1 Introduction

Modelling voters’ linear preferences (aka. rankings) over a set of alternatives as geometric distances is an approach popular in many research fields such as economics [11, 13, 17], political and social sciences [2, 15, 22, 25], and psychology [3, 9]. The idea is to consider the alternatives and voters as points in a d-dimensional space such that

$$\begin{aligned} \text {for each two alternatives, each voter prefers the one that is}~closer ~\text {to her.} \end{aligned}$$
(*)

If the proximity is measured via the Euclidean distance, then (i.e., a collection of distinct linear preference orders specifying voters’ preferences) obeying (*) are called . While the model seems to be canonical, in real life the shortest path between two points may be Manhattan rather than Euclidean. For instance, in urban geography, the alternatives (e.g., a shop or a supermarket) and the voters (e.g., individuals) are often located on grid-like streets. That is, the distance between an alternative and a voter is more likely to be measured according to the Manhattan distance (aka. Taxicab distance or \(\ell _1\)-norm-distance), i.e., the sum of the absolute differences of the coordinates of the alternative and the voter. Similarly to the Euclidean preference notion, we call a preference profile if there exists an embedding for the voters and the alternatives which satisfies condition (*) under the Manhattan distance. Indeed, Manhattan preferences have been studied for a wide range of applications such as facility location [19, 26], group decision making [24], and voting and committee elections [12]. Many voting advice applications, such as the German Wahl-O-Mat [28] and Finnish Ylen Vaalikone [27] use Manhattan distances to measure the distance between a voter and an alternative, indicating that such distances may be perceived as more natural in human decision making.

Despite their practical relevance, Manhattan preferences have attracted far less attention than their close relative Euclidean preferences. Bogomolnaia and Laslier [2] studied how restrictive the assumption of Euclidean preferences is. They showed that every preference profile with n voters and m alternatives is if \(d\ge \min (n,m-1)\). When indifference between alternatives is allowed, the only-if statement holds as well. For \(d=1\), their smallest non preference profile consists of either 3 voters and 3 alternatives or 2 voters and 4 alternatives. For \(d=2\), their smallest non profile consists of either 4 voters and 4 alternatives or 3 voters and 8 alternatives, which is also tight by Bulteau and Chen [5]. To the best of our knowledge, no such kind of characterization result on the preferences exists. For maximally Manhattan preferences, however, Escoffier et al. [16] show that a preference profile for four alternatives can contain up to 19 distinct preference orders. From the computational point of view, it is known that for \(d=1\), deciding whether a given preference profile is Euclidean (and hence Manhattan) can be done in polynomial time [10, 14, 18]. For any fixed \(d \ge 2\), however, testing Euclidean preferences is complete for the complexity class  [23], while it is straightforward to see that the problem for the Manhattan case is contained in NP [21]; note that NP \(\subseteq \!\exists \mathbb {R}\). Nothing about the complexity lower bound is known for Manhattan preferences.

Fig. 1.
figure 1

Boundaries of non (resp. non ) profiles with a given number of voters and alternatives. Each bullet (resp. cross) represents the existence of such a non Euclidean (resp. non Manhattan) profile. (Color figure online)

Our Contribution. In this paper, we aim to close the gap and study how to find a embedding for a given preference profile and what is the smallest dimension for such an embedding. First, we prove that, similarly to the Euclidean case, every preference profiles with m alternatives and n voters is if \(d\ge \min (m-1,n)\) (Theorems 1 and 2). Then, focusing on the two-dimensional case, we seek to determine tight bounds on the smallest number of either alternatives or voters of a non profile. We show that an arbitrary preference profile with n voters and m alternatives is if and only if either \(m\le 3\) (Theorems 2 and 5), or \(n\le 2\) (Theorem 1), or \(n \le 3\) and \(m\le 5\) (Theorem 3 and Proposition 2), or \(n\le 4\) and \(m\le 4\) (Theorem 4 and Proposition 2). Note that this is considerably different than the Euclidean case: There exists a non Euclidean preference profile with \(n=4\) and \(m=4\), while every preference profile with \(n\le 3\) and \(m\le 7\) is Euclidean. The proof for the “only if” part is via presenting forbidden subprofiles (see Definitions 3 to 5), which may be of independent interests for determining preferences. The proof for the “if” part is computer-aided. See Fig. 1 for a summary for \(d=2\).

The paper is organized as follows: Sect. 2 introduces necessary definitions and notations. In Sects. 3 and 4, we present the first positive result and the negative findings, respectively. In Sect. 5, we show the remaining positive results by describing a computer program that finds a Manhattan embedding for every possible preference profile with three voters and five alternatives, or four voters and four alternatives. We conclude with a few future research directions in Sect. 6. Due to space constraints, proofs of results marked with (\(\star \)) are available in the full version [7].

2 Preliminaries

Given a non-negative integer t, we use to denote the set \(\{1,\ldots ,t\}\). Let \(\boldsymbol{x}\) denote a vector of length d or a point in a d-dimensional space, and let i denote an index \(i\in [d]\). We use to refer to the \(i^{\text {th}}\) value in \(\boldsymbol{x}\).

Let \(\mathcal{A}{:}{=}[m]\) be a set of alternatives. A  \(\succ \) of \(\mathcal{A}\) is a linear order (a.k.a. permutation or ranking) of \(\mathcal{A}\); a linear order is a binary relation which is total, irreflexive, and transitive. For two distinct alternatives a and b, the relation means that a is preferred to (or in other words, ranked higher than) b in \(\succ \). An alternative c is the alternative in \(\succ \) if for any alternative \(b\in \mathcal{A}\setminus \{c\}\) it holds that \(c \succ b\). Let \(\succ \) be a preference order over \(\mathcal{A}\). For a subset \(B\subseteq \mathcal{A}\) of alternatives and an alternative c not in B, we use (resp. ) to denote that for each \(b\in B\) it holds that \(b\succ c\) (resp. \(c\succ b\)). A (or in short) \(\mathcal{P}\) specifies the preference orders of a number of voters over a set of alternatives. Formally, , where \(\mathcal{A}\) denotes the set of m alternatives, \(\mathcal{V}\) denotes the set of n voters, and \(\mathcal{R}{:}{=}(\succ _1, \ldots , \succ _n)\) is a collection of n preference orders such that each voter \(v_i\in \mathcal{V}\) ranks the alternatives according to the preference order \(\succ _i\) on \(\mathcal{A}\). We will omit the subscript i from \(\succ _i\) if it is clear from the context. Throughout the paper, if not explicitly stated otherwise, we assume \(\mathcal{P}\) is a preference profile of the form \((\mathcal{A},\mathcal{V},\mathcal{R})\). For notational convenience, for each alternative \(a\in \mathcal{A}\) and each voter \(v_i\in \mathcal{V}\), let denote the rank of alternative a in the preference order \(\succ _i\), which is the number of alternatives which are preferred to a by voter \(v_i\), i.e., \(\textsf{rk}_{i}(a)=|\{b \in \mathcal{A}\mid b\succ _i a\}|\). For instance, if voter \(v_i\) has preference order \(2 \succ _i 3 \succ _i 1 \succ _i 4\), then \(\textsf{rk}_i(3) = 1\).

Given a d-dimensional vector \(\boldsymbol{x}\in \mathbb {R}^{d}\) and an \(\ell _p\)-norm with \(p \ge 1\), let denote the \(\ell _p\)-norm of \(\boldsymbol{x}\), i.e., \(\Vert \boldsymbol{x} \Vert _{p} = (|\boldsymbol{x}[1]|^p+\cdots +|\boldsymbol{x}[d]|^p)^{1/p}\), and let denote the \(\ell _{\infty }\)-norm of \(\boldsymbol{x}\), i.e., \(\Vert \boldsymbol{x} \Vert _{p} = \max \{\boldsymbol{x}[i]\}_{i\in [d]}\). Given two points \(\boldsymbol{u}, \boldsymbol{w}\) in \(\mathbb {R}^{d}\) and \(p \in \{1,2,\infty \}\), we use the \(\ell _p\)-norm of \(\boldsymbol{u}-\boldsymbol{w}\), i.e., , to denote the \(\ell _p\)-distance of \(\boldsymbol{u}\) and \(\boldsymbol{w}\). By convention, we use , , and distances to refer to \(\ell _1\)-, \(\ell _2\)-, and \(\ell _{\infty }\)-distances, respectively.

For \(d=2\), the Manhattan distance of two points is equal to the length of a shortest path between them on a rectilinear grid. Hence, under Manhattan distances, a is a square rotated at a \(45^{\circ }\) angle from the coordinate axes. The intersection of two Manhattan-circles can range from two points to two segments as depicted in Fig. 2.

Fig. 2.
figure 2

The intersection (in red) of two circles under the Manhattan distance in \(\mathbb {R}^{2}\) can be two points, one point and one line segment, one line segment, or two line segments. (Color figure online)

Basic Geometric Notation. Throughout this paper, we use lower case letters in boldface to denote points in a space. Given two points \(\boldsymbol{q}\) and \(\boldsymbol{r}\), we introduce the following notions: Let denote the set of points which are contained in the (smallest) rectilinear bounding box of points \(\boldsymbol{q}\) and \(\boldsymbol{r}\), i.e., \(\textsf{BB}(\boldsymbol{q},\boldsymbol{r}){:}{=}\{\boldsymbol{x}\in \mathbb {R}^{d}\mid \min \{\boldsymbol{q}[i],\boldsymbol{r}[i]\} \le \boldsymbol{x}[i] \le \max \{\boldsymbol{q}[i],\boldsymbol{r}[i]\} \text { for all } i\in [d]\}\). The (bisector in short) between two points \(\boldsymbol{q}\) and \(\boldsymbol{r}\) with respect to a norm \(\ell _p\) is a set \(\textsf{H}_p(\boldsymbol{q},\boldsymbol{r})\) of points which each have the same distance to both \(\boldsymbol{q}\) and \(\boldsymbol{r}\). Formally, \(\textsf{H}_{p}(\boldsymbol{q},\boldsymbol{r}){:}{=}\{\boldsymbol{x}\in \mathbb {R}^{d}\mid \Vert \boldsymbol{x}-\boldsymbol{q} \Vert _p = \Vert \boldsymbol{x}-\boldsymbol{r} \Vert _p\}\). In a d-dimensional space, a bisector of two points under the Manhattan distance (i.e., \(\ell _1\)-norm) can itself be a d-dimensional object, while a bisector under Euclidean distances is always \((d-1)\)-dimensional; see e.g., Fig. 3 (right).

The Two-Dimensional Case. In a two-dimensional space, the vertical line and the horizontal line crossing any point divide the space into four non-disjoint quadrants: the north-east, south-east, north-west, and south-west quadrants. Given a point \(\boldsymbol{q}\), we use \(\textsf{NE}(\boldsymbol{q})\), \(\textsf{SE}(\boldsymbol{q})\), \(\textsf{NW}(\boldsymbol{q})\), and \(\textsf{SW}(\boldsymbol{q})\) to denote these four quadrants. Formally, , , , and .

Fig. 3.
figure 3

The bisector (in green) between points u and v under the Manhattan distance. The green lines and areas extend to infinity. (Color figure online)

Embeddings. The Euclidean (resp. Manhattan) representation models the preferences of the voters over the alternatives using the Euclidean (resp. Manhattan) distance. A shorter distance indicates a stronger preference. For technical reason, we also introduce the \(\ell \)-max preferences which are based on the \(\ell _{\infty }\)-distances.

Definition 1

( , , and   embeddings). Let \(\mathcal{P}{:}{=}(\mathcal{A}, \mathcal{V}{:}{=}\{v_1, \ldots , v_n\}, \mathcal{R}{:}{=}(\succ _1, \ldots , \succ _n))\) be a profile. Let \(E:\mathcal{A}\cup \mathcal{V}\rightarrow \mathbb {R}^{d}\) be an embedding of the alternatives and the voters. For each \((\varLambda , p)\in \{(\) , 1), ( , 2), ( , \(\infty )\}\), a voter \(v_i \in V\) is called with respect to embedding E if for each two alternatives \(a,b\in \mathcal{A}\) it holds that

$$\begin{aligned} a \succ _i b \text { if and only if } \Vert E(a)-E(v_i) \Vert _p < \Vert E(b)-E(v_i) \Vert _p. \end{aligned}$$

Embedding E is called a (resp. , ) embedding of profile \(\mathcal{P}\) if each voter in V is (resp. , ) wrt. E. A preference profile is (resp. , ) if it admits a (resp. , ) embedding.

To characterize necessary conditions for profiles, we need to define several notions which describe the relative orders of the points in each axis.

Definition 2

(BE- and EX-properties). Let \(\mathcal{P}\) be a profile containing at least 3 voters called uvw and let E be an embedding for \(\mathcal{P}\). Then, E satisfies

  • the -property if \(E(v) \in \textsf{BB}(E(u),E(w))\) (see the illustration in the first row of Fig. 4) and

  • the -property if there exists (ij) with \(\{i,j\}=\{1,2\}\) such that

    $$\begin{aligned}&\min \{E(v)[i], E(w)[i]\} \le E(u)[i] \le \max \{E(v)[i], E(w)[i]\} \quad \text { and } \\&\min \{E(u)[j], E(v)[j]\} \le E(w)[j] \le \max \{E(u)[j], E(v)[j]\} \end{aligned}$$

     (see the illustrations in the last two rows of Fig. 4).

If E does not satisfy (vuw)-BE-property (-EX-property) we say it violates (vuw)-BE-property (resp. -EX-property).

For brevity’s sake, by symmetry, we omit voters u and w and just speak of the -property (resp. -property) if uvw are the only voters contained in \(\mathcal{P}\).

Fig. 4.
figure 4

Two possible embeddings illustrating the properties in Definition 2 (the numbering will be used in some proofs). (BE) means “between” while (EX) “external”.

Note that any embedding for three voters uvw must satisfy u-, v- or w-\(\textsf{EX}\)-property or u-, v- or w-\(\textsf{BE}\)-property, although it may satisfy more than one of these (consider for example three voters at the same point). However, each of these embeddings satisfying the (vuw)-\(\textsf{BE}\)-property (resp. (vuw)-\(\textsf{EX}\)-property) forbids certain types of preference profiles. The following two configurations describe preferences whose existence precludes an embedding from satisfying either \(\textsf{BE}\)-property or \(\textsf{EX}\)-property for some voters, as we will show in Lemmas 3 and 4.

Definition 3

(BE-configurations). A profile \(\mathcal{P}\) with 3 voters uvw and 3 alternatives abx is a if the following holds:

$$\begin{array}{lll} u:b \succ _u x \succ _u a, &{} \quad v:a \succ _v x \succ _v b, &{}\quad w:b \succ _w x \succ _w a.\\ \end{array}$$

Definition 4

(EX-configurations). A profile \(\mathcal{P}\) with 3 voters uvw and 6 alternatives xabcde (cde not necessarily distinct) is a if the following holds:

$$\begin{array}{llll} u:&{}\quad a \succ _u x \succ _u b, &{}\quad c\succ _u x, &{}\quad d\succ _u x\\ v:&{}\quad \{a,b\} \succ _v x, &{}\quad &{}\quad x \succ _v \{d,e\},\\ w:&{}\quad b \succ _w x \succ _w a, &{}\quad c\succ _w x, &{}\quad e\succ _w x. \end{array}$$

Example 1

Consider two profiles \(\mathcal{Q}_{1}\) and \(\mathcal{Q}_{2}\) which satisfy the following:

$$\begin{array}{llll} \mathcal{Q}_{1}:&{} v_1:1 \succ _1 2 \succ _1 3, \qquad \qquad \quad &{} v_2:3 \succ _2 2 \succ _2 1, \qquad \qquad \quad &{} v_3:3 \succ _3 2 \succ _3 1\\ \mathcal{Q}_{2}:&{} v_1 :\{1,2\} \succ _1 3 \succ _1 4, &{} v_2 :\{1,4\} \succ _2 3 \succ _2 2, &{} v_3 :\{2, 4\} \succ _3 3 \succ _3 1.\\ \end{array}$$

Clearly, \(\mathcal{Q}_{1}\) is a \((v_1,v_2,v_3)\)-\(\textsf{BE}\)-configuration. Further, one can verify that \(\mathcal{Q}_{2}\) contains a \((v_1,v_2,v_3)\)-, \((v_2,v_1,v_3)\)-, and \((v_3,v_1,v_2)\)-\(\textsf{EX}\)-configuration, by setting \((a,b,x,c,d,e){:}{=}(1,2,3,4,4,4)\), \((a,b,x,c,d,e){:}{=}(1,4,3,2,2,2)\), and (abxcd, \(e){:}{=}(2,4,3,1,1,1)\), respectively.

The next configuration is a restriction of the worst-diverse configuration. The latter is used to characterize the so-called single-peaked preferences [1].

Definition 5 (All-triples worst-diverse configuration)

A profile \(\mathcal{P}\) is an if for every triple of alternatives \(\{x, y, z\} \subseteq \mathcal{A}\) there are three voters \(u, v, w \in \mathcal{V}\) which form a worst-diverse configuration, i.e., their preferences satisfy \(\{x, y\} \succ _u z\), \(\{x, z\} \succ _v y\), and \(\{y, z\} \succ _w x\).

3 Manhattan Preferences: Positive Results

In this section, we show that for sufficiently high dimension d, i.e., \(d\ge \min (n, m-1)\), a profile with n voters and m alternatives is always . The same result holds for profiles [2]. The idea of the proof for n voters is similar to the proof for preferences in [2]. The proof for \(m+1\) voters is more different from -case. Whilst the proof for -case relies on abstract geometric properties, it is relatively straightforward to give a full concrete construction on the case.

Theorem 1

Every profile with n voters is .

Proof

Let \(\mathcal{P}=(\mathcal{A}, \mathcal{V}, (\succ _i)_{i\in [n]})\) be a profile with m alternatives and n voters \(\mathcal{V}\) such that \(\mathcal{A}=\{1,\ldots ,m\}\). The idea is to first embed the voters from \(\mathcal{V}\) onto n selected vertices of an n-dimensional hypercube, and then embed the alternatives such that each coordinate of an alternative reflects the preferences of a specific voter. More precisely, define an embedding \(E:\mathcal{A}\cup \mathcal{V}\rightarrow \mathbb {Z}\) such that for each voter \(v_i\in \mathcal{V}\) and each coordinate \(z\in [n]\), we have \(E(v_i)[z]{:}{=}-m\) if \(z=i\), and \(E(v_i)[z]{:}{=}0\) otherwise.

It remains to specify the embedding of the alternatives. To ease notation, for each alternative \(j\in \mathcal{A}\), let denote the maximum rank of the voters towards j, i.e., . Further, let denote the index of the voter who has maximum rank over j; if there are two or more such voters, then we fix an arbitrary one. That is, . Then, the embedding of each alternative \(j\in \mathcal{A}\) is defined as follows:

$$\begin{aligned}&\forall z\in [n]:E(j)[z] {:}{=}{\left\{ \begin{array}{ll} \textsf{rk}_z(j) - \textsf{mk}_j, &{} \text { if } z \ne \hat{n}_j,\\ M + 2\textsf{rk}_z(j) + \sum \limits _{k\in [n]} (\textsf{rk}_{k}(j)-\textsf{mk}_j), &{} \text { otherwise.} \end{array}\right. } \end{aligned}$$

Herein, M is set to a large but fixed value such that the second term in the above definition is non-negative. For instance, we can set \(M{:}{=}n\cdot m\). Notice that by definition, the following holds for each alternative \(j\in \mathcal{A}\).

$$\begin{aligned} -m \le \textsf{rk}_z(j) - \textsf{mk}_j&\le 0, \quad \text { and } \end{aligned}$$
(1)
$$\begin{aligned} M + 2\textsf{rk}_z(j) +\sum \limits _{k\in [n]} (\textsf{rk}_{k}(j)-\textsf{mk}_j)&\ge M \!- \!n \!\cdot \! m \ge 0. \end{aligned}$$
(2)

Thus, for each \(i\in [n]\) and \(j\in [m]\), it holds that

$$\begin{aligned} |E(j)[i] - E(v_i)[i]|&{\mathop {=}\limits ^{(1),(2)}} E(j)[i] + m, \end{aligned}$$
(3)
$$\begin{aligned} \Vert E(j) \Vert _1&= \sum _{z\in [n]\setminus \{\hat{n}\}}|\textsf{rk}_{z}(j)-\textsf{mk}_j| + |M+2\textsf{rk}_{\hat{n}}(j) + \sum _{k\in [n]}(\textsf{rk}_k(j)-\textsf{mk}_j)\nonumber \\&{\mathop {=}\limits ^{(1),(2)}} \sum _{z\in [n]\setminus \{\hat{n}_j\}}\!\!(-\textsf{rk}_z[j]+\textsf{mk}_j) + M + 2\textsf{rk}_{\hat{n}_j}(j) + \sum _{k\in [n]}(\textsf{rk}_k[j]-\textsf{mk}_j) \nonumber \\&{\mathop {=}\limits ^{\textsf{mk}_j=\textsf{rk}_{\hat{n}_j}(j)}} M + 2\textsf{rk}_{\hat{n}}(j) \end{aligned}$$
(4)

Now, in order to prove that this embedding is we show that the Manhattan distance between an arbitrary voter \(v_i\) and an arbitrary alternative j is linear in the rank value \(\textsf{rk}_i(j)\). By definition, this distance is:

$$\begin{aligned} \Vert E(v_i)-E(j) \Vert _1&= \!\!\!\sum \limits _{k\in [n]}\!\! |E(j)[k]-E(v_i)[k]| = |E(j)[i]-E(v_i)[i]| + \!\!\!\sum \limits _{k\in [n]\setminus \{i\}} \!\!|E(j)[k]|\nonumber \\&{\mathop {=}\limits ^{(3)}} E(j)[i] + m + \Vert E(j) \Vert _1 - |E(j)[i]|. \end{aligned}$$
(5)

We distinguish between two cases.

Case 1: \(i \ne \hat{n}_j\). Then, it follows that \(\Vert E(v_i)-E(j) \Vert _1 {\mathop {=}\limits ^{(5)}} m + E(j)[i] + \Vert E(j) \Vert _1 - |E(j)[i]| {\mathop {=}\limits ^{(1),(4)}} m + 2(\textsf{rk}_i(j)-\textsf{mk}_j) + M + 2\textsf{rk}_{\hat{n}_j}(j) {\mathop {=}\limits ^{\textsf{rk}_{\hat{n}}(j)=\textsf{mk}_j}} m + M + 2\textsf{rk}_i(j)\).

Case 2: \(i = \hat{n}_j\). Then, by definition, it follows that

\(\Vert E(v_i)-E(j) \Vert _1 {\mathop {=}\limits ^{(5)}} m + E(j)[i] + \Vert E(j) \Vert _1 - |E(j)[i]| {\mathop {=}\limits ^{(2)}} m + \Vert E(j) \Vert _1 {\mathop {=}\limits ^{(4)}} m + M + 2\textsf{rk}_{\hat{n}_j}(j)\).

In both cases, we obtain that \(\Vert E(v_i)-E(j) \Vert _1 = m+M+2\textsf{rk}_{i}(j)\), which is linear in the ranks, as desired.    \(\square \)

By Theorem 1, we obtain that any profile with two voters is . The following example provides an illustration.

Example 2

Consider profile \(\mathcal{P}_{1}\) with 2 voters and 5 alternatives: \(v_1:1 \succ 2 \succ 3 \succ 4 \succ 5\) and \(v_2:5 \succ 4 \succ 3 \succ 1 \succ 2\). By the proof of Theorem 1, the maximum ranks and the voters with maximum rank, and the embedding of the voters and alternatives is as follows, where \(M{:}{=}n\cdot m = 10\); see Fig. 5a for an illustration.

figure ck
Fig. 5.
figure 5

(a): Illustration for Example 2. (b): Illustration for Example 3; the circles are with respect to \(v_2\) and \(v_6\), respectively.

Theorem 2

Every profile with \(m+1\) alternatives is .

Proof

Let \(\mathcal{P}=(\mathcal{A}, \mathcal{V}, (\succ _i)_{i\in [n]})\) be a profile with \(m+1\) alternatives and n voters \(\mathcal{V}\) such that \(\mathcal{A}=\{1,\ldots ,m+1\}\). The idea is to first embed the alternatives from \(\mathcal{A}\) onto \(m+1\) selected vertices of an m-dimensional hypercube, and then embed the voters such that the distances from each voter to the alternatives increase as the preferences decrease. More precisely, define an embedding \(E:\mathcal{A}\cup \mathcal{V}\rightarrow \mathbb {N}_0\) such that alternative \(m+1\) is embedded in the origin coordinate, i.e., \(E(m+1)=(0)_{z\in [m]}\). For each alternative \(j\in [m]\) and each coordinate \(z\in [m]\), we have \(E(j)[z]{:}{=}2m\) if \(z=j\), and \(E(j)[z]{:}{=}0\) otherwise.

Then, the embedding of each voter \(v_i\in \mathcal{V}\) is defined as follows: \(\forall j\in [m]:\)

$$\begin{aligned} E(v_i)[j]&{:}{=}{\left\{ \begin{array}{ll} 2m - \textsf{rk}_i(j), &{} \text { if } \textsf{rk}_i(j) < \textsf{rk}_i(m+1),\\ m -\textsf{rk}_i(j), &{} \text { if } \textsf{rk}_i(j) > \textsf{rk}_i(m+1). \end{array}\right. } \end{aligned}$$

Observe that \(0\le E(v_i)[j] \le 2m\). Before we show that E is for \(\mathcal{P}\), let us establish a simple formula for the distance between a voter and an alternative.

Claim 1

(\(\star \)). For each voter \(v_i\in \mathcal{V}\) and each alternative \(j\in \mathcal{A}\), we have

$$\begin{aligned} \Vert E(v_i)-E(j) \Vert _1 = {\left\{ \begin{array}{ll} \Vert E(v_i) \Vert _1 + 2(m-E(v_i)[j]), &{} \text { if } j \ne m+1,\\ \Vert E(v_i) \Vert _1, &{} \text { otherwise. } \end{array}\right. } \end{aligned}$$

Now, we proceed with the proof. Consider an arbitrary voter \(v_i\in \mathcal{V}\) and let \(j,k\in [m+1]\) be two consecutive alternatives in the preference order \(\succ _i\) such that \(\textsf{rk}_i(j) = \textsf{rk}_i(k)-1\). We distinguish between three cases.

Case 1: \(\textsf{rk}_i(k) < \textsf{rk}_i(m+1)\) or \(\textsf{rk}_i(j) > \textsf{rk}_i(m+1)\). Then, by Claim 1 and by definition, it follows that \(\Vert E(v_i)-E(j) \Vert _1 - \Vert E(v_i)-E(k) \Vert _1 = 2(E(v_i)[k]-E(v_i)[j]) = 2 (\textsf{rk}_i(j) - \textsf{rk}_i(k)) < 0\).

Case 2: \(\textsf{rk}_i(k) = \textsf{rk}_i(m+1)\), i.e., \(k=m+1\) and \(E(v_i)[j]=2m-\textsf{rk}_i(j)\). Then, by Claim 1 and by definition, it follows that \(\Vert E(v_i)-E(j) \Vert _1 - \Vert E(v_i)-E(k) \Vert _1 = 2(m - E(v_i)[j]) =2\textsf{rk}_i(j) - 2m < 0\). Note that the last inequality holds since \(\textsf{rk}_i(j)=\textsf{rk}_i(k)-1 < m\).

Case 3: \(\textsf{rk}_i(j) = \textsf{rk}_i(m+1)\), i.e., \(j=m+1\) and \(E(v_i)[k]=m-\textsf{rk}_i(k)\). Then, by Claim 1 and by definition, it follows that \(\Vert E(v_i)-E(j) \Vert _1 - \Vert E(v_i)-E(k) \Vert _1 = -2(m - E(v_i)[k]) = - 2\textsf{rk}_i(k) < 0\). Note that the last inequality holds since \(\textsf{rk}_i(k)=\textsf{rk}_i(j)+1 > 0\).

Since in all cases, we show that \(\Vert E(v_i)-E(j) \Vert _1 - \Vert E(v_i)-E(k) \Vert _1 < 0\), E is indeed for \(\mathcal{P}\).    \(\square \)

Theorem 2 implies that any profile for 3 alternatives is . The following example illustrates the corresponding Manhattan embedding.

Example 3

The following profile \(\mathcal{P}_{2}\) with 6 voters and 3 alternatives is .

$$\begin{array}{lll} v_1:1 \succ 2 \succ 3,\quad \quad &{} v_2 :1 \succ 3 \succ 2, \quad \quad &{} v_3:2 \succ 1 \succ 3,\\ v_4:2 \succ 3 \succ 1, &{} v_5:3 \succ 1 \succ 2, &{} v_6:3 \succ 2 \succ 1. \\ \end{array}$$

One can check that the embedding E given in Fig. 5b is for \(\mathcal{P}_{3}\).

4 Manhattan Preferences: Negative Results

In this section, we consider minimally non profiles. We show that for \(n\in \{3,4,5\}\) voters, the smallest non profile has \(9-n\) alternatives (Theorems 3 to 5). Before we show this, we first go through some technical but useful statements for preference profiles in Sect. 4.1. Then, we show the proofs of the main results in Sects. 4.2 to 4.4. For brevity’s sake, given an embedding E and a voter \(v\in \mathcal{V}\) (resp. an alternative \(a\in \mathcal{A}\)), we use boldface  (resp. \(\boldsymbol{a}\)) to denote the embedding E(v) (resp. E(a)).

4.1 Technical Results

Lemma 1

Let \(\mathcal{P}\) be a profile and let E be a embedding for \(\mathcal{P}\). For any two voters rs and two alternatives xy the following holds: (i) If \(r,s:y \succ x\), then \(\boldsymbol{x}\notin \textsf{BB}(\boldsymbol{r}, \boldsymbol{s})\). (ii) If \(r:x \succ y\) and \(s:y\succ x\), then \(\boldsymbol{s}\notin \textsf{BB}(\boldsymbol{r}, \boldsymbol{x})\).

Proof

Let \(\mathcal{P}\), E, rs, and xy be as defined. Both statements follow from using simple calculations and the triangle inequality of Manhattan distances.

For Statement (i), suppose, towards a contradiction, that \(r,s:y \succ x\) and \(\boldsymbol{x}\in \textsf{BB}(\boldsymbol{r}, \boldsymbol{s})\). By the definition of Manhattan distances, this implies that \(\Vert \boldsymbol{r}-\boldsymbol{x} \Vert _1+\Vert \boldsymbol{x}-\boldsymbol{s} \Vert _1 = \Vert \boldsymbol{r}-\boldsymbol{s} \Vert _1\). By the preferences of voters r and s we infer that \(\Vert \boldsymbol{s}-\boldsymbol{y} \Vert _1+\Vert \boldsymbol{r}-\boldsymbol{y} \Vert _1 < \Vert \boldsymbol{s}-\boldsymbol{x} \Vert _1+\Vert \boldsymbol{r}-\boldsymbol{x} \Vert _1 = \Vert \boldsymbol{r}-\boldsymbol{s} \Vert _1\), a contradiction to the triangle inequality of \(\Vert \cdot \Vert _1\).

For Statement (ii), suppose, towards a contradiction, that \(r:x \succ y\) and \(s:y\succ x\) and \(\boldsymbol{s}\in \textsf{BB}(\boldsymbol{r}, \boldsymbol{x})\). By the definition of Manhattan distances, this implies that \(\Vert \boldsymbol{r}-\boldsymbol{x} \Vert _1=\Vert \boldsymbol{r}-\boldsymbol{s} \Vert _1 + \Vert \boldsymbol{s}-\boldsymbol{x} \Vert _1\). By the preferences of voters r and s we infer that \(\Vert \boldsymbol{r}-\boldsymbol{s} \Vert _1+\Vert \boldsymbol{s}-\boldsymbol{y} \Vert _1< \Vert \boldsymbol{r}-\boldsymbol{s} \Vert _1+\Vert \boldsymbol{s}-\boldsymbol{x} \Vert _1 = \Vert \boldsymbol{r}-\boldsymbol{x} \Vert _1 < \Vert \boldsymbol{r}-\boldsymbol{y} \Vert _1\), a contradiction to the triangle inequality of \(\Vert \cdot \Vert _1\).    \(\square \)

The following is a summary of coordinate differences wrt. the preferences.

Observation 1

(\(\star \)). Let profile \(\mathcal{P}\) admit a embedding E. For each voter s and each two alternatives xy with \(s:x\succ y\), the following holds:

  1. (i)

    If \(\boldsymbol{y}\in \textsf{NE}(\boldsymbol{s})\), then \(\boldsymbol{y}[1]+\boldsymbol{y}[2]>\boldsymbol{x}[1]+\boldsymbol{x}[2]\).

  2. (ii)

    If \(\boldsymbol{y}\in \textsf{NW}(\boldsymbol{s})\), then \(-\boldsymbol{y}[1]+\boldsymbol{y}[2]>-\boldsymbol{x}[1]+\boldsymbol{x}[2]\).

  3. (iii)

    If \(\boldsymbol{y}\in \textsf{SE}(\boldsymbol{s})\), then \(\boldsymbol{y}[1]-\boldsymbol{y}[2]>\boldsymbol{x}[1]-\boldsymbol{x}[2]\).

  4. (iv)

    If \(\boldsymbol{y}\in \textsf{SW}(\boldsymbol{s})\), then \(-\boldsymbol{y}[1]-\boldsymbol{y}[2]>-\boldsymbol{x}[1]-\boldsymbol{x}[2]\).

The next technical lemma excludes two alternatives to be put in the same quadrant region of some voters.

Lemma 2

Let profile \(\mathcal{P}\) admit a embedding E. Let rst and xy be 3 voters and 2 alternatives in \(\mathcal{P}\), respectively. The following holds.

  1. (i)

    If \(r:x\succ y\) and \(s:y\succ x\), then for each \(\varPi \in \{\textsf{NE}\), \(\textsf{NW},\textsf{SE},\textsf{SW}\}\) it holds that if \(\boldsymbol{x}\in \varPi (\boldsymbol{s})\), then \(\boldsymbol{y}\notin \varPi (\boldsymbol{r})\).

  2. (ii)

    If \(r,t:x\succ y\), \(s:y \succ x\), \(\boldsymbol{r}\in \textsf{SW}(\boldsymbol{s})\), and \(\boldsymbol{t}\in \textsf{NE}(\boldsymbol{s})\), then for each \(\varPi \in \{\textsf{NW}, \textsf{SE}\}\) it holds that if \(\boldsymbol{x}\in \varPi (\boldsymbol{s})\), then \(\boldsymbol{y}\notin \varPi (\boldsymbol{s})\).

Proof

Let \(\mathcal{P},E,r,s,t,x,y\) be as defined. The first statement follows directly from applying Observation 1. Hence, we only consider the case with \(\varPi =\textsf{NW}\). For a contradiction, suppose that \(\boldsymbol{x}\in \textsf{NW}(\boldsymbol{s})\) and \(\boldsymbol{y}\in \textsf{NW}(\boldsymbol{r})\). Since \(r:x\succ y\) and \(s:y\succ x\), by Observation 1(ii), we have \(\boldsymbol{y}[2]-\boldsymbol{y}[1]> \boldsymbol{x}[2]-\boldsymbol{x}[1] > \boldsymbol{y}[2]-\boldsymbol{y}[1]\), a contradiction.

Statement (ii): We only show the case with \(\varPi =\textsf{NW}\) as the other case is symmetric. For a contradiction, suppose that \(\boldsymbol{x}, \boldsymbol{y}\in \textsf{NW}(\boldsymbol{s})\). Since \(r,t:x \succ y\), \(s:y \succ x\), \(\boldsymbol{x}\in \textsf{NW}(\boldsymbol{s})\), by the first statement, we have \(\boldsymbol{y}\notin \textsf{NW}(\boldsymbol{r})\cup \textsf{NW}(\boldsymbol{t})\). However, since \(\boldsymbol{y}\in \textsf{NW}(\boldsymbol{s})\), it follows that \(\boldsymbol{y}\in \textsf{BB}(\boldsymbol{r},\boldsymbol{t})\), a contradiction to Lemma 1(i).    \(\square \)

The next two lemmas specify the relation between a \(\textsf{BE}\)-configuration and the \(\textsf{BE}\)-property, and between a \(\textsf{EX}\)-configuration and the \(\textsf{EX}\)-property, respectively.

Lemma 3

(\(\star \)). If a profile contains a (vuw)-BE-configuration, then no embedding satisfies the (vuw)-BE-property.

Lemma 4

(\(\star \)). If a profile contains a (vuw)-EX-configuration, then no embedding satisfies the (vuw)-EX-property.

4.2 The Case with 3 Voters and 6 Alternatives

Using Lemmas 3 and 4, we prove Theorem 3 with the help of Example 4.

Example 4

The following profile \(\mathcal{P}_{3}\) with 3 voters and 6 alternatives is not .

\(\small {\begin{array}{l} v_1:1 \succ 2 \succ 3 \succ 4 \succ 5 \succ 6, \quad v_2:1 \succ 4 \succ 6 \succ 3 \succ 5 \succ 2, \quad v_3:6 \succ 5 \succ 2 \succ 3 \succ 1 \succ 4. \end{array}}\)

Theorem 3

There exists a non profile with 3 voters and 6 alternatives.

Proof

Consider profile \(\mathcal{P}_{3}\) given in Example 4. Suppose, towards a contradiction, that E is a embedding for \(\mathcal{P}_{3}\). Since each embedding for 3 voters must satisfy one of the two properties in Definition 2, we distinguish between two cases: there exists a voter who is embedded inside the bounding box of the other two, or there is no such voter.

Case 1: There exists a voter \(v_i\), \(i\in [3]\), such that E satisfies the \(v_i\)-\(\textsf{BE}\)-property. Since \(\mathcal{P}\) contains a \((v_1,v_2,v_3)\)-\(\textsf{BE}\)-configuration wrt. \((a,b,x)=(2,6,5)\), by Lemma 3 it follows that E violates the \(v_1\)-\(\textsf{BE}\)-property. Analogously, since \(\mathcal{P}\) contains a \((v_2,v_1,v_3)\)-\(\textsf{BE}\)-configuration regarding \(a=4,b=2,x=3\), and \((v_3,v_1,v_2)\)-\(\textsf{BE}\)-configuration with \(a=5,b=1,x=3\), neither does E satisfy the \(v_2\)-\(\textsf{BE}\)-property or the \(v_3\)-\(\textsf{BE}\)-property.

Case 2: There exists a voter \(v_i\), \(i\in [3]\), such that E satisfies the \(v_i\)-\(\textsf{EX}\)-property. Now, consider the subprofile \(\mathcal{P}'\) restricted to the alternatives 1, 2, 3, 6. We claim that this subprofile contains an \(\textsf{EX}\)-configuration, which by Lemma 4 precludes the existence of such a voter \(v_i\) with the \(v_i\)-\(\textsf{EX}\)-property: First, since \(\mathcal{P}'\) contains a \((v_3,v_1,v_2)\)-\(\textsf{EX}\)-configuration (setting \((u,v,w){:}{=}(v_1,v_3,v_2)\) and \((x,a,b,c,d,e)=(3,2,6,1,1,1)\)), by Lemma 4, it follows that E violates the \(v_3\)-\(\textsf{EX}\)-property. In fact, \(\mathcal{P}'\) also contains a \(v_2\)-\(\textsf{EX}\)-configuration (setting \((u,v,w){:}{=}(v_1,v_2,v_3)\) and \((x,a,b,c,d,e)=(3,1,6,2,2,2)\)) and a \(v_1\)-\(\textsf{EX}\)-configuration (setting \((u,v,w){:}{=}(v_2,v_1,v_3)\) and \((x,a,b,c,d,e)=(3,1,2,6,6,6)\)). By Lemma 4, it follows that E violates the \(v_2\)-\(\textsf{EX}\)-property and the \(v_1\)-\(\textsf{EX}\)-property.

Summarizing, we obtain a contradiction for E.    \(\square \)

4.3 The Case with 4 Voters and 5 Alternatives

We prove Theorem 4 for instead of for preferences since the reasoning for is simpler and more intuitive. It is, however, possible to follow similar steps for preferences and obtain an analogous proof. The following proposition allows us to extend any result we obtain of the (non-) existence of embeddings to embeddings and vice versa. The same claim has been made by Escoffier et al. [16, Proposition 2].

Proposition 1

[20]. There is a natural isometry between \(\mathbb {R}^{2}\) under \(\ell _1\)-norm and \(\mathbb {R}^{2}\) under \(\ell _{\infty }\)-norm.

We first prove that any profile with at least 5 alternatives which contains an all-triples worst-diverse configuration is not . Then we proceed to show that the example below with 4 voters and 5 alternatives is such a profile.

Example 5

The following profile \(\mathcal{P}_{4}\) with 5 alternatives contains an all-triples worst-diverse configuration and will be shown to be not .

$$\begin{array}{ll} v_1:1 \succ 2 \succ 3 \succ 4 \succ 5, \qquad \qquad &{} v_2 :1 \succ 2 \succ 3 \succ 5 \succ 4, \\ v_3:1 \succ 4 \succ 5 \succ 3 \succ 2, &{} v_4 :2 \succ 4 \succ 5 \succ 3 \succ 1. \end{array}$$

To do this, we utilize the following two lemmas:

Lemma 5

(\(\star \)). Let \(\mathcal{P}\) admit a embedding E. If \(\boldsymbol{z}\in \textsf{BB}(\boldsymbol{x}, \boldsymbol{y})\), then there is no voter v satisfying \(\{x, y\} \succ _v z\).

Lemma 6

(\(\star \)). For any set \(\mathcal{S}\) of 5 points in \(\mathbb {R}^{2}\), there must exist three distinct points \(\boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z}\in \mathcal{S}\) such that \(\boldsymbol{z}\in \textsf{BB}(\boldsymbol{x}, \boldsymbol{y})\).

Now, we are ready to show our second main result.

Theorem 4

There exists a non profile with 4 voters and 5 alternatives.

Proof

Suppose, towards a contradiction, that we have a profile \(\mathcal{P}\) with at least 5 alternatives \(\{a, b, c, d, e\}\) which contains an all-triples worst-diverse configuration and is with a embedding E.

As we have 5 alternatives, by Lemma 6 there must be a triple \(\{ x, y, z\} \subset \{a, b, c, d, e\}\) such that \(\boldsymbol{z}\in \textsf{BB}(\boldsymbol{x}, \boldsymbol{y})\). This together with Lemma 5 implies that no voter v can satisfy \(\{x, y\} \succ _v z\). However, this is a contradiction to our assumption that \(\mathcal{P}\) contains an all-triples worst-diverse configuration. Therefore we cannot have a profile \(\mathcal{P}\) with at least 5 alternatives which contains an all-triples worst-diverse configuration and has a embedding E.

One can verify that profile \(\mathcal{P}_{4}\) given in Example 5 with 5 alternatives and 4 voters contains an all-triples worst-diverse configuration, and is not : The alternatives 1, 2, 4, and 5 are ranked last by voters \(v_4, v_3, v_2\), and \(v_1\), respectively. Therefore we can pick the corresponding voters for every triple involving only the alternatives 1, 2, 4 and 5. It is straightforward to verify that there is a worst-diverse configuration for every triple of alternatives involving 3 as well. Thus we have shown that there is a profile with 4 voters and 5 alternatives that is not . By Proposition 1 it is also not .    \(\square \)

4.4 The Case with 5 Voters and 4 Alternatives

The proof of Theorem 5 will be based on the following example.

Example 6

Any profile \(\mathcal{P}_{5}\) satisfying the following is not .

$$\begin{array}{lll} v_1:1 \succ 2 \succ 3 \succ 4, &{} \quad v_2:1 \succ 4 \succ 3 \succ 2, &{} \quad v_3:\{2, 4\} \succ 3 \succ 1,\\ v_4:3 \succ 2 \succ 1 \succ 4, &{} \quad v_5:3 \succ 4 \succ 1 \succ 2. \end{array}$$

Before we proceed with the proof, we show a technical but useful lemma.

Lemma 7

(\(\star \)). Let \(\mathcal{P}\) be a profile with 4 voters uvwr and 4 alternatives a, b, c, d satisfying the following:

$$ \begin{array}{lll} u:\{a,b\} \succ c \succ d,&v:\{b,d\} \succ c \succ a, w:\{a,d\} \succ c \succ b,&r:c \succ \{a,b\} \succ d. \end{array}$$

If E is a embedding for \(\mathcal{P}\) with , then .

Theorem 5

(\(\star \)). There exists a non profile with 5 voters and 4 alternatives.

Proof sketch

We show that profile \(\mathcal{P}_{5}\) given in Example 6 is not . Suppose, towards a contradiction, that \(\mathcal{P}_{5}\) admits a embedding E. First, we observe that one of \(v_1,v_2,v_3\) is embedded inside the bounding box defined by the other two since the subprofile of \(\mathcal{P}_{5}\) restricted to voters \(v_1,v_2\) and \(v_3\) is equivalent to profile \(\mathcal{Q}_{2}\) which, by Lemma 4, violates the \(\textsf{EX}\)-property (for each of \(v_1\),\(v_2\), and \(v_3\), respectively). Hence, we distinguish between two cases.

Case 1: or . Note that these two subcases are equivalent in the sense that if we exchange the roles of alternatives 2 and 4, i.e., \(1\mapsto 1\), \(3 \mapsto 3\), \(2\mapsto 4\), and \(4 \mapsto 2\), we obtain an equivalent (in terms of the Manhattan property) profile where the roles of voters \(v_1\) and \(v_2\) (resp. \(v_4\) and \(v_5\)) are exchanged. Hence, it suffices to consider the case of . W.l.o.g., assume that and (see Fig. 6a). Then, by Lemma 7 (setting \((u,v,w,r){:}{=}(v_1,v_2,v_3,v_4)\)), we obtain that . This implies that and .

Fig. 6.
figure 6

Illustrations of possible embeddings for the proof of Theorem 5: Left: . (6a): . (6b): . (6c): , such that , , , .

By the preferences of \(v_4,v_2,v_3\) regarding alternatives 2 and 1, and by Lemma 1, it follows that and . Similarly, regarding the preferences over 3 and 1, it follows that . By Lemma 2(ii) (considering the preferences of \(v_1,v_2\) and \(v_3\) regarding alternatives 2 and 3), we further infer that either and or and . Without loss of generality, assume that and .

By the preferences of \(v_3\) and \(v_2\) (resp. \(v_4\) and \(v_2\)) regarding 1 and 3 and by Lemma 2(i), it follows that (resp. ). By prior reasoning, we have that . However, this is a contradiction due to the preferences of \(v_4\) and \(v_2\) (resp. \(v_3\) and \(v_2\)) regarding 1 and 2: By Lemma 2(ii), it follows that .

Case 2: . Without loss of generality, assume that and ; see Fig. 6b. Then, by Lemma 7 (setting \((u,v,w,r){:}{=}(v_1,v_3,v_2,v_4)\) and \((u,v,w,r){:}{=}(v_2,v_3,v_1,v_5)\), respectively), we obtain that and . This implies that

(6)

By applying Lemmas 1 and 2 repeatedly, we will infer that alternatives 1, 2, 3, 4 are embedded to the northwest, southwest, southeast, and northeast of \(v_3\), respectively. Moreover, the embeddings of voters \(v_1\), \(v_2\), \(v_3\), and \(v_4\) are as specified in Fig. 6c. Now, since \(v_2\) and \(v_4\) (which are on the opposite “diagonal” of \(v_3\)) both have \(1\succ 4\) and \(3 \succ 2\), while \(v_3:4 \succ _3 1\) and \(2 \succ _3 3\), the bisector between alternatives 1 and 4 and the one between alternatives 2 and 3 must “cross” twice. Similarly, due to \(v_1\) and \(v_5\), and \(v_3\), the bisector between alternatives 1 and 2 and the one between alternatives 3 and 4 “cross” twice. This is, however, impossible. The details of the remaining proof can be found in the full version [7].

5 Embeddings

In this section, we identified the following positive result through exhaustive embedding all the possible preference profiles in the two-dimensional space.

Proposition 2

If \((n,m)=(3,5)\) or \((n,m)=(4,4)\), then each profile with at most n voters and at most m alternatives is .

Proof

Since the Manhattan property is monotone, to show the statement, we only need to look at profiles which have either 3 voters and 5 alternatives, or 4 voters and 4 alternatives. We achieve this by using a computer program employing the CPLEX solver that exhaustively searches for all possible profiles with either 3 voters and 5 alternatives, or 4 voters and 4 alternatives, and provide a embedding for each of them. Since the CPLEX solver accepts constraints on the absolute value of the difference between any two variables, our computer program is a simple one-to-one translation of the constraints given in Definition 1, without any integer variables. Peters [21] has noted this formulation for embeddings. The same program can also be used to show that the preference profiles from the Examples 4, 5 and 6 do not admit a embedding.

Following a similar line as in the work of Chen and Grottke [6], we did some optimization to significantly shrink the search space on all profiles: We only consider profiles with distinct preference orders and we assume that one of the preference orders is \(1 \succ \ldots \succ m\). Hence, the number of relevant profiles with n voters and m alternatives is \(\left( {\begin{array}{c}m!-1\\ n-1\end{array}}\right) \). For \((n,m) = (3,5)\) and \((n,m)=(4,4)\), we need to iterate through 7021 and 1771 profiles, respectively. We implemented a program which, for each of these produced profiles, uses the IBM ILOG CPLEX optimization software package to check and find a embedding. The verification is done by going through each voter’s preference order and checking the condition given in Definition 1. All generated profiles, together with their embeddings and the distances used for the verification, are available at https://owncloud.tuwien.ac.at/index.php/s/s6t1vymDOx4EfU9.    \(\square \)

6 Conclusion

Motivated by the questions of how restricted preferences are, we initiated the study of the smallest dimension sufficient for a profile to be . We provided algorithms for larger dimension d and forbidden subprofiles for \(d=2\).

This work opens up several future research directions. One future research direction concerns the characterization of profiles through forbidden subprofiles. Such work has been done for other restricted preference domains such as single-peakedness [1], single-crossingness [4], and 1-Euclideanness [8]. Another research direction is to establish the computational complexity of determining whether a given profile is . To this end, let us mention that profiles cannot be characterized via finitely many finite forbidden subprofiles [8], but they can be recognized in polynomial time [10, 14, 18]. As for \(d\ge 2\), recognizing profiles becomes notoriously hard (beyond NP) [21]. This stands in stark contrast to recognizing preferences, which is in NP. For showing NP-hardness, our forbidden subprofiles may be useful for constructing suitable gadgets. Finally, it would be interesting to see whether assuming preferences can lower the complexity of some computationally hard social choice problems.