Centre de Referència en Economia Analítica

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1 Centre de Reerència en Economia Analítica Barcelona Economics Working Paper Series Working Paper nº 166 A Theorem on Preerence Aggregation Salvador Barberà July, 2003

2 A THEOREM ON PREFERENCE AGGREGATION Salvador Barberà * Universitat Autònoma de Barcelona Barcelona Economics WP nº 166 This Drat: July, 2003 Abstract I present a general theorem on preerence aggregation. This theorem implies, as corollaries, Arrow s Impossibility Theorem, Wilson s extension o Arrow s to non-paretian aggregation rules, the Gibbard-Satterthwaite Theorem and Sen s result on the Impossibility o a Paretian Liberal. The theorem shows that these classical results are not only similar, but actually share a common root. The theorem expresses a simple but deep act that transcends each o its particular applications: it expresses the tension between decentralizing the choice o aggregate into partial choices based on preerences over pairs o alternatives, and the need or some coordination in these decisions, so as to avoid contradictory recommendations. * This paper is based on the second part o my invited lecture at the 5th meeting o the Social Choice and Welare Society (Alicante, June 29 - July 2, 2000). I am grateul to Michael Florig, Jean Fraysse, Matthew Jackson, Cristina Rata, Peter Vida and especially to Carmen Beviá or their comments and help. I thank the editor o Social Choice and Welare and one anonymous reeree or their careul comments, and Bernard Monjardet or extensive historical remarks, rom which I have drawn reely. Financial support rom the Spanish Ministry o Education and Culture through grant PB , rom the Spanish Ministry o Science and Technology through 1

3 1. Introduction I present a general theorem on preerence aggregation. The substance o the theorem is to exhibit the tension between two general principles. One principle would consider the possibility o decomposing the choice o a social aggregate into a set o partial decisions, each one based on pairwise comparisons o alternatives. The other principle would require that no agent plays a dominant role in the aggregation process. I show that, in the presence o some other requirements, decomposable rules can only avoid contradictory recommendations in the presence o the strongest coordination device: a single agent being determinant at all times on all aspects o the global decision. The theorem implies, as corollaries, Arrow's Impossibility Theorem, Wilson's extension o Arrow's theorem to non-paretian aggregation rules, the Gibbard-Satterthwaite Theorem and Sen's result on the Impossibility o a Paretian Liberal. Hence, one contribution o the paper is to provide a common ramework that includes as particular cases all the dierent classes o aggregation rules that these classic theorems reer to. This allows us to understand the many connections that had already been noted among the substance and also among the proos o these dierent results. In particular, it clariies that the connections between Arrow's and Gibbard's results go deeper than any parallelism in their proos; both come rom a common root. It also allows us to note the intimate connection between other results, like Arrow's and Sen's, which had passed less noticed (with some signiicant exceptions that I reer to later). Arrow s and Wilson s theorems consider social welare unctions. The Gibbard- Satterthwaite theorem applies to social choice unctions, and Sen s result is expressed or social decision unctions. All o these unctions have a common domain: the proiles o individual preerences on some set o alternatives. All o them map these proiles into some type o related object, which we call an aggregate. Aggregates are dierent objects in each o these ormulations. In Arrow s and Wilson s ramework, an aggregate is a preerence order satisying the same properties that are required rom individual preerences. In Sen s ormulation, an aggregate is an acyclic binary relation (thus not necessarily transitive when social indierences are allowed). In the context where the Gibbard-Satterthwaite theorem applies, aggregates are simply alternatives. My theorem reers to preerence aggregation rules. These are unctions which map proiles o preerences on alternatives into some set o unspeciied objects, to be called aggregates. They include all o the above as special cases. Speciically, the theorem concentrates on the class o preerence aggregation rules that I call local. All o the classical results I have mentioned reer to special cases o local aggregation. It is thus important to clariy the concept immediately. Local aggregation rules are based on the idea that dierent parts o the inormation contained in preerence proiles can be used in making partial choices among aggregates, and that these partial choices eventually determine the one aggregate which corresponds to each grant BEC , and rom the Generalitat o Catalonia through grant SGR is grateully acknowledged. 2

4 proile. Not all preerence aggregation rules admit a description in these terms, but many do. Formal deinitions come in Section 2, but we should discuss immediately the meaning o local aggregation. When given an aggregation rule, it may be hard to ind out what is the algorithm, the procedure or the set o considerations which lead to associate a certain aggregate to a given preerence proile. But here is one possible way to do it, which characterizes local aggregation: (1) First, identiy some pieces o inormation that will be considered relevant. In our ormulation o local rules, we'll consider that the set o comparisons made by the agents between each pair o alternatives are the pieces o relevant inormation. (2) Second, associate a set o aggregates to each piece o relevant inormation. We can consider this set as the collection o aggregates that are admissible, given the inormation at hand. For example, i aggregators are binary relations, (like those we obtain rom majority rule), we can associate to each set o individual comparisons between x and y the set o all binary relations which rank x as preerred to y, or the set o all those which rank y over x. As another example, consider social choice unctions, which aggregate preerences and choose one alternative to be the aggregate. In that case, a local rule could associate a set o alternatives to each piece o relevant inormation, and interpret that the elements in each o these sets are the potential aggregates whose choice is not precluded by the inormation available. (3) Third, describe the social aggregate as the result o combining the use o all the relevant pieces o inormation. Speciically, the chosen aggregate should be the one (and only one, in our ormulation), which belongs to all the sets o non-excluded aggregates obtained rom using all the available pieces o relevant inormation. They can be a binary relation, obtained as the intersection o sets o binary relations, or an alternative, resulting rom intersecting sets o alternatives, or any other abstract object which belongs to the intersection o sets o objects in the relevant class o aggregates. Limiting attention to local rules is somewhat restrictive. Many aggregation rules one can think o, like the Borda counts and other point voting rules, do not adjust to the structure required by this condition. But we shall see that a theorem on local rules is able to cover substantial ground. More speciically, the theorem is about local and restricted aggregation rules. As already pointed out, the aggregates obtained rom a local rule can be described as combinations o objects that satisy some eatures, but not all combinations o eatures are necessarily acceptable to orm an aggregate. For example, the eature o ranking x over y is acceptable or aggregates which are orders. So is ranking y over z. And so is ranking z over x. These three eatures would be compatible i the binary relation which results rom aggregation was not restricted any urther, as with majority rule. But the three eatures together cannot be met by an aggregate binary relation, i we urther require this relation to satisy transitivity, as Arrow or Wilson do, or acyclicity, as in Sen. Similarly, admissible eatures o an aggregate in the Gibbard-Satterthwaite setting are that the aggregate is not x, that it is not y, and so on, or any possible alternative. But the union o all these eatures cannot be met by any aggregate, because only the empty set, which is not an acceptable aggregate, can meet them all. Again in Section 2, I provide a ormal deinition or a local rule to be restricted, which is enough to capture all the limitations in the choice o an aggregate that is imposed by our classical reerence theorems, in each o the cases. 3

5 As pointed out by Bernard Monjardet in private correspondence, this view o an aggregation process is not only natural but has a long tradition. I quote rom Monjardet s generous comments: "In his Memoire Recherches sur la loi de croissance de I'homme published in 1832 the statistician and social scientist Quetelet proposed to deine the "mean man" o a population o men described by several measurable characteristics by taking the means on each characteristics. This conception was severely criticized by people like the mathematician Cournot which remarked (In Exposition de la théorie des chances, Paris 1843) that this mean man could be nothing else that an impossible man. Indeed one meets a similar "paradox" by considering the triangle obtained by taking the 3 means o the lengths o the corresponding sides o a set o rectangular triangles; this triangle is not necessary rectangular. More generally Cournot writes: "Lorsqu'on applique la détermination des moyennes aux diverses parties d'un système compliqué, il aut bien prendre garde que ces valeurs moyennes peuvent ne pas se convertir: en sorte que l'état du système dans lequel tous les elements prendraint à la ois le valeurs moyennes déterminées séparément pour chacun d'eux serait un état impossible" 1. For Cournot, a 19 th century mathematician, a mean is any mathematical mean deined on numerical values. But more generally any p-ary operation deined on a set and satisying some properties (like idempotence) can be called a mean and the Cournot text can be applied to any aggregation procedure which is "local" in the sense that it is obtained by 1) Decomposing the set o complex objects (to aggregate) into sets o simple objects. 2) Applying mean operation to each o the sets. 3) Recomposing a complex object rom obtained simple objects. The "eet Condorcet" (or paradox o voting) is another amous example o the diiculty met by such a procedure and the relation between these various aggregation paradoxes and the Arrow's theorem has been developed in the (unortunately not suiciently read) 1952 paper by Guilbaud. Let me just add that in the present ormulation I have been completely agnostic about the nature o the object to be called an aggregate. This radical decision makes it hard to give explicit meaning to attractive notions like that o a mean, or that o closeness. Otherwise, I eel very comortable and happy to work within a model with such an old and distinguished tradition. Within the realm o local and restricted aggregators, the theorem expresses a tension between two eatures which are essential in the description o a rule: these are its lexibility, and the way in which it distributes among agents the power to inluence its result. We say that a rule is lexible i two conditions hold: one, that or each pair o alternatives, the rule's outcome is sometimes sensitive to changes in the relative positions o these two alternatives alone; and two, that this sensitivity should not be conditional on the position o any third 1 Compare with my previous remark that "since not all combinations o characteristics or an aggregate are jointly compatible, the ully decentralized choices o characteristics or aggregators across dierent pairs o alternatives may be contradictory". 4

6 alternative. We say that one agent concentrates all power under a rule i only the preerences o this agent (when they are strict) determine the social aggregate. I prove that i a preerence aggregation rule is local and restricted, then either it is not lexible or else it gives all decision power to a single individual. From this theorem we can derive as corollaries all the above mentioned classical results. Providing a common ramework or such disparate theorems is interesting per se. In some cases (Arrow and Wilson), the close connection was obvious rom the start. Yet, the present ormulation allows or a better understanding o the role played by the Pareto condition in Arrow, and in particular reinorces Wilson s message that this role is quite limited. In the case o Arrow and Gibbard-Satterthwaite, the existence o a close connection has been very well understood or years, but claiming that both results are a corollary o a unique theorem is a stronger and novel statement. The connection established here between Sen s result and the rest may be more problematic because Sen's model and its interpretation are themselves subject to debate (Salles, 2000). At any rate, Saari (1998) had emphasized the relationship between Sen s and Arrow s result, by using a dierent line o argument. We discuss all these connections later, in Section 3. I do not want to leave the reader with the impression that all I do here is to orce old results into a common rame or the sake o it. I already pointed out that the theorem has content and meaning o its own: the tensions between locality, restriction and lexibility can only be solved by an extreme distribution o decision power. Moreover, this content is clearly relected in the structure o its proo, which ollows the essential line o two proos o Arrow s and Gibbard-Satterthwaite s theorem that I provided some twenty years ago (Barberà, 1980, 1983a, 1983b). These separate proos are now blended into a single one, which reveals the essence o the diiculty at hand. Local aggregation requires the decomposition o our choice o aggregate into partial choices, each o which must be exclusively dependent on the relative positions o some pair o alternatives. Although possibly centralized among agents, decisions are decentralized in terms o the inluence exerted by dierent pairs o alternatives in determining which aggregates are ruled out as possible candidates. Since not all combinations o characteristics or an aggregate are jointly compatible, the ully decentralized choices o characteristics or aggregators across dierent pairs o alternatives may be contradictory. Some connections must be ensured between the choices made or some pairs and the choices made by some others. It turns out that, or any given preerence proile, and as long as rules are lexible, this restricts the ability to inluence the choice o aggregate to a single agent. Moreover, this local concentration o power extends to become global: only a single agent can make a dierence, at any preerence proile (as long as this agent holds strict preerences). The need or coordinated action becomes extreme: only one agent can make any dierence at all! Beore I inish this Introduction, let me comment on some related papers. The connections between Arrow's theorem and the issue o manipulability had been remarked (see Vickrey, 1960) even beore Gibbard's ormulation o his celebrated theorem. Gibbard's proo relies on Arrow's, and establishes a connection that was later stressed by Satterthwaite (1975), Mueller and Satterthwaite (1977) and many other authors. One should point out that the possibility o deining a nondictatorial Arrowian Social Welare Function on a given domain 5

7 o preerences, or that o deining a nondictatorial strategy-proo social choice unction on the same domain are logically independent questions: there are domains where one o the two objectives can be achieved and not the other. But the connection is tight or universal domains, the ones considered here, and also or the domains o single-peaked preerences (Barberà, Gul and Stacchetti, 1993). For universal domains, the present theorem oers a convincing explanation o why so many parallels have been observed between the two classics: there is a general statement that covers both! Regarding proos, Arrow's own (Arrow 1963), which has been reined by many other authors (see Sen, 1970, or a very nice version), relies on a careul analysis o the global decision power o coalitions: it is irst observed that, under Arrow's conditions, i a coalition is decisive or a pair it must be decisive or all pairs; it is then proven that i a coalition is decisive and can be broken into subcoalitions, one o these subcoalitions must also be decisive; inally, observing that the grand coalition is decisive allows to prove the theorem. In the preceding argument, decisiveness is the ability to determine the outcome o aggregation at all preerence proiles: a coalition is either decisive or not. As already mentioned, Gibbard's (1973) early proo, and many that ollowed, used a construct that allowed to prove that result by indirectly appealing to Arrow's theorem. There exist many proos o the Gibbard- Satterthwaite theorem, using dierent approaches. Those by Schmeidler and Sonnenschein (1978), Batteau, Blin and Monjardet (1981) or Barberà and Peleg (1990) did constitute real innovations by the time they appeared. Recent interest on the subject is obvious: three simple proos o the theorem have been published recently (Benoit, 2000; Sen, 2001; and Reny, 2001). One o them (Reny, 2001) highlights the parallelism between the proos o Arrow and (one o the) proos o Gibbard-Satterthwaite. A very recent paper by Eliaz (2001) makes the point that both theorems are corollaries o a larger result. In that sense, the paper by Eliaz emphasizes, as my paper does, the act that there is a deeper connection between these results than just a parallelism. Other than that, however, the paper by Eliaz and the present paper proceed quite dierently, both in the ormal model and in the type o proo. My strategy o proo here ollows the line o my already mentioned papers in the early eighties (Barberà 1980, 1983a, 1983b), which were the irst to exploit the role o pivotal voters. The proo starts by considering the ability o agents to change the outcomes o the aggregation process at a given preerence proile: it concentrates on a local property. Then I proceed to show that, under the conditions o the paper, i one agent can aect the outcome o the aggregation process at some proile, it must be able to aect it at any proile. The paper is organized as ollows. First, I present notation, ormal deinitions and a statement o the theorem in a simpliied ramework (Section 2). Then I discuss the relationship between the general theorem and the classical results (Section 3). Section 4 concludes, with a discussion o the possibility or a more general version o the theorem and its proo. 6

8 2. A simple version o the Theorem Let N = { 1,2,..., n} be a inite set o agents ( n 2). Let X = { x, y,... } be a inite set o alternatives ( X 3). Let A = { a 1 a,... } be a set o aggregates., 2 A 2 stands or the power set o A. A is the set o singletons in 2. D stands or the set o distinct pairs o alternatives. Let P be the set o linear orders 2 (complete, antisymmetric, transitive binary relations) on X. Elements o P are called preerences. n Elements o P are preerence proiles. Preerence proiles are denoted by P, P ', etc. Given a preerence proile P, and any set S o alternatives, denote by P(S) the proile o preerences on S such that, or all i N, P i (S) is the restriction o i s preerence P i to the set S. In particular, we denote by P() the preerence proile on the pair x, y induced by the proile P on X. A Alternatives x and y are contiguous in Pi i or all z X, z { x, y}, zpi x zpi y. Alternatives x and y are contiguous in P i they are contiguous in Pi, or all i N. Proiles P where two alternatives are contiguous play an important role in my setup, because they can be changed to other proiles P ' where P( ) P'( ) while keeping P ( zw) = P' ( zw) or all other pairs z, w other than x, y. Proiles P and P ' are -equivalent i P ( ) = P'( ). Otherwise, say that P and P ' ' dier on x, y. Proiles P and P only dier on x,y i they dier on x, y, and they are zw - equivalent or any pair z, w other than x, y. Clearly, i P and P ' only dier on x, y it must be that x and y are contiguous in the preerences o all agents, both in P and P '. or all Proiles P and ' P ' dier on i i P. They only dier on i i P and i P i ' i P i j i. They only dier on x,y or i, i they only dier on i and they only dier on ' P j = P j x, y. Contiguity o a pair o alternatives is a special case o what I will call connectedness o a set o alternatives with respect to a preerence. Formally, B X is connected in Pi i only i or all x, y B, z A \ B, zpix zpi y. B is connected in P i it is connected or all Pi 's. 2 Notice that I assume preerences to be strict (no indierences are allowed among dierent alternatives, by antisymmetry). This simpliies the exposition. 7

9 n An aggregation rule (on the universal domain P ) is a unction : P n A assigning a single aggregate to each preerence proile 3. P n and An aggregation rule is local i or each pair { 2 A \, such that ( P) = Ι g ( P) (1) or each x, y X, P, P ', x, y D, ' ' [ P( ) = P ( )] [ g ( P) g ( P )], x, y } D, there exist unctions = g : that is, g is a unction o x and y 's relative position in a proile, or any proile. (2) the range r o consists o two distinct sets o aggregates T, T, or all x, y X g 4. g 2 1 A Locality is a demanding requirement on aggregation rules. Locality requires that the global decision on which aggregator to attach to any given preerence proile should be decomposable into partial decisions, one or each pair o alternatives, depending only on the ranking o this pair o alternatives in the proile 5. Each partial decision attached to one pair o alternatives is a set o aggregators, and the global decision is the unique aggregator belonging to all o the sets selected, or each pair o alternatives, given the preerence proile. Locality 3 Notice that I model the unction as choosing a singleton set in A, rather than an alternative in A. This is to keep as ormal as possible. 4 This second requirement could be relaxed to cover several additional interesting cases. In particular, I could allow or the image o some s to consist o a single value. This would leave room to include the constant g unctions (and others) among the local we could also allow or more than two sets in the range. This would extend the reach o the theorem s applications to cover, among others, the case o Arrowian aggregation when social preerences can express indierence between pairs o alternatives. And it would require a careul modiication o the way to deine restricted aggregators (see my concluding remarks). I have chosen to keep this somewhat restrictive requirement (2) or expositional clarity. 5 O course, this is extremely close to the idea that the aggregation rule satisies Arrow's Independence o Irrelevant Alternatives. But I preer to reer to it by another term because (1) it applies to a larger context, and (2) it implies the description o aggregation as the intersection o sets, which is neither necessary nor implied by Arrow's ormulation. 8

10 also requires that the set o aggregators g (P) associated to each -pair at proile P should only depend on the relative ranking o x and y in the preerences o the agents, at that proile. Finally, the range o the partial decision unctions is restricted to a limited number o values. In the present ormulation, the ranges o these unctions are restricted to be dichotomic. The rules g associated to a local aggregation rule are called 's partial selection rules. For any range o g P, (P) is either T or T. I denote by g 1 g which is not chosen at P. 2 [ g ] comp P) ( the element in the Beore we proceed, let us see how this ormulation applies to the rameworks o other classical results. Consider the simple version o Arrow s theorem where the preerences o agents, as well as the preerences or society (the aggregates) are required to be linear orders. First notice that under the assumption o Independence o Irrelevant Alternatives, the social order between any two alternatives x, y, at any preerence proile P, will be determined only by the ranking o x and y in the preerences o individuals at P. Hence, given a preerence proile and two alternatives, x, y, one o the two possibilities x preerred to y or y preerred to x will be excluded. The set o all orders o alternatives ranking x and y in the nonexcluded position will be the image o the unction g (P) or each P. Indeed, the image (P) will be the order respecting the nonexcluded positions or all pairs. Hence, it is clear that any IIA social welare unction can be written as a local aggregation unction, where the aggregate is the intersection o sets o nonexcluded orders; a similar reasoning applies in the ramework considered by Wilson. The analysis o strategy proo rules as local aggregators builds on two essential remarks already made by Gibbard (1973). One is that, when all agents agree that all alternatives in a set S are preerred by all agents to all alternatives in X \ S, then a strategyproo social choice unction must choose rom S. (I assume here that the set o alternatives equals the range o a strategy-proo social choice unction. I not, drop the alternatives not in the range and start again, with no loss o generality). This implies, in particular, unanimity: an alternative must be chosen i it is unanimously considered to be the best by all agents. The second remark is that, i an alternative x is chosen when x and y are the best two alternatives or all agents, then y cannot be chosen at any preerence proile where the order o these two alternatives is preserved. Building on these two remarks, notice that the outcome at each proile can be written in the orm demanded by a local aggregate. For each given proile P, and each pair x, y, determine which one o the two alternatives is deinitely a noncandidate to be the aggregate at this proile. For this, check which one o the two alternatives would not be elected at the proile P() obtained rom P and where, ceteris paribus, x and y become the top two alternatives rom all agents. 9

11 Now, any strategy-proo social choice unction can be written as a local aggregation unction, where or each pair o alternatives x, y, either the set X \ x or the set X \ y are retained (the one not containing the alternative which is eliminated by pairwise comparisons). The aggregate corresponding to each proile must clearly be expressed as the intersection o all the sets o non-excluded alternatives, or each pairwise comparison. Having illustrated the basic eatures o local aggregation rules, let us now go back to study some urther requirements. Given a local aggregation rule, and a set o pairs D D deine Ι x, y D G( P, D) = g ( P). We can interpret G( P, D) as the set o aggregates which are not excluded on the basis o the comparisons o pairs in D. Clearly, with this notation, G(P, D ) = (P), or all P. We say that is -sensitive at P i comp ( P) = G (P, D) G( P, D \ { x, y} ) g ( P). Thus, is -sensitive at P i a change in the value o the -partial selection would change the value o the global selection o aggregate. The notion o sensitivity is introduced because it is not always the case that the change in one o the partial decisions associated to one pair o alternatives can change the global outcome. This depends on the type o aggregate we consider. Since we want to cover dierent types o aggregates, and provide a result that is independent o the type o aggregate under consideration, we cannot be more speciic on the connections between changes in the partial decisions and eventual changes in the aggregate. In Arrow's ramework, where each partial decision determines the ranking o a pair o alternatives within the overall ranking o alternatives, any change in one partial decision is translated into a change in the aggregate. Yet, in Gibbard's ramework, the same is not true. Starting rom some preerence proile, you may change the partial recommendation on a pair x, y, and yet see a third alternative z be elected beore and ater the change: the aggregation rule would not then be -sensitive at that proile. We say that is restricted i, or all x, y, z X, and all proiles P where is - sensitive and yz-sensitive, there exist sets and in the range o and g or which G( P, D \ { y} T r x,, { z} yz T s y, ) T yz r T A. s g yz 10

12 To better understand this assumption, let us again go back to examples. In Arrow's ramework, or each unction, the range consists o two possible outcomes, each o which g is a set o orders: the orders with x over y, on the one hand, and the orders with y over x, on the other. The act that aggregates must be orders implies that our aggregation unction must be restricted: indeed, take any proile and any triple x, y, z. I at the initial proile P we have x over y, then the set o orders with x over y appears in the intersection o values deining the aggregate at P. Then, the set o orders with y over z, and the set o orders with yz g xz g z over x, both in the ranges o and, respectively, will have an empty intersection with the rest o sets o orders deining the value o at P. Likewise, i at P we had that y stood over x, then the choices o sets o orders with x over z and z over y would have led to the empty intersection required by restriction. Thus, the condition called restriction in our general theorem corresponds in Arrow s theorem to the requirement o transitivity o the social aggregate. The social choice unctions in Gibbard-Satterthwaite are also restricted local aggregators: indeed, or each pair o alternatives, the ranges o the unctions consist o two well-deined sets: all alternatives except x and all alternatives except y. Notice that, or any triple x, y, z o alternatives, and any proile P where is -sensitive and also yz - sensitive, the intersection o all the partial selections at P, other than those or and yz, must contain x, y and z. But then, the choice o X \ y rom the range o, coupled with the choice o X \ y, as well, rom the range o g, would leave us with both x and z in the overall intersection. Since only singletons, not two-element sets o aggregates, can be in the range o, we have identiied a restriction to be satisied by the social choice unctions considered in the Gibbard-Satterthwaite setup. Hence, a rule is restricted i not all partial choices or dierent pairs are compatible with an acceptable global choice, when these partial choices do make a dierence. A restricted local aggregation rule is thus one or which the choices o the dierent sets o aggregates associated to each pair o alternatives is not completely ree: some combinations o potential choices would not be able to produce any aggregate at all. Hence, some coordination among partial outcomes will be necessary in order to obtain one and only one aggregator or each particular proile. We say that is lexible i or some triple o alternatives x, y, z X, there is some proile P at which z} (1) The set { x, y, is connected in the preerences o all agents at P, (2) is -sensitive and also yz -sensitive at P, xz comp (3) G( P,D\{ x, y},{ x, z}) ( g ( P)) ( g ( P)) G( P,D\{ x, y},{ x, z} ) ( g yz g g ( P)) comp ( g xz ( P)) comp, 11

13 P and y, z,{ x, z} yz xz comp g ( P)) ( g ( P)) G(,D\{ } ) ( P G(,D\{ x, y},{ z} x, ) ( g yz ( P)) comp ( g xz ( P)) comp, In words, condition 3 requires that 's -sensitivity and yz-sensitivity at P would be xz preserved by changes in the value o g. Flexibility encompasses two requirements. One is that, or each pair o alternatives, their relative position alone can sometime make a dierence. The other, complementary requirement, is that i the relative positions o two pairs make a dierence at the same proile, then the relative position o any third alternative does not aect the rule s sensitivity with regard to any o these two pairs. In Arrow's ramework, lexibility is an implication o the Pareto axiom and IIA. In Gibbard's and Satterthwaite's, the connection requires more care and it is discussed below, in section 3.4. Agent i concentrates all power under rule i, or all preerence proiles P, P ' and all pairs x, y, be written as ' [ P ( ) P ( ) ] g ( P) g ( P' ) =. i i = Notice that, then, we can write each partial unction as g P ), and that then can P ), or all P. ( i ( i Also notice that a dictator (with strict preerences) concentrates all power, but that an anti-dictator also does, and so do any agents whose preerences are the only relevant inputs to determine the outcome o an aggregation rule. Theorem I an aggregation rule is local, lexible and restricted, then there is a single agent i who concentrates all the power under. Proo Let be local, lexible and restricted. We irst provide some useul deinitions. We say that agent i is -pivotal at preerence proile P i P ' such that P and P ' only dier in or agent i and such that ( P) ( P' ). (Clearly, i agent i is -pivotal at P, he is also -pivotal at P '.) 12

14 Remark (and Remark that i P and P ' only dier in or agent i, and is -sensitive at P P ' ), then i is -pivotal at P (and P ' ). Lemma 1 The proo is organized through a sequence o simple Lemmas. I i is -pivotal at proile P, P ' only diers rom P in j 's ranking o x, } and ( P ) = ( P' ), then i is still -pivotal at P '. { z, w} { y Proo Just notice that, since ( P) = ( P' ) and i is -pivotal, it must be that (P, D \ x, y ) = G( P' D \ { x, y}). G { }, Lemma 2 and There cannot be any proile P at which two dierent agents i and j are -pivotal yz -pivotal, respectively, with x and y contiguous in P and y and z contiguous in P. i j Proo P i Suppose there was, and that i, j were the preerences (with x and y contiguous in P, and y and z contiguous in ) such that ( P, P ) ( P), ( P, P ) ( P). Then, P j we can write the images (P), (P, ), ( P P ), ( P ij P ) as r i P i P G( P, D \ { y} i x,, { y, z} i j, j,, P i j g r yz s ) T T, or all possible choices o T in the range o and T in the range o ( use Lemma 1). Hence, one o these our possible outcomes cannot be an aggregate, because is restricted. Lemma 3 There is an agent who concentrates all power under rule. yz s j j yz g 13

15 Proo Since is lexible, P and x, y, z X such that (1) the set { x, y, z} is connected in the preerences o all agents at P, and (2) is -sensitive and also yz -sensitive at P. '' Notice that, since is -sensitive at P there must exist P ', P such that they only dier on { x, y} in i and g ( P' ) g ( P" ). Similarly, there must exist P ~, Pˆ and j such that yz ~ they only dier on { y, in j, and g yz (Pˆ) g (P). z} Since x, y and z are connected, it is possible to ind a proile P where x, y and y} are still connected, where { x, are contiguous in i y, z are contiguous in P j, and i is -pivotal, while j is yz -pivotal. We use here condition 3 in the deinition o lexibility, since in such proile the relative position o xz or zy may be dierent that at the original P. Only one o these two needs to be changed in order to meet the requirements. To see that, notice that i an agent is -pivot when x and y are contiguous, the agent is still pivot when its preerences over x and y are switched. P, { } But now we have the conditions o Lemma 1. Hence, only one agent can be pivotal at this proile, and it is pivotal or two dierent pairs. From this point on, we can start switching the preerences o any agent or dierent pairs o contiguous alternatives including any one o the three alternatives x, y or z in question. No such change alter the outcome i it happens or any agent other than i, since that would lead us to the situation o Lemma 2. Hence, at some proile we shall eventually get a pivot or the pair, and this pivot will necessarily be i again. That general argument can be applied repeatedly to conclude that the same agent i must be pivot at some proile or all pairs o alternatives and hence concentrate all the power under. z 3. Some applications and corollaries In what ollows, I present a sketchy discussion o why the new theorem does indeed cover the setups or which dierent results were initially ormulated, and how the particular conditions under which the classical theorems hold are particular versions o those imposed here. 14

16 3.1. Arrow's ramework and Theorem We have already discussed why Arrowian Social Welare unctions are local and restricted. Notice now that the Pareto condition imposed by Arrow implies lexibility. Indeed, or proiles where all agents agree that x is preerred to y, x must be preerred to y socially, and the reverse must hold when all agents rank y over x. By changing the preerences one by one, reversing the order o x and y, starting rom a proile where x and y are contiguous and all agents preer x to y, and moving to one where they all preer y to x, it must be that one agent is -pivotal at some o the proiles in this sequence. This can be done or other pairs yz. Moreover, -pivotality and yz -pivotality can be preserved in proiles where the conditions o lexibility do hold. It is thus obvious that an Arrowian social welare unction must be a local, restricted and lexible aggregator: hence, all the decision power must be concentrated in an agent. Moreover, again by Pareto, this agent obtains the ranking x over y whenever he preers x to y : this is the dictator Wilson s theorem The initial ramework is the same as Arrow s: IIA implies locality and transitivity implies that the aggregation unction is restricted. Unlike in Arrow s ormulation, lexibility does not come rom Pareto, but rom the range condition requiring that x should be ranked above y or some proile, and below y or some other. The same construction as beore, but having as extremes two such proiles with x and y contiguous, shows the existence o -pivots or all pairs. Hence, one agent always concentrates all decision power: but this agent may be getting always the same order among pairs o alternatives that he expresses, or the reverse. Because Pareto does not necessarily apply, the individual who concentrates the power may be a dictator or an anti-dictator. Other combinations, where the agent might be a dictator over some pairs, and an anti-dictator over other pairs are excluded by restriction. For example, i some pairs were ordered according to the preerences o one agent and others in the opposite direction than the preerences o the same agent, this could easily lead to cycles. On the other hand, we may consider the constant unction as an example o an aggregation rule which satisies all o our conditions except or lexibility and locality 6. This is the other type o aggregator obtained by Wilson, again violating the Pareto condition. 6 I we had chosen to use a more general deinition o locality as discussed in Aizerman and Aleskerov (1995), then the constant unction could be accommodated among the local ones, while still violating lexibility. 15

17 3.3. Sen s impossibility o a Paretian liberal The essential remark regarding Sen s result is that, it postulates explicitly as desiderata the two conditions which are stated by our theorem to be incompatible under local and restricted rules. One o them is the Pareto condition: we have already seen that this condition, or a universal domain o preerences, implies that the aggregation unction must be lexible. The other condition imposed by Sen is that at least two agents must be able to determine the strict ranking o two alternatives each (these two alternatives describe social states which are identical except or some eatures over which the agent in question is entitled to have ull control. This is the condition o (minimal) liberalism, a condition which is in open contradiction with the possibility o one agent holding all the decision power under an aggregation rule. The interpretation o this ability to determine the ranking o some alternatives is subject to debate, because Sen is reerring to rights, and their connection to choices is not an obvious one. In the present paper I limit mysel to comments about the ormal structure in Sen's ormulation, not about its meaning(s). Since Sen requires two conditions which are incompatible or any local and restricted rule, it is enough or us to argue that he is considering, in his theorem, rules that can be written as members o this class. No explicit argument is provided by Sen regarding the general structure o the unctions he considers: hence, some characteristics o these unctions or some special proiles might escape the description (and hence the ull orce) o our theorem. But remember that the theorem contains two statements: one is about the impossibility o allocating pivotal rights to more than one individual at the same proile (the local aspect, step one o the proo), and the other is the extension o this local argument to a global one. We only need the local result to understand Sen s impossibility. When it applies, the Pareto principle allows us to ascertain that the aggregate must be among the class o binary relations respecting the agents unanimous ranking o a pair. Likewise, the rankings o pairs associated to the private domain o an agent depend only (by the assumption o minimal liberalism) on the interested agent s ranking o these pairs. Hence, the aggregate or proiles where Pareto applies and agents have strict preerences over the pairs o their exclusive concern must be in the intersection o those sets o aggregates respecting the above conditions, as in a local aggregation rule. Now, the aggregate is required by Sen to be an acyclic rule, and this imposes a restriction on the choice o aggregates, which cannot be respected at proiles where Pareto applies and both agents with rights can exert them The Gibbard-Satterthwaite theorem I have already discussed why a strategy-proo social choice unction is a local and restricted aggregator. Moreover, strategy-prooness guarantees that, whenever the range o the rule contains at least three alternatives, then there must be proiles where some agent is pivotal or more than one pair. My 1983 proo o the Gibbard-Satterthwaite theorem establishes (step g) the ollowing consequence o strategy-prooness: i no individual is ever pivotal under or more than two alternatives, then the range o consists o at most two alternatives. Moreover, the construction in Gibbard s proo that determines whether each alternative is 16

18 excluded or is not at a proile can only depend on the positions o pairs o alternatives, when is strategy-proo. This gives us the second part o lexibility. We can now reer to the general theorem to obtain dictatorship. 4. Final Remarks I have proven the theorem or the case where the range o each partial selection rule consists o two sets only. The proo is suicient to cover the Gibbard-Satterthwaite theorem. It is also suicient to cover the versions o the rest o the theorems we mention (Arrow, Wilson, Sen) where society's preerences are required to be strict. But since these theorems also hold when societies (as well as individuals) are allowed to express indierences, it would be interesting to discuss how our theorem should be extended to cover the case o three-valued (or many-valued) ranges. The same basic ideas apply, but urther deinitions and some delicate distinctions must be made. I leave this extension or another paper. REFERENCES Aizerman, M. and F. Aleskerov (1995): "Theory o Choice", North Holland, Elsevier Science, Amsterdam. Arrow, K. (1963): "Social Choice and Individual Values", 2nd edition, Wiley: New York. Barberà, S. (1980): "Pivotal Voters: A New Proo o Arrow's Theorem", Economics Letters 6, Barberà, S. (1983a): "Strategy-Prooness and Pivotal Voters: A Direct Proo o the Gibbard- Satterthwaite Theorem", International Economic Review 24, 2, Barberà, S. (1983b): "Pivotal Voters: A Simple Proo o Arrow's Theorem", in P.K. Pattanaik and M. Salles (Eds.), Social Choice and Welare, North Holland. Barberà, S., Gul, F. and E. Stacchetti (1993): "Generalized Median Voter Schemes and Committees", Journal o Economic Theory 61, Barberà, S. and Peleg, B. (1990): "Strategy-proo Voting Schemes with Continuous Preerences", Social Choice and Welare 7, Batteau, P., Blin, J-M. and B. Monjardet (1981): "Stability o Aggregation Procedures, Ultrailters and Simple Games", Econometrica 40, Benoit, J-P. (2000): "The Gibbard-Satterthwaite Theorem: A Simple Proo", Economics Letters 69,

19 Eliaz, K. (2001): "Arrow's Theorem and the Gibbard-Satterthwaite Theorem Are Special Cases o a Single Theorem", mimeo, Tel-Aviv University. Gibbard, A. (1973): "Manipulation o Voting Schemes: A General Result", Econometrica 41, Guilbaud, G. Th. (1952): Les Théories de l Intérêt Général et le Problème Logique de l Aggrégation, Economie Appliquée 5, English translation in P.F Lazarseld and N.W. Henry (Eds.) Readings in Mathematical Social Sciences (Science Research association Inc.-Chicago, 1965) pp Mueller, E. and A. Satterthwaite (1977): "The Equivalence o Strong Positive Association and Strategy-Prooness", Journal o Economic Theory 14, Reny, P.J. (2001): "Arrow's Theorem and the Gibbard-Satterthwaite Theorem: A Uniied Approach", Economics Letters 70, Saari, D. (1998): Connecting and Resolving Sen s and Arrow s Theorems, Social Choice and Welare 15, Salles, M. (2001): "Amartya Sen: Droits et Choix Social", Revue Economique 81-3, Satterthwaite, M. (1975): "Strategy-Prooness and Arrow's Conditions: Existence and Correspondence Theorems or Voting Procedures and Social Welare Functions", Journal o Economic Theory 10, Schmeidler, D. and H. Sonnenschein (1978): "Two Proos o the Gibbard-Satterthwaite Theorem on the Possibility o Strategy-Proo Social Choice Functions", in H.W. Gottinger and W. Leinellner (eds.), Decision Theory and Social Ethics, Issues in Social Choice, , D. Reidel Publishing Company, Dordrecht, Holland. Sen, A.K. (1970): "The Impossibility o a Paretian Liberal", Journal o Political Economy 78, Sen, A. (2001): "Another Direct Proo o the Gibbard-Satterthwaite Theorem", Economics Letters 70, (2001). Vickrey, W.S. (1969): "Utility, Strategy and Social Decision Rules", Quarterly Journal o Economics 74, Wilson, R. (1972): "Social Choice without the Pareto Principle", Journal o Economic Theory 5,