VOTING BY VETO: MAKING THE MUELLER-MOULIN ALGORITHM MORE VERSATILE

Transcription

1 DAN S. FELSENTHAL AND MOSHt~ MACHOVER SEQUENTIAL VOTING BY VETO: MAKING THE MUELLER-MOULIN ALGORITHM MORE VERSATILE ABSTRACT. This paper shows that a relatively easy algorithm for computing the (unique) outcome of a sophisticated voting procedure called sequential voting by veto (SVV) applies to a more general situation than considered hitherto. According to this procedure a sequence of n voters must select s out of m + s options (s > 0, m I> n I> 2). The ith voter, when his turn comes, vetoes k i options (ki >/1, P. k~ = m). The s remaining non-vetoed options are selected. Every voter is assumed to be fully informed of all other voters' total (linear) preference orderings among the competing options, as well as of the order in which the veto votes are cast. This algorithm was proposed by Mueller (1978) for the special case where s and the k~ are all equal to 1, and extended by Moulin (1983) to the somewhat more general case where the k~ are arbitrary but s is still 1. Some theoretical and practical issues of voting by veto are discussed. Keywords: Algorithm, Pareto optimality, veto, voting methods. 1. INTRODUCTION Mueller (1978) proposed a voting procedure for dividing some good among n persons (n ~> 2). This procedure, called sequential voting by veto (SVV), consists of two stages: (i) Each of the n persons makes a division proposal; (ii) Each of these persons, in a predetermined sequence, casts a veto vote against one of the n + 1 proposals (the additional proposal being the status quo, where no-one gets any portion of the good). The order in which the votes are cast is randomly determined, and the single proposal that is left unvetoed wins and is implemented. Mueller argued that this procedure has essentially three advantages. First, it produces a unique winning proposal regardless of the voters' proposals and the order of voting. Second, if voters act rationally, the selected proposal must be Pareto optimal and will not constitute any voter's least preferred proposal. Third, the procedure is fair because although the winning proposal may depend not only on the type of proposals made but also on the order of voting, each voter has an Theory and Decision 33: , Kluwer Academic Publishers. Printed in the Netherlands.

2 224 DAN S. FELSENTHAL AND MOSHt~ MACHOVER equal chance to be located at any particular place in this order. Consequently, if the order in which the voters are to cast their veto-votes is made known only after the proposals have been stated, all voters have an incentive to state their proposals sincerely, as well as to rank sincerely the other voters' proposals. The last mentioned advantage is especially noteworthy because there are only two other known non-dictatorial voting procedures in which voters have no incentive to misrepresent their preferences among the competing options. First, the procedure in which the s selected options must be unanimously supported by all the voters. Second, the procedure whereby the s selected options are picked at random from among the s top preferences of each voter (cf. Pattanaik, 1978). However, if no unanimity is reached under the former procedure, the result will be either a vacuum where no option is selected, or an indefinite maintenance of the status quo (which may be Pareto inferior). The latter procedure, in contrast, albeit both Pareto optimal and fair- in the sense that it provides every voter with an equal chance of getting his top s preferences selected- may nevertheless result in selecting the bottom preference(s) of an absolute majority of voters. Mueller presented an algorithm for determining the winning proposal, given the sequence of voting. He showed that when n = 2 then the voter who moves first has an advantage, because the second voter will be forced to veto the status quo, provided the first voter's proposal allots to him some positive portion of the good. (In fact, this result for n = 2 is readily seen to extend to the somewhat more general situation discussed in the next paragraph.) But when n >t 3 the voter who moves first may no longer have an advantage. However, although Mueller's algorithm is correct, his proof of this fact has some minor lacunae, and relies directly on an incompletely explicated notion of rationality. Moulin (1981, 1983: ) extended Mueller's idea of sequential voting by veto to any situation in which n voters have to select one out of n + 1 options, and the voters have complete information of all other voters' total (linear) preference orderings among the competing options, as well as of the sequence in which the voters are to cast their veto votes. He also provided a rigorous proof of the correctness of Mueller's algorithm in this case. Moreover, as Moulin observes, this result easily extends to the seemingly more general case where there

3 SEQUENTIAL VOTING BY VETO 225 are m + 1 options (m t> n), and each voter is allowed to veto a number of options. However, both Mueller and Moulin address themselves only to situations in which a single option must be selected. Indeed, as observed by Lijphart and Grofman (1984), and by Felsenthal and Maoz (1992), despite the multitude of real life instances where more than one option must be selected- for example, electing a committee or selecting a jury- the public choice literature to date has been almost exclusively preoccupied with situations involving single winners. The purpose of this paper is to extend and generalize the Mueller- Moulin result for the SVV procedure to a situation in which n voters must select s out of m + s options (s > 0, m ~> n/> 2). The reader may compare our results with those of Banks (1985). 2. PRELIMINARY DEFINITIONS AND CONSIDERATIONS The most typical problem considered in the mathematical theory of social choice has the following general pattern. A finite number of agents or voters are presented with a finite set X of possible outcomes or options. Each voter V comes equipped with an order of preference P among the options. More precisely, P is normally assumed to be a total ordering of X, or at least a total pre-ordering, allowing indifferences. A procedure is then specified, according to which the voters collectively are to choose one single option out of the set X. Each voter is assumed to behave 'optimally'-that is, roughly speaking, in such a way that the option eventually chosen will be as high as possible (subject to the rules of the specified procedure) in his own order of preference. (For brevity, we refer to the voters using the masculine gender; so 'his' is used as short for 'his or her', etc.) If such behaviour on the part of all the voters leads to the eventual choice of a unique option, then the latter is regarded as the solution of the problem. In reality there are of course situations where what is required is to choose not one but a package of s members out of a set X of options, where s > 1. This is the case, for example, when a jury of 12 are to be selected out of a panel of jurors. Now, if each voter has a definite order of preference (a total ordering, or at least a total pre-ordering) among all s-element subsets

4 226 DAN S. FELSENTHAL AND MOSHt~ MACHOVER of X, then the case s > 1 can be reduced to a problem of the usual type, in which s = 1, by the obvious device of re-defining the term option to mean an s-element subset of X rather than an individual member of X. There are, however, situations where it is unrealistic to assume that each voter possesses a total ordering (or even pre-ordering) of the s-element packages, but only an ordering among individual options. It is often tacitly assumed that in such situations the case s > 1 is not amenable to the usual kind of mathematical treatment. On the face of it, this seems inescapable: if an s-element subset of X is to be selected, but a voter is not able to compare any two such subsets and say which of them he prefers (or profess indifference between them) then apparently there is no way to define what is meant by an optimal strategy for that voter. However, we wish to point out that this pessimism may be over-hasty. Indeed, there are selection procedures for which there exist strategies that are optimal in a very strong sense - so strong in fact as to make them independent of a total ordering of all the s-element subsets of X. For the sake of illustration, consider the following trivial example. Suppose there is just one voter - a dictator - who has to select s out of a set X of s + 2 options. The voter is assumed to have a total ordering P of X. It is intuitively obvious that the optimal thing for the voter to do is to choose the set A comprising all members of X except the two he prefers least. Here it is not necessary for the voter to be able to compare any two s-element subsets of X in order to recognize that the particular s-element subset A is 'better' than any other. We shall show that a similar result obtains also in a much more general set-up. DEFINITION 1. An n-ary sequential veto-voting scheme (briefly: an n-ary scheme) is an ordered (n + 1)-tuple {X; P1, P2, - 9 9, Pn), where n is a positive integer, X is a finite set with at least n + 1 members, called the options of the scheme, and (for each i = 1, 2,..., n) Pi is a total ordering of a finite set that includes X. Note that there would have been no loss in generality had we assumed that P~ is a total ordering of X itself; but for technical reasons it is

5 SEQUENTIAL VOTING BY VETO 227 convenient to allow Pi to be a total ordering of any finite superset of X. Note also that the orderings are purely qualitative, and involve no numerical cardinalities or weights. We write [P : x > y] as short for 'x is higher than y in the ordering P'. Similarly, we write [P : x > y] as short for 'x is higher than y in the ordering P or is equal to y'. If A and B are any two sets, then we denote by A - B the set of all members of A that do not belong to B. If B = {b} (that is, if B has b as its sole member), we write A\b as short for A - B. Thus A\b is the set of all members of A that are different from b. Informally, we interpret the scheme (X;P1, P2,-.., Pn} as follows. There are n persons, called voters, V1, V2,..., Vn. The statement [Pi: x > y] means that voter Vii prefers option x to option y. (Note that the Pi are assumed to be strict orderings rather than pre-orderings, so that indifferences between distinct options are not allowed. The reason for this somewhat unrealistic assumption is that, as we shall see, the mathematical tools used here break down in the presence of indifferences.) We assume that each voter is fully informed of the order of voting and of the other voters' orders of preference. Now let us suppose that each voter in turn is invited to veto an option. First, V 1 vetoes one member of X, say xl. Next, V 2 vetoes one of the remaining members of X, say x 2. And so on. Finally, when the turn of V n arrives, n - 1 options have been vetoed by the preceding voters; so V~ vetoes one of the remaining members of X, say x,,. The ordered n-tuple {x 1, x 2..., x n } of distinct members of X is called a veto sequence, and the set of remaining options, X- {x I, x2,..., x,,}, is considered to be selected by this sequence. Recall that the total number of options in an n-ary scheme is n + s, where s is some positive integer; thus, a veto sequence selects a set of s options. Informally speaking, we wish to say that a veto sequence (X 1, X2,..., Xn) is optimal for the scheme <X; P1, P2,. 9 9, P,,> if, for each i, the voter Vii chooses his vetoed option x~ in a 'rational' way; that is, in such a way that-provided the voters following him also behave rationally-the set of options that will be selected by the sequence is ranked as highly as possible on his preference scale. The set of options selected by an optimal sequence will be called a solution of the scheme (X; P1, P2," 9 9, P~}.

6 228 DAN S. FELSENTHAL AND MOSHt~ MACHOVER This informal characterization invites two immediate objections. First, if s > 1 how is a given voter supposed to compare two different s-element subsets of X? If P is a total ordering of the set X, there are several alternative 'reasonable' ways of defining an ordering among the s-element subsets of X. By 'reasonable' we mean here compatible with P. Second, more generally, the concept of rationality is perhaps somewhat contentious and vague, and its uncritical use is notoriously paradox-ridden. We get round the first objection by imposing the following very strong sufficient condition under which a given voter is to prefer one set of options over another. DEFINITION 2. Let P be a total ordering of X and let A and B be two subsets of X. We shall say that A is P-superior to B if B is obtained from A by removing from A one option and substituting for it an option b such that [P: a > b] for every a in A. Suppose P is the order of preference of a voter; the idea here is that if A is P-superior to B then by any reasonable criterion this is a sufficient (though by no means necessary!) condition for the voter to prefer A over B. Note that if A has just one member, A = {a}, then A is P-superior to B if and only if B = {b}, for some b such that [P: a >b]. We deal with the second objection by giving a formal definition of the notions optimal sequence and solution. This definition, which proceeds by recursion on n, makes no reference to rationality. Of course, our intention is to capture and explicate in this definition the informal notion of rationality appropriate to the present context. We leave it to the reader to decide to what extent we have succeeded. DEFINITION 3. First, let n = 1. If {X; P} is a unary scheme, let x be the lowest (least preferred) member of X in the ordering P. We take (x) to be the only optimal sequence for {X; P), and we take X~ to be the [unique] solution of IX; P). Next, assume (as induction hypothesis) that for any n-ary scheme

7 SEQUENTIAL VOTING BY VETO 229 (X; Pa,/2, 9 9 9, Pn ) we have defined the notions of optimal sequence and solution. Now let (X; P, Pa, P2,.. 9, Pn) be an (n + 1)-ary scheme. By a sub-optimal sequence for (X; P, P1, P2, 9-9, P~ ) we mean any (n + 1)-tuple (x, x 1, x2,..., x,) of distinct options such that the n-tuple (x 1, x2,..., x n ) is an optimal sequence for the n-ary scheme (X\x; P1, P2,..., P~)' By an optimal sequence for (X; P, P1, P2,..., Pn} we mean a sub-optimal sequence such that the set of options selected by it is the same as or P-superior to the set of options selected by every other sub-optimal sequence. A set of options selected by an optimal sequence for (X; P, Pa, P2,. 9 9, P,) is called a solution of that scheme. This concludes the definition. It follows at once from Definition 3 that a scheme cannot have more than one solution. Indeed, for a unary scheme the definition explicitly prescribes a unique solution. And for an (n+l)-ary scheme (X; P, P~, P2, 9 9 9, Pn ), if both A and B are solutions (each selected by an optimal sequence), then according to Definition 3 we must have A = B, otherwise A and B would have to be P-superior to each other, which is impossible. On the other hand, it is not immediately obvious from Definition 3 that each scheme does have a solution. For the special case of an n-ary scheme with exactly n + 1 options (that is, s = 1) it is quite easy to establish, by induction on n, the existence of at least one optimal sequence, and hence of a [unique] solution. We shall not bother to present this proof here, because in the next section we shall prove the same fact for every scheme. Definition 3 may, at first reading, seem a bit intimidating; but the idea behind it is really quite simple. Let us explain this by using informal induction on n. In a unary scheme, there is just one voter and s + 1 options, of which s are to be selected. The voter, if he is to behave rationally, must surely veto the option he prefers least, leaving the remaining options selected. In an (n + 1)-ary scheme, if the initial voter vetoes an arbitrary

8 230 DAN S. FELSENTHAL AND MOSHI~ MACHOVER option x, he leaves behind for the remaining voters an n-ary scheme, with the remaining options X\x (all except the one he has just vetoed). A sub-optimal sequence (x, xl, x2,..., xn) is then supposed (by the informal induction hypothesis) to represent a pattern of voting in which those remaining n voters all behave as rationally as possible, given that the initial voter has chosen (rationally or otherwise) to veto x. If an optimal sequence exists - and we shall prove that indeed there is always at least one-then it represents a voting pattern in which those remaining n voters behave as rationally as possible, and the initial voter achieves the selection of a set of options that is equal or P-superior to any set selected by every other sub-optimal sequence. Such a sequence therefore represents a voting pattern in which all voters, including the initial one, behave as rationally as possible. However, we must stress again that although Definition 3 is intended to capture the idea of rational strategy as applied to a sequential veto-voting scheme, the definition itself makes no mention of rationality. It must be pointed out that the existence of an optimal sequence depends crucially on the assumption that the voters' orders of preference do not admit indifferences. To see this, suppose for a moment we were to drop this condition, and consider the (generalized) 3-ary scheme ({a, b, c, d}; P, Q, R}, where P, Q and R are pre-orderings, allowing indifferences: [P: a>b-c>d], [Q:c>a-d>b] and [R : c > b > a- d]. Here - denotes indifference, so that for example the first voter is indifferent between b and c but prefers both to d and prefers a most of all. An easy informal argument shows that if the second and third voters behave rationally, and if the first voter vetoes a or b or d, then c will be selected; but if the first voter vetoes c, then the second voter will surely veto b, leaving the third voter with the dilemma of having to choose between a and d. Thus the first voter is faced with a choice between two possible alternatives: first, he can ensure that c will be selected; second, he can ensure that either a or d will be selected but without being able to determine which. Without some additional extraneous criterion (for example, cardinal preference ordering) these two alternatives are mutually incomparable from the first voter's point of view: he can neither prefer one to the other nor

9 SEQUENTIAL VOTING BY VETO 231 express indifference between them. Therefore there is no reasonable way of extending the definition of optimal sequence to such a generalized scheme that would guarantee the existence of at least one such sequence. It is for this reason that we must assume that the voters' preferences are strictly ordered. Definition 3 produces (as will be shown later) a unique solution, but not necessarily a unique optimal sequence. Indeed, there are schemes for which there are several such sequences. For example, consider the 2-ary scheme ~{a, b, c}; P, Q), where [P: a >b >c] and [Q : b >c > a]. Here s = 1, so just a single option is to be selected. The first voter cannot get his favourite option a selected, because if he does not veto a then the second voter surely will. The best he can do is to get b selected. This he can achieve by vetoing either a (in which case the second voter will veto c) or c (whereupon the second voter will veto a). In fact, using Definition 3 it is easy to see that this scheme has {b} as solution, and both /a, c) and {c, a) as optimal sequences. Note that in this example the first voter behaves quite rationally if he vetoes his favourite option, a; if he does so he will still get b selected, which is the best result he can achieve. True, this voter has an alternative: he will also achieve the same outcome if he vetoes his least favoured option, c. This raises the question whether there are cases where a voter, if he wishes to obtain the best possible result, may have no choice but to veto his most favoured surviving option. This seems to perverse, that one is led to the conjecture that such cases cannot exist. This conjecture turns out to be correct; its proof, which is not quite trivial, will be presented in the next section. Definition 3 provides us with an algorithm whereby the solution and all the optimal sequences of any given scheme can in principle be computed. However, this algorithm is highly inefficient, and becomes impracticable even for fairly small n. In order to solve a 5-ary scheme, this algorithm requires the solutions of six 4-ary schemes; each of the latter requires the solutions of five 3-ary schemes; each of these requires the solutions of four 2-ary schemes, each of which requires the solutions of three unary schemes. As n increases, the length of the computation needed to solve an n-ary scheme grows at a roughly exponential rate.

10 232 DAN S. FELSENTHAL AND MOSHI~ MACHOVER 3. CANONICAL SEQUENCES The following definition is essentially (except for minor differences of terminology and notation) due to Mueller (1978). DEFINITION 4. For any given n-ary scheme (X; Pl, P2,' we define a unique canonical sequence (Yl, Y2,. 9 9, Y~). The Yi (for i = 1, 2,..., n) are distinct options determined by the following rule, which proceeds by recursion on n- i. Let 1 ~<i <~n, and suppose (as induction hypothesis) that for all j such that i <j ~< n the options yj have already been determined. Then we define Yi to be the lowest (least preferred) member of X - {yj : i < j ~< n} in the ordering Pi. In other words, the canonical sequence (Yl,Y2,...,Y~} of (X; P1, P2,. 9 9, P,) is obtained by taking y, to be the option least "", Pn), preferred by the last voter V~ ; then Yn-1 is taken as the option which the penultimate voter V,_~ least prefers among the remaining options X~y, ; and so on, going backwards. Finally, Yl is taken as the option that the first voter V1 least prefers of the s + 1 remaining options X- (Y2, 9 9 9, Yn-1, Y.}" Informally speaking, the idea behind this definition is that the canonical sequence is a veto sequence that results from the following rational consideration. Since Yn is the last voter's least preferred option, none of the n - 1 preceding voters need bother to veto this option, because they know that, no matter what they do, the last voter is bound to veto it, if it survives until it is his turn to vote. So, as far as they are concerned, this option is as good as dead and can be ignored. If they all think along these reasonable lines, then Yn will indeed survive until the turn of the last voter arrives, and then he will indeed have to veto it. A similar reasoning is now applied to the option that the penultimate voter least prefers among the remaining options X~y n, leading to the conclusion that none of the previous n- 2 voters will bother to veto it, and therefore the penultimate voter will indeed veto it. And so on, backwards, until we reach the first voter. As far as he is concerned, n - 1 options are as good as dead and may be ignored; and so he will consider only the s + 1 remaining options, and will veto whichever of these he prefers least.

11 SEQUENTIAL VOTING BY VETO 233 Mueller (1978) discusses only the case s = 1 (that is, n-ary schemes with n + 1 options). He offers what amounts to a version of the informal argument we have presented in the preceding paragraph as proof that if all voters behave as rationally as possible, the single option selected by the canonical sequence will in fact be selected, and the canonical sequence therefore represents, in this sense, voting behavior that is as beneficial as possible. However, the argument as presented by him has some minor lacunae, and does not generalize directly to the general case where the number s of options to be selected is an arbitrary positive integer. Moreover, it invokes an incompletely explicated notion of rationality. He does point out that the canonical sequence is not necessarily the only one that leads to the best possible result, but does not offer a complete characterization of what we have called optimal sequences. Moulin (1983: ) presents a formal proof of a theorem that essentially states- again only in the case s--1- that the canonical sequence is optimal in the sense defined above. The rationale behind the definition of optimal sequences (Definition 3) is a positive one: each voter V~ vetoes any option that leaves the following voters with an (n- i)-ary scheme such that, if they too follow the same reasoning, the set of options that will finally be selected is the best possible one from V[s point of view. In contrast, the definition of the canonical sequence follows a negative rationale: each voter decides not to bother to veto an option that he knows will be vetoed by a subsequent voter. Since both approaches seem reasonable, it is natural to conjecture that they should lead to the same end result. We shall soon prove that this is indeed the case. Our proof-a modified version of the argument used by Moulin (1983)- will be formal, and will not refer to rationality. Thus, rather than simply assuming the thesis that Definitions 3 and 4 reflect equally rational strategies, our proof will lend it some corroboration. LEMMA. Let (X; P, P1, P2,. 9 9, Pn) be an (n + 1)-ary scheme whose canonical sequence is (y, Yl, Y2,..., Yn). Let x by any option in X and let (x 1, x2,..., x~ } be the canonical sequence of (X\x;P1,P2...,Pn). Then for each i=o, 1,2,...,n the set {xj : i <j <~ n} contains all those members of the set {yj : i <j <. n} that are different from x.

12 234 DAN S. FELSENTHAL AND MOSHt~ MACHOVER Proof. We note that since both {yj : i <j ~< n} and {xj : i <j ~< n) have exactly the same number of members, our Lemma means that if x is not in {yj : i<j~<n} then {xj : i<j~<n} is the same as (yj : i<j~< n}; but if x is in (yj: i<j<-n} then {xj: i<j<~n} is obtained from {yj : i <j ~< n} by replacing x with some other member of X. Note also that our Lemma does not claim that if the set {xj : i <j n} contains a particular Yk, then Yk must be equal to x~. Rather, Yk may be some other xj. We shall prove our Lemma by 'backward induction' on i, which amounts to ordinary induction on n -i. (A similar method is used by McKelvey and Niemi, 1978: 16.) For i = n the claim of the Lemma is vacuously true, because both {yj: n <j~<n} and (xj: n <j~<n} are empty. Next, we assume - as induction hypothesis - that our claim holds for a given positive i, where i <~ n. We shall show that it must then hold for i- 1 as well. Since (y, Yl, Y2,..., Y,) is the canonical sequence of (X;P, P1, P2,.. 9, P, ), it follows from Definition 4 that (1) y i is the least member of the set X- { y j : i < j <~ n} in the ordering Pi. We also note that our induction hypothesis implies that (2) XXx - {xj : i <j <~ n} is a subset of X - {yj : i <j <~ n}. Indeed, any member of X\x - {xj : i <j <~ n} must be an option that differs from x and does not belong to {xj:i<j<~n}; hence by the induction hypothesis this option cannot belong to {yj : i <j ~< n} and must be a member of X- {y~ : i <j ~< n}. Now, by the induction hypothesis, all members of {yj: i <j" ~<n} that are different from x belong to {xj : i <j ~< n}; hence they certainly belong to the larger set (xj : i - 1 <j ~< n}. So, in order to show that the claim of our Lemma holds for i - 1 as well, it is enough to establish that Yi itself - the only member of {yj : i - 1 <]" ~< n} that is not in {yj : i<j ~<n} - belongs to the set {xj : i- 1 <j~<n} if it differs from x. So let us suppose that Yi indeed differs from x; otherwise there is nothing to prove.

13 ...,P.). SEQUENTIAL VOTING BY VETO 235 If Yi is already in {xj:i<j<~n}, then it certainly belongs to the larger set {xj: i- 1 <j ~<n}, and we are done. On the other hand, if Yi is not in {xj : i <j ~< n}, then - since it also differs from x - it belongs to set X\x - {xj : i <j <~ n}. It now follows from (1) and (2) that yi must be the least member of the set X\x- {xj : i < j <~ n} in the ordering Pi. But (x~,xz,...,x,) is the canonical sequence of (XXx;P~, P2, 9 9 9, P,, ) ; so by Definition 4 x i is the smallest member of this very set X\x-{xj: i<j<~n} in that same ordering. Hence xi =Yi, so yi belongs to {xj : i - 1 <] ~< n}, as claimed. THEOREM. The canonical sequence of any n-ary scheme (X; P~, P2,..., P,} is optimal for that scheme. Hence the set of options selected by this sequence is the (unique) solution of (X; P1, P2, Proof. We proceed by induction on n. For n = 1 the theorem is readily verified. Now let us assume, as induction hypothesis, that the theorem holds for all n-ary schemes. Let us consider an (n + 1)-ary scheme (X; P, P1, /)2, '", P,), whose canonical sequence is (y, Yl, Y2,.-., Y,). First, we establish that (y, Yl, Y2,-.., Y,) is sub-optimal for {X; P, P1, P2,.. 9, P, ). By Definition 3, this will follow if we show that the sequence {yl,y2,...,y,) is optimal for (X\y; P~, P2,..., P,). To this end, we use our Lemma. Let {u~, u2,..., un) be the canonical sequence of (X\y; P1, P2,..., Pn ). In this particular case, the Lemma reads: For each i = O, l, 2,..., n the set {uj : i<j<~n} contains all those members of the set {yj : i <j <~ n} that are different from y. But in fact all the members of {yj : i <j <~ n} do differ from y, because (Y, Yl, Y~... y,) is canonical and hence by Definition 4 all its members are distinct. Therefore we simply have: (1) {uj: i<j<~n}=(yj: i<j<~n} for each i = O, 1, 2,..., n.

14 236 DAN S. FELSENTHAL AND MOSHt~ MACHOVER Hence it is easy to verify that (2) yi=ui for i=l, 2,...,n. (To see that Yi = ui holds for a particular positive i, use (1) for i - 1 and i.) Therefore we have {Ul, u2,..., u,) = (YI, Y2,..., Y~); so (Yl, Y2,''",Y~) is the canonical sequence of (X\y; P1, P2,..., P~), and hence (by the induction hypothesis) an optimal sequence for that scheme. Thus our claim that (y, yt, Y2, ''", Y~) is sub-optimal is now established. Next, we denote by A the set selected by our canonical sequence: (3) A=X- {Y, Yl, Y2,''', Yn}" By Definition 4, y is the least member of X - {Yl, Y2,. 9 9, Y~} in the ordering P; therefore (4) [P: a>y] for every aina. In accordance with Definition 3, we must compare A with the sets selected by all other sub-optimal sequences for (X; P, P1, P2,..., Pn ). Let (x, x 1, x 2,..., xn) be such a sequence, and let B be the set of options selected by it: B =X-{x, xl, x2,..., xn}, which means the same as (5) B =X\x- {Xl, x2,..., Xn}. In order to prove that (y, Yl, Y2, 9 9 9, Yn) is optimal, it only remains to show that A is either equal or P-superior to B. Since (x, xl,x 2...,xn) is sub-optimal for (X; P, P1, P2,..., Pn), it follows from Definition 3 that (xl, x2,..., xn) must be an optimal sequence for (X\x; P1, Pz,..., Pn). Therefore (5) means that B is a solution of (X\x; P1, P2..., Pn). In fact, as remarked earlier, B must be the unique solution of that n-ary scheme. By the induction hypothesis it now follows that B is the set selected by the canonical sequence of (X\x; P1, P2..., Pn)" Therefore there

15 9..,pn). SEQUENTIAL VOTING BY VETO 237 is no loss of generality in assuming that the sequence/x~, x2, 9.., x,, ) which features in (5) is itself the canonical sequence of IX\x; P~, P2, We can now apply our Lemma to this canonical sequence. For i = 0, the Lemma can be written The set {xj : 1 <~j <~ n} contains all those members of the set (yj : 1 <~j <~ n} that are different from x. From (5) and the Lemma above it follows that none of the yj (for j = 1, 2,..., n) can belong to B. Indeed, if yj is not x, then by the re-stated Lemma it belongs to {xj: l<~j<~n}, and so by (5) cannot belong to B. On the other hand, if yj is x, then- again by (5)-it cannot be in B. We can now compare A and B. By (3) and (5), both sets have the same number of members. If B is not the same as A, then B must contain an option that does not belong to A. Since, as we have just seen, the yj do not belong to B, it follows from (3) that the only option that can possibly belong to B but not to A is y. Thus, B is either equal to A or is obtained from A by substituting y for some member of A, in which case (4) shows that A is P-superior to B. We have therefore completed the proof that /Y, Yl, Y2,. - 9, Y~) is an optimal sequence for IX; P, P1, P2,.-., P~). In the course of proving the Theorem we have established the following: COROLLARY. If the (n + 1)-tuple (y, y~, Y2,..., Y,7) is the canonical sequence of the (n + 1)-ary scheme IX; P, P1, P2..., P,), then {Yl, Y2,..., Yn) is the canonical sequence of IX\y; P~, P2,... Pn). It can now be shown that if the voters of an n-ary scheme {X; P1, P2..., P~) follow the canonical sequence in imposing their vetoes, then (each of them achieves the best result possible for him and) no voter vetoes his most preferred surviving option. Indeed, from Definition 4 it follows at once that the first voter does not veto his most preferred option, since he vetoes his least favoured of s + 1 options. By

16 238 DAN S. FELSENTHAL AND MOSH[~ MACHOVER the Corollary it now follows by an easy inductive argument that no voter vetoes his most favoured surviving option. This confirms the conjecture we made earlier. We also note that Definition 4 provides us with a very efficient algorithm for finding the canonical sequence and (in view of the Theorem) the solution of any given n-ary scheme. The length of computation using this algorithm is roughly proportional to n 2. The results obtained above can readily be generalized to sequential veto-voting schemes of a more general kind than that admitted in Definition 1. Instead of allowing each voter to veto just one option, we could allow the ith voter to veto (not necessarily consecutively) k i options, where k i is any positive integer, and the total number of options is greater than s ki. This generalized scheme can be reduced to an equivalent ordinary scheme: for each i we can replace the ith voter by k~ 'clones', all having the same order of preference as him, who are allowed to veto one option each. 4. DISCUSSION The optimal sequences- and in particular the canonical sequence- of an SVV scheme represent instances of sophisticated voting behaviour, in the sense that each voter utilizes information concerning the preferences of other voters who have yet to vote, and bases his own voting on assumptions as to how they are going to vote. Sophisticated voting is to be contrasted with sincere voting. In particular, the sincere sequence of an n-ary scheme {X; P1, P2, 9.., Pn) is defined to be the n-tuple (Yl, Y2,' 9 9, Yn) obtained by taking Yl to be the option least favoured by the first voter V 1 ; then Y2 is taken as the option which the second voter V 2 least favours among the remaining options X\yl; and so on. Finally, Yn is taken as the option that the last voter Vn least prefers of the s + 1 remaining options X - {y~, Y2, 9 9 9, Yn-1}" It is reasonable to assume that under SVV voters will follow the sincere sequence if no voter is informed of the preferences of subsequent voters (but each voter knows which options have been vetoed by previous voters). Observe (cf. Moulin, 1983: 138) that (Yl,Y2,...,Yn) is the

17 SEQUENTIAL VOTING BY VETO 239 sincere sequence of the scheme (X; Px, Pz, 9 9 -, P.) if and only if its reverse, the n-tuple (y~, Y,7-1,..., Yl), is the canonical sequence of the reverse scheme (X;Pn,Pn 1,''',P1)" Given the voters' preference orderings of the set of competing options - whether this set is exogenously dictated or formulated by the voters themselves-the outcome of both sincere and sophisticated SVV may depend on the order of voting. Moreover, because in reality the quality of information may not approximate the completeness needed for sophisticated voting under any procedure (Ordeshook and Palfrey, 1988), sophisticated voting under SVV is expected to be limited to relatively small electorates- for which SVV is eminently suitable and the assumption of complete information is quite realistic. Regardless of the size of the electorate for which SVV may be applicable, the social desirability of the SVV procedure is independent of whether voters act sincerely or sophisticatedly. This is so because from a collective point of view the sincere and sophisticated outcomes (where they differ) are equally desirable, since both guarantee what the SVV procedure sets out to achieve; namely, that the selected option(s) will be Pareto optimal and that no voter will obtain his least preferred option. Thus in contrast to majoritarian voting procedures, whose possible outcomes must be evaluated using external normative criteria (for example, whether a Condorcet winner, if one exists, is selected), the normative criterion for evaluating the outcome under SVV is built into the procedure itself. It is true, of course, that some voters may obtain a more or less preferred outcome under SVV, depending on the order of voting. However, as that order itself is determined randomly, a significant advantage of SVV is that the outcome cannot be deliberately manipulated by some interested party, say a chairperson, as it can be under the common amendment and successive procedure (cf. Farquharson, 1969; Jung, 1990). Nevertheless, it would be interesting to determine in future experimental research whether voters tend to act sophisticatedly or sincerely under SVV when they have complete information; as well as whether, if they do act sophisticatedly, they follow the algorithm discussed in this paper or reach the sophisticated outcome by using a different cognitive model.

18 240 DAN S. FELSENTHAL AND MOSHI~ MACHOVER REFERENCES Banks, J. S.: 1985, 'Sophisticated Voting Outcomes and Agenda Control', Social Choice and Welfare 1, Farquharson, R.: 1969, Theory of Voting, New Haven, CT.: Yale University Press. Felsenthal, D. S. and Maoz, Z.: 1992, 'Normative Properties of Four Single-Stage Multi-Winner Electoral Procedures', Behavioral Science 37, Jung, J. P.: 1990, 'Black and Farquharson on Order-of-Voting Effects', Social Choice and Welfare 7, Lijphart, A. and Grofman, B.: 1984, 'Choosing an Electoral System', in: Lijphart, A. and Grofman, B. (Eds.), Choosing an Electoral System: Issues and Alternatives, New York: Praeger. McKelvey, R. D. and Niemi, R. G.: 1978, 'A Multistage Game Representation of Sophisticated Voting for Binary Procedures', Journal of Economic Theory 18, Moulin, H.: 1981, 'Prudence Versus Sophistication in Voting Strategy', Journal of Economic Theory 24, Moulin, H.: 1983, The Strategy of Social Choice, Amsterdam: North-Holland. Mueller, D. C.: 1978, 'Voting by Veto', Journal of Public Economics 10, Ordeshook, P. C. and Palfrey, T. R.: 1988, 'Agendas, Strategic Voting, and Signaling with Incomplete Information', American Journal of Political Science 32, Pattanaik, P.: 1978, Strategy and Group Choice, Amsterdam: North-Holland. Dan S. Felsenthal University of Haifa, Israel. Mosh6 Machover King's College, London, U.K.