Strategy-proof social choice on multiple singlepeaked domains and preferences for parties

Transcription

1 Strategy-proof social choice on multiple singlepeaked domains and preferences for parties Alexander Reffgen August 26, 2011 Abstract The concept of single-peaked domains is generalized to multiple single-peaked domains, where the set of alternatives is equipped with several underlying orderings with respect to which a preference can be single-peaked. We provide a complete characterization of the strategy-proof social choice functions on multiple single-peaked domains. We show also in the framework of a spatial voting model for party elections that multiple single-peaked domains are appropriate to represent preferences over parties. Keywords: Single-peaked preferences, Strategy-proofness, Party preferences, Efficiency, Degree of manipulability. The author would like to thank John Weymark, Hans Peters, Dolors Berga, Lars-Gunnar Svensson, Tommy Andersson, Håkan Holm, and Philipp Wichardt for very helpful comments. Financial support from The Jan Wallander and Tom Hedelius Foundation is gratefully acknowledged. Alexander Reffgen. Department of Economics, Lund University, Box 7082, SE Lund, Sweden. Fax: alexander.reffgen@nek.lu.se 1

2 1 Introduction The central concern in the theory of strategy-proof social choice is to investigate the conditions under which it is possible to construct voting procedures that are incentive compatible in the sense that voters can never gain from strategic misrepresentation of their preferences. In general, this problem can not be solved in a satisfactory way: if there are three or more eligible alternatives, then by the Gibbard-Satterthwaite theorem (Gibbard 1973; Satterthwaite 1975) only the dictatorial voting procedures are strategy-proof. A crucial prerequisite for this result is, however, the unrestricted domain assumption, which means that voters are allowed to hold any conceivable preference. At first glance, this is a plausible assumption because the exact shape of a voter s preference is private information, and therefore it is strictly speaking impossible to classify some preference for certain as infeasible. But at the same time, accepting the unrestricted domain assumption also means not taking into account the fact that preferences do not come up in an arbitrary way, but are normally the result of a deliberate mental process, and it is therefore likely that the domain of preferences exhibits some structure (see, for instance, Sen (1970, p. 165)). This objection has inspired a large number of studies that consider restricted preference domains with different structures and investigate which voting procedures are strategy-proof on these domains. Thereby, it turned out that some domains admit large classes of non-dictatorial strategy-proof voting procedures (e.g., Moulin (1980) and Barberà et al. (1991)), while other domains lead back to the same conclusion as in the Gibbard-Satterthwaite theorem (e.g., Aswal (2003)). Among all restricted preference domains, the domain of single-peaked preferences, introduced by Black (1948), has been especially influential. For single-peaked domains, the structure of preferences comes up as a consequence of the nature of the available alternatives. A necessary prerequisite for single-peaked preferences is that the available alternatives have a common characteristic that makes it possible to order these alternatives in an objective way in a line from, say, the left to the right. Given this ordering, a voter forms his preference by first choosing a most preferred alternative, and the further he then moves away from his top to either the left or the right, the less preferred are the alternatives. Single-peaked preferences are reasonable in a wide range of applications. To begin with, it is likely that preferences are single-peaked when the value of a quantitative variable must be chosen, e.g., the height of a particular tax rate, the wage rate of labor, or the rate of CO2 emission (examples of this type were studied in Black (1948)). Further, when the alternatives are qualitative, there may also be a 2

3 natural ordinal ordering of the alternatives which then again induces single-peaked preferences; e.g., this is the case when the voting concerns what civil rights to assign to a certain group in the society, and the alternatives are ordered on a scale from no rights via basic democratic rights such as freedom of speech and franchise to full rights including membership in social security and pension system. More generally, in many political decisions the available alternatives can be ordered on an ideological left right scale, and therefore single-peaked preferences also play an important role in political economy; for example, an extensive social security system and economic regulations are traditionally considered as left-wing politics, while free markets and a broad protection of private property are considered as rightwing politics. A particularly important example of political votings are of course elections involving parties or candidates for official positions, and also in this context it is common to assume a left right ordering and single-peaked preferences (see, for example, the seminal work by Downs (1957) on electoral competition). A crucial assumption for single-peaked preference domains is, however, that all voters apply the same underlying ordering of the available alternatives when they form their single-peaked preferences. This is almost undisputable in the case of quantitative variables, and it is also reasonable in many cases of qualitative variables with a clear ordinal ordering. But when it comes to political alternatives, there is often no equally convincing objective ordering, and hence the assumption of one common underlying ordering accepted by all voters may be unjustified. In particular for the case of political parties, there is empirical evidence casting doubt on the hypothesis that voters preferences are single-peaked with respect to a common ordering of the parties: in several studies performed in the U.S. and Germany, it turns out that for any given left right ordering of the available parties there is normally a large proportion of voters whose preferences are not single-peaked with respect to that ordering (e.g., Niemi and Wright (1987), Feld and Grofman (1988), Pappi and Eckstein (1998)). Instead, it appears to be more likely that voters have different perceptions of how to place political alternatives on a left right scale, but given their personal understanding of the ordering of the alternatives, they still form single-peaked preferences. To describe this preference structure formally, we introduce in this article the concept of a multiple single-peaked preference domain. By this, we mean the union of several (ordinary) single-peaked domains which are constructed with respect to different underlying orderings of the alternatives. Multiple single-peaked domains cover a wide range of domains: at the one extreme, when there is only one underlying ordering, there are of course the usual single-peaked domains, and, at 3

4 the other extreme, the unrestricted domain is also a multiple single-peaked domain. The main purpose of this article is to provide a complete characterization of the strategy-proof voting procedures on multiple single-peaked domains. For singlepeaked domains, Moulin (1980) has shown that a voting procedure is strategy-proof if and only if it is a generalized median voter scheme. Since multiple single-peaked domains consist of several single-peaked domains, a strategy-proof voting procedure on a multiple single-peaked domain must be such that the restriction of the voting procedure to each of the single-peaked domains that belongs to the multiple domain is a generalized median voter scheme. However, a voting procedure that is a generalized median voter scheme on one single-peaked domain is not necessarily a generalized median voter scheme on another single-peaked domain, because the two domains are based on different orderings of the available alternatives, and therefore the voting procedure is not necessarily strategy-proof on the second domain either. Thus, to obtain a strategy-proof voting procedure on the multiple single-peaked domain, we must find a coordination condition to ensure that a voting procedure, which is a generalized median voter scheme on one single-peaked domain belonging to the multiple domain, is also a generalized median voter scheme on any other single-peaked domain that belongs to the multiple domain. In Theorem 1, we show that the following condition is necessary and sufficient for a voting procedure on a multiple single-peaked domain to be strategy-proof: When we compare the different orderings that generate the single-peaked domains belonging to the multiple domain, we can divide the set of alternatives into a left and a right component where all orderings coincide, and a middle component where the orderings differ. A strategy-proof voting procedure must, to begin with, be dictatorial on the middle component. On the left component, the voting procedure applies first a generalized median voter scheme for all voters except the middle component dictator, and, comparing the resulting alternative with the top alternative of the dictator, the voting procedure then chooses the alternative that is closest to the middle component. A similar procedure applies to the right component. Intuitively, the class of strategy-proof voting procedures on a multiple single-peaked domain thus falls in between the class of all generalized median voter schemes, which are the strategy-proof rules on a traditional single-peaked domain, and the class of dictatorial rules, which are the only rules that are strategy-proof on the unrestricted domain. So far, we have argued somewhat loosely that traditional single-peaked domains are not sufficient to represent preferences over parties, but that these preferences belong rather to a multiple single-peaked domain. In the second part of this article, 4

5 we present therefore a more thorough motivation for the use of multiple singlepeaked domains to represent preferences over parties. A first thought on the nature of parties already reveals that parties differ in one important respect from most of the other examples of single-peaked domains mentioned above: In general, parties are not single-decision alternatives, but can be considered as bundles of alternatives from different political dimensions, such as economic policy or social and civil rights policy, which they would implement if elected (see, e.g., Austen-Smith (1996)). To derive the structure of preferences over parties from the nature of parties as bundles of alternatives, we construct a spatial voting model in which parties are represented as elements in a multi-dimensional set of political alternatives. Imposing a weak local structure on voters preferences over the multi-dimensional set of political alternatives, we show then that voters preferences over parties constitute a multiple single-peaked domain (Proposition 1), where the exact number and shape of the underlying orderings depend intimately on how the parties are located in the multi-dimensional set. This article is organized as follows: Section 2 introduces the social choice model. Section 3 provides a complete characterization of the strategy-proof voting procedures on multiple single-peaked domains. In Section 4, we show that multiple single-peaked domains are appropriate to represent preferences over parties. Section 5 considers the trade-off between strategy-proofness, anonymity, and efficiency on multiple single-peaked domains. All proofs are collected in the appendix. 2 The social choice model and some important results from the literature 2.1 The basic model The basic formal framework of this study is as follows: Let N = {1,...,n} be a finite society of n individuals, who consider a finite set A of m social alternatives. The individuals have complete, transitive, and asymmetric 1 preferences over the alternatives in A, and the preference of individual i is denoted by P i. Given P i, the kth ranked alternative (1 k m) in P i is denoted by r k (P i ), which means that #{a A; ap i r k (P i )} = k 1. The set of all complete, transitive, and asymmetric 1 A preference P on A is complete if for all a,a A, a a, either apa or a Pa; P is transitive if apa and a Pa implies apa for all a,a,a A; finally, P is asymmetric if, for all a,a A, apa implies that a Pa does not hold. 5

6 preferences over A is called the unrestricted domain over A, and it is denoted by Σ A. The set of all admissible preferences, referred to as the preference domain, is denoted by Ω, and Ω is normally a strict subset of Σ A. If P Ω and B A, then the restriction of P to B is that preference P B on B which satisfies bp B b if and only if bpb for all b,b B. The domain of preferences obtained when every P Ω is restricted to B is denoted by Ω B = {P B ; P Ω}. The preferences of the n individuals in the society are collected in a (preference) profile (P 1,...,P n ) Ω n. In order to focus on individual i s preference, a profile (P 1,...,P n ) can be rewritten as (P i,p i ), where P i Ω n 1 thus denotes the profile of all individuals except individual i. A social choice function (SCF) is a mapping f : Ω n A that assigns to every profile (P 1,...,P n ) Ω n a unique social choice a A. For a given SCF f, the set of all a A that can be attained by f is called the range of f and is denoted by R f, i.e., R f = {a A; a = f (P 1,...,P n ) for some (P 1,...,P n ) Ω n }. If R f = A, then f is surjective. An SCF f : Ω n A has the tops-only property if the choice of f depends only on the top alternatives in every profile, i.e., f (P 1,...,P n ) = f (P 1,...,P n) for all (P 1,...,P n ) Ω n and (P 1,...,P n) Ω n such that r 1 (P i ) = r 1 (P i ) for all i N. Further, an SCF f : Ω n A is manipulable at the profile (P 1,...,P n ) Ω n if there is some i N and some P i Ω such that f (P i,p i)p i f (P i,p i ). If f cannot be manipulated at any admissible profile, then f is strategy-proof. Finally, f : Ω n A is dictatorial on B A if there is some i N, called the dictator on B, such that f (P i,p i ) = r 1 (P i ) for all (P i,p i ) Ω n with r 1 (P i ) B. If f is dictatorial on A, then f is shortly called dictatorial. In this basic model, the reference point in the theory on strategy-proof social choice is the well-known Gibbard-Satterthwaite theorem (Gibbard 1973; Satterthwaite 1975), which states the general incompatibility of strategy-proofness and nondictatorship. The Gibbard-Satterthwaite theorem. Let A be a finite set of at least three alternatives. A surjective SCF f : Σ n A A is strategy-proof if and only if f is dictatorial. 2.2 Social choice on single-peaked domains Given the basic model above, we now introduce a structure for the preference domain Ω which generalizes the structure of single-peaked preferences. To this end, we recall first the definition of single-peaked preferences, starting with the concept of an underlying ordering of A: An ordering S of A is a complete, transitive, and asymmetric binary relation on A, and if asb for a,b A, we say that a is to 6

7 the left of b, or equivalently, that b is to the right of a. The set of all orderings of A is denoted by Σ A, exactly as for preferences. For a given ordering S Σ A, denote by ρ k (S) the kth alternative in A from the left (1 k m), i.e., ρ k (S) is the unique alternative in A that satisfies #{a A; asρ k (S)} = k 1. The total ordering ρ 1 (S)Sρ 2 (S)S... Sρ m (S) of A by S is in short represented as a vector S = [ρ 1 (S) ρ 2 (S)... ρ m (S)]. If S Σ A, define the inverse ordering S 1 of S by setting as 1 b if and only if bsa for all a,b A. Further, for B A, define the maximum max S (B) of B with respect to S to be the unique alternative b B which satisfies bs b for all b B \ { b}. Similarly, the minimum min S (B) of B with respect to S is that alternative b B which satisfies bsb for all b B \ {b}. In particular, note that min S (A) = ρ 1 (S) and max S (A) = ρ m (S). To every S Σ A, we associate a ternary betweenness relation B S on A by setting B S (a,b,c) for a,b,c A if and only if either asbsc or csbsa, and if B S (a,b,c) we say that b is between a and c. Further, for a,b A with asb or a = b, the set [a,b] S {x A; x {a,b} or B S (a,x,b)} is called the (closed) interval of a and b with respect to S; we also define the two half-closed intervals ]a,b] S [a,b] S \{a} and [a,b[ S [a,b] S \{b}. Finally, the projection of x A on an interval [a,b] S with respect to S, denoted by π S [a,b] S (x), is defined as follows: π S [a,b] S (x) = x if x [a,b] S ; π S [a,b] S (x) = a if xsa; and π S [a,b] S (x) = b if bsx. Given some ordering S Σ A, a preference P Σ A is single-peaked with respect to S if B S (r 1 (P),a,b) implies apb for all a,b A. Equivalently, P is single-peaked with respect to S if B S (a,b,c) implies bpa or bpc for all triples a,b,c A. The first of these two definitions gives a global characterization of single-peaked preferences: if the alternatives in A are set out in a line, then a utility function representing P is upward sloping towards the top of P, and downward sloping afterwards. 2 The second definition can instead be seen as a local characterization: it considers one triple at a time and requires that the alternative that is in the middle according to S must not be ranked worst in the triple by P. 3 It is easily checked that these two definitions are equivalent in the given framework where A is one-dimensional. If the set of alternatives is multi-dimensional, however, then this equivalence no longer holds, which is explained in Section 4. Finally, the set of all P Σ A that are single-peaked with respect to some given ordering S Σ A is called the single-peaked domain with respect to S and denoted by Ω S. When preferences belong to a single-peaked domain, then strategy-proofness 2 In this way, single-peaked preferences were introduced in Black (1948). 3 This requirement is a special case of the more general condition of value-restricted preferences introduced in Sen (1966). 7

8 and non-dictatorship do no longer exclude each other, but instead there is a large class of strategy-proof and non-dictatorial SCFs. The following result by Moulin (1980), which is the reference point in the theory of strategy-proof social choice on single-peaked domains, provides a complete characterization of the strategy-proof tops-only SCFs on single-peaked domains. Moulin (1980, Proposition 3) An SCF f : Ω n S A is strategy-proof and has the tops-only property if and only if there exists a family {a T } T N of fixed alternatives from A such that S f (P 1,...,P n ) = min { max S {a T,r 1 (P j )} } for all (P 1,...,P n ) Ω n S. (2.1) T N j T The minmax SCFs in (2.1) are known as generalized median voter schemes in the literature. These SCFs can be interpreted as follows. Every coalition T N of individuals proposes one alternative from A using a max rule, and the scheme then selects the smallest of these proposed alternatives. Thereby, the family {a T } T N of fixed alternatives can be considered as preset votes for all T N, and a coalition T can only deviate from a T if some i T moves the vote to the right. In this sense, a T delimits the possibilities of coalition T to influence the location of the social choice, and, in particular, the more a T is to the right in A, the less power has T to influence f. For a concrete example, consider a dictatorial, and hence strategy-proof SCF f : Ω n S A. To represent f as a generalized median voter scheme, let î N be the dictator for f. If a {î} = ρ 1 (S) and a T = ρ m (S) for all T N with T {î}, then it is easily checked that the righ-hand side in (2.1) reduces to r 1 (Pî). Choosing {a T } T N in this way, all power is thus assigned to the singleton coalition {î}, and no power to any other coalition. From a normative perspective, an important subclass of the generalized median voter schemes consists of the schemes that treat all individuals equally and that hence are called anonymous. Formally, an SCF f : Ω n A is anonymous if f (P 1,...,P n ) = f (P σ(1),...,p σ(n) ) for all permutations σ of N and all profiles (P 1,...,P n ) Ω n. A generalized median voter scheme is obviously anonymous if a T = a T whenever #T = #T. A well-known example of an anonymous strategyproof SCF on a single-peaked domain Ω S is the median rule, which assigns to every profile (P 1,...,P n ) Ω n S the median of the n top alternatives r 1(P i ), i N, in the profile. 4 In fact, Moulin (1980) showed that every anonymous strategy-proof SCF 4 The median med(x 1,...,x n ) of n alternatives x 1,...,x n is the alternative that ends up in the middle when these alternatives are ordered according to S, i.e., if x = med(x 1,...,x n ), then #{x i ; x i 8

9 f : Ω n S A that satisfies the tops-only property can be represented as the median of the n top alternatives of the individuals in N and n + 1 fixed alternatives in A. The characterization of the strategy-proof SCFs in (2.1) is based on the assumption that all individuals in the society apply the same ordering S when they form their single-peaked preferences. As discussed in the introduction, this may be a reasonable assumption in many situations, but in other cases it is more likely that the individuals form their single-peaked preferences along different orderings of A. This leads to the following central definition of this article. Definition 1. Let S = {S 1,...,S q } be a family of q orderings S τ Σ A, 1 τ q. The multiple single-peaked domain Ω S with respect to S is Ω S q τ=1 Ω S τ. Further, Ω S is called non-trivial if there exist S,S S such that S S and S 1 S. A multiple single-peaked domain consists of a number of ordinary single-peaked domains which are constructed with respect to different orderings of A. Thus, if P Ω S, then P is single-peaked with respect to at least one S S. Note that if Ω S is not non-trivial, then either #S = 1 or S = {S,S 1 } for some S Σ A, and in both cases, Ω S is an ordinary single-peaked domain. This is obvious if #S = 1, and if S = {S,S 1 }, this follows from the fact that B S (a,b,c) for a,b,c A if and only if B S 1(a,b,c), which means Ω S = Ω S 1. The class of multiple single-peaked domains covers a wide range of domains, which is illustrated in the following example. Example 1. At the one extreme, if #S = 1, the ordinary single-peaked domains turn out as a special case of multiple single-peaked domains. At the other extreme, also the unrestricted domain is a special case of multiple single-peaked domains which is obtained for instance if S consists of all possible orderings of A, i.e., if S = Σ A. For a non-trivial example of a multiple single-peaked domain in between an ordinary single-peaked domain and the unrestricted domain, consider the set A = {a,b,c,d}. Suppose that the individuals in the society consider unanimously a to be the leftmost and d to be the rightmost alternative, but that there is disagreement on how to order the two middle alternatives b and c. Then, an individual may either use the ordering S 1 = [a b c d ] or S 2 = [a c b d ] to form his single-peaked preference, and the preference domain is Ω {S1,S 2 }. For a slightly more complicated example, again let A = {a,b,c,d}, and suppose now that there is consensus in the society that A can be divided into a left block consisting of a and b, and a right block [ρ 1 (S), x] S } n/2 and #{x i ; x i [ x,ρ m (S)] S } n/2. Further, if n is odd, then the median rule can be represented as a generalized median voter scheme with a T = ρ 1 (S) if #T = (n+1)/2 and a T = ρ m (S) of #T (n + 1)/2. 9

10 consisting of c and d, but there is no agreement on how to order these two blocks internally. In this case, the collection S generating the domain Ω S consists of the four orderings S 1 = [a b c d ], S 2 = [b a c d ], S 3 = [a b d c], and S 4 = [b a d c]. As a matter of fact, it turns out that Ω S = Σ A. Remark 1. An alternative way to interpret multiple single-peaked domains is as follows. If S is the underlying ordering of an ordinary single-peaked domain Ω S, then the associated betweenness relation B S is complete in the sense that for every triple of distinct alternatives from A, there is always exactly one alternative that is between the other two, and this alternative cannot be ranked worst in the triple. On the contrary, a multiple single-peaked domain can no longer be associated with a complete betweenness relations. For example, if Ω S is generated by S 1 = [a b c d ] and S 2 = [a c b d ], then b is unambiguously between a and d, but for the three alternatives a, b, and c, there is no unique middle alternative and every alternative can be ranked worst in the triple. Allowing for several underlying orderings can thus be seen as a relaxation of the assumption of a complete betweenness relation. In this context, we want to mention that an important alternative generalization of single-peaked domains, as derived from an abstract betweenness relation, has been considered in Nehring and Puppe (2007b). The deep analysis in that article is too different from our approach to motivate a detailed comparison of the two models, but it should nevertheless be pointed out that they are logically independent. 5 3 Strategy-proof social choice on multiple single-peaked domains We now provide a complete characterization of the strategy-proof SCFs on multiple single-peaked domains. It turns out that given a collection S = {S 1,...,S q } of orderings of A, the class of strategy-proof SCFs on Ω S depends on how similar the orderings in S are. In particular, to what extent the orderings in S coincide at their left and right ends will be of importance. To make this precise, we introduce 5 For the interested reader, it can be mentioned that this logical independence can be established easily by means of examples. To begin with, for many of the examples of rich single-peaked domains considered in Nehring and Puppe (2007b) it is almost obvious that they cannot be represented as multiple single-peaked domains. For a simple converse example, it is straightforward to check that the multiple single-peaked domain Ω S generated by the two orderings S 1 = [a b c d ] and S 2 = [b a d c] does not satisfy the closure condition in Theorem 1 in Nehring and Puppe, and therefore, cannot be represented as a rich single-peaked domain in the sense of Nehring and Puppe. 10

11 first a way to decompose the set of alternatives A into three components, namely a left component A L S, a middle component AM S, and a right component AR S, such that A = A L S AM S AR S, and all orderings in S coincide on the left and on the right component, while there is some disagreement on how to order the alternatives in the middle component. Thereby, we must take into account the fact that if S and S are inverse orderings to each other, then the single-peaked domains Ω S and Ω S are actually identical because B S (a,b,c) if and only if B S (a,b,c) for all a,b,c A. When we analyze the similarity of the orderings in S, we must therefore direct these orderings in a similar way, because the class of strategy-proof SCFs on Ω S depends of course only on dissimilarities among the orderings in S that enlarge the preference domain Ω S, but not on dissimilarities that depend on different directions of the orderings, and that hence do not have any impact on Ω S. We now present the formal construction of the decomposition of A: (1) Consider first the case when there exist two orderings S,S S such that { ρ1 (S),ρ m (S) } { ρ 1 (S ),ρ m (S ) } =. (3.1) This means that none of the endpoints of S and S coincide, even if we take the inverse of one of the two orderings. Then there cannot be some common left or right component for all orderings in S, and therefore we set A L S = AR S = and A M S = A. (2) Consider now the case when there exist no two orderings in S for which (3.1) occurs. Then, replacing some of the orderings in S with their inverse orderings if necessary, we can assume that ρ 1 (S) = ρ 1 (S ) for all S,S S. If now all orderings in S are identical, set A L S = A and AM S = AR S =. Otherwise, there exists a unique integer l with 1 l < m such that ρ k (S) = ρ k (S ) for all k with 1 k l and all S,S S, and ρ l+1 (S) ρ l+1 (S ) for some S,S S. Then, take some S S and set A L S = l k=1 {ρ k (S)}, and note that A L S does not depend on the particular choice of S. We distinguish two cases: (i) If the orderings in S do not have a common right endpoint, i.e., if ρ m (S) ρ m (S ) for some S,S S, set A R S = and AM S = A \ AL S. (ii) If (i) is not the case, then all orderings in S have a common right endpoint, i.e., ρ m (S) = ρ m (S ) for all S,S S. Then there exists a unique integer r with l < r m such that ρ k (S) = ρ k (S ) for all k with r k m and all S,S S, but ρ r 1 (S) ρ r 1 (S ) for some S,S S. Take some S S and set A R S = m k= r {ρ k (S)}, and note that A R S does not depend on the particular choice of S. Finally, set A M S = A \ (AL S AR S ). 11

12 The triple (A L S,AM S,AR S ) is called the maximal common decomposition of A with respect to S. The sets A L S, AM S, and AR S are pairwise disjoint, and AL S AM S AR S = A. The maximal decomposition of A is unique up to a possible inversion in the sense that if (A L S,AM S,AR S ) and (ĀL S,ĀM S,ĀR S ) are two maximal common decompositions of A, then A M S = ĀM S and either AL S = ĀL S and AR S = ĀR S, or AL S = ĀR S and AR S = ĀL S. Note also that all orderings in S, some of them if necessary replaced by their inverse orderings, order the alternatives in A L S and AR S in exactly the same way, and if al A L S, am A M S, and ar A R S, then al Sa M Sa R for all S S. We now consider the maximal common decomposition in two concrete examples. Example 2. Consider the set A = {x 1,x 2,x 3,x 4,x 5 }, and suppose that A is equipped with the two orderings S 1 = [x 1 x 2 x 3 x 4 x 5 ] and S 2 = [x 2 x 1 x 3 x 5 x 4 ]. In this case, the sets of endpoints {ρ 1 (S 1 ),ρ 5 (S 1 )} and {ρ 1 (S 2 ),ρ 5 (S 2 )} are disjoint, which means that the maximal common decomposition of A with respect to S = {S 1,S 2 } is (A L S,AM S,AR S ) = (,A, ). Example 3. As a second example, now let A = {x 1,...,x 7 } be a set of seven alternatives, and suppose that S = {S 1,S 2,S 3 } consists of the following three orderings: S 1 = [x 1 x 2 x 3 x 4 x 5 x 6 x 7 ], S 2 = [x 7 x 5 x 6 x 4 x 3 x 2 x 1 ], S 3 = [x 1 x 2 x 3 x 5 x 4 x 6 x 7 ]. Here, the sets of endpoints {ρ 1 (S 1 ),ρ 7 (S 1 )}, {ρ 1 (S 2 ),ρ 7 (S 2 )}, and {ρ 1 (S 3 ),ρ 7 (S 3 )} coincide for all three orderings, and hence both A L S and AR S are non-empty. To find the maximal common decomposition of A with respect to S, replace first S 2 by its inverse ordering S2 1 = [x 1 x 2 x 3 x 4 x 6 x 5 x 7 ] to make the three orderings have the same left endpoint. Then it is easily checked that A L S = {x 1,x 2,x 3 }, A M S = {x 4,x 5,x 6 }, and A R S = {x 7}. Using the maximal common decomposition of A with respect to S, we can now give a complete characterization of all strategy-proof surjective SCFs on Ω S. Theorem 1. Let A be a finite set of m alternatives, and suppose that Ω S is a non-trivial multiple single-peaked domain over A. Further let (A L S,AM S,AR S ) be a maximal common decomposition of A with respect to S. Then a surjective SCF f : Ω n S A is strategy-proof if and only if π S ( [r 1 (Pî),a] GL (P S î ) ) if r 1 (Pî) Ā L f (Pî,P î ) = r 1 (Pî) if r 1 (Pî) Ā M (3.2) π[a,r S ( 1 (Pî)] GR (P S î ) ) if r 1 (Pî) Ā R 12

13 for all (Pî,P î ) Ω n S, where (1) î N, (2) S S, (3) ĀM = [a,a] S for some a A L S {ρ 1(S)} and a A R S {ρ m(s)}, and further Ā L = A L S \ ĀM and Ā R = A R S \ ĀM, and (4) G L : Ω n 1 S A and G R : Ω n 1 S A are generalized median voter schemes, defined for all individuals except individual î, with ρ 1 (S) R GL and ρ m (S) R GR, i.e., there are two families {a G L T } T N\{î} and {a G R T } T N\{î} of fixed alternatives in A with a G L T = ρ 1(S) for some T N \ {î} and a G R = ρ m (S) such that G L (P î ) = min S T N\{î} and G R (P î ) = min S T N\{î} { max S {a G L j T T,r 1(P j )} } for all P î Ω n 1 S, { max S {a G R j T T,r 1(P j )} } (3.3) for all P î Ω n 1 S. Theorem 1 characterizes the strategy-proof surjective SCFs on a multiple singlepeaked domain as follows. To begin with, A is divided into three parts Ā L, Ā M, and Ā R, which are intervals with respect to some ordering S S. The condition Ā M = [a,a] S, where a A L S {ρ 1(S)} and a A R S {ρ m(s)}, means that the left endpoint a of Ā M belongs to A L S if AL S is non-empty, and the right endpoint a belongs to A R S if AR S is non-empty; if AL S is empty, then a equals ρ 1(S), i.e., the left endpoint of A M S with respect to S, and if AR S is empty, a equals the right endpoint ρ m(s) of A M S. Thus, ĀM always contains A M S, and, if possible, ĀM continues to at least one alternative in A L S and AR S. Given ĀM, the set Ā L (Ā R ) is that part of A L S (AR S ) which is not covered by Ā M. If f now is a strategy-proof surjective SCF, then some individual î must be a dictator on Ā M. Further, if individual î reports a top alternative r 1 (Pî) that does not belong to Ā M, say r 1 (Pî) Ā L, then the choice of f is determined by first applying a generalized median voter scheme G L to all individuals with the exception of individual î, and the resulting alternative G L (P î ) is then projected on the interval between r 1 (Pî) and a; in particular, if G L (P î ) is between r 1 (Pî) and a, then G L (P î ) is the final outcome. A similar procedure applies when r 1 (Pî) belongs to the right part Ā R. Thus, individual î has no dictatorial power on Ā L and Ā R, but he has nevertheless large influence on the location of the social choice even if his top alternative belongs either to Ā L or Ā R. Finally, the two conditions ρ 1 (S) R GL and ρ m (S) R GR, which are equivalent to a G L T = ρ 1(S) for some T N \ {î} and = ρ m (S), are necessary and sufficient for f to be surjective on Ā L and Ā R. The a G R following example illustrates how Theorem 1 works in two different cases. Example 4. Consider again the two maximal common decompositions from Examples 2 and 3. In Example 2, we found (A L S,AM S,AR S ) = (,A, ), so in this case only the dictatorial SCFs f : Ω n S A are strategy-proof and surjective. Thus, combining the two single-peaked domains Ω S1 and Ω S2 turns the possibility result for singlepeaked domains depicted in Moulin (1980) into a dictatorial result. 13

14 In Example 3, both A L S = {x 1,x 2,x 3 } and A R S = {x 7} are non-empty. Since the dictatorial part Ā M of a strategy-proof SCF f : Ω n S A must cover AM S = {x 4,x 5,x 6 } and continue into A L S and AR S if possible, f must be dictatorial at least on {x 3,x 4,x 5,x 6,x 7 }. In particular, the non-dictatorial right part Ā R is empty here. A strategy-proof and not completely dictatorial SCF f : Ω n S A can now for example be constructed as follows: Let some î N be a dictator on {x 3,...,x 7 }. If r 1 (Pî) {x 1,x 2 }, calculate the median med(p î ) of the top alternatives in P î Ω n 1 S (assume here for simplicity that n 1 is odd so that med(p î ) is well-defined); if this median belongs to A L S, choose that alternative of r 1(Pî) and med(p î ) which is closest to x 3, and if med(p î ) / A L S, choose x 3. Remark 2. Compared with the characterization of strategy-proof SCFs in Moulin (1980), the important contribution of Theorem 1 is of course that it applies for a more general class of preference domains. The assumptions in these two results differ, however, slightly in one more respect. While Moulin (1980) assumes that the SCF in addition to strategy-proofness also satisfies the tops-only property, Theorem 1 requires strategy-proofness and surjectivity. A priori, surjectivity is a weak normative requirement guaranteeing voters sovereignty, while the tops-only property is a strong limitation on which information on voters preferences may be used by the SCF. However, in the proof of Theorem 1, we show that on multiple singlepeaked domains, strategy-proofness and surjectivity actually imply the tops-only property. Note also that Theorem 1 does not apply to traditional single-peaked domains, i.e., when S contains only one ordering. The SCFs in (3.2) are of course strategyproof even if #S = 1, but the characterization is no longer complete in this case because (3.2) cannot represent for example the anonymous strategy-proof SCFs which exist on single-peaked domains, e.g., the median rule. Instead, if S = {S}, the complete characterization of all surjective strategy-proof SCFs f : Ω n S A is given by Moulin s characterization in (2.1) with the additional requirement that a T = ρ 1 (S) for some T N and a = ρ m (S) to ensure surjectivity. Remark 3. As a special case, Theorem 1 covers the Gibbard-Satterthwaite theorem because if S contains all possible orderings of A, then Ω n S equals the unrestricted domain Σ A and (A L S,AM S,AR S ) = (,A, ), so by (3.2), a strategy-proof surjective SCF on Σ A must be dictatorial. Furthermore, from the perspective of the literature on maximal domains admitting non-dictatorial strategy-proof SCFs (see, e.g., Barberà (2011, p. 817)), Theorem 1 shows how the possibility result of anonymous 14

15 strategy-proof social choice in Moulin (1980) turns gradually into the dictatorial result of the Gibbard-Satterthwaite theorem when the preference domain is extended successively from a single-peaked domain to the unrestricted domain. Remark 4. Theorem 1 characterizes the strategy-proof SCFs on multiple singlepeaked domains under the additional requirement of surjectivity. But Theorem 1 can also be used to derive the functional form of an arbitrary strategy-proof SCF on Ω S, i.e., also when range restrictions are allowed. To see this, suppose that f : Ω n S A is a strategy-proof SCF with range R f = B A. Note first that f (P 1,...,P n ) must only depend on how the preferences in (P 1,...,P n ) rank the alternatives in B because if P i,p i Ω S are such that P i B = P i B, then P i and P i rank f (P i,p i ) and f (P i,p i) in the same way for all P i ΩS n 1, so strategy-proofness requires f (P i,p i ) = f (P i,p i). Hence, it is possible to construct a well-defined SCF ˆf : (Ω S ) n B B by assigning to ( P 1,..., P n ) (Ω S ) n B the outcome f ( P 1,..., P n ) = f (P 1,...,P n ), where (P 1,...,P n ) Ω n S is some profile with P i B = P i for all i N. Then, ˆf is (1) defined on a multiple single-peaked domain because (Ω S ) B = Ω (S B ), (2) strategy-proof because f is strategy-proof, and (3) surjective by construction. Then, ˆf can be characterized by Theorem 1, using the maximal common decomposition of B with respect to S B. In a final step, the functional form of f is obtained via the relationship f (P 1,...,P n ) = ˆf (P 1 B,...,P n B ) for all (P 1,...,P n ) Ω n S. Remark 5. The ordering S S used in the construction of Ā M, the projections in (3.2), and the minmax rules in (3.3) can be replaced by any other ordering S S, holding everything else in Theorem 1 fixed, because such a replacement will only affect the representation of f, but not f itself. The reason for this is that the functional form of f depends only on S on those parts of A where all orderings in S coincide. This is obvious in (3.2), and in (3.3), the choice of for example G L is only of importance for f if G L (P î ) Ā L, and if this is the case, then G L (P î ) will not change if S is replaced by some other ordering S S since all orderings in S coincide on Ā L (a similar statement applies for G R ). However, replacing S by some other S S may change the choices of G L and G R on A M S, where the orderings in S differ. Note also that neither G L nor G R are in general strategy-proof, because a generalized median voter scheme with respect to S is not necessarily a generalized median voter scheme with respect to some other S S, and will hence be manipulable when preferences from Ω S are admitted. Remark 6. A similar characterization of strategy-proof SCFs as a combination of dictatorial rules and generalized median voter schemes is obtained in Schummer 15

16 and Vohra (2002). Their result differs, however, both motivationally and formally from Theorem 1: Schummer and Vohra consider the problem of choosing a strategyproof location on a network, which is represented as a continuous graph in a Euclidean space, when the individuals have quadratic single-peaked preferences, i.e., the ranking of an alternative depends only on the Euclidean distance to the top alternative. In this setup, Schummer and Vohra show that a strategy-proof SCF must be dictatorial on the circular parts and a generalized median voter scheme on the cycle-free parts of the graph. 4 Preferences over parties In this section, we show in the framework of a spatial voting model that preferences over parties can be represented by multiple single-peaked domains. Thereby, our starting point is the assertion that the political sphere normally covers several political dimensions such as economic policy, civil rights policy, security policy, etc., and most political parties take a position in every dimension. Identifying every party with its positions in the different political dimensions, parties can be considered as multi-dimensional alternatives. In addition, the political system usually has the following two properties which are crucial for the structure of party preferences. First, in every political dimension, the possible positions a party can take can be categorized by an ideological left right scale. For example, for the political dimension of socioeconomic policy, equalizing redistribution of income is a typical leftist position, while restrictive redistribution in favor of protection of private property is a typical rightist position. Similarly, concerning economic policy, an extensive regulation of markets would be classified as leftist and minimal regulatory policy as rightist. Based on these left right scales, it is reasonable that preferences are single-peaked within every political dimension, provided that all other dimensions are kept fixed. Moreover, the ideological left right scale can be used in some cases also to order combinations of positions from different dimensions. For example, the combination (high income redistribution, high market regulation) is certainly to the left of the combination (no income redistribution, no market regulation). On the other hand, the two combinations (high income redistribution, no market regulation) and (no income redistribution, high market regulation) cannot be compared unambiguously on a left right scale. For the structure of preferences, we assume that whenever a collection of policy combinations can be ordered unambiguously on a left right scale, then preferences are single-peaked on this collection. Second, 16

17 only a small proportion of all possible policy combinations is normally represented by parties, i.e., the party system is sparse in the multi-dimensional set of all possible political alternatives. This means that preferences over policy combinations are not identical to preferences over parties, but the latter are obtained by restricting the former to the prevailing party system. We now formalize the reasoning from the preceding paragraph. Thereby, we restrict ourselves to the case when there are exactly two political dimensions along which parties locate their positions; this will be sufficient to gain insight into the structure of preferences over parties, while adding further dimensions would only make the analysis more technical, but lead to the same qualitative results. So let X 1 = {x 11,...,x 1m1 } and X 2 = {x 21,...,x 2m2 } be two finite sets of m 1 respectively m 2 alternatives, which are referred to as the two political dimensions. The sets X 1 and X 2 are equipped with left right orderings S X1 and S X2, respectively. The set of political alternatives is the product set X = X 1 X 2. The two orderings S X1 and S X2 induce a (partial) ordering S X on X by (x 1,x 2 )S X (y 1,y 2 ) def [x 1 S X1 y 1 and x 2 S X2 y 2 ] or [x 1 S X1 y 1 and x 2 = y 2 ] or [x 1 = y 1 and x 2 S X2 y 2 ]. This means that x = (x 1,x 2 ) is to the left of y = (y 1,y 2 ) according to S X if at least one coordinate of x is to the left of the corresponding coordinate of y, and the other coordinate of x either coincides with or is also to the left of the corresponding coordinate of y. Note that S X is transitive and asymmetric, but not complete in all non-trivial cases (i.e., when m 1 2 and m 2 2), because if x 1 S X1 y 1 (where x 1,y 1 X 1 ) and x 2 S X2 y 2 (where x 2,y 2 X 2 ), then neither (x 1,y 2 )S X (x 2,y 1 ) nor (x 2,y 1 )S X (x 1,y 2 ). Denote the betweenness relation associated with S X by B X, i.e., for all x,y,z X, we have B X (x,y,z) if and only if xs X ys X z or zs X ys X x. Definition 2. A complete, transitive, and asymmetric preference P on X is singlepeaked with respect to B X if B X (x,y,z) implies ypx or ypz for all x,y,z X. The domain of all P Σ X that are single-peaked with respect to B X is called the singlepeaked domain with respect to B X, and it is denoted by Γ X. The domain Γ X consists of preferences with a variety of different shapes, which is illustrated by the following example. Example 5. Let X 1 = {x 1,x 2,x 3 } and X 2 = {y 1,y 2,y 3 }, and assume that X 1 is ordered by S X1 = [x 1 x 2 x 3 ], while X 2 is equipped with the ordering S X2 = [y 1 y 2 y 3 ]. 17

18 Consider the following three preferences on X = X 1 X 2, where the number corresponding to the pair (x i,y j ) indicates the rank r k (P) of (x i,y j ) in P: P 1 x 1 x 2 x 3 y y y P 2 x 1 x 2 x 3 y y y P 3 x 1 x 2 x 3 y y y It is straightforward to check that these three preferences are single-peaked with respect to B X. Moreover, the shape of these preferences can be interpreted in different ways: Preference P 1 can be represented by the utility function u(x i,y j ) = 3i j for (x i,y j ) X, which means that P 1 can be thought of as being represented by a utility plane that attains its maximum on X at (x 1,y 1 ) from which it slopes downward in both directions towards (x 3,y 3 ). Preference P 2 has its top at the center (x 2,y 2 ) X and can be represented by a hat-shaped utility function on X. On the contrary, preference P 3 has rather a saddle shape since P 3 is -shaped on the line from (x 1,y 1 ) to (x 3,y 3 ), but -shaped from (x 1,y 3 ) to (x 3,y 1 ). A party system on X is a subset A X. For a given party system A, the domain Γ X induces a domain Γ A (Γ X ) A of preferences over parties. Further, S X induces a left right ordering (S X ) A on A. In general, (S X ) A will not be complete, but by the Szpilrajn extension theorem, there is always at least one complete ordering S Σ A that is compatible with (S X ) A. 6 The set S A = {S Σ A ; S is compatible with (S X ) A } is thus non-empty, and S A is called the set of all party orderings of A. Example 6. Suppose that X 1 = {x 1,x 2,x 3,x 4 } and X 2 = {y 1,y 2,y 3,y 4 } are two political dimensions, equipped with the left right orderings S X1 = [x 1 x 2 x 3 x 4 ] and S X2 = [y 1 y 2 y 3 y 4 ], and X = X 1 X 2 is the set of political alternatives. If the party system on X is A = {(x 1,y 1 ),(x 2,y 2 ),(x 3,y 3 ),(x 4,y 4 )}, then (x 1,y 1 )S X (x 2,y 2 )S X (x 3,y 3 )S X (x 4,y 4 ), and hence the restriction (S X ) A of S X to A is complete in this case. This means that (S X ) A is the unique party ordering of A, i.e., S A = {(S X ) A }. 6 If B is a set of alternatives, and S is a transitive and asymmetric, but not necessarily complete ordering of B, then a complete, transitive, and asymmetric ordering S of B is compatible with S if bsb implies b Sb for all b,b B. The Szpilrajn extension theorem (Szpilrajn, 1930) states that for every transitive and asymmetric ordering S of B there exists at least one complete ordering S of B compatible with S. 18

19 On the other hand, if A = {(x 1,y 1 ),(x 2,y 3 ),(x 3,y 2 ),(x 4,y 4 )}, then (x 1,y 1 )S X (x 2,y 3 ), (x 1,y 1 )S X (x 3,y 2 ), (x 1,y 1 )S X (x 4,y 4 ), (x 2,y 3 )S X (x 4,y 4 ) and (x 3,y 2 )S X (x 4,y 4 ), but (x 2,y 3 ) and (x 3,y 2 ) cannot be ordered by S X. Thus, (S X ) A is incomplete in this case. There are two complete orderings of A that are compatible with (S X ) A, namely S = [(x 1,y 1 ) (x 2,y 3 ) (x 3,y 2 ) (x 4,y 4 )] and S = [(x 1,y 1 ) (x 3,y 2 ) (x 2,y 3 ) (x 4,y 4 )], and the set of party orderings of A is S A = {S,S }. The structure of the domain of preferences over parties is described by the following result. Proposition 1. Γ A = Ω SA. The main message of Proposition 1 is that if voters preferences over political alternatives are single-peaked with respect to B X, then the ordinary single-peaked domains are in general too narrow to represent preferences over parties, but the domain of preferences over parties is instead a multiple single-peaked domain. Furthermore, the orderings needed to generate the multiple single-peaked domain are given by S A. We conclude this section by comparing our results with another approach to multi-dimensional single-peaked preferences, which has been very influential in the literature. Definition 2 can be considered as a generalization of single-peaked preferences to a multi-dimensional setup based on a local characterization of singlepeakedness in the spirit of Sen (1966), since only one triple from X is considered at a time. In one dimension, single-peakedness can equivalently be described by the property that alternatives become less preferred the further one moves away in either direction from the top alternative of a preference, which is the definition used in Black (1948). A generalization of this global characterization of singlepeakedness to multi-dimensional sets of alternatives has been introduced in Barberà et al. (1993). Using a geometric betweenness relation 7, in contrast to the ideological betweenness relation in our approach, Barberà et al. (1993) define a preference P Σ X to be generalized single-peaked if xpy whenever x is between r 1 (P) and y. Denote the domain of all P Σ X which are single-peaked in this sense by ˆΓ X. 7 Formally, the betweenness relation in Barberà et al. (1993) is defined as follows: Assuming that X 1 and X 2 are integer intervals, Barberà et al. define first the L 1 -norm of x = (x 1,x 2 ) X 1 X 2 by x = x 1 + x 2. Given the L 1 -norm, y X is between x X and z X if x z = x y + y z. Geometrically, this means that y belongs to the minimal box containing x and z. 19