- Email: [email protected]
Social welfare functions generating social choice rules that are invulnerable to manipulation
Mathematical Social Sciences 51 (2006) 81 – 89 www.elsevier.com/locate/econbase
Social welfare functions generating social choice rules that are invulnerable to manipulation Donald E. Campbell a,*, Jerry S. Kelly b a
Department of Economics and Program in Public Policy, PO Box 8795, The College of William and Mary Williamsburg, VA 23187-8795, United States b Department of Economics, Syracuse University, United States Received 1 April 2003; received in revised form 1 September 2003; accepted 1 July 2005 Available online 12 October 2005
Abstract Any social welfare function f—whether transitive-valued or not—induces a social choice function g as follows: If at profile p in the domain of f there is a feasible alternative x that is bstrictly greater thanQ every other feasible alternative according to f( p) then we set g( p) = x. The domain of g is the set of all such profiles p, whether or not f( p) is acyclic. We specify a condition on a social welfare function that is necessary and sufficient for the induced social choice rule to be invulnerable to manipulation by any individual or coalition, and we generalize to an arbitrary collection of coalitions—including the family of singleton coalitions. D 2005 Elsevier B.V. All rights reserved. Keywords: Majority rule; Manipulation; Non-reversal JEL classification: D70; D71
1. Introduction Majority rule can be viewed as a social welfare function that selects a binary relation as a function of individual preferences. For a carefully chosen set of profiles, majority rule can be viewed as a social choice function that selects a single alternative as a function of individual preferences. For instance, for any set of profiles on which there is a strong Condorcet winner at each profile, the majority rule relation identifies a single bbest alternativeQ, and thus can also be *Corresponding author. Tel.: +1 757 221 2383; fax: +1 757 221 1175. E-mail address: [email protected] (D.E. Campbell). 0165-4896/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2005.07.002
82
D.E. Campbell, J.S. Kelly / Mathematical Social Sciences 51 (2006) 81–89
interpreted as a social choice rule on that set of profiles. (Alternative x is a strong Condorcet winner at profile p if for every other feasible alternative y the number of individuals who strictly prefer x to y at p exceeds the number of individuals who strictly prefer y to x.) This paper considers all social welfare functions. Each can be used in the same way to generate a social choice function on the domain of profiles at which the social ordering induced by the social welfare function has a most-preferred alternative: The social choice function selects that alternative. We present a necessary and sufficient condition, called non-reversal, for the rule to be invulnerable to manipulation by any individual or coalition, and then we generalize the condition to guarantee that no member of any given set of coalitions—for instance, the set of all singletons—can profit by deviating from truthful revelation. Although the following two families of social welfare functions are not exhaustive, they occupy a central place in axiomatic voting theory: [1] On the one hand we have the positional rules, which include the Borda rule as the salient example. A member of this family requires each individual to assign q j points to the alternative ranking in jth place in her preference ordering, with q j N q k if j b k. We could even have different q functions assigned to different individuals. Alternative x ranks above alternative y if the total point score across all individuals is algebraically higher for x than for y. (The Borda rule has each individual using the function q j = j.) Neither the Borda rule nor any other positional rule satisfies non-reversal, except when the number of alternatives is severely restricted. In other words, these procedures are quite vulnerable to manipulation. [2] On the other hand, there is the family of weighted majority rules. A particular member of this set assigns each individual i a weight r i , and alternative x ranks above y if the sum of the weights assigned to the individuals who prefer x to y exceeds the sum of the weights of the individuals who prefer y to x. (If there are n individuals and r 1 = 1 + 1 / n, with r i = 1 for all i p 1, then we have conventional majority rule with person 1 in the role of tie breaker.) Partial weighted majority rule is a special case for which, for some given J o N, the weight r j is zero for all j a J. For instance, when shareholders elect new directors to the board, each individual’s vote is proportional to the fraction of shares that he or she owns, but holders of the firm’s bonds do not get a vote—they have weight zero. The difficulty with weighted majority rule is, of course, that for many profiles there is a cycle that prevents any alternative from ranking above every other in the social ordering. However, because each member of the weighted majority family satisfies our non-reversal condition, within the set of profiles for which a weighted majority winner does emerge, no individual or coalition can benefit by deviating from truthful revelation provided that the deviation does not take us outside of the set of profiles that yields an unambiguous winner. In short, one consequence of our main result is that it makes a distinction between positional rules and weighted majority rules. We will show, however, that the class of social welfare functions satisfying non-reversal is strictly larger than the set of weighted majority rules. We begin with the folk theorem: On any domain in which each profile has a strong Condorcet winner, majority decision (viewed as a social choice rule) is invulnerable to manipulation by any individual or coalition. This claim is true regardless of the completeness and transitivity properties assumed (or not assumed) for the individual relations. (If x is a strong Condorcet winner at p then the set S of individuals who strictly prefer y to x at p cannot precipitate the selection of y by misrepresenting their preference because x defeats y when each member of S truthfully declares a preference for y over x. Coalition S might be able to prevent x from being selected, but it cannot engineer the selection of any alternative that each coalition member prefers to x at profile p.) It must be noted that both Murakami (1968) and Sen (1970) state the result for the more conventional domain of all profiles of complete and transitive individual
D.E. Campbell, J.S. Kelly / Mathematical Social Sciences 51 (2006) 81–89
83
preferences, although Murakami essentially restricts attention to two-element feasible sets X. The point of stating the more general claim is to emphasize that nothing need be assumed about individual preference, beyond the requirement that there be a strong Condorcet winner. It is worth emphasizing that, for both the folk theorem and the generalization to follow, attention is restricted to profiles at which there is an alternative that is strictly greater than any other. We do not enlarge the domain of the induced social choice function to include profiles at which there is a unique alternative x such that no other feasible alternative y is strictly greater than x. In fact, Sen (1970) recognized that invulnerability to manipulation by groups is not a property of the unique Condorcet winner, even within the set of profiles of linear orderings that yield a unique Condorcet winner. Example 1 in Campbell and Kelly (2003a) shows that even a single individual can manipulate majority rule within that domain. Not only does the folk theorem not appeal to any property of individual preference—other than the basic binary comparisons—it does not employ any property of majority rule other than the following non-reversal condition on an arbitrary social welfare function f, mapping some domain into the set of binary relations: If x is strictly greater than y in the social binary relation determined by f at profile p, and q is another profile in the domain of f, then y cannot be strictly greater than x in the social binary relation at q if the only individuals with a different preference at q than at p all strictly prefer y to x at p. It is clear that majority rule has this property for any domain on which there is a strong Condorcet winner at each profile in the domain. For an arbitrary social welfare function f, non-reversal of f is sufficient for the social choice rule g f , generated by selecting the bmost-preferredQ alternative according to f( p), to be invulnerable to manipulation by any individual or group. It is not necessary, however, as the following simple example reveals: Example 1. The feasible set X has three or more members and the domain of f is the set of all profiles of linear orders on X. The linear ordering f( p), the image of f at profile p, is obtained from person 1’s ordering at p by switching the positions of the bottom two alternatives in person 1’s ordering. The induced social choice rule g sets g( p) equal to the alternative at the top of f( p). That is, g( p) selects the top ranking alternative in person 1’s preference ordering. Hence, g is dictatorship of person 1 and thus cannot be manipulated by any coalition, although f does not satisfy non-reversal. Of course, g is also generated by a social welfare function h that satisfies non-reversal; we just set h( p) equal to the preference ordering of person 1. We will show that the social choice rule g generated by an arbitrary social welfare function f is invulnerable to manipulation by any individual or group if and only if g = gV for some social choice rule gV generated by a social welfare function with the non-reversal property. Nonreversal can be viewed as a normative condition but also, as we show in the next section, as the property of majority rule that allows an extension of the folk theorem to the widest possible family of social welfare functions. 2. The model and the results We begin with some elementary and standard notation. X is the given feasible set (finite or infinite) of alternatives, and N is the finite set of individuals. (Our proofs actually work for infinite N, but this case would appear to have limited or no applicability.) B(X) is the set of all binary relations on X, and B C(X) is the set of all complete (and reflexive) members of B(X). We let L(X) denote the set of linear orders on X. A profile p is a member of B(X)N , and it assigns the
84
D.E. Campbell, J.S. Kelly / Mathematical Social Sciences 51 (2006) 81–89
binary relation p(i) to i a N. A social welfare function for domain i K B(X)N is a mapping from i into B C(X). We write x d f( p)y to express the fact that x is strictly greater than y in the binary relation f( p) determined by f at profile p, and we use x d pi y to denote the fact that individual i strictly prefers x to y at p. A social choice rule with domain i is a function from i into X. We say that coalition J can manipulate the social choice rule g: i Y X at p via q if p and q belong to i and, g( q) d pj g( p) for all j a J, and q(i) = p(i) for all i a N\J. Let if denote the set of all profiles p in the domain of f such that there exists x a X satisfying x d f( p)y for all y a X\{x}, and define the social choice rule g f on if by setting g f ( p) = x in that case. We say that g f is generated by f. (If f is majority rule, then g f is the only strategy-proof and non-dictatorial social choice rule on if , Campbell and Kelly, 2003b). As usual, IIA denotes Arrow’s independence axiom, which requires f( p) to agree with f( q) on {x, y} if for each i a N the relation p(i) agrees with q(i) on {x, y}. Weak unanimity requires x d f( p)y for all y distinct from x if every individual strictly prefers x to every other alternative at profile p. 2.1. Non-reversal For all x, y a X and any two profiles p and q in the domain of f, if x d f( p)y then x + f( q)y holds if, for all j a N, q( j) p p( j) implies y d pj x. Weighted majority rule (with non-negative individual weights) satisfies non-reversal. Eliaz (2004) uses a very similar condition to prove a theorem from which both the Arrow and Gibbard–Satterthwaite impossibility theorems are easily derived as special cases. Eliaz begins by pointing out that a social choice function can be viewed as a social welfare function that maps profiles into a special family of orderings in which there is a unique top-ranked alternative, with all the others ranking below in a single indifference class. He then considers a broad class of social aggregators, and proves that only the dictatorial ones satisfy the Pareto criterion and his preference reversal condition. In order to compare his condition and ours, we temporarily adopt the Eliaz assumption that profiles assign only linear orders to individuals. 2.2. The Eliaz reversal condition For all x, y a X and any two profiles p and q in the domain of f, if x d f( p)y then x d f( q)y if, for all j a N, x d pj y implies x d qj y. Our condition applies only when x d pj y implies q( j) = p( j) (not merely x d qj y) and requires x + f( q)y instead of the stronger x d f( q)y. Therefore, the Eliaz condition is more demanding than our non-reversal. The next social choice rule satisfies IIA but violates both the Eliaz condition and our non-reversal property. Example 2. There is an odd number n of individuals. Set x d f( p)y if x d pi y holds for an odd number of i a N, otherwise set y d f( p)x. Therefore, if y d 1p x and y d 2p x, with x d pi y for all i N 2 then x d f( p)y. But if q is the same as p except that x d1q y then y d f( q)x. (Obviously, there is a counterpart for even n.) You can be in the minority but change the outcome in your favor simply by changing the parity of the set of individuals who prefer the alternative that you like least. (We can modify the example to satisfy our condition but not the Eliaz version by setting x above y in the social ordering if an odd number of individuals strictly prefer x to y, but with x and y indifferent to each other in the social ordering when an even number strictly prefer x to y.)
D.E. Campbell, J.S. Kelly / Mathematical Social Sciences 51 (2006) 81–89
85
If there are four or more alternatives then our non-reversal condition is not satisfied by the Borda rule, and it is not satisfied by any positional rule unless the number of alternatives is sufficiently small, as we now demonstrate. Example 3. Suppose that there are at least four alternatives and exactly four individuals. Let the preference profile p have the following structure: Person 1
Person 2
Person 3
Person 4
x z y v
x y v
y x v
y x v
Alternative x has the highest Borda score so will rank higher than y in f( p). However, if individual 4’s ordering changes to one that has y ranked first and x ranked fourth (or lower) then y’s Borda score will exceed that of x and hence y will rank higher than x in the new social ordering. Furthermore, the same switch involving x and y will occur with any positional rule because at the new profile q, alternative y’s score will exceed that of x by q 3 q 4, and of course q 3 N q 4. If there are exactly four alternatives, then at the new profile the preferences are not single peaked because for the triple {x, y, z} every alternative ranks below the other two in at least one person’s preference ordering. The preferences can be single-caved, however, because alternative z never ranks above both x and y for any individual at either the old profile p or the new profile q. Moreover, if there are five or more alternatives then we can put two alternatives other than z between y and x in q(4) and hence satisfy the single-peak condition. (The point of using only single-peaked preferences is that we then remain within the realm in which majority rule works well.) For two or more individuals, the Borda social welfare function satisfies non-reversal if and only of there are less than four alternatives: If x ranks above y at profile p then x’s Borda score exceeds y’s by at least one. However, with three (or fewer) alternatives an individual who prefers y to x can reduce the score differential by at most one by changing her reported preference, and hence cannot cause y to rank strictly above x by doing so. Suppose, however, that there are four or more alternatives and an even number of individuals. Consider the following profile p: Person 1
Person 2
Odd i N 1
Even i N 2
x z y w
y x w z
w y z x
x z y w
Note that the preferences of all i N 2 can be ignored because N\{1,2} can be partitioned into pairs so that the preference ordering of each member of a pair is just the opposite of that of the other member. (If there are more than four alternatives we can add them to the bottom of each individual ordering.) Confining attention to persons 1 and 2, we see that x’s Borda score is highest, followed by y, then z then w. If, however, person 2 reports the ordering with y first, followed by w, z, and x in that order, then y will have the uniquely highest Borda score. We have x above y in the initial social ordering, but y ranks above x after the individual who prefers y to x changes her reported ordering, violating non-reversal.
86
D.E. Campbell, J.S. Kelly / Mathematical Social Sciences 51 (2006) 81–89
Finally, suppose that there are four or more alternatives, an odd number of individuals, and the following initial profile p: Person 1
Person 2
Person 3
Even i N 2
Odd i N 3
x y z w
x y w z
y x w z
w y z x
x z y w
The preferences of all i N 3 can be ignored. With respect to persons 1, 2, and 3 we see that x’s Borda score is highest, followed by y, but if person 3 moves x down to the bottom then y will have the uniquely highest Borda score. Again, non-reversal fails. Before we present our first theorem, we generalize non-reversal in order that our results also deal with cases where we are only concerned with manipulation by individuals. In fact, the new condition applies to any non-empty family C of subsets of N. NR(C): For all x, y a X, and any two profiles p and q in the domain of f, if x d f( p)y then x + f( q)y holds if the set { j a N: q( j) p p( j)} is a member of C and is also a subset of { j a N : y d pj x}. Non-reversal is the condition NR(C) when C is the set of all non-empty subsets of N. If C is the family of singleton subsets of N then invulnerability to manipulation by any member of C is what we usually refer to as strategy-proofness. Given a social welfare function f, we let H f denote the set of social welfare functions h such that if p ih and the restriction of g h to if equals g f . The set H f is non-empty because it contains f. Theorem 1. The social choice function g f :if Y X generated by the social welfare function f: i Y B C (X) is invulnerable to manipulation by any coalition in C if and only if some member of H f satisfies NR(C). Proof. (i) We show that no coalition in C can manipulate g f if some member of H f satisfies NR(C). Suppose that h a H f satisfies NR(C). Suppose that J belongs to C, p and q belong to i, and we have y d pj g f ( p) for all j a J with q(i) = p(i) for all i a N \ J. By definition of if we have g f ( p) d f( p)z for all z a X \ { g f ( p)}. Because g h agrees with g f on if and if p ih , we also have g h ( p) d h( p)z for all z a X\{ g h ( p)}. Then we have g h ( p) + h( q)y by non-reversal of h, and hence y p g h ( q) = g f ( q). Therefore, g f cannot be manipulated by any coalition in C. (ii). We show that some coalition in C can manipulate g f if none of the social welfare functions in H f satisfy NR(C). Define the social welfare function p with domain if by setting g f ( p) d k( p)y and y ~ k( p) z for all y, z a X \{ g f ( p)} and for arbitrary p a if . Then if = ik and g k = g f . Consequently, p belongs to H f and thus k does not satisfy NR(C). Then we can find a coalition J in C, two profiles p and q in ik, and two alternatives x and y in X, such that both x d k( p)y and y d k( q)x hold, although y dpj x for all j a J and q(i) = p(i) for all i a N \ J. But k was defined so that g f (r) = g k(r) for every profile r. Then x d k( p)y and y d k( q)x imply, respectively, g f ( p) = g k( p) = x and g f ( q) = g k( q) = y. Therefore, coalition J can manipulate g f at p via q. 5 Theorem 4–6 on page 78 of Murakami (1968) proves that when X has exactly two alternatives, a simple montonicity condition is equivalent to invulnerability to manipulation by an individual or a group. Assume temporarily that there are three individuals and three alternatives. Then of the 216 profiles of linear individual orderings, there will be 204 for which the majority rule relation is
D.E. Campbell, J.S. Kelly / Mathematical Social Sciences 51 (2006) 81–89
87
itself a linear order, and twelve profiles (based on the two Latin squares) at which there is a voting cycle. In case of a cycle we arbitrarily modify the social ordering so that a specific alternative, say x, is top ranked, with the other two ranked below according to majority rule for the profile in question. This defines a new social welfare function f such that if contains all profiles of linear orders. Theorem 1 implies that this function will not satisfy non-reversal. (Of course, the Gibbard–Satterthwaite Theorem tells us that it can be manipulated.) Consider the standard Latin square profile p:
Person 1
Person 2
Person 3
x y z
y z x
z x y
We have x d f( p)z for this new social welfare function. However, if person 2 were to change her reported preference ordering by moving z to the top then f would have z ranking above x for the new profile, even though person 2 prefers z to x at p and f( p) has x ranking above z. The family of weighted majority rules does not contain all of the social welfare functions satisfying non-reversal. Any constant rule would have that property. The next example identifies a more interesting class of social welfare functions with the non-reversal property. Example 4. The domain of f is B(X)N . Let p be a function that assigns a non-negative number ki (x) to each alternative x for each individual i. For profile p we set x d f( p)y if and only if X X fpi ð xÞ : xdpi ygN fpi ð yÞ; ydpi xg: iaN
iaN
Of course, majority rule has ki (x) = 1 for all i a N and all x a X and, in general, all weighted majority rules have ki (x) = ki ( y) for all x and y in X and arbitrary individual i. If ki (x) = 1 for all i a N and ki ( y) = 0 for all i a N and all y p x, then if is the set of all profiles p p such that for each y a X\{x} there is some i a N such that x d i y. Then we have g f ( p) = x for all p a if . Example 1 demonstrates that g f can be invulnerable to manipulation even if f does not satisfy IIA. (If individual 1 has x ranked first and y second at profile p then x d f( p)y, but if q is the same as p except that we slide x and y down to the bottom of p(1), so that y is last in q(1) and x is second last, then we have y d f( q)x.) However, social welfare functions that do satisfy IIA, such as majority rule, are defined at every profile in B(X)N , even if they are not acyclic-valued at every profile in L(X)N . Therefore, we now investigate the family of social welfare functions satisfying IIA. In general, IIA and invulnerability to manipulation of g f by arbitrary coalitions do not imply non-reversal of f, as the next example shows. Example 5. The domain of f is L(X)N . Choose some alternative x* in X. Set x* d f( p)y for all p y a X\{x*} and all p a L(X)N . For y, z a X \ {x*} set y d f( p)z if and only if z d 1 y. The range of g f is {x*}, a singleton, so g f cannot be manipulated by any coalition. Clearly, f satisfies IIA, but not non-reversal. The social choice rule of Example 5 does not satisfy weak unanimity. Subject to an extremely modest domain assumption, for any social welfare function f satisfying IIA and weak unanimity, invulnerability to manipulation of g f by any member of C implies NR(C).
88
D.E. Campbell, J.S. Kelly / Mathematical Social Sciences 51 (2006) 81–89
Our domain assumption is substantially weaker than the free-pairs-at-the-top (FPT) property. (FPT: For arbitrary x and y and any q a B C ({x, y})N there is a profile p in the domain of f such that the restriction of p to {x, y} is q and a dpi z holds for all a a {x, y} and all z a X \ {x, y}.) Barbera` and Peleg (1990) and Sen (2001) prove that a strategy-proof rule is dictatorial if its domain has the FPT property and its range contains more than two alternatives. Aswal et al. (2003) derive the dictatorship theorem with a significantly weaker condition: Given a pair of alternatives {x, y} there need not be a profile at which x and y are placed arbitrarily in first or second rank for each individual, but there has to be a overlapping sequence of such pairs with x belonging to the first pair and y belonging to the last one. (In fact, slightly more is required, and the domain is assumed to have the form D N , for some set D of linear orderings.) At this stage we merely adopt a free pairs somewhere (FPS) assumption. Formally, for any pair of alternatives {x, y} and any assignment q a B C ({x, y})N of an ordering of {x, y} to each individual, there is a profile p in the domain of f such that the restriction of p to {x, y} is q. (Note that because one member of B({x, y}) is incomplete, profile q may assign no ordering at all to some individual.) If X = {x, y, z} and S is the set of linear orderings on X such that y is not last then S N satisfies FPS but not FPT. Of course, S N is not a dictatorial domain because majority rule yields linear orderings (for n odd) on this domain. We can think of FPS as the boundary between voting models and models of economic policy determination. Economic policies inevitably involve private goods, through their tax implications if for no other reason, and in that realm there will be outcomes x and y such that some individuals never prefer x to y—because x and y are identical except that x leaves the individual with less disposable income available for private goods. We can define a voting model as one for which a fixed amount of government revenue has already been raised, and it remains to determine which public project will be financed with the money. The adoption of both FPS and IIA allows us to assume with impunity that the domain of f is B C (X)N . Note that this is vastly different from the assumption that the domain of g f includes L(X)N . (Majority rule is defined on B C (X)N but it yields a unique winner only on a proper subset. Of course it is not even transitive-valued on L(X)N .) Rather than explicitly assuming FPS we will simply suppose that the social welfare function in question is defined on B C (X)N . But note that our next result is not implied by the Gibbard–Satterthwaite Theorem because the domain of g f can be much smaller than the domain of the social welfare function generating it. Theorem 2. Let C be arbitrary and let f: B C (X) N Y B C (X) satisfy IIA and weak unanimity. Then g f cannot be manipulated by any coalition in C if and only if f satisfies NR(C). Proof. The bifQ statement follows from Theorem 1, so we conclude with the proof of the bonly ifQ statement: Note that IIA and weak unanimity imply that at any profile p for which, for each individual i, x d pi z and y d pi z hold for each z a X \ {x, y} we will have p a if and g f ( p) = x if x d f( p)y, with g f ( p) = y if y d f( p)x. That follows from the fact that weak unanimity and IIA imply a d f( p)z for all a a {x, y} and all z a X \ {x, y}. Now, suppose that f satisfies IIA and weak unanimity, but not NR(C). Then there exist J aC, x, y a X, and profiles p and q in the domain of f such that x d f( p)y and y d f( q)x although y d pj x for all j a J and q(i) = p(i) for all i a N \ J. Choose any profile r in B C (X)N such that the restriction of r to {x, y} agrees with the restriction of p to {x, y}, and, for all i a N, a d ri z for all a in {x, y} and all z in X \ {x, y}. Then x d f(r)y by IIA, and x d f(r)z for all z a X \ {x, y} by IIA and weak unanimity. Therefore, r a if and g f (r) = x. Choose any profile s in B C (X)N such that s(i) = r(i) for all i a N \ J, and for all j a J the restriction of s( j) to {x, y} agrees with the restriction of q( j) to {x, y}, and with a d sj z for all a in {x, y} and all z in X \ {x, y}. We have
D.E. Campbell, J.S. Kelly / Mathematical Social Sciences 51 (2006) 81–89
89
s a if and y d f(s)x and thus g f (s) = y by the first paragraph. Then coalition J can manipulate g f at r via s. 5 When C includes all singleton bcoalitionsQ we can actually replace weak unanimity in the hypothesis of Theorem 2 with the assumption that for every x a X, there is some profile p in the domain of f such that x d f( p)y for all y a X \ {x}. If we start with such a profile and lift x up to the top for one individual at a time, alternative x must be selected at each profile unless some individual has an opportunity to manipulate at some point, establishing weak unanimity. Note that Example 2 satisfies weak unanimity and IIA, but it does not satisfy NR(C) for any C containing at least one odd proper subset of N. In particular, Example 2 does not satisfy our original non-reversal condition. When C is the family of all non-empty subsets of N, Theorem 1 tells us that the social choice function g f generated by social welfare function f cannot be manipulated by any individual or coalition if and only if g f can be generated by some social welfare function satisfying nonreversal, and Theorem 2 tells us that if f satisfies weak unanimity and IIA then it must also satisfy non-reversal if g f cannot be manipulated by any individual or coalition. Finally, we conjecture that a social welfare function satisfies non-reversal—that is, NR(2N \{F })—if and only if it is constant or it belongs to the family of Example 4 for some p. Acknowledgements We are grateful to John Duggan for posing the question addressed in this paper. Also, we thank our two referees for a number of valuable suggestions. We have benefited from an invitation to visit the PhD program in Economics at the University of Athens, where the final version of this paper was prepared: We are indebted to Andreas Papandreou and Yanis Varoufakis, and their students and colleagues, for their hospitality and intellectual stimulation. References Aswal, N., Chatterji, S., Sen, A., 2003. Dictatorial domains. Economic Theory 22, 45 – 62. Barbera`, S., Peleg, B., 1990. Strategy-proof voting schemes with continuous preferences. Social Choice and Welfare 7, 31 – 38. Campbell, D.E., Kelly, J.S., 2003a. Are serial Condorcet rules strategy-proof ? Review of Economic Design 7, 385 – 410. Campbell, D.E., Kelly, J.S., 2003b. A strategy-proofness characterization of majority rule. Economic Theory 22, 557 – 568. Eliaz, K., 2004. Social aggregators. Social Choice and Welfare 22, 317 – 330. Murakami, Y., 1968. The Logic of Social Choice. Routledge and Kegan-Paul, London. Sen, Amartya, 1970. Collective Choice and Social Welfare. Holden-Day, San Francisco. Sen, Arunava, 2001. Another direct proof of the Gibbard–Satterthwaite Theorem. Economics Letters 70, 381 – 385.
Social welfare functions generating social choice rules that are invulnerable to manipulation Donald E. Campbell a,*, Jerry S. Kelly b a
Department of Economics and Program in Public Policy, PO Box 8795, The College of William and Mary Williamsburg, VA 23187-8795, United States b Department of Economics, Syracuse University, United States Received 1 April 2003; received in revised form 1 September 2003; accepted 1 July 2005 Available online 12 October 2005
Abstract Any social welfare function f—whether transitive-valued or not—induces a social choice function g as follows: If at profile p in the domain of f there is a feasible alternative x that is bstrictly greater thanQ every other feasible alternative according to f( p) then we set g( p) = x. The domain of g is the set of all such profiles p, whether or not f( p) is acyclic. We specify a condition on a social welfare function that is necessary and sufficient for the induced social choice rule to be invulnerable to manipulation by any individual or coalition, and we generalize to an arbitrary collection of coalitions—including the family of singleton coalitions. D 2005 Elsevier B.V. All rights reserved. Keywords: Majority rule; Manipulation; Non-reversal JEL classification: D70; D71
1. Introduction Majority rule can be viewed as a social welfare function that selects a binary relation as a function of individual preferences. For a carefully chosen set of profiles, majority rule can be viewed as a social choice function that selects a single alternative as a function of individual preferences. For instance, for any set of profiles on which there is a strong Condorcet winner at each profile, the majority rule relation identifies a single bbest alternativeQ, and thus can also be *Corresponding author. Tel.: +1 757 221 2383; fax: +1 757 221 1175. E-mail address: [email protected] (D.E. Campbell). 0165-4896/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2005.07.002
82
D.E. Campbell, J.S. Kelly / Mathematical Social Sciences 51 (2006) 81–89
interpreted as a social choice rule on that set of profiles. (Alternative x is a strong Condorcet winner at profile p if for every other feasible alternative y the number of individuals who strictly prefer x to y at p exceeds the number of individuals who strictly prefer y to x.) This paper considers all social welfare functions. Each can be used in the same way to generate a social choice function on the domain of profiles at which the social ordering induced by the social welfare function has a most-preferred alternative: The social choice function selects that alternative. We present a necessary and sufficient condition, called non-reversal, for the rule to be invulnerable to manipulation by any individual or coalition, and then we generalize the condition to guarantee that no member of any given set of coalitions—for instance, the set of all singletons—can profit by deviating from truthful revelation. Although the following two families of social welfare functions are not exhaustive, they occupy a central place in axiomatic voting theory: [1] On the one hand we have the positional rules, which include the Borda rule as the salient example. A member of this family requires each individual to assign q j points to the alternative ranking in jth place in her preference ordering, with q j N q k if j b k. We could even have different q functions assigned to different individuals. Alternative x ranks above alternative y if the total point score across all individuals is algebraically higher for x than for y. (The Borda rule has each individual using the function q j = j.) Neither the Borda rule nor any other positional rule satisfies non-reversal, except when the number of alternatives is severely restricted. In other words, these procedures are quite vulnerable to manipulation. [2] On the other hand, there is the family of weighted majority rules. A particular member of this set assigns each individual i a weight r i , and alternative x ranks above y if the sum of the weights assigned to the individuals who prefer x to y exceeds the sum of the weights of the individuals who prefer y to x. (If there are n individuals and r 1 = 1 + 1 / n, with r i = 1 for all i p 1, then we have conventional majority rule with person 1 in the role of tie breaker.) Partial weighted majority rule is a special case for which, for some given J o N, the weight r j is zero for all j a J. For instance, when shareholders elect new directors to the board, each individual’s vote is proportional to the fraction of shares that he or she owns, but holders of the firm’s bonds do not get a vote—they have weight zero. The difficulty with weighted majority rule is, of course, that for many profiles there is a cycle that prevents any alternative from ranking above every other in the social ordering. However, because each member of the weighted majority family satisfies our non-reversal condition, within the set of profiles for which a weighted majority winner does emerge, no individual or coalition can benefit by deviating from truthful revelation provided that the deviation does not take us outside of the set of profiles that yields an unambiguous winner. In short, one consequence of our main result is that it makes a distinction between positional rules and weighted majority rules. We will show, however, that the class of social welfare functions satisfying non-reversal is strictly larger than the set of weighted majority rules. We begin with the folk theorem: On any domain in which each profile has a strong Condorcet winner, majority decision (viewed as a social choice rule) is invulnerable to manipulation by any individual or coalition. This claim is true regardless of the completeness and transitivity properties assumed (or not assumed) for the individual relations. (If x is a strong Condorcet winner at p then the set S of individuals who strictly prefer y to x at p cannot precipitate the selection of y by misrepresenting their preference because x defeats y when each member of S truthfully declares a preference for y over x. Coalition S might be able to prevent x from being selected, but it cannot engineer the selection of any alternative that each coalition member prefers to x at profile p.) It must be noted that both Murakami (1968) and Sen (1970) state the result for the more conventional domain of all profiles of complete and transitive individual
D.E. Campbell, J.S. Kelly / Mathematical Social Sciences 51 (2006) 81–89
83
preferences, although Murakami essentially restricts attention to two-element feasible sets X. The point of stating the more general claim is to emphasize that nothing need be assumed about individual preference, beyond the requirement that there be a strong Condorcet winner. It is worth emphasizing that, for both the folk theorem and the generalization to follow, attention is restricted to profiles at which there is an alternative that is strictly greater than any other. We do not enlarge the domain of the induced social choice function to include profiles at which there is a unique alternative x such that no other feasible alternative y is strictly greater than x. In fact, Sen (1970) recognized that invulnerability to manipulation by groups is not a property of the unique Condorcet winner, even within the set of profiles of linear orderings that yield a unique Condorcet winner. Example 1 in Campbell and Kelly (2003a) shows that even a single individual can manipulate majority rule within that domain. Not only does the folk theorem not appeal to any property of individual preference—other than the basic binary comparisons—it does not employ any property of majority rule other than the following non-reversal condition on an arbitrary social welfare function f, mapping some domain into the set of binary relations: If x is strictly greater than y in the social binary relation determined by f at profile p, and q is another profile in the domain of f, then y cannot be strictly greater than x in the social binary relation at q if the only individuals with a different preference at q than at p all strictly prefer y to x at p. It is clear that majority rule has this property for any domain on which there is a strong Condorcet winner at each profile in the domain. For an arbitrary social welfare function f, non-reversal of f is sufficient for the social choice rule g f , generated by selecting the bmost-preferredQ alternative according to f( p), to be invulnerable to manipulation by any individual or group. It is not necessary, however, as the following simple example reveals: Example 1. The feasible set X has three or more members and the domain of f is the set of all profiles of linear orders on X. The linear ordering f( p), the image of f at profile p, is obtained from person 1’s ordering at p by switching the positions of the bottom two alternatives in person 1’s ordering. The induced social choice rule g sets g( p) equal to the alternative at the top of f( p). That is, g( p) selects the top ranking alternative in person 1’s preference ordering. Hence, g is dictatorship of person 1 and thus cannot be manipulated by any coalition, although f does not satisfy non-reversal. Of course, g is also generated by a social welfare function h that satisfies non-reversal; we just set h( p) equal to the preference ordering of person 1. We will show that the social choice rule g generated by an arbitrary social welfare function f is invulnerable to manipulation by any individual or group if and only if g = gV for some social choice rule gV generated by a social welfare function with the non-reversal property. Nonreversal can be viewed as a normative condition but also, as we show in the next section, as the property of majority rule that allows an extension of the folk theorem to the widest possible family of social welfare functions. 2. The model and the results We begin with some elementary and standard notation. X is the given feasible set (finite or infinite) of alternatives, and N is the finite set of individuals. (Our proofs actually work for infinite N, but this case would appear to have limited or no applicability.) B(X) is the set of all binary relations on X, and B C(X) is the set of all complete (and reflexive) members of B(X). We let L(X) denote the set of linear orders on X. A profile p is a member of B(X)N , and it assigns the
84
D.E. Campbell, J.S. Kelly / Mathematical Social Sciences 51 (2006) 81–89
binary relation p(i) to i a N. A social welfare function for domain i K B(X)N is a mapping from i into B C(X). We write x d f( p)y to express the fact that x is strictly greater than y in the binary relation f( p) determined by f at profile p, and we use x d pi y to denote the fact that individual i strictly prefers x to y at p. A social choice rule with domain i is a function from i into X. We say that coalition J can manipulate the social choice rule g: i Y X at p via q if p and q belong to i and, g( q) d pj g( p) for all j a J, and q(i) = p(i) for all i a N\J. Let if denote the set of all profiles p in the domain of f such that there exists x a X satisfying x d f( p)y for all y a X\{x}, and define the social choice rule g f on if by setting g f ( p) = x in that case. We say that g f is generated by f. (If f is majority rule, then g f is the only strategy-proof and non-dictatorial social choice rule on if , Campbell and Kelly, 2003b). As usual, IIA denotes Arrow’s independence axiom, which requires f( p) to agree with f( q) on {x, y} if for each i a N the relation p(i) agrees with q(i) on {x, y}. Weak unanimity requires x d f( p)y for all y distinct from x if every individual strictly prefers x to every other alternative at profile p. 2.1. Non-reversal For all x, y a X and any two profiles p and q in the domain of f, if x d f( p)y then x + f( q)y holds if, for all j a N, q( j) p p( j) implies y d pj x. Weighted majority rule (with non-negative individual weights) satisfies non-reversal. Eliaz (2004) uses a very similar condition to prove a theorem from which both the Arrow and Gibbard–Satterthwaite impossibility theorems are easily derived as special cases. Eliaz begins by pointing out that a social choice function can be viewed as a social welfare function that maps profiles into a special family of orderings in which there is a unique top-ranked alternative, with all the others ranking below in a single indifference class. He then considers a broad class of social aggregators, and proves that only the dictatorial ones satisfy the Pareto criterion and his preference reversal condition. In order to compare his condition and ours, we temporarily adopt the Eliaz assumption that profiles assign only linear orders to individuals. 2.2. The Eliaz reversal condition For all x, y a X and any two profiles p and q in the domain of f, if x d f( p)y then x d f( q)y if, for all j a N, x d pj y implies x d qj y. Our condition applies only when x d pj y implies q( j) = p( j) (not merely x d qj y) and requires x + f( q)y instead of the stronger x d f( q)y. Therefore, the Eliaz condition is more demanding than our non-reversal. The next social choice rule satisfies IIA but violates both the Eliaz condition and our non-reversal property. Example 2. There is an odd number n of individuals. Set x d f( p)y if x d pi y holds for an odd number of i a N, otherwise set y d f( p)x. Therefore, if y d 1p x and y d 2p x, with x d pi y for all i N 2 then x d f( p)y. But if q is the same as p except that x d1q y then y d f( q)x. (Obviously, there is a counterpart for even n.) You can be in the minority but change the outcome in your favor simply by changing the parity of the set of individuals who prefer the alternative that you like least. (We can modify the example to satisfy our condition but not the Eliaz version by setting x above y in the social ordering if an odd number of individuals strictly prefer x to y, but with x and y indifferent to each other in the social ordering when an even number strictly prefer x to y.)
D.E. Campbell, J.S. Kelly / Mathematical Social Sciences 51 (2006) 81–89
85
If there are four or more alternatives then our non-reversal condition is not satisfied by the Borda rule, and it is not satisfied by any positional rule unless the number of alternatives is sufficiently small, as we now demonstrate. Example 3. Suppose that there are at least four alternatives and exactly four individuals. Let the preference profile p have the following structure: Person 1
Person 2
Person 3
Person 4
x z y v
x y v
y x v
y x v
Alternative x has the highest Borda score so will rank higher than y in f( p). However, if individual 4’s ordering changes to one that has y ranked first and x ranked fourth (or lower) then y’s Borda score will exceed that of x and hence y will rank higher than x in the new social ordering. Furthermore, the same switch involving x and y will occur with any positional rule because at the new profile q, alternative y’s score will exceed that of x by q 3 q 4, and of course q 3 N q 4. If there are exactly four alternatives, then at the new profile the preferences are not single peaked because for the triple {x, y, z} every alternative ranks below the other two in at least one person’s preference ordering. The preferences can be single-caved, however, because alternative z never ranks above both x and y for any individual at either the old profile p or the new profile q. Moreover, if there are five or more alternatives then we can put two alternatives other than z between y and x in q(4) and hence satisfy the single-peak condition. (The point of using only single-peaked preferences is that we then remain within the realm in which majority rule works well.) For two or more individuals, the Borda social welfare function satisfies non-reversal if and only of there are less than four alternatives: If x ranks above y at profile p then x’s Borda score exceeds y’s by at least one. However, with three (or fewer) alternatives an individual who prefers y to x can reduce the score differential by at most one by changing her reported preference, and hence cannot cause y to rank strictly above x by doing so. Suppose, however, that there are four or more alternatives and an even number of individuals. Consider the following profile p: Person 1
Person 2
Odd i N 1
Even i N 2
x z y w
y x w z
w y z x
x z y w
Note that the preferences of all i N 2 can be ignored because N\{1,2} can be partitioned into pairs so that the preference ordering of each member of a pair is just the opposite of that of the other member. (If there are more than four alternatives we can add them to the bottom of each individual ordering.) Confining attention to persons 1 and 2, we see that x’s Borda score is highest, followed by y, then z then w. If, however, person 2 reports the ordering with y first, followed by w, z, and x in that order, then y will have the uniquely highest Borda score. We have x above y in the initial social ordering, but y ranks above x after the individual who prefers y to x changes her reported ordering, violating non-reversal.
86
D.E. Campbell, J.S. Kelly / Mathematical Social Sciences 51 (2006) 81–89
Finally, suppose that there are four or more alternatives, an odd number of individuals, and the following initial profile p: Person 1
Person 2
Person 3
Even i N 2
Odd i N 3
x y z w
x y w z
y x w z
w y z x
x z y w
The preferences of all i N 3 can be ignored. With respect to persons 1, 2, and 3 we see that x’s Borda score is highest, followed by y, but if person 3 moves x down to the bottom then y will have the uniquely highest Borda score. Again, non-reversal fails. Before we present our first theorem, we generalize non-reversal in order that our results also deal with cases where we are only concerned with manipulation by individuals. In fact, the new condition applies to any non-empty family C of subsets of N. NR(C): For all x, y a X, and any two profiles p and q in the domain of f, if x d f( p)y then x + f( q)y holds if the set { j a N: q( j) p p( j)} is a member of C and is also a subset of { j a N : y d pj x}. Non-reversal is the condition NR(C) when C is the set of all non-empty subsets of N. If C is the family of singleton subsets of N then invulnerability to manipulation by any member of C is what we usually refer to as strategy-proofness. Given a social welfare function f, we let H f denote the set of social welfare functions h such that if p ih and the restriction of g h to if equals g f . The set H f is non-empty because it contains f. Theorem 1. The social choice function g f :if Y X generated by the social welfare function f: i Y B C (X) is invulnerable to manipulation by any coalition in C if and only if some member of H f satisfies NR(C). Proof. (i) We show that no coalition in C can manipulate g f if some member of H f satisfies NR(C). Suppose that h a H f satisfies NR(C). Suppose that J belongs to C, p and q belong to i, and we have y d pj g f ( p) for all j a J with q(i) = p(i) for all i a N \ J. By definition of if we have g f ( p) d f( p)z for all z a X \ { g f ( p)}. Because g h agrees with g f on if and if p ih , we also have g h ( p) d h( p)z for all z a X\{ g h ( p)}. Then we have g h ( p) + h( q)y by non-reversal of h, and hence y p g h ( q) = g f ( q). Therefore, g f cannot be manipulated by any coalition in C. (ii). We show that some coalition in C can manipulate g f if none of the social welfare functions in H f satisfy NR(C). Define the social welfare function p with domain if by setting g f ( p) d k( p)y and y ~ k( p) z for all y, z a X \{ g f ( p)} and for arbitrary p a if . Then if = ik and g k = g f . Consequently, p belongs to H f and thus k does not satisfy NR(C). Then we can find a coalition J in C, two profiles p and q in ik, and two alternatives x and y in X, such that both x d k( p)y and y d k( q)x hold, although y dpj x for all j a J and q(i) = p(i) for all i a N \ J. But k was defined so that g f (r) = g k(r) for every profile r. Then x d k( p)y and y d k( q)x imply, respectively, g f ( p) = g k( p) = x and g f ( q) = g k( q) = y. Therefore, coalition J can manipulate g f at p via q. 5 Theorem 4–6 on page 78 of Murakami (1968) proves that when X has exactly two alternatives, a simple montonicity condition is equivalent to invulnerability to manipulation by an individual or a group. Assume temporarily that there are three individuals and three alternatives. Then of the 216 profiles of linear individual orderings, there will be 204 for which the majority rule relation is
D.E. Campbell, J.S. Kelly / Mathematical Social Sciences 51 (2006) 81–89
87
itself a linear order, and twelve profiles (based on the two Latin squares) at which there is a voting cycle. In case of a cycle we arbitrarily modify the social ordering so that a specific alternative, say x, is top ranked, with the other two ranked below according to majority rule for the profile in question. This defines a new social welfare function f such that if contains all profiles of linear orders. Theorem 1 implies that this function will not satisfy non-reversal. (Of course, the Gibbard–Satterthwaite Theorem tells us that it can be manipulated.) Consider the standard Latin square profile p:
Person 1
Person 2
Person 3
x y z
y z x
z x y
We have x d f( p)z for this new social welfare function. However, if person 2 were to change her reported preference ordering by moving z to the top then f would have z ranking above x for the new profile, even though person 2 prefers z to x at p and f( p) has x ranking above z. The family of weighted majority rules does not contain all of the social welfare functions satisfying non-reversal. Any constant rule would have that property. The next example identifies a more interesting class of social welfare functions with the non-reversal property. Example 4. The domain of f is B(X)N . Let p be a function that assigns a non-negative number ki (x) to each alternative x for each individual i. For profile p we set x d f( p)y if and only if X X fpi ð xÞ : xdpi ygN fpi ð yÞ; ydpi xg: iaN
iaN
Of course, majority rule has ki (x) = 1 for all i a N and all x a X and, in general, all weighted majority rules have ki (x) = ki ( y) for all x and y in X and arbitrary individual i. If ki (x) = 1 for all i a N and ki ( y) = 0 for all i a N and all y p x, then if is the set of all profiles p p such that for each y a X\{x} there is some i a N such that x d i y. Then we have g f ( p) = x for all p a if . Example 1 demonstrates that g f can be invulnerable to manipulation even if f does not satisfy IIA. (If individual 1 has x ranked first and y second at profile p then x d f( p)y, but if q is the same as p except that we slide x and y down to the bottom of p(1), so that y is last in q(1) and x is second last, then we have y d f( q)x.) However, social welfare functions that do satisfy IIA, such as majority rule, are defined at every profile in B(X)N , even if they are not acyclic-valued at every profile in L(X)N . Therefore, we now investigate the family of social welfare functions satisfying IIA. In general, IIA and invulnerability to manipulation of g f by arbitrary coalitions do not imply non-reversal of f, as the next example shows. Example 5. The domain of f is L(X)N . Choose some alternative x* in X. Set x* d f( p)y for all p y a X\{x*} and all p a L(X)N . For y, z a X \ {x*} set y d f( p)z if and only if z d 1 y. The range of g f is {x*}, a singleton, so g f cannot be manipulated by any coalition. Clearly, f satisfies IIA, but not non-reversal. The social choice rule of Example 5 does not satisfy weak unanimity. Subject to an extremely modest domain assumption, for any social welfare function f satisfying IIA and weak unanimity, invulnerability to manipulation of g f by any member of C implies NR(C).
88
D.E. Campbell, J.S. Kelly / Mathematical Social Sciences 51 (2006) 81–89
Our domain assumption is substantially weaker than the free-pairs-at-the-top (FPT) property. (FPT: For arbitrary x and y and any q a B C ({x, y})N there is a profile p in the domain of f such that the restriction of p to {x, y} is q and a dpi z holds for all a a {x, y} and all z a X \ {x, y}.) Barbera` and Peleg (1990) and Sen (2001) prove that a strategy-proof rule is dictatorial if its domain has the FPT property and its range contains more than two alternatives. Aswal et al. (2003) derive the dictatorship theorem with a significantly weaker condition: Given a pair of alternatives {x, y} there need not be a profile at which x and y are placed arbitrarily in first or second rank for each individual, but there has to be a overlapping sequence of such pairs with x belonging to the first pair and y belonging to the last one. (In fact, slightly more is required, and the domain is assumed to have the form D N , for some set D of linear orderings.) At this stage we merely adopt a free pairs somewhere (FPS) assumption. Formally, for any pair of alternatives {x, y} and any assignment q a B C ({x, y})N of an ordering of {x, y} to each individual, there is a profile p in the domain of f such that the restriction of p to {x, y} is q. (Note that because one member of B({x, y}) is incomplete, profile q may assign no ordering at all to some individual.) If X = {x, y, z} and S is the set of linear orderings on X such that y is not last then S N satisfies FPS but not FPT. Of course, S N is not a dictatorial domain because majority rule yields linear orderings (for n odd) on this domain. We can think of FPS as the boundary between voting models and models of economic policy determination. Economic policies inevitably involve private goods, through their tax implications if for no other reason, and in that realm there will be outcomes x and y such that some individuals never prefer x to y—because x and y are identical except that x leaves the individual with less disposable income available for private goods. We can define a voting model as one for which a fixed amount of government revenue has already been raised, and it remains to determine which public project will be financed with the money. The adoption of both FPS and IIA allows us to assume with impunity that the domain of f is B C (X)N . Note that this is vastly different from the assumption that the domain of g f includes L(X)N . (Majority rule is defined on B C (X)N but it yields a unique winner only on a proper subset. Of course it is not even transitive-valued on L(X)N .) Rather than explicitly assuming FPS we will simply suppose that the social welfare function in question is defined on B C (X)N . But note that our next result is not implied by the Gibbard–Satterthwaite Theorem because the domain of g f can be much smaller than the domain of the social welfare function generating it. Theorem 2. Let C be arbitrary and let f: B C (X) N Y B C (X) satisfy IIA and weak unanimity. Then g f cannot be manipulated by any coalition in C if and only if f satisfies NR(C). Proof. The bifQ statement follows from Theorem 1, so we conclude with the proof of the bonly ifQ statement: Note that IIA and weak unanimity imply that at any profile p for which, for each individual i, x d pi z and y d pi z hold for each z a X \ {x, y} we will have p a if and g f ( p) = x if x d f( p)y, with g f ( p) = y if y d f( p)x. That follows from the fact that weak unanimity and IIA imply a d f( p)z for all a a {x, y} and all z a X \ {x, y}. Now, suppose that f satisfies IIA and weak unanimity, but not NR(C). Then there exist J aC, x, y a X, and profiles p and q in the domain of f such that x d f( p)y and y d f( q)x although y d pj x for all j a J and q(i) = p(i) for all i a N \ J. Choose any profile r in B C (X)N such that the restriction of r to {x, y} agrees with the restriction of p to {x, y}, and, for all i a N, a d ri z for all a in {x, y} and all z in X \ {x, y}. Then x d f(r)y by IIA, and x d f(r)z for all z a X \ {x, y} by IIA and weak unanimity. Therefore, r a if and g f (r) = x. Choose any profile s in B C (X)N such that s(i) = r(i) for all i a N \ J, and for all j a J the restriction of s( j) to {x, y} agrees with the restriction of q( j) to {x, y}, and with a d sj z for all a in {x, y} and all z in X \ {x, y}. We have
D.E. Campbell, J.S. Kelly / Mathematical Social Sciences 51 (2006) 81–89
89
s a if and y d f(s)x and thus g f (s) = y by the first paragraph. Then coalition J can manipulate g f at r via s. 5 When C includes all singleton bcoalitionsQ we can actually replace weak unanimity in the hypothesis of Theorem 2 with the assumption that for every x a X, there is some profile p in the domain of f such that x d f( p)y for all y a X \ {x}. If we start with such a profile and lift x up to the top for one individual at a time, alternative x must be selected at each profile unless some individual has an opportunity to manipulate at some point, establishing weak unanimity. Note that Example 2 satisfies weak unanimity and IIA, but it does not satisfy NR(C) for any C containing at least one odd proper subset of N. In particular, Example 2 does not satisfy our original non-reversal condition. When C is the family of all non-empty subsets of N, Theorem 1 tells us that the social choice function g f generated by social welfare function f cannot be manipulated by any individual or coalition if and only if g f can be generated by some social welfare function satisfying nonreversal, and Theorem 2 tells us that if f satisfies weak unanimity and IIA then it must also satisfy non-reversal if g f cannot be manipulated by any individual or coalition. Finally, we conjecture that a social welfare function satisfies non-reversal—that is, NR(2N \{F })—if and only if it is constant or it belongs to the family of Example 4 for some p. Acknowledgements We are grateful to John Duggan for posing the question addressed in this paper. Also, we thank our two referees for a number of valuable suggestions. We have benefited from an invitation to visit the PhD program in Economics at the University of Athens, where the final version of this paper was prepared: We are indebted to Andreas Papandreou and Yanis Varoufakis, and their students and colleagues, for their hospitality and intellectual stimulation. References Aswal, N., Chatterji, S., Sen, A., 2003. Dictatorial domains. Economic Theory 22, 45 – 62. Barbera`, S., Peleg, B., 1990. Strategy-proof voting schemes with continuous preferences. Social Choice and Welfare 7, 31 – 38. Campbell, D.E., Kelly, J.S., 2003a. Are serial Condorcet rules strategy-proof ? Review of Economic Design 7, 385 – 410. Campbell, D.E., Kelly, J.S., 2003b. A strategy-proofness characterization of majority rule. Economic Theory 22, 557 – 568. Eliaz, K., 2004. Social aggregators. Social Choice and Welfare 22, 317 – 330. Murakami, Y., 1968. The Logic of Social Choice. Routledge and Kegan-Paul, London. Sen, Amartya, 1970. Collective Choice and Social Welfare. Holden-Day, San Francisco. Sen, Arunava, 2001. Another direct proof of the Gibbard–Satterthwaite Theorem. Economics Letters 70, 381 – 385.