On the Robustness of Majority Rule

Transcription

1 On the Robustness of Majorit Rule Partha Dasgupta * and Eric Maskin ** Januar 1998 Current version: March 2004 Abstract We show that siple ajorit rule satisfies four standard and attractive properties the Pareto propert, anonit, neutralit, and (generic) transitivit on a bigger class of preference doains than (essentiall) an other voting rule. Hence, in this sense, it is the ost robust voting rule. * Facult of Econoics and Politics, Universit of Cabridge ** School of Social Science, Institute for Advanced Stud This research was supported b grants fro the Beijer International Institute of Ecological Econoics and the U.S. National Science Foundation.

2 1. Introduction A voting rule is a ethod for choosing fro a set of social alternatives on the basis of voters stated preferences. Man different voting rules have been studied in theor and used in practice. But far and awa the ost popular ethod has been siple ajorit rule, the rule that chooses alternative over alternative if ore people prefer to than vice versa. There are, of course, good reasons for ajorit rule s 1 popularit. It not onl is attractivel straightforward to use in practice, but satisfies soe copelling theoretical properties, aong the the Pareto propert (the principle that if all voters prefer to and is available, then should not be chosen), anonit (the principle that choices should not depend on voters labels), and neutralit (the principle that the choice between a pair if alternatives should depend onl on the pattern of voters preferences over that pair, not on the alternatives labels). In fact, Ma (1952) establishes that ajorit rule is the unique voting rule satisfing the Pareto propert, anonit, and neutralit, and a fourth propert called positive responsiveness. 2. We shall coe back to Ma s characterization below. 1 For convenience, we will oit the odifier siple when it is clear that we are referring to siple ajorit rule rather to an of the an variants, such as the superajorit rules. 2 A voting rule is positivel responsive if whenever alternative is chosen (perhaps not uniquel) for a given specification of voter s preferences and the onl change that is then ade to these preferences is to ove up in soe voter s preference ordering, then becoes uniquel chosen. 1

3 But ajorit rule has a well-known flaw, discovered b the Marquis de Condorcet (1785) and illustrated b the Parado of Voting (or Condorcet Parado): it can generate intransitive choices. Specificall, suppose that there are three voters 1, 2, 3, three alternatives,, z, and that the profile of voters preferences is as follows: z z z (i.e., voter 1 prefers to to z, voter 2 prefers to z to, and voter 3 prefers z to to ). Then, as Condorcet noted, a two-thirds ajorit prefers to, to z, and z to, so that ajorit rule fails to select an alternative. Despite the theoretical iportance of the Condorcet Parado, there are iportant cases in which ajorit rule avoids intransitivit. Most faousl, when alternatives can be arranged linearl and each voter s preferences are single-peaked in the sense that his utilit declines onotonicall as one oves awa fro his favorite alternative, then, following Black (1948), ajorit rule is transitive for (alost) all 3 profiles of voters preferences. Alternativel, suppose that, for ever three alternatives, there is one that no voter ranks in the iddle. This propert, which is a special case of value restriction (see Sen 1966, Inada 1969, and Sen and Pattanaik 1969), sees 3 We clarif what we ean b alost all in section 2. 2

4 to have held in recent French presidential elections, where the Gaullist and Socialist candidates have not engendered uch passion, but the National Front candidate, Jean-Marie Le Pen, has inspired either revulsion or adiration, i.e., the vast ajorit of voters rank hi either last or first, but not in between. Whether or not this pattern of preferences has been good for France is open to debate, but it is certainl good for ajorit rule: value restriction, like single-peakedness, ensures transitivit (alost alwas). So, ajorit rule works well in the sense of satisfing the Pareto propert, anonit, neutralit and generic transitivit for soe doains of voters preferences but not for others. A natural question to ask is how its perforance copares with that of other voting rules. Clearl, no voting rule can work well for all doains; this conclusion follows iediatel fro the Arrow ipossibilit theore 4 (Arrow, 1951). But we ight inquire whether there is a voting rule that works well for a bigger class of doains than does ajorit rule. 5 4 Our forulation of neutralit (see section 3) which is, in fact, the standard forulation (see Sen, 1970 or Capbell and Kell, 2002) incorporates (i) Arrow s independence of irrelevant alternatives, the principle that the choice between two alternatives should depend onl on voters preferences for those two alternatives and not on their preferences for other alternatives and (ii) setr with respect to alternatives, the principle that peruting the alternatives in voters preferences should perute social choices in the sae wa. 5 It is eas to find voting rules that satisf three out of our four properties on all doains of preferences. For eaple, ajorit rule and an of its variants, e.g., two-thirds ajorit rule (which dees two alternatives as sociall indifferent unless one garners at least a two-thirds ajorit against the other), satisf Pareto, anonit, and neutralit on an doain. Siilarl, rank-order voting (see below) satisfies Pareto, anonit, and transitivit on an doain. 3

5 We show that the answer to this question is no. Specificall, we establish (see our Theore) that if a given voting rule F works well on a doain of preferences, then ajorit rule works well on that doain too. Conversel, if F differs fro ajorit rule 6, there eists soe other doain on which ajorit rule works well but F does not. Thus ajorit rule is essentiall uniquel the voting rule that works well on the ost doains; it is, in this sense, the ost robust voting rule. 7 Indeed, this gives us a characterization of ajorit rule different fro the one provided b Ma (1952) (see the corollar to the Theore). Notice that, unlike Ma, we appeal to no onotonicit condition (Ma requires positive responsiveness), but instead invoke aial robustness. Our Theore is closel related to a result obtained in Maskin (1995), but greatl iproves on that earlier result. Maskin s proposition requires two strong assuptions, one of which is particularl unpalatable. The first assuption is that the nuber of voters be odd. This is needed because Maskin (1995) deands transitivit for all preference profiles drawn fro a given doain (oddness is also needed for uch of the earl work on ajorit rule, e.g., Inada, 1969). And, as we will see below, 6 More accuratel, the hpothesis is that F differs fro ajorit rule for a regular preference profile belonging to a doain on which ajorit rule works well. 7 More precisel, an other aiall robust voting rule can differ fro ajorit rule onl for irregular profiles on an doain on which it works well (see the corollar to our Theore). 4

6 even when preferences are single-peaked, intransitivit is possible if the population splits eactl between two alternatives; an odd nuber of voters prevents this fro happening. To capture the idea that such a split is unlikel, we will work with a large nuber of voters and ask onl for generic transitivit. (Forall, we shall work with a continuu of voters, but it will becoe clear that we could alternativel deal with a finite nuber b defining generic transitivit to ean transitive with sufficientl high probabilit. In that case, we would not need to ipose oddness (a strong assuption, since it presuabl holds onl half the tie). Second, Maskin (1995) invokes the restrictive assuption that the voting rule F being copared with ajorit rule satisfies Pareto, anonit, and neutralit on an doain. This is quite undesirable because, although it accoodates certain ethods (such as the superajorit rules and the Pareto etension rules), it rules out such voting rules as the Borda count, pluralit voting, approval voting, and runoff voting. These are the ost coon alternatives in practice to siple ajorit rule, et fail to satisf neutralit on the unrestricted doain. We show that this assuption can be eliinated altogether. We proceed as follows. In section 2, we set up the odel. In section 3, we define our five properties, Pareto, anonit, neutralit, independence 5

7 of irrelevant alternatives, and generic transitivit forall. We also characterize when rank-order voting a ajor copetitor of ajorit rule satisfies all these properties. In section 4, we establish a lea, closel related to a result of Inada (1969), that characterizes when ajorit rule is genericall transitive. We use this lea in section 5 to establish our ain Theore on ajorit rule. We obtain our alternative to Ma s (1950) characterization as a corollar. Finall, in section 6 we discuss a few etensions. 2. The Model Our odel is in ost respects a standard social-choice fraework. Let X be the set of social alternatives (including alternatives that a turn out to be infeasible). For technical convenience, we take X to be finite with cardinalit ( 3). The possibilit of individual indifference often akes technical arguents in the social-choice literature a great deal essier (see for eaple, Sen and Pattanaik, 1969). We shall sipl rule it out b assuing that individual voters preferences can be represented b strict orderings (of course, with onl a finite nuber of alternatives, the assuption that a voter is not eactl indifferent between an two alternatives does not see ver strong). If R is a strict ordering, then, for an alternatives, X with, the notation " R " denotes is strictl 6

8 preferred to in ordering R. 8 Let R X be the set of all logicall possible strict orderings of X. We shall tpicall suppose that voters preferences are drawn fro soe subset 1 2 R R X. For eaple, for soe ordering (,,, ) of the social alternatives, R consists of single-peaked preferences (relative to this ordering) if, for all R R, whenever R i i + 1 for soe i, then R j j + 1 for all j > i, and whenever i+ 1Ri for soe i, then j+ 1R for all j < i. For the reason entioned in the Introduction (and elaborated on below), we shall suppose that there is a continuu of voters indeed b points in the unit interval [ 0,1 ]. A profile R on R is a apping R :0,1 [ ] R, where R ( i ) is voter i s preference ordering. Hence, profile R is a specification of the preferences of all voters. We shall use Lebesgue easure µ as our easure of the size of j voting blocs. 9 Given alternatives and and profile R, let { } (, ) = µ R( ) q i i. R 8 Forall, a strict ordering is a binar relation that is refleive, coplete, transitive, and antisetric (antisetr eans that if R and, then it is not the case that R ). 9 Because Lebesgue easure is not defined for all subsets of [ 0,1 ], we will restrict attention to profiles R such that, for all and,{ i R ( i) } is a Borel set. Call these Borel profiles. 7

9 Then qr (, ) is the fraction of the population preferring to in profile R. Let C be the set of refleive, coplete, binar relations (not necessaril transitive or strict) on X. A voting rule F is a apping that, for each profile R on RX (strictl speaking, we ust liit attention to Borel profiles see footnote 9 but henceforth we will not eplicitl state this qualification), assigns a relation F ( R ) C. F(R) can be interpreted as the social preference relation corresponding to R under F. More specificall, for an profile R and an alternatives, X, the notation F( R ) denotes that is sociall weakl preferred to under F ( R ). If both F( R ) and F( R ), we shall sa that is sociall indifferent to and denote this b F ( ) R. Finall, the notation " F( R ) " denotes that is not sociall weakl preferred to, given F and R. Hence, if F( R ) and F( R ), we shall sa that is sociall strictl preferred to under F ( R ), which we will usuall denote b F ( ) R., and R For eaple, suppose that F is siple ajorit rule. Then, for all, 8

10 ( ) if and onl if R(, ) R(, ) F R q q, i.e., is sociall weakl preferred to provided that the proportion of voters preferring to is no less than the proportion preferring to. As another eaple, consider rank-order voting. Given R R X, let vr ( ) be if is the top-ranked alternative of R, 1 if is second-ranked, and so on. That is, a voter with preference ordering R assigns points to her favorite alternative, 1 points to her net favorite, etc. Thus, given 1 profile R v ( ) ( ) dµ ( i) is alternative s rank-order score (the total nuber, 0 R i of points assigned to ) or Borda count. If for all,, and R, RO F is rank-order voting, then, 1 1 ( R ) ( ) ( ) µ ( ) ( ) ( ) µ ( ) 0 R i 0 R i RO F if and onl if v d i v d i, i.e., is sociall weakl preferred to if s Borda count eceeds that of. Speaking in ters of social preferences a see soewhat indirect because the Introduction depicted a voting rule as a wa of aking social choices. Because, as noted at the beginning of the section, the set of feasible alternatives a not be known in advance, we cannot sipl ake a voting rule a apping fro profiles to outcoes; the designated outcoe ight turn out to be infeasible. However, we could define a voting rule as a apping that to each profile R on RX assigns a choice function C ( ), which, 9

11 for each subset Y subset CY ( ) X (where Y is the available or feasible set), selects a Y ( CY ( ) consists of the optial alternatives in Y). 10 However, partl because it is less cubersoe working with preference relations than choice functions, there is a tradition going back to Arrow (1951) of taking the forer route. Furtherore, it is well known that there is a close connection between the two approaches. 11 In our setting, we shall take the stateent and are sociall indifferent to ean if is chosen and is also available, then ust be chosen too. Siilarl, is sociall strictl preferred to should be interpreted as if is available, then is not chosen. 3. The Properties We are interested in four standard properties that one a wish a voting rule to satisf. Pareto Propert on R: For all R on R and all, i, R ( i), then F( R ) and F( R ), i.e., F ( ) R. X with, if, for all In words, the Pareto propert requires that if all voters prefer to, then societ should also (strictl) prefer to. Virtuall all voting rules 10 Indeed, we took this approach in an earlier version of the paper 11 See, for eaple, Arrow (1959). 10

12 used in practice satisf this propert. In particular, ajorit rule and rankorder voting satisf it on the unrestricted doain R X. Anonit on R: Suppose that π :0,1 [ ] [ 0,1] is a easure-preserving perutation of [ 0,1 ] (b easure-preserving we ean that, for all Borel ( ) sets T [ 0,1, ] µ ( T) = µ π( T) ). If, for all on R ( ) π π R ( i) = R π ( i) for all i, then F( ) = F( ) R R. R, π R is the profile such that In words, anonit sas that social preferences should depend onl on the distribution of voters preferences and not on who has those preferences. Thus if we perute the assignent of voters preferences b π, social preferences should reain the sae. The reason for requiring that π be easure-preserving is to ensure that the fraction of voters preferring to be the sae for π R as it is for R. Anonit ebodies the principle that everbod s vote should count equall. 12 It is obviousl satisfied on R X b both ajorit rule and rankorder voting. Neutralit on R: For all profiles R and R on R and all alternatives,, w, z, if ( ) if and onl if ( ) R i wr i z for all i 12 Indeed, it is soeties called voter equalit (see Dahl, 1989). 11

13 then and ( R) if and onl if ( R ) F wf z ( ) if and onl if ( ) F R zf R w. In words, neutralit requires that the social preference between and should depend onl on the set of voters preferring and on that preferring, and not on what the alternatives and actuall are. As noted in the Introduction, this (standard) version of neutralit ebodies independence of irrelevant alternatives, the principle that the social preference between and should depend onl on voters preferences between and, and not on preferences entailing an other alternative: Independence of Irrelevant Alternatives (IIA) on R: For all profiles R and R on Rand all alternatives and, if ( ) if and onl if ( ) R i R i for all i, then ( ) if and onl if ( ) F R F R, and ( ) if and onl if ( ) F R F R. Clearl, ajorit rule satisfies neutralit on the unrestricted doain R X. Rank-order voting violates neutralit on R X because, as is well known, 12

14 it violates IIA on that doain. However, it satisfies neutralit on an doain R on which quasi-agreeent holds. Quasi-agreeent on R: Within each triple { z,, } X, there eists an alternative, sa, such that either (a) for all R, and all R, R and zr R R Rz R ; or (c) for all R R, either RRz or zrr. ; or (b) for In other words, quasi-agreeent holds on doain R if, for an triple of alternatives, all voters with preferences in R agree on the relative ranking of one of these alternatives: either it is best within the triple, or it is worst, or it is in the iddle. Lea 1: RO F satisfies neutralit on R if and onl if quasi-agreeent holds on R. Proof: See appendi. A binar relation C C is transitive if for all z,, X, C and Cz ipl that Cz. Transitivit deands that if is weakl preferred to and is weakl preferred to z, then should be weakl preferred to z. We shall define transitivit of a voting rule F as follows: Transitivit on R: F ( R ) is transitive for all profiles R on R. For our results on ajorit rule, we will, in fact, not require transitivit for all profiles in R but onl for alost all. To otivate this weaker requireent, let us first observe that, as entioned in the 13

15 Introduction, single-peaked preferences do not guarantee that ajorit rule is transitive for all profiles. Specificall, suppose that < < z and consider the profile [ 0, 1 ) [ 1,1] z 2 2 That is, we are supposing that half the voters (those fro 0 to 1 2 ) prefer to to z and that the other half (those fro 1 2 to 1) prefer to z to. Note that these preferences are certainl single-peaked relative to the linear arrangeent, < < z. However, the social preference relation under ajorit rule for this profile is not transitive: is sociall indifferent to, is sociall strictl preferred to z, et z is sociall indifferent to. We can denote the relation b: z z. Nevertheless, this intransitivit is a knife-edge phenoenon - - it requires that eactl as an voters prefer to as to, and eactl as an prefer to z as prefer z to. Thus, there is good reason for us to overlook it as pathological or irregular. And, because we are working with a continuu of voters, there is a foral wa in which we can do so, as follows. 14

16 Let S be a subset of (0, 1). A profile R on R is regular with respect to S (which we call an eceptional set) if, for all alternatives and, ( ) q, S. R That is, a regular profile is one for which the proportions of voters preferring one alternative to another all fall outside the specified eceptional set. Generic Transitivit on R: There eists a finite eceptional set S such that, for all profiles R on R that are regular with respect to S, F ( R) is transitive. In other words, generic transitivit requires that social preferences be transitive onl for regular profiles, ones where the preference proportions do not fall into soe finite eceptional set. For eaple, as Lea 2 below iplies, ajorit rule is genericall transitive on a doain of single-peaked preferences because if the eceptional set consists of the single point 1 2 i.e., S = { 1 2} social preferences are then transitive for all regular profiles. In view of the Condorcet parado, ajorit rule is not genericall transitive on doain genericall transitive on with eceptional set S R X. B contrast, rank-order voting is not onl R X but full transitive (i.e., genericall transitive = φ ). We shall sa that a voting rule works well on a doain R if it satisfies the Pareto propert, anonit, neutralit, and generic transitivit on that 15

17 doain. Thus, in view of our previous discussion, ajorit rule works well, for eaple, on a doain of single-peaked preferences, whereas rank-order voting works well on a doain with quasi-agreeent. Although we are considering onl generic transitivit, we could easil accoodate generic versions of the other conditions too without changing an of the conclusions (indeed, we did eactl that in an earlier draft of this paper). The reason for concentrating onl on transitivit is that, to our knowledge, no coonl-used voting rule has nongeneric failures ecept with respect that propert. 4. Generic Transitivit and Majorit Rule We will show below (see the Theore) that ajorit rule works well on ore doains than (essentiall) an other voting rule. To establish this result, it will be useful to have a characterization of precisel when ajorit rule works well, which aounts to asking when ajorit rule is genericall transitive. We alread suggested in the previous section that a single-peaked doain ensures generic transitivit. And we noted in the Introduction that the sae is true when the doain satisfies the propert that, for ever triple of alternatives, there is one that is never in the iddle. But these are onl 16

18 sufficient conditions for generic transitivit; what we want is a condition that is both sufficient and necessar. To obtain that condition, note that, for an three alternatives,, z, there are si logicall possible strict orderings, which can be sorted into two Condorcet ccles 13 : z z z z z z ccle 1 ccle 2 We shall sa that a doain R satisfies the no-condorcet-ccle propert 14 if it contains no Condorcet ccles. That is, for ever triple of alternatives, at least one ordering is issing fro each of ccles 1 and 2 (ore precisel for each triple { z,,, } there do not eist orderings R, R,R in R that, when restricted to { z,,, } generate ccle 1 or ccle 2). Lea 2: Majorit rule is genericall transitive on doain R if and onl if R satisfies the no-condorcet-ccle propert. 15 Proof: If there eisted a Condorcet ccle in R, then we could reproduce the Condorcet parado. Hence, the no-condorcet-ccle propert is clearl necessar. 13 We call these Condorcet ccles because the constitute preferences that give rise to the Condorcet parado 14 Sen (1966) introduces this condition and calls it value restriction. 15 For the case of an odd and finite nuber of voters, Inada (1969) establishes that the no-condorcet-ccle propert is necessar and sufficient for ajorit rule to be transitive. 17

19 To show that it is sufficient, we ust deonstrate, in effect, that the Condorcet parado is the onl thing that can interfere with ajorit rule s generic transitivit. To do this, let us suppose that F is not genericall transitive on doain R. Then, in particular, if we let S = { 1 2} there ust eist a profile R on R that is regular with respect to { 1 2 } but for which F ( R ) is intransitive. That is, there eist,, z such that ( R) ( R) ( R ), with at least one strict preference. But because R F F zf is regular with respect to { 1 }, F ( ) 2 (1) ( ) 1 q > R, 2, X R iplies that that is, over half the voters prefer to. Siilarl, F ( R ) z iplies that (2) ( ) 1 q z > R, 2, eaning that over half the voters prefer to z. Cobining (1) and (2), we conclude that there ust be soe voters in R who prefer to to z, i.e., (3) z R.16 B siilar arguent, it follows that z z, R. Hence, Rcontains a Condorcet ccle, as was to be shown. 16 To be precise, forula (3) sas that there eists an ordering in R in which is preferred to and is preferred to z. However, because F satisfies IIA we can ignore the alternatives other than z.,, 18

20 Q.E.D. It is eas to check that a doain of single-peaked preferences satisfies the no-condorcet-ccle propert. Hence, Lea 2 iplies that ajorit rule is genericall transitive on such a doain. The sae is true of the doain we considered in the Introduction in connection with French elections. 5. The Robustness of Majorit Rule We can now state our ain finding: Theore: Suppose that voting rule F works well on doain R. Then, ajorit rule well on doain F works well on R too. Conversel, suppose that F works R. Then, if there eists profile respect to F s eceptional set, such that (4) F( ) F ( ) there eists a doain R on which R R, R on R, regular with F works well, but F does not. Reark: Without the requireent that the profile R for which F and F differ belongs to a doain on which ajorit rule works well, the converse assertion above would be false. In particular, consider a voting rule that coincides with ajorit rule ecept for profiles that contain a Condorcet ccle. It is eas to see that such a rule works well on an doain for which ajorit rule does because it coincides with ajorit rule on such a doain. 19

21 Proof: Suppose first that F works well on R. If, contrar to the theore, F does not work well on R, then, fro Lea 2, there eists a Condorcet ccle in R: z (5), z, z R, for soe,, z X. Let S be the eceptional set for F on R. Because S is finite (b definition of generic transitivit), we can find an integer n such that, if we divide the population into n equal groups, an profile for which all the voters in each particular group have the sae ordering inr ust be regular with respect to S. 1 Let 0, n be group 1, ( 1, 2 n n n be group 2,, and ( 1 n,1 be group n. Consider a profile R 1 on R such that all voters in group 1 prefer to and all voters in the other groups prefer to. That is, the profile is (7) 1 2 n. Fro (5), such a profile eists on R. Fro neutralit (ipling IIA), the social preference between and under F ( R 1 ) does not depend on voters preferences over other alternatives. 20

22 There are three cases: either (i) is sociall strictl preferred to under F( R 1 );( ii) is sociall indifferent to under F( R 1 ); or ( iii) is sociall strictl preferred to under F ( R 1 ). Case (i): F ( R1 ) Consider a profile 1 R on R in which all voters in group 1 prefer to to z; all voters in group 2 prefer to z to ; and all voters in the reaining groups prefer z to to. That is, n (8) R 1 =. z z z z Notice that, in profile R 1, voters in group 1 prefer to z and that all other voters prefer z to. Hence, neutralit and the case (i) hpothesis ipl that z ust be sociall strictl preferred to under F ( R 1 ) 17, i.e., (9) F ( R1 ) z. Observe also that, in R 1, voters in group 2 prefer to and all other voters prefer to. Hence fro anonit and neutralit and the case (i) 17 This is not quite right because we are not specifing how voters rank alternatives other than,, and z. But fro IIA, these other alternatives do not atter for the arguent. 21

23 hpothesis, we conclude that ust be sociall strictl preferred to under F ( R 1 ), i.e., (10) F ( R1 ). Now (9), (10), and generic transitivit ipl that z is sociall strictl preferred to under F ( R 1 ), i.e., (11) F ( R1 ) z. But (8), (11), and neutralit ipl for an profile such that n, z z z z z ust be sociall strictl preferred to. Hence, fro neutralit, for an profile R 2 on Rsuch that (12) n, ust be sociall strictl preferred to, i.e., (13) F ( R2 ). 22

24 That is, we have shown that if is sociall strictl preferred to when just one out of n groups prefers to (as in (7)), then is again sociall strictl preferred to when two groups out of n prefer to (as in (12)). Now choose R 2 on R so that n (14) R 2 =. z z z z z Arguing as above, we can use (12) (14) to show that is sociall strictl preferred to if three groups out of n prefer to. Continuing iterativel, we conclude that is strictl sociall preferred to even if n 1 groups out of n prefer to, which, in view of neutralit, violates the case (i) hpothesis. Hence case (i) is ipossible. Case (ii): F ( R1 ) But fro the case (i) arguent, case (ii) leads to the sae contradiction as before. Hence we are left with Case (iii): ( R1 ) F Consider a profile ˆR on R such that ˆ 1 n 1 n R=. z z z 23

25 Fro anonit, neutralit and the case (iii) hpothesis, we conclude that is sociall indifferent to and is sociall indifferent to z under F ( R ˆ ), i.e., (15) and (16) F ( R ˆ ) F ( R ˆ ) z.. But the Pareto propert iplies that is sociall strictl preferred to z under F ( R ˆ ), which together with (15) and (16) contradicts generic transitivit. We conclude that case (iii) is ipossible too, and so after all, as claied. which R R F ust work well on R Turning to the converse, suppose that there eists doain R on F works well. If F does not work well on R too, we can take = to coplete the proof. Hence, assue that F works well on eceptional set S and that there eists regular profile F ( R ) F ( R ). Because F ( R ) and F ( ) with (17) 1 α > α, and alternatives, differentl fro F ( R ) X such that q (, ) = 1 α. Fro (17), we have R on R with R such that R differ, there eist α ( 0,1) R and F ( ) R ranks and 24

26 F ( R ). We thus infer that (18) F( ) Because F is neutral on R. orderings R, R R such that (19) R and R, we can assue that R. Furtherore, because F is anonous on R consists of just two R, we can write R as (20) [ 0, α) [ α,1] R =, R R so that voters between 0 and α have preferences R, and those between α and 1 have R. Let us assue for the tie being that F satisfies the Pareto propert, anonit, and neutralit on the unrestricted doain z {, } and profile R such that R X. Consider (21) R = [ 0, α) [ α,1 α) [ 1 α,1] z z z. 18 Then fro (18)-(21), anonit, and neutralit, we have 18 We have again left out the alternatives other than,, z, which we are entitled to do b IIA. To ake atters siple, assue that the orderings of R are all the sae for these other alternatives. Suppose z,,. furtherore that, in these orderings,,, z are each preferred to an alternative not in { } 25

27 (22) ( R ) ( R ) F and F z. Fro the Pareto propert, we have (23) F ( R ). z But, b construction, R is regular with respect to F s eceptional set. Thus, (22) and (23) together ipl that F violates generic transitivit on zz R= z,,. Yet, fro Lea 2, iplies that R is a doain on which F is genericall transitive on R, which F works well but F does not. Thus, we are done in the case in which F alwas satisfies the Pareto propert, anonit and neutralit. However, if F does not alwas satisf these properties, then we can no longer infer (22) fro (18)-(21), and so ust argue in a different wa. Consider R and R of (19). Suppose first that there eists alternative z X such that (24) zr and zr. Let w be the alternative iediatel below z in ordering R. If w be the strict ordering that is identical to R ecept that w and z are now, let R interchanged (so that wr z ). B construction of R, the doain { R, R, R } does not contain a Condorcet ccle, and so, fro Lea 2, F works well 26

28 on this doain. Hence, we can assue that F works well on this doain too (otherwise, we are done). Notice that neutralit of F and (18) then ipl that if we replace R b R in profile R (to obtain profile R ) we ust have (25) F( R ). Now, if w is the alternative iediatel below z in R and w, we can perfor the sae sort of interchange as above to obtain R and R and so conclude that F and F work well on { R, R, R } and that (26) F( ) R. B such a succession of interchanges, we can, in effect, ove z downward while still ensuring that F and F work well on the corresponding doains and that the counterparts to (18), (25) and (26) hold. The process coes to end, however, once the alternative iediatel below z in R (or R, R, etc.) is. Furtherore, this ust happen after finitel an interchanges (since X is finite). Hence, we can assue without loss of generalit that w = (i.e., that is iediatel below z in R ). Let R be the strict ordering that is identical to R ecept that and z (which we are assuing are adjacent in R ) are now interchanged. Fro Lea 2, F works well on R= { R, R, R }, and we can suppose that F does 27

29 too (otherwise, we are done). Hence, fro the sae arguent we used for R above, we can conclude that (27) ( R ) and ( R ) and F F z (28) F ( R ) z, where R is the profile [ 0, α) [ α,1 α) [ 1 α,1] R R R, contradicting the generic transitivit of case where (24) holds. Net, suppose that there eists z X such that (29) Rz and R z. F on R. Thus, we are done in the But this case is the irror iage of the case where (24) holds. That is, just as in the previous case we generated R with (30) R zr through a finite succession of interchanges in which z oves downwards in R, so we can now generate R satisfing (30) through a finite succession of interchanges in which z oves upwards in R. If we then take R= { R, R, R }, we can furtherore conclude, as when (24) holds, that F 28

30 and F work well on R. But, paralleling the arguent for R, we can show that and ( R ) and ( R ) F zf F ( R ), z where R is the profile ipling that F ( R ) [ 0, α) [ α,1 α) [ 1 α,1], R R R is intransitive. This contradicts the conclusion that F works well on R, and so again we are done. such that Finall, suppose that there eists z X (31) zr and R zr. As in the preceding case, we can ove z upwards in R through a succession of interchanges. Onl this tie, the process ends when z and are interchanged to generate ˆR such that (32) zrˆ Rˆ. As in the previous cases, we can conclude that F and { R, R, R ˆ }. Take R ˆ such that F work well on 29

31 Then, as in the arguents about [ 0, α) [ α,1 α) [ 1 α,1] ˆ R =. R Rˆ R R and R, we infer that ( ˆ F R ) is intransitive, a contradiction of the conclusion that F works well on { R, R, R ˆ }. This copletes the proof when (31) holds. The reaining possible cases involving z are all repetitions or irror iages of one or another of the cases alread treated. As a siple illustration of Theore 1, let us see how it applies to rank-order voting. If X = { z,, }, Lea 1 iplies that eaple, on the doain Q.E.D. RO F works well, for, z z. And, as Theore 1 guarantees, F also works well on this doain, since it obviousl does not contain a Condorcet ccle. Conversel, on the doain z (*) R=, z, z, F RO ( R) F ( R ) for an profile R in which the proportion of voters with ordering z is α, the proportion with ordering z is β and (**) 1< 2α < β

32 (if (**) holds, then F RO and F rank and differentl). But, fro Lea 2, F works well on R given b (*). Hence, fro Lea 1, R constitutes a doain on which Theore. F works well but RO F does not, as guaranteed b the We alread entioned Ma s (1952) characterization of ajorit rule in the Introduction. In view of our Theore, we can provide an alternative characterization. Specificall, call two voting rules F and F genericall the sae on doain R if there eists a finite set S ( 0,1) such that ( R) = ( R ) for all R on R for which R (, ) F F q S. Call F aiall robust if there eists no other voting rule that (i) works well on ever doain on which F works well and (ii) works well on soe doain on which F does not work well. The Theore iplies that ajorit rule is essentiall uniquel the aiall robust voting rule: Corollar: Majorit rule is aiall robust, and an other aiall robust voting rule F is genericall the sae as ajorit rule on an doain on which F or ajorit rule works well. 6. Etensions The setr inherent in neutralit is often a reasonable and desirable propert we would presuabl want to treat all candidates in a presidential election the sae. However, there are also circustances in 31

33 which it is natural to favor certain alternatives. The rules for changing the U.S. Constitution are a case in point. The have been deliberatel devised so that, at an tie, the current version of the Constitution the status quo is difficult to revise. In related work (see Dasgupta and Maskin, 2004), we show that when neutralit is replaced b the weaker condition of IIA (and the requireent that ties be broken consistentl is also iposed), then unaniit rule with an order of precedence 19 (the rule according to which is chosen over if it precedes in the order of precedence, unless everbod prefers to ) supplants ajorit rule as the ost robust voting rule. We have assued throughout that voting rules ust satisf anonit; this is part of the definition of working well. But in practice there are an circustances in which voters are, for good reason, not treated equall. Think, for instance, of the weighted voting sste used b the council of the European Union, where ore populous eber nations have larger weights. Such eaples suggest that it is worthwhile eaining what becoes of our results when anonit is relaed. Now, if we were to eliinate anonit altogether as a requireent, nothing resebling our Theore would continue to hold; instead, a 19 For discussion of this voting rule in a political setting see Buchanan and Tullock (1962). 32

34 dictatorship (in which one particular voter s preferences deterine social preferences) would now be the ost robust voting rule, since it satisfies neutralit, the Pareto propert, and transitivit on the unrestricted doain R X. However, eploring what would happen if we replaced anonit with weaker conditions sees useful. Consider, for eaple, the properties of voting-bloc responsiveness: Voting-Bloc Responsiveness on R: For an V [ 0,1] with ( V ) 0 eist profiles R and R on R such that R( i) = R ( i) for all i V but ( ) ( ) F R F R. µ >, there In words, voting-bloc responsiveness requires that ever bloc of voters of positive size can soeties affect the social ranking. The condition is clearl satisfied b an voting rule for which the Pareto propert and anonit hold. But it also holds for an non-anonous voting rules, such weighted ajorit rule, defined as follows: Given a positivevalued, Lebesgue-easurable function w on [ 0,1], w F is weighted ajorit w rule with weight w, if for all alternatives,, and profiles, F ( ) and onl if R R if wid ( ) µ ( i) { R ( ) } i j j wid ( ) µ ( i) { R( ) } i j j. 33

35 Analogous to our Theore, it can be shown (see Dasgupta and Maskin, 1998) that if a voting rule satisfies the Pareto propert, neutralit, generic transitivit, and voting-bloc responsiveness on a doain R then, for an w, w F also satisfies those properties on R. We conjecture that the converse w holds too. That is, if, for all w, F( ) F ( ) doain R where doain R on which R R for a regular profile R on w F satisfies these four properties, then there eists a w F satisfies all the properties, but F does not. Another interesting etension to consider is strategic voting. It has long been known that there is a close connection between the proble of defining reasonable social preferences on a doain of preferences and that finding voting rules iune fro strategic anipulation b voters (see Maskin 1979 and Kalai and Muller 1977). Because we have assued a continuu of voters, sincere voting is autoaticall copatible with individual incentives for an voting rule in which a single voter s ordering akes no difference for social preferences. But the sae is not true for coalitions (voting blocs). We conjecture that a counterpart to our Theore can be derived when independence of irrelevant alternatives is replaced with the requireent that a voting rule be coalitionall strateg-proof. 34

36 Appendi Lea 1: For an doain R, quasi-agreeent holds on R. RO F satisfies neutralit on R if and onl if Proof: Assue first that quasi-agreeent holds on R. We ust show that RO F satisfies neutralit on R. Consider profiles R and R on R and alternatives,, w, and z such that (A1) ( ) if and onl if ( ) We ust show that R i wr i z for all i. RO RO (A2) ( R) if and onl if ( R ) and F wf z RO RO (A3) ( ) if and onl if ( ) If, for all i, R ( ) F R zf R w. i, then because RO F satisfies the Pareto propert, we have RO F ( R) and RO F ( R ) w z, in accord with (A2) and (A3). Assue, therefore, that if we let and { R( ) } and R ( ) { } I = i i I = j j w { ( ) } and z ( ) { } I = iwr i z I = jr j w,

37 then I, I, I, and I are nonept. w z We clai that (A4) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) for all and R i Ri = Rj R j. v v v v i I j I Now, (A4) holds because, if there eist i I and z X such that R ( i ) z, then quasi-agreeent iplies Siilarl, we have R( i) R( j) for all i I and for all j I. z z (A5) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) for all and i i = j j w z. R R R R v w v z v z v w i I j I But fro (A4) and (A5) and the definition of as required. RO F, we obtain (A2) and (A3), Net, suppose that quasi-agreeent does not hold on doain R. Then there eist alternatives,, z and orderings RR R, such that (A6) R z and 2

38 (A7) R z. Fro (A6) and (A7) we have (A8) vr ( ) vr( ) < vr ( ) vr ( ) (A9) vr ( ) vr( z) v R ( z) vr ( ) Choose Then fro (A8) and (A9) >. [ 0, 1 2) [ 1 2,1] R =. R R (A10) RO F ( R) z. But, b construction, we have, for all i, and ( ) if and onl if ( ) R i R i z ( ) if and onl if ( ) R i zr i. Thus, if neutralit held we should have which contradicts (A10). RO ( ) if and onl if ( ) RO F R zf R, Q.E.D. 3

39 References Arrow, K.J. (1951), Social Choice and Individual Values (New York: John Wile). Arrow, K.J. (1959), Rational Choice Functions and Orderings, Econoica, 26. Black, D. (1948), On the Rationale of Group Decision-aking, Journal of Political Econo, 56(1), Buchanan, J. and G. Tullock (1962), The Calculus of Consent (Ann Arbor: Universit of Michigan Press). Capbell, D. and J. Kell (2002), Ipossibilit Theores in the Arrovian Fraework, in K. Arrow, A. Sen, and K. Suzuura, Handbook of Social Choice and Welfare, vol. 1, (Asterda: Elsevier), pp Condorcet (1785), Essai sur l application de l analse à la pluralité des voi, (Paris: Iprierie Roale). Dahl, R. (1989), Deocrac and Its Critics, (New Haven: Yale Universit Press). Dasgupta, P. and E. Maskin (1998), On the Robustness of Majorit Rule, ieo. Dasgupta, P. and E. Maskin (2004) On the Robustness of Unaniit Rule, ieo. Inada, K. (1969), On the Siple Majorit Rule Decision, Econoetrica, 37(3), Kalai, E. and E. Muller (1977), Characterization of Doains Aditting Nondictatorial Social Welfare Functions and Nonanipulable Voting Procedures, Journal of Econoic Theor, 16:

40 Maskin, E. (1979), Fonctions de Préférence Collective Définies sur des Doaines de Préférence Individuelle Souis à des Constraintes, Cahiers du Seinaire d'econoétrie, (Paris: Centre National de la Recherche Scientifique), pp Maskin, E. (1995), Majorit Rule, Social Welfare Functions, and Gae Fors, in K. Basu, P.K. Pattanaik and K. Suzuura, eds., Choice, Welfare and Developent (Oford: Clarendon Press). Ma, K. (1952), Set of Necessar and Sufficient Conditions for Siple Majorit Decisions, Econoetrica, 20 (4), Sen, A. (1966), A Possibilit Theore on Majorit Decisions, Econoetrica, 34(2), Sen, A. (1970), Collective Choice and Social Welfare, (San Francisco: Holden Da). Sen, A. and P. Pattanaik (1969), Necessar and Sufficient Conditions for Rational Choice under Majorit Decision, Journal of Econoic Theor, 1(2),