Characterization of Domains Admitting Strategy-Proof and Non-Dictatorial Social Choice Functions: A Reappraisal Francesca Busetto∗ Giulio Codognato† February 2007

Abstract We show, with an example, that the theorem on the characterization of domains admitting strategy-proof and non-dictatorial social choice functions by Kalai and Muller (1977) does not hold when the set of alternatives is infinite. Then, we introduce a class of social choice functions for which it is possible to restore the validity of the theorem also in this case. Journal of Economic Literature Classification Number: D71.

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Introduction

Kalai and Muller (1977) contains the first published characterization of domains of preferences for which there exists a strategy-proof and non-dictatorial social choice function.1 In particular, these authors showed that the domains ∗

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degli Studi di Udine, Via Tomadini 30, 33100 Udine, Italy. 1 Maskin (1976) proposed a similar characterization in his unpublished Ph.D. thesis.

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admitting strategy-proof and non-dictatorial social choice functions coincide with those where there exist non-dictatorial social welfare functions. We provide an example which shows that Kalai and Muller’s characterization theorem does not hold when the set of alternatives in infinite. Then, we define a class of social choice functions for which the theorem is true with both finite and infinite sets of alternatives. Moreover, we state it in terms of a notion of rationality which is the natural extension of individual rationality to social choice functions. Incidentally, we amend Kalai and Muller’s proof eliminating a logical imprecision. Finally, we extend Kalai and Muller’s analysis establishing the conditions under which there exists a one-to-one correspondence between strategy-proof, non-dictatorial social choice functions and non-dictatorial social welfare functions. In particular, we stress the role of the hypothesis that a social welfare function is monotone.

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Notation and definitions

Given a set S, let |S| denote the cardinality of S. Let I be any initial finite subset of the natural numbers with at least two elements and let |I| be denoted by n. Elements of I are called individuals. Let A be a set such that |A| ≥ 3. Elements of A are called alternatives. Let A be the set of all the non-empty subsets of A. Elements of A are called feasible sets. Let P be the set of all the complete, transitive, and antisymmetric binary relations on A. Elements of P are called preference orderings. Let Ω denote a nonempty subset of P. Elements of Ω are called admissible preference orderings. Given a feasible set X ∈ A, two admissible preference orderings P, P 0 ∈ Ω are said to agree on X whenever, for all x, y ∈ X, xP y if and only if xP 0 y. Let Ωn denote the n-fold cartesian product of Ω. Elements of Ω are called preference profiles. Given P ∈ Ωn and Pi0 ∈ Ω, P \ Pi0 denotes the preference profile (P1 , . . . , Pi−1 , Pi0 , Pi+1 , . . . , Pn ). Given X ∈ A, two preference profiles P, P0 ∈ Ωn are said to agree on X if, for all i ∈ I, Pi and Pi0 agree on X. A Social Welfare Function (SWF) on Ω is a function w : Ωn → P. 2

w is Pareto Optimal (PO) if, for all P ∈ Ωn and for all x, y ∈ A, xPi y, for all i ∈ I, implies xw(P)y. w is Independent of Irrelevant Alternatives (IIA) if, for all P, P0 ∈ Ωn and for all x, y ∈ A, P, P0 agree on {x, y} implies w(P) and w(P0 ) agree on {x, y}. w is Monotone (M) if, for all P, P0 ∈ Ωn and for all x, y ∈ A, xw(P)y implies xw(P0 )y whenever xPi y implies xPi0 y, for all i ∈ I. w is dictatorial if there exists an individual d ∈ I such that, for all P ∈ Ωn and for all x, y ∈ A, xPd y implies xw(P)y. w is Non-Dictatorial (ND) if it is not dictatorial. A Social Choice Function (SCF) on Ω is a function f : Ωn × A → A such that, for all P ∈ Ωn and for all X ∈ A, f (P, X) ∈ X.2 f is Pareto Optimal (PO) if, for all P ∈ Ωn , for all X ∈ A, and for all x, y ∈ X, xPi y, for all i ∈ I, implies f (P, X) 6= y. f is Independent of Non-Optimal Alternatives (INOA) if, for all P ∈ Ωn and for all X, Y ∈ A such that Y ⊂ X, f (P, X) ∈ Y implies f (P, X) = f (P, Y ).3 f is manipulable by an individual i ∈ I at P ∈ Ωn and X ∈ A via Pi0 ∈ Ω if f (P \ Pi0 , X)Pi f (P, X). f is manipulable if it is manipulable by some individual i ∈ I at some P ∈ Ωn and X ∈ A via some Pi0 ∈ Ω. f is Strategy-Proof (SP) if it is not manipulable. f is dictatorial if there exists an individual d such that, for all P ∈ Ωn , X ∈ A, f (P, X)Pd y, for all y ∈ X. f is Non-Dictatorial (ND) if it is not dictatorial. A SWF on Ω, w, is said to underly a SCF on Ω, f , if, for all P ∈ Ωn and for all X ∈ A, f (P, X)w(P)y, for all y ∈ X. A SCF on Ω, f , is Rational (R) if there exists a SWF on Ω, w, which underlies it.4 2

As pointed out by Blin and Satterthwaite (1978), this definition of a SCF, due to Karni and Schmeidler (1976), is different from the definition proposed by Gibbard (1973) and Satterthwaite (1975), according to which the only argument of a SCF is represented by preference profiles. 3 The notion of a INOA social choice function was introduced by Karni and Schmeidler (1976). 4 The notion of a R social choice funcion was introduced by Blin and Satterthwaite (1978).

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Characterization theorems

We start by reproducing two theorems on non-dictatorial SWF, shown by Kalai and Muller. The first theorem states the following. Theorem 1. There exists a SWF on Ω, w0 , which is PO, IIA, and ND for n = 2 if and only if there exists a SWF on Ω, w00 , which is PO, IIA, and ND for n > 2. Kalai and Muller’s second theorem concerns the characterization of domains admitting SWF which are PO, IIA, and ND. In order to state it, we need some additional definitions. Let T = {(x, y) ∈ A×A : x 6= y}, T R = {(x, y) ∈ T : there exist no P, P 0 ∈ Ω such that xP y and yP 0 x}, and N T R = T \ T R. A set R ⊂ T is closed under decisiveness implication if, for every two pairs (x, y), (x, z) ∈ N T R, the following two conditions are true. DI1. If there exist P, P 0 ∈ Ω with xP yP z and yP 0 zP 0 x, then DI1a. (x, y) ∈ R implies that (x, z) ∈ R, DI1b. (z, x) ∈ R implies that (y, x) ∈ R. DI2. If there exists P ∈ Ω with xP yP z, then DI2a. (x, y) ∈ R and (y, z) ∈ R imply that (x, z) ∈ R, 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 ⊂ R ⊂ T , R 6= T R, R 6= T , which is closed under decisiveness implication. Then, Kalai and Muller’s second theorem can be enunciated as follows. Theorem 2. Ω is decomposable if and only if there exists a SWF on Ω, w, which is PO, IIA, and ND for n ≥ 2. We are now ready to introduce a third theorem proven by Kalai and Muller on the characterization of domains admitting SCF which are PO, INOA, SP, and ND. These authors showed that these domains are the same allowing for the existence of SWF which are PO, IIA, and ND. Theorem 3. Ω is decomposable if and only if there exists a SCF on Ω, f , which is PO, INOA, SP, and ND for n ≥ 2. Kalai and Muller’s proof of Theorem 3 proceeds as follows. They first showed that, if there exists a SCF on Ω, f , which is PO, INOA, SP, and ND for n ≥ 2, then Ω is decomposable. To this end, they defined a 4

function w : Ωn → P such that, for all P ∈ Ωn and for all x, y ∈ A, xw(P)y if and only if f (P, {x, y}) = x. Then, they showed that w is a SWF on Ω which is PO, IIA, and ND for n ≥ 2. It follows, by Theorem 2, that Ω is decomposable. As regards Kalai and Muller’s proof that the SWF w introduced above is IIA, we have to point out a logical problem. These authors showed this statement by contradiction, using the argument that, if w is not IIA, then there exist two profiles P, P0 ∈ Ωn , two alternatives x, y ∈ A, and an individual i ∈ I such that Pj = Pj0 , for j ∈ I with j 6= i, P and P0 agree on {x, y} but w(P) and w(P0 ) do not agree on {x, y}. However, this is not the right negation of IIA, which is in fact the following. If w is not IIA, then there exist two profiles P, P0 ∈ Ωn and two alternatives x, y ∈ A such that P and P0 agree on {x, y} but w(P) and w(P0 ) do not agree on {x, y}. We provide now an amended version of this part of Kalai and Muller’s proof, using an argument proposed by Karni and Schmeidler (1976). We shall show that, since the SCF on Ω, f , is SP, then f (P, {x, y}) = f (P0 , {x, y}), for all P, P0 ∈ Ωn and for all x, y ∈ A such that P and P0 agree on {x, y}. We show this last statement by contradiction. Suppose that there exist two preference profiles P, P0 ∈ Ωn and two alternatives x, y ∈ A such that P and P0 agree on {x, y}, f (P, {x, y}) = x, and f (P0 , {x, y}) = y. Therefore, there 0 exists an individual i ∈ I such that f (P1 , . . . , Pi , Pi+1 , . . . , Pn0 , {x, y}) = x 0 0 and f (P1 , . . . , Pi−1 , Pi , . . . , Pn , {x, y}) = y. If yPi x, then f is manipulable 0 by i at (P1 , . . . , Pi , Pi+1 , . . . , Pn0 ) and {x, y} via Pi0 , a contradiction. If xPi y, then f is manipulable by i at (P1 , . . . , Pi−1 , Pi0 , . . . , Pn0 ) and {x, y} via Pi , a contradiction. This implies that w is IIA. Let us consider now the proof that, if Ω is decomposable, then there exists a SCF on Ω, f , which is PO, INOA, SP, and ND for n ≥ 2. In this part, Kalai and Muller used the following result, which we introduce as a proposition. Proposition 1. If there exists a SCF on Ω, f 0 , which is PO, INOA, SP, and ND for n = 2, then there exists a SCF on Ω, f 00 , which is PO, INOA, SP, and ND for n > 2. Proof. Let f 0 be a SCF on Ω which is PO, INOA, SP, and ND for n = 2. Let f 00 : Ωn × A → A be a function such that, for each P ∈ Ωn and for each X ∈ A, f 00 (P1 , . . . , Pn , X) = f 0 (P1 , P2 , X). It is straightforward to verify that f 00 is a SCF on Ω which is PO, INOA, SP, and ND for n > 2. 5

Kalai and Muller’s argument proceeds as follows. As Ω is decomposable, by Theorem 2 there exists a SWF on Ω, w0 , which is PO, IIA, and ND for n = 2. Then, these authors defined a function f 0 : Ω2 × A → A such that, for each P ∈ Ω2 and for each X ∈ A, f 0 (P, X)w0 (P)y, for all y ∈ X. They showed that f 0 is a SCF on Ω which is PO, INOA, SP, and ND for n = 2. Therefore, by Proposition 1, this implies that there exists a SCF on Ω, f 00 , which is PO, INOA, SP, and ND for n > 2 Nevertheless, we can show that the statement in Theorem 3, as it was formulated by Kalai and Muller, is false. In fact, we shall provide an example which shows that, if Ω is decomposable, there may not exist any SCF on Ω, f , which is PO, INOA, SP, and ND for n ≥ 2 when |A| = ∞. Before presenting the example, we introduce a further proposition. Proposition 2. If there exists a SCF on Ω, f , which is PO, INOA, SP, and ND, then there exists a SWF on Ω, w, which is PO, IIA, ND and which underlies f . Proof. Let f be a SCF on Ω which is PO, INOA, SP, and ND and, as in the proof of Theorem 3, let w : Ωn → P be a function such that, for all P ∈ Ωn and for all x, y ∈ A, xw(P)y if and only if f (P, {x, y}) = x. We already know that w is PO, IIA, and ND. Suppose now that w does not underly f . Then, there exist P ∈ Ωn , X ∈ A, and y ∈ X such that yw(P)x, where x = f (P, X). But, since f is INOA, we have f (P, {x, y}) = x, a contradiction. This implies that w underlies f . We can now provide the example which contradicts the statement of Theorem 3 when |A| = ∞. Example. Let |A| = ∞ and Ω = {P, P 0 }, where P is such that, for all x ∈ A, there exists y ∈ A which satisfies yP x, and P 0 6= P . Then, Ω is decomposable and there exists no SCF on Ω, f , which is PO, INOA, SP, and ND. Proof. We first show that there exists a SWF on Ω, w0 , which is PO, IIA, and ND for n = 2. Let w0 : Ω2 → P be a function such that w0 (P, P ) = P , w0 (P, P 0 ) = P , w0 (P 0 , P ) = P , and w0 (P 0 , P 0 ) = P 0 . It is immediate to verify that w0 is a SWF on Ω which is PO, IIA, and ND for n = 2. Then, Theorem 2 implies that Ω is decomposable. Moreover, Theorem 1 implies that there exists a SWF on Ω, w00 , which is PO, IIA, and ND for n > 2. Now, let P denote a preference profile such that Pi = P , for all i ∈ I, and let W 6

be the set - which we have shown to be nonempty - of all the SWF on Ω which are PO, IIA, and ND. Then, PO implies that, for all w ∈ W and for all x ∈ A, there exists y ∈ A such that yw(P)x. Therefore, there exists no w ∈ W which can underly any f . But then, the contrapositive of Proposition 2 implies that there exists no SCF on Ω which is PO, INOA, SP and ND. The Example shows that Theorem 2 and Theorem 3 are, in a sense, asymmetric, as the former holds when |A| ≤ ∞ whereas the latter holds only when |A| < ∞. In order to restore the validity of Theorem 3 when |A| = ∞, we propose the following new definition of a SCF. Let F denote the set of all the nonempty and finite subsets of A. A SCF on Ω is a function f ∗ : Ωn × F → A such that, for all P ∈ Ω and for all X ∈ F, f ∗ (P, X) ∈ X. f ∗ will inherit, mutatis mutandis - i.e., when A is replaced by F - all the properties previously introduced on f . It is straightforward to verify that Kalai and Muller’s Theorem 3 holds for f ∗ also when |A| = ∞. Moreover, by using Proposition 2 and Theorem 3 with reference to f ∗ , it is immediate to show the following result, which establishes a logical equivalence between the notions of INOA and R. Proposition 3. A SCF on Ω, f ∗ , is INOA if and only if it is R. Proposition 3 allows us to state a version of Theorem 3 for |A| ≤ ∞ which generalizes the notion of individual rationality to a SCF. This gives a formal content to what verbally asserted by Kalai and Muller with reference to their Theorem 3 (see p. 457): “Our definition of a non-manipulable voting procedure assumes a certain rationality condition.”5 Theorem 30 . Ω is decomposable if and only if there exists a SCF on Ω, f ∗ , which is PO, R, SP, and ND. We can now extend Kalai and Muller’s analysis, showing some further results on SWF which are not only PO, IIA, and ND but also M. Following Maskin (1976), we first prove that the decomposability of Ω also assures the existence of a SWF on Ω which is PO, IIA, M, and ND. 5

We refer to Blin and Satterthwaite (1978) for a deep discussion of the rationality of a SCF as a generalization of individual rationality.

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Proposition 4. If Ω is decomposable, then there exists a SWF on Ω, w, which is PO, IIA, M, and ND for n ≥ 2. Proof. Let Ω be decomposable. Then, by Theorem 2, there exists a SWF on Ω, w0 , which is PO, IIA, and ND for n = 2. Suppose that w0 is not M. Then, there exist P, P0 ∈ Ωn and x, y ∈ A which satisfy xPi0 y, for all i ∈ I such that xPi y, xw0 (P)y, and yw0 (P0 )x. Consider first the case where xPi y, for all i ∈ I. Then, xw0 (P0 )y since w0 is PO, a contradiction. Consider now the case where xP1 y and yP2 x. Then, either xP10 y and yP20 x, or xP10 y and xP20 y. In the first case, xw0 (P0 )y since w0 is IIA whereas, in the second case, xw0 (P0 )y since w0 is PO. In both cases, we have a contradiction. Finally, the same contradiction arises in the case where yP1 x and xP2 y. The proof that w0 is M is then concluded. Now, let w : Ωn → P be a function such that, for each P ∈ Ωn , w(P1 , . . . , Pn , X) = w0 (P1 , P2 , X). It is straightforward to verify that w is a SWF on Ω which is PO, IIA, M, and ND for n > 2. The last proposition permits us to state a logical equivalence between the existence of SWF which are PO, IIA, M, and ND and the existence of SCF which are PO, INOA (R), SP, and ND. Proposition 5. There exists a SWF on Ω, w, which is PO, IIA, M, and ND if and only if there exists a SCF on Ω, f ∗ , which is PO, INOA (R), SP, and ND. The proof is a straightforward consequence of Proposition 3, Proposition 4 and Theorem 30 . We notice that Proposition 5 holds non-vacuously if and only if Ω is decomposable. We conclude by proving that there is a one-to-one correspondence `a la Satterthwaite (see Satterthwaite (1975)) between the set of the SWF which are PO, IIA, M, and ND and the set of the SCF which are PO, INOA (R), SP, and ND. Theorem 4. (i) If w is a SWF on Ω which is PO, IIA, M, and ND, then there exists a unique SCF on Ω, f , which is PO, INOA (R), SP, and ND and which underlies it; (ii) if f ∗ is a SCF on Ω which is PO, INOA (R), SP, and ND, then there exists a unique SWF on Ω, w, which is PO, IIA, M, and ND and which underlies it. Proof. (i) Let w be a SWF on Ω which is PO, IIA, M, and ND and let f ∗ : Ωn × F → A be a function such that, for each P ∈ Ωn and for each 8

X ∈ F, f (P, X)w(P)y, for all y ∈ X. By using Theorem 3 and Proposition 3, it is immediate to show that f ∗ is PO, INOA (R), and ND. Moreover, the fact that Ω consists of complete, transitive and antisymmetric binary relations on A and F consists of all the nonempty and finite subsets of A implies straightforwardly that f ∗ is the unique SCF which w underlies. Suppose now that f ∗ is not SP. Then, there exist i ∈ I, P ∈ Ωn , X ∈ A, Pi0 ∈ Ω such that f ∗ (P\Pi0 , X)Pi f ∗ (P, X). Let f ∗ (P, X) = x and f ∗ (P\Pi0 , X) = y. Therefore, f ∗ (P, {x, y}) = x and f ∗ (P \ Pi0 , {x, y}) = y since f ∗ is INOA and xw(P)y and yw(P \ Pi0 )x since w underlies f ∗ . Consider first the case where yPi0 x. Then, xw(P \ Pi0 )y as w is IIA, a contradiction. Consider now the case where xPi0 y. Then, yw(P)x as w is M, a contradiction. We can thus conclude that f ∗ is SP. Finally, suppose that f ∗ is not ND. Then, there exists an individual d such that, for all P ∈ Ωn and x, y ∈ A, xPd y implies f ∗ (P, {x, y}) = x and, since w underlies f ∗ , we have xw(P)y, a contradiction. This implies that f ∗ is ND. (ii) Let f ∗ be a SCF on Ω which is PO, INOA (R), SP, and ND and let w : Ωn → P be a function such that, for all P ∈ Ωn and for all x, y ∈ A, xw(P)y if and only if f ∗ (P, {x, y}) = x. We already know, from the proof of Proposition 2 and Theorem 3, that w is a SWF on Ω which is PO, IIA, and ND and which underlies f ∗ . Suppose that w is not the unique SWF on Ω which underlies f ∗ . Then, there exist a SWF on Ω, w0 , a preference profile P ∈ Ωn , and a pair of alternatives x, y ∈ A such that xw(P)y and yw0 (P)x. But then, the definition of w implies that yw0 (P)f (P, {x, y}), a contradiction. Hence, w is the unique SWF on Ω which underlies f ∗ . Finally, suppose that w is not M. Then, there exist P, P0 ∈ Ωn and x, y ∈ A which satisfy xPi0 y, for all i ∈ I such that xPi y, xw(P)y, and yw(P0 )x. Consider first the case where xPi0 y, for all i ∈ I. Then, xw(P0 )y as w is PO, a contradiction. Consider now the case where, for all i ∈ I, xPi0 y if and only if xPi y. Then, xw(P0 )y since w is IIA, a contradiction. Finally, consider the logical complement of the two previous cases. Let P00 be a preference profile obtained by replacing Pi with Pi0 in P for each i ∈ I such that xPi y. Then, xw(P00 )y as w is IIA. Moreover, there exists an individual i ∈ I such that yPi00 x, 00 0 00 0 , . . . , Pn00 )x , Pi0 , Pi+1 , . . . , Pn00 )y, and yw(P10 , . . . , Pi−1 , Pi00 , Pi+1 xw(P10 , . . . , Pi−1 00 0 , . . . , Pn00 , {x, y}) = x , Pi00 , Pi+1 and then, by the definition of w, f ∗ (P10 , . . . , Pi−1 00 0 , . . . , Pn00 , {x, y}) = y. But then, f ∗ is manipula, Pi0 , Pi+1 and f ∗ (P10 , . . . , Pi−1 00 00 0 0 , . . . , Pn00 ) and {x, y} via Pi0 , a contradiction. ble by i at (P1 , . . . , Pi−1 , Pi , Pi+1 The proof that w is M is then concluded.

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This theorem shows that, for every Ω ∈ P, the cardinality of the set of the SWF on Ω which are PO, IIA, M, and ND is equal to the cardinality of the set of the SCF on Ω which are PO, INOA (R), SP, and ND.

References [1] Blin J.M., Satterthwaite M.A. (1978), “Individual decisions and group decisions,” Journal of Public Economics 10, 247-267. [2] Gibbard A. (1973),“Manipulation of voting schemes: a general result,” Econometrica 41, 587-601. [3] Kalai E., Muller E. (1977), “Characterization of domains admitting nondictatorial social welfare functions and nonmanipulable voting procedures,” Journal of Economic Theory 16, 457-469. [4] Karni E., Schmeidler D. (1976), “Independence of nonfeasible alternatives, and independence of nonoptimal alternatives,” Journal of Economic Theory 12, 488-493. [5] Maskin E. (1976), Social choice on restricted domains, Ph.D. Thesis, Harvard University. [6] Satterthwaite M.A. (1975), “Strategy-proofness and Arrow’s conditions: existence and correspondence theorems for voting procedures and social welfare functions,” Journal of Economic Theory 10, 187-217.

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