On The Robustness of Majority Rule

Transcription

1 On The obustness of Majorit ule The Harvard counit has ade this article openl available. Please share how this access benefits ou. Your stor atters. Citation Published Version Accessed Citable Link Ters of se Dasgupta, Partha, and Eric Maskin On The obustness of Majorit ule. Journal of the European Econoic Association 6 (5) (Septeber): doi: /jeea doi: /jeea April 29, :00:21 PM EDT This article was downloaded fro Harvard niversit's DASH repositor, and is ade available under the ters and conditions applicable to Open Access Polic Articles, as set forth at (Article begins on net page)

2 On the obustness of Majorit ule and ule b Consensus Partha Dasgupta * and Eric Maskin ** Januar 1998 Current version: Septeber 2003 Abstract We show that siple ajorit rule satisfies the Pareto propert, anonit, neutralit, and (generic) transitivit on a bigger class of preference doains than an other voting rule. If we replace neutralit in the above list of properties with independence of irrelevant alternatives, then the corresponding conclusion holds for unaniit rule (rule b consensus). * acult of Econoics and Politics, niversit of Cabridge ** School of Social Science, Institute for Advanced Stud and Departent of Econoics, Princeton niversit This research was supported b grants fro the Beijer International Institute of Ecological Econoics and the.s. National Science oundation. We thank Salvador Barberá, rançois Maniquet, and Willia Thoson for helpful coents on an earlier version.

3 1. Introduction A voting rule is a ethod for choosing fro a set of social alternatives on the basis of voters 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) 2. 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 1 or convenience, we will oit the odifier siple when it is clear that we are referring to siple ajorit rule rather to the an variants, such as the superajorit rules. 2 In fact, Ma (1952) established that ajorit rule is the unique voting rule satisfing the Pareto propert, anonit, and neutralit, and a fourth propert called positive responsiveness if alternative is chosen (perhaps not uniquel) for a given configuration of voters preferences and the onl change that is then ade to those preferences is to ove up in soe voters preference ordering, is now uniquel chosen. Without positive responsiveness, there are an voting rules including all the superajorit rules that satisf the properties. We shall coe back to Ma s characteriation in section 5. 1

4 three voters 1, 2, 3, three alternatives,,, and that voters preferences are as follows: (i.e., voter 1 prefers to to, voter 2 prefers to to, and voter 3 prefers to to ). Then, as Condorcet noted, a two-thirds ajorit prefers to, to, and 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 in both directions fro his favorite alternative, then, following Black (1948), ajorit rule is transitive for (alost) all 3 configurations of voters preferences. Alternativel, suppose that, for ever three alternatives, there is one that no voter ranks in the iddle. This propert, called liited agreeent (see Inada 1969, Sen and Pattanaik 1969), sees to have held in recent rench presidential elections, where the Gaullist and Socialist candidates have not inspired uch passion, but the National ront candidate, Jean-Marie Le Pen, has attracted either revulsion or adiration, i.e., everbod ranks hi either first or last. Whether or not this pattern of preferences has been good for rance is open to debate, but it is certainl good for ajorit rule: liited agreeent, like single-peakedness, ensures transitivit (alost alwas). 3 We clarif what we ean b alost all in section 2. 2

5 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 We show that the answer to this question is no. Specificall, we establish (Theore 1) that if a given voting rule works well on a doain of preferences, then ajorit rule works well on that doain too. Conversel, if differs fro ajorit rule 6, there eists soe other doain on which ajorit rule works well and does not. Thus essentiall ajorit rule is uniquel the voting rule that works well on the ost doains; it is, in this sense, the ost robust voting rule. This propert can be viewed as a characteriation of ajorit rule copleenting the one given b Ma (1952) (for ore on this, see the discussion and corollar following Theore 1). Theore 1 strengthens a result obtained in Maskin (1995). That earlier proposition requires two rather strong auiliar assuptions: 4 Our forulation of neutralit (see section 3) which is, in fact, the standard forulation (see Sen, 1971) incorporates 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. We could instead have decoposed neutralit into two separate properties: (i) setr with respect to alternatives, and (ii) independence of irrelevant alternatives. 5 It is eas to find voting rules that satisf three out of our four properties on all doains of preferences. or eaple, two-thirds ajorit rule (which dees two alternatives as sociall indifferent unless one garners at least a two-thirds ajorit against the other) satisfies Pareto, anonit, and neutralit on an doain. Siilarl, rank-order voting (see below) satisf Pareto, anonit, and generic transitivit on an doain. 6 More precisel, the hpothesis is that differs fro ajorit rule in soe open neighborhood of preference configurations belonging to a doain on which ajorit rule works well. 3

6 The first is that the nuber of voters be odd. This assuption is needed because Maskin (1995) deands transitivit for all preference configurations drawn fro a given doain. And as we will see below, even when preferences are single-peaked, intransitivit is possible if the population splits eactl between two preference orderings; an odd nuber of voters prevents this fro happening. To capture the idea that such a split is unlikel, we will work with a continuu of voters and ask onl for generic transitivit. Second, to prove the second half of the proposition, Maskin (1995) akes the strong assuption that the voting rule being copared with ajorit rule satisfies Pareto, anonit, and neutralit on an doain. We show that this assuption can be dropped. Although treating all alternatives alike as neutralit entails is a natural constraint in an political and econoic settings, it is not alwas a reasonable assuption. or eaple, there are cases in which we a wish to treat the status quo differentl fro other alternatives. or that reason, it is of soe interest to investigate which voting rule works best when neutralit is replaced b the weaker assuption of independence of irrelevant alternatives. Our second ajor finding (Theore 2) establishes that, in this odified scenario (where we also ipose a ild tie-break consistenc requireent), unaniit rule with an order of precedence is uniquel the ost robust voting rule. To define this rule, fi an ordering of the alternatives, interpreted as the order of precedence. Then, between two alternatives, the rule will choose the one earlier in the ordering unless voters unaniousl prefer the other alternative. naniit rule with an order of precedence thus corresponds 4

7 to the sequential protocol that a coittee ight follow were it not willing to replace the status quo with another alternative ecept b consensus. We proceed as follows. In section 2, we set up the odel. In section 3, we define our four properties, Pareto, anonit, neutralit, and generic transitivit forall. We also characterie 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 Sen and Pattanaik (1969) that characteries when ajorit rule is genericall transitive. We use this lea in section 5 to establish our ain result on ajorit rule. inall, we prove the corresponding result for unaniit rule in section The Model Our odel is in ost respects a standard social-choice fraework. Let X be the set of social alternatives. or 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 are drawn fro a set of strict orderings, where is a subset of set of all logicall possible strict orderings of X. or an ordering X and an X, the alternatives, X, the notation denotes is preferred to in ordering. or eaple, if we can arrange the social alternatives fro least to greatest, i.e., 1 < 2 < <, 7 then consists of single-peaked preferences (relative to this 7 We are using the ters least and greatest figurativel. All we ean is that there is soe linear order of the alternatives, e.g., how the line up on the left-right ideological spectru. 5

8 arrangeent) if, for all, whenever i i + 1 for soe i, then j j + 1 for all j > i, and whenever i+ 1i for soe i, then j+ 1 j for all j < i. or 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 on is a apping :0,1 [ ], where ( i ) is voter i s preference ordering. Hence, profile is a specification of the preferences of all voters. We shall use Lebesgue easure µ as our easure of the sie of voting blocs. Given alternatives and and profile, let Then q (, ) { } (, ) = µ ( ) q i i. is the fraction of the population preferring to in profile. Let C be the set of coplete, binar relations (not necessaril transitive or strict) on X. A voting rule is a apping that, for each profile, assigns a relation ( ) C. () can be interpreted as the social preference relation corresponding to under. More specificall, for an profile and an alternatives, X, the notation ( ) denotes that is sociall weakl preferred to under ( ) if is chosen and is also available, then is chosen too. or eaple, suppose that is siple ajorit rule. Then, ( ) ( ) ( ) if and onl if q, q,.. This eans that 6

9 As another eaple, consider rank-order voting. Given (strict) ordering of X, let ( ) v be if is the top-ranked alternative of, 1 if is second-ranked, and so on. That is, a voter with preference ordering assigns points to her favorite alternative, 1 1 points to her net favorite, etc. Thus, given profile v ( ) ( ) dµ ( i) is, 0 i alternative s rank-order score (the total nuber of points assigned to ) or Borda count. If O is rank-order voting, then 1 1 ( ) ( ) ( ) µ ( ) ( ) ( ) µ ( ) 0 i 0 i O if and onl if v d i v d i. The notation ( ) " " denotes that is not sociall weakl preferred to, given and. Hence, if ( ) and ( ) preferred to under ( ), which we will usuall denote b ( ). And if both ( ) and ( ) denote this b 3. The Properties satisf. ( )., we shall sa that is sociall strictl, we shall sa that is sociall indifferent to and We are interested in four standard properties that one a wish a voting rule to Pareto Propert on : or all on and all, X, if, for all i, ( i), then ( ) and ( ), i.e., 7

10 ( ). In words, the Pareto propert requires that if all voters prefer to, then societ should also (strictl) prefer to. Virtuall all voting rules used in practice satisf this propert. In particular, ajorit rule and rank-order voting satisf it on the unrestricted doain X. Anonit on : Suppose that π :0,1 [ ] [ 0,1] is a easure-preserving perutation of [ 0,1 ] (b easure-preserving we ean that, for all T [ 0,1, ] µ ( T) µ ( π( T) ) = ). If, for all, π π is the profile such that ( i) = π ( i) π ( ) for all i, then ( ) = ( ). 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 as it is for. Anonit ebodies the principle that everbod s vote should count equall. It is obviousl satisfied on X b both ajorit rule and rank-order voting. Neutralit on : or all profiles and on and all alternatives,, w,, if then ( ) if and onl if ( ) i w i for all i and ( ) if and onl if ( ) w 8

11 ( ) if and onl if ( ). In words, neutralit requires that the social preference between and should depend onl on the proportions of voters preferring and 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 : or all profiles and on and all alternatives and, if ( ) if and onl if ( ) i i for all i, then ( ) if and onl if ( ). Clearl, ajorit rule satisfies neutralit on the unrestricted doain X. ankorder voting violates neutralit on X because, as is well known, it violates IIA on that doain. However, it satisfies neutralit on an doain on which quasi-agreeent holds. Quasi-agreeent on : Within each triple {,, } X, there eists an alternative, sa, such that either (a) for all, and ; or (b) for all, and ; or (c) for all, either or. 9

12 In other words, quasi-agreeent holds on doain if, for an triple {,, }, anbod with preference in either agrees, that, sa, is best aong the triple, or that is worst, or that is in the iddle. Lea 1: O satisfies neutralit on if and onl quasi-agreeent holds on. Proof: See appendi. An ordering is transitive if for all,, X, and ipl that. Transitivit deands that if is weakl preferred to and is weakl preferred to, then should be weakl preferred to. Transitivit on : ( ) is transitive for all profiles on. or our results on ajorit rule we will, in fact, not require transitivit for all profiles in but onl for alost all. To otivate this weaker requireent, let us first observe that, as entioned in the introduction, single-peaked preferences do not guarantee that ajorit rule is transitive for all profiles. Specificall, suppose that < < and consider the profile [ 0, 1 ) [ 1,1] 2 2 That is, we are supposing that half the voters (those fro 0 to 1 2 ) prefer to to and that the other half (those fro 1 2 to 1) prefer to to. Note that these preferences are certainl single-peaked relative to the linear arrangeent, < <. However, the social preference relation under ajorit rule for this profile is not transitive: is sociall indifferent to, is sociall strictl preferred to, et is sociall indifferent to. We can denote the relation b: 10

13 . 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 as prefer 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. Let S be a subset of (0, 1). A profile on is regular with respect to S (which we call an eceptional set) if, for all alternatives and, ( ) q, S. 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 : There eists a finite eceptional set S such that, for all profiles on that are regular with respect to S, ( ) is transitive. In other words, generic transitivit requires onl that social preferences be transitive for regular profiles, ones where the preference proportions do not fall into soe finite eceptional set. or eaple, 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. doain In view of the Condorcet parado, ajorit rule is not genericall transitive on X. B contrast, rank-order voting is not onl genericall transitive on full transitive (i.e., genericall transitive with eceptional set S = φ ). X but 11

14 We shall sa that a voting rule works well on a doain if it satisfies the Pareto propert, anonit, neutralit, and generic transitivit on that doain. Thus, in view of our previous discussion, ajorit rule works well on a doain of single-peaked preferences, whereas rank-order voting works well on a doain with quasi-agreeent. 4. Generic Transitivit and Majorit ule We will show below (Theore 1) that ajorit rule works well on or ore doains than (essentiall) an other voting rule. To establish this result, it will be useful to have a characteriation of precisel when ajorit rule works well, which aounts to asking when ajorit rule is genericall transitive. We have alread seen 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 liited agreeent. But single-peakedness and liited agreeent are onl 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,,, there are si logicall possible strict orderings, which can be sorted into two Condorcet ccles 8 : ccle 1 ccle 2 We shall sa that a doain satisfies the no-condorcet-ccle propert 9 if it contains no Condorcet ccles. That is, for each triple of alternatives at least one ordering is issing 8 We call these Condorcet ccles because the constitute preferences that give rise to the Condorcet parado 9 Sen and Pattanaik (1969) refer to this condition as etreal restriction. 12

15 fro each of ccles 1 and 2 (ore precisel for each triple {,, }, there do not eist orderings {,, } in that, when restricted to {,, }, generate ccle 1 or ccle 2). Lea 2: Majorit rule is genericall transitive on doain if and onl if satisfies the no-condorcet-ccle propert. 10 Proof: If there eisted a Condorcet ccle in, then we could reproduce the Condorcet parado. Hence, the no-condorcet-ccle propert is clearl necessar. 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 is not genericall transitive on doain. Then, in particular, if we let S = { 1 2} there ust eist a profile on that is regular with respect to { 1 2 } but for which ( ) ( ) ( ) ( ) is intransitive. That is, there eist,, X such that, with at least one strict preference. But because is regular with respect to { 1 }, ( ) 2 (1) ( ) 1 q >, 2, iplies that that is, over half the voters prefer to. Siilarl, ( ) (2) ( ) 1 q >, 2, iplies that eaning that over half the voters prefer to. Cobining (1) and (2), we conclude that there ust be soe voters in who prefer to to, i.e., (3) or the case of an odd and finite nuber of voters, Sen and Pattanaik (1969) establish that the no- Condorcet-ccle propert is necessar and sufficient for ajorit rule to be transitive. 13

16 B siilar arguent, it follows that,. Hence, contains a Condorcet ccle, as was to be shown. Q.E.D. 5. The obustness of Majorit ule To establish our ain finding about ajorit rule, we need one final concept. { } Given profile and ordering let w( ; ) = i ( i ) =. That is, ( ; ) w is the proportion of voters in profile with ordering. or ε > 0, let the ε -neighborhood of which we will denote b N ε ( ) be the set of profiles whose proportions differ fro those of b less than ε : { ; ; ε for all } ( ) ( ) ( ) N = w w <. ε Theore 1: Suppose that voting rule works well on doain. Then, ajorit rule works well on too. Conversel, suppose that differs fro on soe open neighborhood. More precisel, assue that there eist a profile which works well) and ε > 0 such that (4) ( ) ( ) for all N ε ( ) Then, there eists a doain on which eark: Without the requireent that. (on a doain works well, but does not. on belong to a doain on which ajorit rule works well, the converse assertion above would be false. In particular, consider a voting 11 To be precise, forula (3) sas that there eists an ordering in in which is preferred to and is preferred to. However, because satisfies IIA we can ignore the alternatives other than.,, 14

17 rule that coincides with ajorit 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. Proof: Suppose first that works well on. If, contrar to the theore, work well on, then, fro Lea 2, there eists a Condorcet ccle in : does not (5),,. Let S be the eceptional set for on. Because S is finite (b assuption), 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 in ust be regular with respect to S. 1 Let 0, n 1 2 be group 1, ( n, n be group 2,, and ( n 1 n,1 be group n. Consider a profile 1 on 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. ro (5), such a profile eists on. ro neutralit (ipling IIA), the social preferences ( 1 ) do not depend on voter s preferences over other alternatives. There are three cases either (i) is sociall strictl preferred to under ( 1 );( ii) is sociall indifferent to under ( ) preferred to under ( 1 ). 1 ; or ( iii) is sociall strictl Case (i): ( 1 ) 15

18 Consider a profile 1 on in which all voters in group 1 prefer to to ; all voters in group prefer to to ; and all voters in the reaining groups prefer to to. That is, 12 (8) 1 = n Notice that, in profile 1, voters in group 1 prefer to and that all other voters prefer to. Hence, neutralit and the case (i) hpothesis ipl that ust be sociall strictl preferred to under ( 1 ), i.e., (9) ( 1 ) Observe also that, in 1, voters in group 2 prefer to and all other voters prefer to. Hence fro anonit and neutralit and the case (i) hpothesis, we conclude that ust be sociall strictl preferred to under ( 1 ), i.e., (10) ( 1 ). Now (9), (10), and generic transitivit ipl that is sociall strictl preferred to under ( 1 ), i.e., (11) ( 1 ) 12 This is not quite right because we are not specifing how voters rank alternatives other than,, and. But fro IIA, these other alternatives do not atter. 16

19 But (8), (11), and neutralit ipl for an profile such that n, ust be sociall strictl preferred to. Hence, fro neutralit, for an profile 2 on such that (12) n, ust be sociall strictl preferred to, i.e., (13) ( 2 ) 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 2 on so that n (14) 2 =. 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 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): ( 1 ) 17

20 But fro the case (i) arguent, case (ii) leads to the sae contradiction as before. Hence we are left with Case (iii): ( 1 ) Consider a profile ˆ on such that ˆ 1 n 1 n =. ro anonit, neutralit and the case (iii) hpothesis, we conclude that is sociall indifferent to and is sociall indifferent to under ( ˆ ), i.e., (15) ( ˆ ). and (16) ( ˆ ) But the Pareto propert iplies that is sociall strictl preferred to under ( ˆ ), which together with (15) and (16) contradicts generic transitivit. We conclude that case (iii) is ipossible too, and so. ust work well on after all, as claied. Turning to the converse, suppose that there eist doain and ε > 0 such that works well on, profile on and (4) holds. ro (4), we can partition, voters into n equal group and assign everone within a group the sae ordering in such a wa that the resulting profile on is regular and ( ) ( ). urtherore, because is assued to work well on, we can assue that does too (otherwise, 18

21 we can take = ). Hence, fro neutralit of, we can assue that consists of just two orderings and. More forall, there eist integers n and k with (17) n k> k, alternatives, X and orderings, with (18) and such that if we take (19) 1 k k+ 1 n =, then is weakl sociall preferred to under ( ) (20) ( ) Notice that ( ) ( ).., i.e., To give the idea of the proof, let us assue for the tie being that satisfies the Pareto propert, anonit, and neutralit on the unrestricted doain X. Consider {, } and profile such that (21) 1 k k+ 1 n k n k+ 1 n =. 13 Then fro (18)-(21), anonit, and neutralit, we have (22) ( ) ( ) and. ro the Pareto propert, we have 13 We have again left out the alternatives other than,,, which we are entitled to do b IIA. To ake atters siple, assue that the orderings of are all the sae for these other alternatives. Suppose,,. furtherore that, in these orderings,,,, are each preferred to an alternative not in { } 19

22 (23) ( ). But, b construction, is regular with respect to s eceptional set. Thus, (22) and (23) together ipl that violates generic transitivit on =,,. Yet, fro Lea 2 (see also footnote 11), is a doain on which is genericall transitive on, which iplies that works well but does not. Thus, we are done in the case in which alwas satisfies the Pareto propert, anonit and neutralit. However, if does not alwas satisf these properties, then we can no longer infer (22) fro (18)-(21), and so ust argue less directl (although we shall still ake use of the sae basic idea). Consider and of (18). Suppose first that there eists alternative X such that (24) and. Let w be the alternative iediatel below in ordering. If w, let be the strict ordering that is identical to ecept that w and are now interchanged (so that w ). B construction of, the doain {,, } ccle, and so, fro Lea 2, does not contain a Condorcet works well on this doain. Hence, we can assue that works well on this doain too (otherwise, we are done). Notice that neutralit of and (20) then ipl that if we replace b in profile ust have (25) ( ). (to obtain profile ) we 20

23 Now, if w is the alternative iediatel below in and w, we can perfor the sae sort of interchange as above to obtain work well on {,, } and that (26) ( ). and and so conclude that and B such a succession of interchanges, we can, in effect, ove downward while still ensuring that and work well on the corresponding doains and that the counterparts to (20), (25) and (26) holds. The process coes to end, however, once the alternative iediatel below in (or,, etc.) is. urtherore, 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 in ). Let be the strict ordering that is identical to ecept that and (which we are assuing are adjacent in ) are now interchanged. ro the above arguent, { } works well on =,, and we can suppose that does too. Hence, fro the sae arguent we used for above, we can conclude that (27) ( ) and ( ) and (28) ( ), where is the profile 1 k k+ 1 n k n k+ 1 n. 21

24 But (27) and (28) generate the sae contradiction we obtained following (23). Thus, we are done in the case where (24) holds. Net, suppose that there eists X such that (29) and. But this case is the irror iage of the case where (24) holds. That is, just as in the previous case we generated with (30) through a finite succession of interchanges in which oves downwards in, so we can now generate satisfing (30) through a finite succession of interchanges in which oves upwards in. If we then take {,, } as when (24) holds, that, we can show that and where =, we can furtherore conclude, and work well on. But, paralleling the arguent for ( ) and ( ) ( ), is the profile ipling that ( ) on, and so again we are done. 1 k k+ 1 n k n k+ 1 n, is intransitive. This contradicts the conclusion that works well inall, suppose that there eists X 22

25 such that (31) and. As in the preceding case, we ove upwards in through a succession of interchanges. Onl this tie, the process when and are interchanged to generate ˆ such that (32) ˆ ˆ. As in the previous cases, we can conclude that and work well on {,, ˆ }. Take ˆ such that Then, as in the arguents about ˆ 1 k k + 1 n k n k + 1 n =. ˆ ˆ and, we infer that ( ˆ ) is intransitive, a contradiction of the conclusion that works well on {,, ˆ }. This copletes the proof when (31) holds. The reaining possible cases involving are all repetitions of one or another of the cases alread treated. Q.E.D. We have alread entioned Ma s (1952) characteriation of ajorit rule (see footnote 2). In view of Theore 1, we can provide an alternative characteriation. Specificall, call two voting rules and genericall the sae on doain if the set of profiles on for which ( ) = ( ) is open and dense (put another wa, we are requiring that if ( ) ( ) ( ) ( ) for all ε ( ) then there should not eist ε > 0, such that N ). Call aiall robust if there eists no voting 23

26 rule that (i) works well on ever doain on which works well and (ii) works well on soe doain on which does not work well. Theore 1 iplies: Corollar: An aiall robust voting rule is genericall the sae as ajorit rule. 6. naniit ule 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 an circustances in which it is natural to favor certain alternatives. The rules for changing the.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. Accordingl, let us rela neutralit and just ipose IIA. We will require the following additional weak condition on voting rules: Tie-break Consistenc: Given voting rule, there eists an ordering, X and all on for which q (, ) = q (, ) ( ) if and onl if., we have such that, for all Tie-break consistenc requires that in situations where the population splits between two alternatives, the tie be broken (or not broken as the case a be) consistentl in the sense that it be done transitivel (note that, given IIA, the onl aspect of the condition that is restrictive is the stipulation that be an ordering which entails transitivit). That is, if is chosen over when the population splits between and, and is chosen over when the population splits between and, then should be chosen over when the population splits between and. Observe that because the 24

27 likelihood that the population will split eactl is ver low, tie-break consistenc is not a terribl deanding condition. Let be a strict ordering of X. We shall denote unaniit rule with order of precedence b and define it so that, for all profiles and all alternatives and, ( ) if and onl if either ( i) for all i [ 0,1] ( ) and ( ) or else there eist j such that j. That is, between and, the alternative earlier in the order of precedence will be chosen unless voters unaniousl prefer the other alternative. can be ipleented b the following procedure. Begin with alternative 1 as the status quo (where 1 2 ). At each stage (there are 1 in all), copare the current status quo with the net alternative in the order. If everone prefers this net alternative, then it becoes the new status quo; otherwise, the old status quo reains in place. We shall sa that a voting rule works satisfactoril on a doain if it satisfies the Pareto propert, anonit, IIA, and transitivit on. 14 Just as Leas 1 and 2 characterie when rank-order voting and ajorit rule work well, Lea 3 tells us when unaniit rule with an order of precedence works satisfactoril: Lea 3: naniit rule with order of precedence works satisfactoril on doain if and onl if, for all triples {,, } with 14 There is an obvious sense in which to work satisfactoril is a less deanding requireent than to work well, since the forer iposes onl IIA rather than the stronger condition, neutralit. Note, however, that working satisfactoril requires eact transitivit, whereas working well onl generic transitivit. 25

28 (33), there do not eist both and in such that (34). Proof: Suppose that, for soe triple {,, } satisfing (33), there eist and in satisfing (34). Consider profile ˆ such that ˆ = [ 0, 1 ) [ 1,1] 2 2. Because and voters fro 1 2 to 1 prefer to, we have (35) ( ˆ ). Siilarl, we have (36) ( ˆ ). But because everone prefers to, we have ( ˆ ), 26

29 which together with (35) and (36) contradicts transitivit. We conclude that if (34) holds, then a necessar condition for issing fro. to work satisfactoril on is that either or be Conversel, suppose that does not work satisfactoril on. Because this voting rule alwas satisfies Pareto, anonit, and IIA, there ust eist {,, } satisfing (33) and a profile (37) or such that either ( ) (38) ( ). Suppose first that (38) holds. Then, fro (33), we ust have ( ) ( ) for all [ 0,1] i i i, which contradicts the hpothesis that ( ) b assuption,, we infer that (39) ( i) for all i [ 0,1] Because ( ) and ( ).. Hence, (37) ust hold. Then because,, there ust eist i and i such that (40) ( ) and ( ) But (39) and (40) ipl: i i. 27

30 ( i ) ( i ) and Hence, when (37) holds, that not both and belong to is also a sufficient condition for to work satisfactoril on. Q.E.D. We can now establish our second ajor result: Theore 2: Suppose that satisfies tie-break consistenc. There eists a strict ordering such that for all doains on which works satisfactoril, works satisfactoril on too. urtherore, if there eist a doain on which works satisfactoril and profile on such that ( ) ( ), then there eists a doain on which works satisfactoril but does not. Proof: Given voting rule, let be the corresponding tie-break ordering prescribed b tie-break consistenc. Choose a strict ordering consistent with strict ordering such that, for all, X (41) if then Consider {,, } with., i.e., let be a (42) and suppose that works satisfactoril on doain. ro Lea 3, works satisfactoril on provided that whenever and are two strict orderings such that 28

31 (43) and, then not both and can belong to. Thus, to establish the first assertion of the Theore, it suffices to show that if (43) holds, either and ust be issing fro. Suppose to the contrar that,. Consider the profile ˆ on such that (44) ˆ = [ 0, 1 2) [ 1 2,1] ro (41) and (42) we have (45) (although the rankings in (45) a not be strict). Hence, fro (44) and (45), tie-break consistenc iplies that (46) ( ˆ) ( ˆ). But because everone in ˆ prefers to, the Pareto propert gives us ( ˆ ), which, together with (46), eans that ( ˆ ) is not transitive, a contradiction. Thus the first assertion of the theore is indeed established. To prove the converse, consider profile and doain (47) is on such that (48) works satisfactoril on 29

32 and (49) ( ) ( ). If there are ultiple such and choose a pair (, ) and alternatives, to solve (50) a q (, ) subject to (47) - (49) and (51) ( ) and ( ) ro (51), we have (52).. Let be the opposite of, i.e., for all, if and onl if. Let be the alternative just below in ordering (if is the lowest alternative in, the arguent is ver siilar). Let be the ordering that coincides with ecept that and are interchanged. inall, let ˆ be the ordering that coincides with ecept that and are interchanged. It is a atter of straightforward verification to check that, for all {,, ˆ } and all,, if, then we have neither 30

33 (53) nor (54), which, fro Lea 3, iplies that is transitive on {,, } =. We know, fro (41) and (52), that. There are two cases. Case I: Because, (41) iplies that (55). Consider the profile ro (55), we have = 1 1 [ ) [ ] 1 0, 2 2,1 1 (56) ( ). ro the Pareto propert, we have 1 ( ). inall, fro the Case I hpothesis, we have (57) ( 1 ). 31

34 1 But cobining (55) (57) we conclude that ( ) is intransitive, and so, if Case I holds, we can take = to coplete the proof. Case II: ecall fro our choice of that ( ) and ( ) Thus, in view of (52) and the Case II hpothesis, we cannot have 1 (, ) = (, ) = q q. We ust therefore have either (58) q (, ) > q (, ) or (59) q (, ) < q (, ) 2. rank and differentl. Suppose first that (58) holds. Because works satisfactoril on, we can assue that does too (otherwise, we can take = and we are done). Hence, if is a profile on such that (60) q (, ) = q (, ), anonit and neutralit of ipl that (61) ( ). Let be the ordering that coincides with ecept that interchanged. One can verif echanicall that for all {,, ˆ, },,, if and are and all 32

35 then we do not have (62). Hence, fro Lea 3, works satisfactoril on ˆ {,,, ˆ, } =, and so we can assue that the sae is true of. Hence, if is a profile on ˆ satisfing (60), we can infer (61). Consider 2 = 2 such that Because q 2 (, ) = q (, ) 2 (63) ( ) ro the Pareto propert 1 1 [ 0, ) 2 q ( 2 )) ( ) q,,,,1., the above arguent iplies that (64) 2 ( ). urtherore, because 2 (, ) 2 (65) ( ) 1 q = 2, (41) and the fact that ipl that. 2 But (63) (65) contradict the transitivit of ( ) (58) holds., and so we can take = ˆ when 33

36 1 inall, assue that (59) holds. If there eists β < 2 and a profile on ˆ such that q = β (66) (, ) and (67) ( ), consider profile 3 such that ( )) ( ) ( ) β ) ( ) 0,,,,,,,1 3 q q q q β + + = ˆ Because 3 (, ) that q = β, (66) and (67) ipl 3 (68) ( ). Because q 3 (, ) = q (, ) 3 (69) ( ). Now, 3 (, ) (, ) fact that, we have q = q + β and so fro the choices of,, and and the 3 ( ), we ust have (70) 3 ( ). 3 But (68) (70) contradict the transitivit of ( ). 34

37 1 Thus assue that, for all β < 2 and profiles on ˆ q = β (71) (, ) we have (72) ( ) 1 If there eists ( 0, ),. (73) (, ) and 2 with δ and profile on ˆ such that q δ = (74) ( ) then consider profile ro the Pareto propert,, 4 such that [ δ) [ δ ] 4 0,,1 =. (75) 4 ( ) ro (71) and (72), we have 4 (76) ( ). ro (73) and (74), we have (77) ( ) 4. 4 But (75) (77) contradict the transitivit of ( ). So we conclude that, for all 1 ( 0, ) δ, if on ˆ satisfies (73), then 2 35

38 (78) ( ) inall, consider profile. 5 such that ( )) ( ) 0, q, q,,1 5 =. ro the Pareto propert, we have (79) 5 ( ). Because q 5 (, ) = q (, ) 5 (80) ( ) inall, because 5 (, ) q <. 1 2, we have, (73) and (75) ipl (81) 5 ( ). 5 Now, (79) (81) contradict the transitivit of ( ), and so we can take = ˆ. Q.E.D. 36

39 eferences Arrow, K.J. (1951), Social Choice and Individual Values (New York; John Wile). Second ed., Black, D. (1948), On the ationale of Group Decision-aking, Journal of Political Econo, 56(1), Inada, Ken-ichi (1969), On the Siple Majorit ule Decision, Econoetrica, 37(3), Maskin, E. (1995), Majorit ule, Social Welfare unctions, and Gae ors, in K. Basu, P.K. Pattanaik and K. Suuura, 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. (1970), Collective Choice and Social Welfare, (San rancisco: Holden Da). Sen, A. and P. Pattanaik (1969), Necessar and Sufficient Conditions for ational Choice under Majorit Decision, Journal of Econoic Theor, 1(2),