives D4,1 = {abcd, bacd, bcad, cbad, bcda, cbda, cdba, dcba}, D4,2 = {abcd, bacd, bcad, cbad, bcda, cbda, bdca, dbca}.

with their median graphs shown in Figs. 2 and 3, respectively. D4,1 is the only maximal Black’s single-peaked domain. Proof. Let A = {a, b, c , d} and D ⊂ L(A) be a maximal Arrow’s single-peaked domain. By Lemma 4.1, up to an isomorphism, we assume that Term(D) = {a, d}. By Theorem 1(e) we know that there are two extremal linear orders and without loss of generality we may assume that one of them is abcd. By Theorem 1(a) D contains 24−1 = 8 linear orders. Let us list all eight linear orders of D as columns of a 4 × 8 table. Then D satisfies bN{a,b,c } 3 since cN{a,b,c } 3 is not satisfied for the abcd. By Theorem 1(b) we know that D is copious hence bN{a,b,c } 3 leads to D containing the following linear orders abcd, bacd, bcad, cbad. Moreover, Lemma 4.2 states that the two terminals can be swapped, so bcda and cbda should be also in D. Hence we obtain the table a b c d

b a c d

b c a d

c b a d

c b d a

b c d a

a

a

Now in the third row of this table we cannot have more than two elements that did not appear in the last row (or they will form a 9 Ballester and Haeringer (2011) call it worst-restricted.

A. Slinko / Journal of Mathematical Economics 84 (2019) 166–175

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triple where every element happens to be last). Hence the only two options to complete the table are as follows: a b c d

b a c d

b c a d

c b a d

c b d a

b c d a

c d b a

d c b a

a b c d

b a c d

b c a d

c b a d

c b d a

b c d a

b d c a

d b c a

These are indeed Arrow’s single-peaked domains D4,1 and D4,2 and we have already met them in Examples 3 and 4. □ 6.2. Five alternatives Let A = {a, b, c , d, e} and let us assume that Term(D) = {d, e}. The idea of the construction is to put two isomorphic copies of a maximal Arrow’s single-peaked domain {a, b, c }—without loss of generality let us assume that it satisfies bN{a,b,c } 3—each of them appended by two terminals d and e (but in different order). Then try extending each of the isomorphic copies to a maximal Arrow’s single-peaked domain on four alternatives. Hence we start with the half-filled table

K

e

e

I e

e

a b c d e

b a c d e

b c a d e

c b a d e

a b c e d

b a c e d

b c a e d

c b a e d

Fig. 5. Graph of the Arrow’s single-peaked domain D5,1 (Case 1.1).

L

d

J d

d

d

Initially, all focus is on areas I and J of the table. Due to the neverbottom property we can have no more than two alternatives different from d and e in the second to last row of the table. Moreover, all alternatives in I must be the same and so is in J. Case 1. Both areas I and J are occupied by the same alternative. Due to the choice of bN{a,b,c } 3, it can be either a or c. Up to an isomorphism, we can assume that this alternative is a. Then, we have the following possibilities:

Case Case Case Case

1.1 1.2 1.3 1.4

K

L

cN{b,c ,d} 3 cN{b,c ,d} 3 bN{b,c ,d} 3 bN{b,c ,d} 3

bN{b,c ,e} 3 cN{b,c ,e} 3 bN{b,c ,e} 3 cN{b,c ,e} 3

Fig. 6. Graph of the Arrow’s single-peaked domain D5,2 (Case 1.2).

Let us consider these cases one by one. Case 1.1. We have the table of the domain D5,1 d c b a e

c d b a e

c b d a e

b c d a e

a b c d e

b a c d e

b c a d e

c b a d e

a b c e d

b a c e d

b c a e d

c b a e d

b c e a d

c b e a d

b e c a d

e b c a d

and its graph is presented in Fig. 5. Case 1.2. The table of the domain D5,2 in this case would be d c b a e

c d b a e

c b d a e

b c d a e

a b c d e

b a c d e

b c a d e

c b a d e

a b c e d

b a c e d

b c a e d

c b a e d

b c e a d

c b e a d

c e b a d

e c b a d

with the graph depicted in Fig. 6.

with the graph shown in Fig. 7.

Case 1.3. In this case the table of domain D5,3 would be d b c a e

b d c a e

c b d a e

b c d a e

a b c d e

b a c d e

b c a d e

c b a d e

a b c e d

Fig. 7. Graph of the Arrow’s single-peaked domain D5,3 (Case 1.3).

b a c e d

b c a e d

c b a e d

b c e a d

c b e a d

b e c a d

Case 1.4. This domain is isomorphic to D5,1 in Case 1.1 under the isomorphism that swaps d and e and fixes all other alternatives. e b c a d

Case 2. Areas I and J are occupied by different alternatives. These must be a and c since, if b occupies one of them, we would be able to find a triple, namely {a, b, c }, which does not satisfy never-bottom condition. So we get the following possibilities:

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A. Slinko / Journal of Mathematical Economics 84 (2019) 166–175

Fig. 8. Graph of the Arrow’s single-peaked domain D5,4 (Case 2.1).

Fig. 9. Graph of the Arrow’s single-peaked domain D5,5 (Case 2.2).

Case Case Case Case

2.1 2.2 2.3 2.4

Fig. 10. Graph of the Arrow’s single-peaked domain D5,6 (Case 2.3).

K

L

cN{b,c ,d} 3 cN{b,c ,d} 3 bN{b,c ,d} 3 bN{b,c ,d} 3

aN{a,b,e} 3 bN{a,b,e} 3 bN{a,b,e} 3 aN{a,b,e} 3

the graph of which is shown in Fig. 10. Case 2.4. The table in this case would be

Case 2.1. The table of D5,4 would be d c b a e

c d b a e

c b d a e

b c d a e

a b c d e

b a c d e

b c a d e

c b a d e

a b c e d

b a c e d

b c a e d

c b a e d

a b e c d

b a e c d

a e b c d

e a b c d

This is the maximal Black’s single-peaked domain with two completely reversed orders with the graph shown in Fig. 8.

d b c a e

b d c a e

b c d a e

c b d a e

a b c d e

b a c d e

b c a d e

c b a d e

a b c e d

b a c e d

b c a e d

c b a e d

b a e c d

a b e c d

a e b c d

e a b c d

This domain is isomorphic to D5,5 in Case 2.2. under the map which swaps a and c and also d and e. 7. Complexity considerations

Case 2.2. We get the domain D5,5 with the table d c b a e

c b d a e

c d b a e

b c d a e

a b c d e

b a c d e

b c a d e

c b a d e

a b c e d

b a c e d

b c a e d

c b a e d

a b e c d

b a e c d

b e a c d

e b a c d

the graph of which is shown in Fig. 9.

N = {xN{a,b,c } 3 | {a, b, c } ⊆ A, x ∈ {a, b, c } }

Case 2.3. We get the domain D5,6 with the table d b c a e

b d c a e

c b d a e

b c d a e

a b c d e

b a c d e

b c a d e

c b a d e

a b c e d

b a c e d

b c a e d

c b a e d

a b e c d

In Example 5 we saw that some sets of never-bottom conditions are inconsistent and do not define a Condorcet domain. Naturally, the question arises: does there exist a polynomial-time algorithm that, given a complete set of never-bottom conditions, determines if this set is actually consistent? We suggest the following algorithm. Let

b a e c d

b e a c d

e b a c d

be a set of never-bottom conditions on a set of alternatives A. The key idea in constructing a linear order satisfying all never-bottom conditions from N is to look recursively for terminal elements. We proceed as follows. The algorithm gradually builds an order ≻ and reduces subsets A′ ⊆ A and N ′ ⊆ N . Start with a trivial

A. Slinko / Journal of Mathematical Economics 84 (2019) 166–175

order ≻ on an empty set of alternatives ∅ and with A′ = A and N ′ = N . The step of the algorithm is as follows: 1. If |A′ | = 2 output ‘‘N is consistent’’, prepend both elements of A′ to ≻ in any order and halt. Alternatively calculate the set X = {x ∈ A′ | ∀ a, b ∈ A′ it holds that xN{x,a,b} 3 ∈ / N ′ }; 2. If X is empty, then output ‘‘there is no order’’ and halt. Alternatively, if x belongs to this set, prepend x to the order ≻, remove x from A′ , remove all never conditions with participation of x from N ′ and go to step 1. We justify this algorithm as follows. After selection of an alternative x all never-bottom conditions yN{x,y,z } , with y ̸ = x, are removed because y and z will be higher in the constructed order, hence it will automatically satisfy yN{x,y,z } . If it halts when |A′ | ≥ 3, then there are at least three alternatives and none of them can be placed lower than the rest of them. So the subset N ′ ⊆ N is clearly inconsistent. On the other hand, it is easy to see that, if this algorithm outputs a linear order, then it satisfies all never conditions from N . To prove this suppose that a ≻ b ≻ c but one of the never conditions from N is not satisfied for {a, b, c }. This can be only cN{a,b,c } 3. By the construction of ≻ alternative c must be added earlier than b and a but this could not happen due to cN{a,b,c } 3 ∈ N . Clearly it runs in polynomial time. It is worth noting that analogous question for never-middle domains is NP-complete which follows from Opatrny (1979) and Guttmann and Maucher (2006). 8. Conclusion and future work Maximal Condorcet domains represent a compromise which allows to a society to always have transitive collective preferences and, under this constraint, provide voters with as much individual freedom as possible. The domain of single-peaked preferences introduced by Black is one of those and, although it is not the largest, it is by far the most popular in modelling the spectrum of opinions in a society in the context of political thought. One of the hallmark features of Black’s single peakedness is the existence of two completely reversed orders (the existence of which is also called maximal width condition). Not every society, however, is as tolerant to satisfy this requirement and in many countries dissidents are prosecuted (e.g., in the USSR). We show that abandonment of the maximal width condition while retaining connectedness and minimal richness leads to a larger class of Arrow’s single-peaked Condorcet domains which are characterised by these two conditions. We show the existence of two nonisomorphic maximal Arrow’s single-peaked Condorcet domains for four alternatives and six for the case of five alternatives. Arrow’s single-peaked domains have a nice recursive structure and also have two extremal orders whose top and bottom preferences are reversed. A future work will concern with classification of Condorcet domains defined by never-middle conditions. Danilov and Koshevoy (2013) showed that this class is very interesting. In particular, for every number of alternatives it contains maximal Condorcet domains consisting only of four linear orders.

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