Nondictatorial Arrovian Social Welfare Functions and Integer Programs

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Nondictatorial Arrovian Social Welfare Functions and Integer Programs Francesca Busetto∗ Giulio Codognato† December 2010

Abstract We provide a respecification of an integer program characterization of Arrovian social welfare functions introduced by Sethuranam et al. (2003). By exploiting this respecification, we give a new and simpler proof of Theorem 2 in Kalai and Muller (1977). Journal of Economic Literature Classification Number: D71.

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Introduction

In two pathbreaking papers, Sethuraman. et al. ((2003), (2006)) proposed a characterization of Arrovian social welfare functions in terms of an integer program. As remarked by these authors, integer programming is a powerful analytical tool, which makes it possible to derive in a systematic and simple way many of the known results on those functions. In particular, it permits one to reconsider the fundamental issues concerning the characterization of domains admitting nondictatorial Arrovian social welfare functions. Kalai and Muller (1977) provided the first complete characterization of these domains. In their Theorem 1, they showed that there exists a n-person nondictatorial Arrovian social welfare function for a given domain if and only if there exists a 2-person nondictatorial Arrovian social welfare function for ∗

Dipartimento di Scienze Economiche, Universit` a degli Studi di Udine, Via Tomadini 30, 33100 Udine, Italy. † Dipartimento di Scienze Economiche, Universit` a degli Studi di Udine, Via Tomadini 30, 33100 Udine, Italy.

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the same domain. In their Theorem 2, they gave the domain characterization, based on the notions of decisiveness implication and decomposability. Taking inspiration from Kalai and Muller, Sethuraman et al. (2003) built up an integer program which incorporates a reformulation of the condition of decisiveness implication. By using this integer program, they provided a simplified version of Kalai and Muller’s Theorem 1. In this paper, we proceed along the way opened by Sethuraman et al. (2003). We propose an amended version of the integer program they used to reformulate Kalai and Muller’s Theorem 1, as we show that it is not completely specified and contains logically redundant constraints. Then, by exploiting our integer program, we give a new and much simpler proof of Kalai and Muller’s Theorem 2. We do this in two steps. We first show that Kalai and Muller’s decisiveness implication condition itself contains logical redundancies, so we propose a simpler definition of decomposability which eliminates them. This enables us to prove that decomposable domains admit nondictatorial solutions to our simplified integer program. Then, Kalai and Muller’s Theorem 2 can be straightforwardly obtained as a corollary of this theorem.

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Notation and definitions

Let E be any initial finite subset of the natural numbers with at least two elements and let |E| be the cardinality of E, denoted by n. Elements of E are called agents. Let E be the collection of all subsets of E. Given a set S ∈ E, let S c = E \ S. Let A be a set such that |A| ≥ 3. Elements of A are called alternatives. Let A2 denote the set of all ordered pairs of alternatives. Let Σ be the set of all the complete, transitive, and antisymmetric binary relations on A, called preference orderings. Let Ω denote a nonempty subset of Σ. An element of Ω is called admissible preference ordering and is denoted by p. We write xpy if x is ranked above y under p. A pair (x, y) ∈ A2 is called trivial if xpy, for all p ∈ Ω. Let T R denote the set of trivial pairs. A pair (x, y) ∈ A2 is called vacuously trivial if ypx, for all p ∈ Ω. Let V T R denote the set of vacuously trivial pairs. A pair (x, y) ∈ A2 is called nontrivial if it is neither trivial nor vacuously trivial.

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Let N T R denote the set of nontrivial pairs.1 Let Ωn denote the n-fold Cartesian product of Ω. An element of Ωn is called preference profile and is denoted by P = (p1 , p2 , . . . , pn ), where pi is the preference ordering of individual i ∈ E. A Social Welfare Function (SWF) on Ω is a function f : Ωn → Σ. A SWF on Ω, f , satisfies Pareto Optimality (PO) if, for all (x, y) ∈ A2 and for all P ∈ Ωn , xpi y, for all i ∈ E, implies xf (P)y. A SWF on Ω, f , satisfies Independence of Irrelevant Alternatives (IIA) if, for all (x, y) ∈ N T R and for all P, P0 ∈ Ω, xpi y if and only if xp0i y, and, ypi x if and only if yp0i x, for all i ∈ E, implies, xf (P)y if and only if xf (P0 )y, and, yf (P)x if and only if yf (P0 )x. An Arrovian Social Welfare Function (ASWF) on Ω is a SWF on Ω, f , which satisfies PO and IIA. An ASWF on Ω, f , is dictatorial if there exists an individual j ∈ E such that, for all (x, y) ∈ N T R and for all P ∈ Ωn , xpj y implies xf (P)y. f is nondictatorial if it is not dictatorial. Given a pair (x, y) ∈ A2 and a set S ∈ E, let dS (x, y) denote a variable such that dS (x, y) ∈ {0, 1}. An Integer Program (IP) on Ω consists of a set of linear constraints on variables dS (x, y), for all (x, y) ∈ A2 and for all S ∈ E. A solution to an IP on Ω, d, is a solution to its linear constraints. A solution to an IP on Ω, d, is dictatorial if there exists an individual j ∈ E such that, for all (x, y) ∈ N T R and for all S ∈ E with j ∈ S, dS (x, y) = 1. d is nondictatorial if it is not dictatorial. An ASWF on Ω, f , and a solution to an IP on Ω, d, are said to correspond if, for each (x, y) ∈ N T R and for each S ∈ E, xf (P)y if and only if dS (x, y) = 1, yf (P)x if and only if dS (x, y) = 0, for all P ∈ Ωn such that xpi y, for all i ∈ S, and ypi x, for all i ∈ S c . Finally, consider the following definition, introduced by Kalai and Muller (1977) to characterize the domains admitting nondictatorial ASWF. A set R ⊆ A2 is closed under decisiveness implication if, for every two pairs (x, y), (x, z) ∈ N T R, the following two conditions are satisfied. DI1. If there exist p, q ∈ Ω for which xpypz and yqzqx, then DI1a. (x, y) ∈ R implies that (x, z) ∈ R, DI1b. (z, x) ∈ R implies that (y, x) ∈ R. DI2. If there exists a preference ordering p ∈ Ω for which xpypz, then DI2a. (x, y) ∈ R and (y, z) ∈ R imply that (x, z) ∈ R, 1

We adopt the convention that all pairs (x, x) ∈ A2 are trivial.

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DI2b. (z, x) ∈ R implies that (y, x) ∈ R or (z, y) ∈ R. Ω is decomposable if there exists a set R such that (T R ∪ V T R) $ R $ A2 , which is closed under decisiveness implication.

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Arrovian social welfare functions and integer programs

The first formulation of an IP on Ω was proposed by Sethuraman et al. (2003). It consists of the following set of constraints, which we will call IP0: dS (x, y) = 1,

(i)

for all (x, y) ∈ T R and for all S ∈ E such that S 6= ∅; dE (x, y) = 1,

(ii)

dS (x, y) + dS c (y, x) = 1,

(iii)

for all (x, y) ∈ A2 ; for all (x, y) ∈ N T R and for all S ∈ E; dA∪U ∪V (x, y) + dB∪U ∪W (y, z) + dC∪V ∪W (z, x) ≤ 2,

(iv)

for all triples of distinct alternatives x, y, z ∈ A, and for all disjoint and possibly empty sets A, B, C, U, V, W ∈ E, whose union includes all agents and which satisfy the following conditions (hereafter referred to as Conditions (∗)): A 6= ∅ only if there exists p ∈ Ω such that xpzpy, B 6= ∅ only if there exists p ∈ Ω such that ypxpz, C 6= ∅ only if there exists p ∈ Ω such that zpypx, U 6= ∅ only if there exists p ∈ Ω such that xpypz, V 6= ∅ only if there exists p ∈ Ω such that zpxpy, W 6= ∅ only if there exists p ∈ Ω such that ypzpx. This formulation of an IP turns out to be not completely specified. In particular, it does not explicitly assign a value to variables dS (x, y), for all (x, y) ∈ V T R and for all S ∈ E. On the other hand, such specification is needed, for instance, in the proof of Claim 2 in Sethuraman et al. (2003), 4

where a zero value must be implicitly associated with this kind of variables. Moreover, IP0 does not explicitly specify a value for d∅ (x, y), for all (x, y) ∈ T R, whereas a zero value is implicitly associated by Sethuraman et al. (2003) with these variables in various arguments of their paper. We propose here a slightly amended version of IP0, which overcomes these problems without substantially affecting the results obtained by Sethuraman et al. (2003). It consists of the following set of constraints, which we will call IP1: dS (x, y) = 1, (1) and for all (x, y) ∈ T R and for all S ∈ E such that S 6= ∅; d∅ (x, y) = 0,

(2)

dS (x, y) = 0,

(3)

for all (x, y) ∈ T R; for all (x, y) ∈ V T R and for all S ∈ E; dE (x, y) = 1,

(4)

dS (x, y) + dS c (y, x) = 1,

(5)

for all (x, y) ∈ N T R; for all (x, y) ∈ N T R and for all S ∈ E; dA∪U ∪V (x, y) + dB∪U ∪W (y, z) + dC∪V ∪W (z, x) ≤ 2,

(6)

for all triples of distinct alternatives x, y, z ∈ A and for all disjoint and possibly empty sets A, B, C, U, V, W ∈ E whose union includes all agents and which satisfy Conditions (∗). Sethuraman et al. (2003) showed that there exists a one-to-one correspondence between the set of the ASWF on Ω and the set of the solutions d to IP0 on Ω. The following theorem establishes an analogous result for IP1. We omit the proof here, as it is obtained by simply adapting to our respecification the arguments used by Sethuraman et al. (2003). Theorem 1. Given an ASWF on Ω, f , there exists a unique solution to IP1 on Ω, d, which corresponds to f . Given a solution to IP1 on Ω, d, there exists a unique ASWF on Ω, f , which corresponds to d. Taking inspiration from Kalai and Muller (1977), Sethuraman et al. (2003) built up a further IP on Ω - which we will call IP2 - where constraint 5

(iv) in IP0 (and thus constraint (6) in our respecification) is replaced by the following set of constraints: dS (x, y) ≤ dS (x, z),

(7)

dS (z, x) ≤ dS (y, x),

(8)

for all triples of distinct alternatives x, y, z ∈ A and for all S ∈ E such that there exist p, q ∈ Ω which satisfy xpypz and yqzqx; dS (x, y) + dS (y, z) ≤ 1 + dS (x, z),

(9)

dS (z, y) + dS (y, x) ≥ dS (z, x),

(10)

for all triples of distinct alternatives x, y, z ∈ A and for all S ∈ E such that there exists p ∈ Ω which satisfies xpypz. Constraints (7) and (8) translate, in terms of variables ds (x, y), Kalai and Muller’s conditions DI1a and DI1b. In their Claim 1, Sethuraman et al. (2003) showed that they are special cases of (iv) (and, thus, of (6)). Constraints (9) and (10) translate Kalai and Muller’s conditions DI2a and DI2b. In their Claim 2, Sethuraman et al. (2003) showed that also these constraints are special cases of (iv) (and of (6)). Now, we will prove that the set of constraints (7)-(10) exhibits problems of logical dependence. More precisely, the following proposition shows that one of the constraints (7) and (8) is redundant. Proposition 1. d is a solution to (1)-(5) and (7) if and only if it is a solution to (1)-(5) and (8). Proof. Let d be a solution to (1)-(5) and (7). Let x, y, z ∈ A be a triple of distinct alternatives. Suppose that there exist p, q ∈ Ω for which xpypz and yqzqx and that dS (z, x) > dS (y, x), for some S ∈ E. Then, dS (z, x) = 1, dS (y, x) = 0. But then, dS c (x, z) = 0, dS c (x, y) = 1. This implies that dS c (x, y) > dS c (x, z), a contradiction. Therefore d is a solution to (1)-(5) and (8). Conversely, let d be a solution to (1)-(5) and (8). Let x, y, z ∈ A be a triple of distinct alternatives. Suppose that there exist p, q ∈ Ω for which xpypz and yqzqx and that dS (x, y) > dS (x, z), 6

for some S ∈ E. Thus, dS (x, y) = 1, dS (x, z) = 0. But then, dS c (y, x) = 0, dS c (z, x) = 1. This implies that dS c (z, x) > dS c (y, x), a contradiction. Therefore, d is a solution to (1)-(5) and (7). Moreover, the following proposition shows that one of the constraints (9) and (10) is redundant. Proposition 2. d is a solution to (1)-(5) and (9) if and only if it is a solution to (1)-(5) and (10). Proof. Let d be a solution to (1)-(5) and (9). Let x, y, z ∈ A be a triple of distinct alternatives. Suppose that there exists p ∈ Ω for which xpypz and that dS (z, y) + dS (y, x) < dS (z, x), for some S ∈ E. Thus, dS (z, y) = 0, dS (y, x) = 0, and dS (z, x) = 1. But then, dS c (y, z) = 1, dS c (x, y) = 1, and dS c (x, z) = 0. This implies that dS c (x, y) + dS c (y, z) > 1 + dS c (x, z), a contradiction. Therefore, d is a solution to (1)-(5) and (10). Conversely, let d be a solution to (1)-(5) and (10). Let x, y, z ∈ A be a triple of distinct alternatives. Suppose that there exists p ∈ Ω for which xpypz and that dS (x, y) + dS (y, z) > 1 + dS (x, z), for some S ∈ E. Thus, dS (x, y) = 1, dS (y, z) = 1, and dS (x, z) = 0. But then, dS c (y, x) = 0, dS c (z, y) = 0, and dS c (z, x) = 1. This implies that dS c (z, y) + dS c (y, x) < dS c (z, x), a contradiction. Therefore, d is a solution to (1)-(5) and (9). Proposition 1 and 2 have the consequence that we can focus on a different IP on Ω - which we will call IP3 - consisting of the logically independent constraints (1)-(5), (7), (9). We will consider now the relationships between IP1 and IP3. The following result is a straightforward consequence of Sethuraman et al.’s Claims 1 and 2, mentioned above.

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Proposition 3. If d is a solution to IP1 on Ω, then it is a solution to IP3 on Ω. Moreover, the following result shows that the converse of Proposition 3 holds - and IP1 and IP3 coincide - when n = 2. An analogous equivalence between IP0 and IP2 for n = 2 was provided, without an explicit proof, by Sethuraman et al. (2003). Proposition 4. Let n = 2. If d is a solution to IP3 on Ω, then it is a solution to IP1 on Ω. Proof. Let n = 2. Let d be a solution to IP2 on Ω. Then, d satisfies (1)-(5), (7), (9). Suppose that there is a triple of distinct alternatives x, y, z ∈ A, and disjoint and possibly empty sets A, B, C, U, V, W ∈ E whose union includes all agents and which satisfy Conditions (∗), such that dA∪U ∪V (x, y) + dB∪U ∪W (y, z) + dC∪V ∪W (z, x) = 3. The case where the set {1, 2} coincides with one of the sets A, B, C, U, V, W and the case where two of the sets A, B, C are nonempty lead to a straightforward contradiction. Therefore, consider the following case: A 6= ∅, B = ∅, C = ∅. Suppose, without loss of generality, that A = {1}. Thus, we must have W = {2}. But then, there exists p ∈ Ω for which ypzpx. Hence, d{2} (y, z) + d{2} (z, x) > 1 + d{2} (y, x), contradicting (9). The cases where A = ∅, B 6= ∅, C = ∅ and A = ∅, B = ∅, C 6= ∅ lead, mutatis mutandis, to the same contradiction. Now, consider the case where U 6= ∅, V 6= ∅. Suppose, without loss of generality, that U = {1} and V = {2}. Thus, there exist p, q ∈ Ω for which xpypz and zqxqy. But then, d{2} (z, x) > d{2} (z, y), contradicting (7). The cases where V 6= ∅, W = 6 ∅ and U 6= ∅, W 6= ∅, lead, mutatis mutandis, to the same contradiction. Hence, d is a solution to IP1 on Ω.

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Nondictatorial Arrovian social welfare functions and integer programs

Kalai and Muller (1977) were the first who provided a complete characterization of the domains of preference orderings which admit nondictatorial 8

ASWF. They did this by means of two theorems. In their Theorem 1, they showed that, for a given domain, there exists a nondictatorial ASWF for n > 2, if and only if, for the same domain, there exists a nondictatorial ASWF for n = 2. In their Theorem 2, they gave the domain characterization, based on the notion of decomposability we have reproduced in Section 2. Sethuraman et al. (2003) provided a simplified version of Kalai and Muller’s Theorem 1 in terms of IP0. The arguments of their proof can be straightforwardly re-expressed in terms of IP1. Then, Sethuraman et al.’s result can be restated as follows. Theorem 2. There exists a nondictatorial solution to IP1 on Ω, d, for n = 2, if and only if there exists a nondictatorial solution to IP1 on Ω, d∗ , for n > 2. In this section, we go forward along the line opened by Sethuraman et al. (2003), giving a new and simpler proof of Kalai and Muller’s Theorem 2 in terms of IP1. In order to obtain this result, we preliminarily show the following proposition and theorem. The proposition states that Kalai and Muller’s notion of decomposability of a domain Ω can be reformulated in terms of the existence of two sets of pairs (x, y) ∈ A2 such that only conditions DI1a and DI2a need to be satisfied, for all triples of distinct alternatives (rather than for all couples of nontrivial pairs, as in the original definition). This reformulation of the notion of decomposability implies a logical redundancy of conditions DI1b and DI2, which parallels the redundancy of constraints (8) and (10), proven in Propositions 2 and 3. On the basis of this result, the theorem establishes that the decomposability of Ω is equivalent to the existence of a nondictatorial solution to IP2 on Ω, for n = 2. Proposition 5. Ω is decomposable if and only if there exist two sets R1 and R2 such that, for all (x, y) ∈ N T R, (x, y) ∈ R1 if and only if (y, x) ∈ / R2 . 2 Moreover, (T R ∪ V T R) $ Ri $ A , i = 1, 2, and, for all triples of distinct alternatives x, y, z ∈ A, DI1a and DI2a are satisfied with respect to Ri , i = 1, 2. Proof. Let Ω be decomposable. Then, there exists a set R which is closed under decisiveness implication and satisfies (T R ∪ V T R) $ R $ A2 . Let R1 = R. By Lemma 4 in Kalai and Muller, there exists a set R2 which is closed under decisiveness implication and is such that, for all (x, y) ∈ N T R, (x, y) ∈ R1 if and only if (y, x) ∈ / R2 , and (T R ∪ V T R) $ R2 $ A2 . Let x, y, z ∈ A be a triple of distinct alternatives and consider, without 9

loss of generality, the set R1 . Suppose that there exist p, q ∈ Ω for which xpypz and yqzqx. Then, (x, y) and (x, z) are nontrivial. Consequently, DI1a is satisfied for x, y, z with respect to R1 , as it is satisfied for (x, y) and (x, z) with respect to R. Now, suppose that there exists p ∈ Ω for which xpypz. Moreover, suppose that (x, y), (y, z) ∈ R1 and (x, z) ∈ / R1 . Therefore, (z, x) ∈ N T R and (z, x) ∈ R2 . If (x, y) ∈ N T R, then DI2a is satisfied for x, y, z with respect to R1 , as it is satisfied for (x, y) and (x, z) with respect to R. Consequently, (x, z) ∈ R1 , a contradiction. Now, consider the case where (x, y) ∈ T R. Then, the fact that (x, z) ∈ N T R implies that there exists q ∈ Ω for which zqxqy. Moreover, (z, y) ∈ R2 , as the triple z, x, y satisfies DI1a with respect to R2 . Therefore, (y, z) ∈ / R1 , a contradiction. Then, DI2a is satisfied for the triple x, y, z with respect to R1 . Conversely, suppose that there exist two sets R1 and R2 such that, for all (x, y) ∈ N T R, (x, y) ∈ R1 if and only if (y, x) ∈ / R2 , (T R∪V T R) $ Ri $ A2 , i = 1, 2, and, for all triples of distinct alternatives x, y, z ∈ A, DI1a and DI2a are satisfied with respect to Ri , i = 1, 2. Consider the set R1 . Then, it is immediate to verify that, for every two pairs (x, y), (x, z) ∈ N T R, DI1a and DI2a are satisfied with respect to R1 . Now, consider two pairs (x, y), (x, z) ∈ N T R and suppose that there exist p, q ∈ Ω for which xpypz and yqzqx. Moreover, suppose that (z, x) ∈ R1 and (y, x) ∈ / R1 . Then, (x, y), (x, z) ∈ R2 , as DI1a is satisfied for x, y, z with respect to R2 . But then, (z, x) ∈ / R1 , a contradiction. Therefore, DI1b is satisfied for (x, y) and (x, z) with respect to R1 . Now, consider two pairs (x, y), (x, z) ∈ N T R and suppose that there exists p ∈ Ω for which xpypz. Moreover, suppose that (z, x) ∈ R1 , (y, x), (z, y) ∈ / R1 . Then, (x, y), (y, z) ∈ R2 . Consequently, (x, z) ∈ R2 , as DI2a is satisfied for x, y, z with respect to R2 , a contradiction. Therefore, DI2b is satisfied for (x, y) and (x, z) with respect to R1 . Hence, R1 is closed under decisiveness implication. Moreover, (T R ∪ V T R) $ R1 $ A2 . Therefore, Ω is decomposable. Theorem 3. There exists a nondictatorial solution to IP3 on Ω, d, for n = 2, if and only if Ω is decomposable. Proof. Let d be a nondictatorial solution to IP3 on Ω, for n = 2. Let R1 = T R ∪ V T R ∪ {(x, y) ∈ N T R : d{1} (x, y) = 1} and R2 = T R ∪ V T R ∪ {(x, y) ∈ N T R : d{2} (x, y) = 1}. Then, for all (x, y) ∈ N T R, (x, y) ∈ R1 if and only if (y, x) ∈ / R2 , as d satisfies (5). Moreover, (T R∪V T R) $ Ri $ A2 , i = 1, 2, as d is nondictatorial. Let x, y, z ∈ A be a triple of distinct alternatives and consider, without loss of generality, the set R1 . Suppose that there exist p, q ∈ Ω for which xpypz and yqzqx. Consider (x, y) and 10

suppose that (x, y) ∈ R1 . Thus, d{1} (x, z) = 1, as d satisfies (7). But then, (x, z) ∈ R1 . Now, suppose that there exists p ∈ Ω for which xpypz. Consider (x, y), (y, z) and suppose that (x, y), (y, z) ∈ R1 . Thus, d{1} (x, z) = 1, as d satisfies (9). But then, (x, z) ∈ R1 . Therefore, DI1a and DI2a are satisfied for x, y, z with respect to Ri , i = 1, 2. Hence, Ω is decomposable, by Proposition 5. Conversely, suppose that Ω is decomposable. Then, by Proposition 5, there exist two sets R1 and R2 such that, for all pairs (x, y) ∈ N T R, (x, y) ∈ R1 if and only if (y, x) ∈ / R2 , (T R ∪ V T R) $ Ri $ A, i = 1, 2, and, for all triples of distinct alternatives x, y, z ∈ A, DI1a and DI2a are satisfied with respect to Ri , i = 1, 2. Determine d as follows. Given (x, y) ∈ T R and S ∈ E, assign a value to dS (x, y) according to (1) and (2). Given (x, y) ∈ V T R and S ∈ E, assign a value to dS (x, y) according to (3). For each (x, y) ∈ N T R, let d∅ (x, y) = 0, dE (x, y) = 1, and d{i} (x, y) = 1, if and only if (x, y) ∈ Ri ; otherwise, d{i} (x, y) = 0, for i = 1, 2. Then, d satisfies (4) and (5) as, for each (x, y) ∈ N T R, (x, y) ∈ R1 if and only if (y, x) ∈ / R2 . Consider a triple of distinct alternatives x, y, z ∈ A. Suppose that there exist p, q ∈ Ω for which xpypz and yqzqx and that dS (x, y) > dS (x, z), for some S ∈ E. Then, S 6= ∅, S 6= E, dS (x, y) = 1, and dS (x, z) = 0. Suppose, without loss of generality, that S = {1}. Therefore, (x, y) ∈ R1 and (x, z) ∈ / R1 , a contradiction. But then, d satisfies (7). Suppose that there exists p ∈ Ω for which xpypz and that dS (x, y) + dS (y, z) > 1 + dS (x, z), for some S ∈ E. Then, S 6= ∅, S 6= E, dS (x, y) = 1, dS (y, z) = 1, and dS (x, z) = 0. Suppose, without loss of generality, that S = {1}. Therefore, (x, y), (y, z) ∈ R1 and (x, z) ∈ / R1 , a contradiction. But then, d satisfies (9). Hence, d is a solution to IP3 on Ω. Moreover, d is nondictatorial as (T R ∪ V T R) $ Ri $ A, i = 1, 2. Then, Theorem 2 in Kalai and Muller (1977) can be stated as a straightforward corollary of Theorem 3. Corollary. There exists a nondictatorial ASWF on Ω, f , for n ≥ 2 if and only if Ω is decomposable. Proof. It is an immediate consequence of Propositions 3 and 4 and Theorems 1 and 2. 11

References [1] Kalai E., Muller E. (1977), “Characterization of domains admitting nondictatorial social welfare functions and nonmanipulable voting procedures,” Journal of Economic Theory 16, 457-469. [2] Sethuraman J., Teo C.P., Vohra R.V. (2003), “Integer programming and Arrovian social welfare functions,” Mathematics of Operations Research 28, 309-326. [3] Sethuraman J., Teo C.P., Vohra R.V. (2006), “Anonymous monotonic social welfare functions,” Journal of Economic Theory 128, 232-254.

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Integer Programming and Nondictatorial Arrovian ...