www.elsevier.nl / locate / econbase
Representation in majority tournaments
a ,* b
´ Gilbert Laffond , Jean Laine a
´ ´ ´
Laboratoire d’Econometrie, Conservatoire National des Arts et Metiers, 2 Rue Conte, 75003 Paris, France
b
´
ENSAI and CREST – Laboratoire de Statistique et Modelisation, Campus de Ker Lann, Rue Blaise Pascal, 36170 Bruz, France
Received 1 February 1997; received in revised form 1 June 1998; accepted 1 December 1998
Abstract
The paper presents a general setting for studying majority-based collective decision procedures where the electorate is divided into constituencies according to an equal-representation principle. It generalizes the well-known Referendum Paradox to the non-dichotomous choice case, and shows that all Condorcet choice rules are sensitive to the design of the apportionment of the electorate, in the sense that final outcomes may entirely differ from those prevailing when there is a single constituency. Direct and representative democratic systems thus lead to mutually inconsistent collective decisions. 2000 Elsevier Science B.V. All rights reserved.
Keywords: Representation; Majority voting; Tournament choice
1. Introduction
Representational democracy refers to multi-step collective choice procedures: in the first step, individuals are allocated among committees or constituencies, each one sending one or several representatives in charge of their constituents’ interests in the higher levels; then representatives are themselves allocated among committees, such a process going on until some decision is finally taken. In the case where collective decisions rest on the majority rule, the formal structure of such a representational system in the dichotomous choice case has been investigated by Murakami (1966); Fishburn (1971); Fine (1972). These studies complete May’s characterization of simple majority rule (see May, 1952) by considering the representative (or indirect) majority rule obtained by repeated applications of the simple majority decision rule. Representational
*Corresponding author.
democracy is defined ‘‘as a hierarchy of voting procedures, each of which may be called a council.. Every individual casts a ballot or ballots in one council or councils. A decision in each council is represented in a higher council whose decision is, in turn, represented in a still higher council and so on’’ (Murakami, 1966, pp. 710–711). This axiomatic study involves no restriction on the way committees are initially designed: in particular, some individuals may belong to several committees, and committees may be of different sizes.
This paper is restricted to the study of two-step procedures, where, in the first step, the electorate is partitioned into constituencies, each one sending a single representative in charge of her constituents’ interests within the final choice made by representatives in the second step. Moreover, it tackles the question of how to design a representational system in a ‘consistent’ way.
A possible approach to such a consistency concept is the following: let us consider the number of representatives as a fixed parameter R, and suppose that the number C of constituencies is imposed by the polity; the problem is then to design a ‘satisfactory’ mechanism which allocates the R representatives among constituencies. ‘Satisfactory’ may here rest on basic constitutional principles such as the fact that representatives should be allocated among constituencies according to their relative population sizes. Such an approach reduces to the ‘fractional problem’ which arises when the ratio C /R is not an integer: how to allocate representatives when the exact egalitarian apportionment is impossible? The reader may refer to Balinsky and Young (1975), (1982) and the references quoted there for details.
In this paper, we follow a different approach. We still consider the number of constituencies as exogenously given, and we suppose that the apportionment of the electorate among them satisfies the above equal representation principle. Our purpose is to compare the way direct (referendum-type) and representational voting systems behave for some given preference profile in the electorate. In other words, we are interested in the study of referenda outcomes in representational democracies. It is well known that, in the case of a dichotomous collective choice problem, it may well happen that the majority of voters favors an opinion whereas the majority of representatives its negation. Such a situation is closely related to the so-called ‘Ostrogorski Paradox’, and is
1
introduced as the ‘Referendum Paradox’ in Nurmi (1998). The following simple example illustrates this potential inconsistency: let us consider a 9-voter electorate facing h0,1j as choice set; the next matrix summarizes the individual asymmetric preferences (where a cell labelled 0 corresponds to a voter preferring 0 to 1):
0 1 1
0 0 1
0 1 0
Then an overall majority prefers outcome 0 to outcome 1, which means that 0 is the
1
referendum winner. Now, if the apportionment is designed according to the three columns, and if each constituency is represented by a single representative who votes consistently with the majority of her constituents, then 1 defeats 0 under representational democracy. This paradox, and close versions of it, have been extensively studied by Ostrogorski (1903); Anscombe (1976); Daudt and Rae (1976); Deb and Kelsey (1987); Nermuth (1992); Nurmi (1998). It describes the well-known sensitivity of representa-tional democratic outcomes to the design of apportionments.
The primary goal of the present paper is to extend the analysis of the referendum paradox to the case of more than two outcomes, and where the collective decision procedure is based on the simple majority rule. Formally, we consider a finite set N of n voters (where n is odd) who have to select one or several outcomes among a finite set X. Each voter has preferences over X represented by a complete linear order. In a direct
2
democratic system, or referendum, individual preferences induce a complete asymmetric binary relation on X, called a tournament T, which states that outcome x dominates outcome y whenever more than one half of voters prefer x to y. However, the referendum problem does not reduce to identifying a maximal element of X for T : it is well-known that such a maximal element, called a Condorcet winner, may not exist, for nothing precludes T to be cyclic. Therefore, choosing from a tournament entails no natural procedure or solution. A tournament solution is formalized as a multi-valued mapping S from the set of all possible tournaments on X, to X. A rather large literature is devoted to the search for Condorcet-consistent tournament solutions (i.e. those solutions which select the Condorcet winner whenever it exists) tournament solutions. This line of research has been drawn either on axiomatic grounds (see, for example, Moulin, 1986; Dutta, 1988; Schwartz, 1990; Laffond et al., 1995, 1996), or on strategic grounds (see
3
Banks, 1985; Fisher and Ryan, 1992, 1995a,b; Laffond et al., 1993, 1994). To summarize, for any preference profile aggregated into a tournament T, S(T ) represents the referendum outcome for some collective choice procedure S.
This setting is easily adapted to a representational democracy. Suppose that one has to divide the electorate N into K mutually disjoint subsets of odd and equal cardinality N ,k
4
1#k#K, where K is exogenous. The fact that each constituency contains an odd
number of voters precludes ties in pairwise majority comparisons of outcomes. Therefore, voters’ preferences in constituency Nk lead to a tournament Tk on X. Furthermore, let S be the prevailing choice concept in the society. We introduce here two specific methods for selecting the final set of winners given the family hT ,...,T1 Kj of tournaments. The first one is called the ‘one-shot method’ and is defined as follows: the representative of constituency N reports her constituents’ preferences by announcing T .k k
Then, two outcomes x and y are finally compared according to the majority rule applied to all representatives. Formally, x will defeat y in the society if x is preferred to y in more than 0.5K constituencies. Since n and alluNku are odd integers, this defines a new tournament T on X, called tournament among representatives, which is used to defineA
2
Referendum naturally refers to the dichotomous case. For simplicity, we keep the word for the general case study.
3
Laslier (1997) offers a complete presentation of this literature. 4
the final set S(T ) of winners, called the representational or indirect outcome. ThisA
method is a natural extension of the dichotomous case illustrated above, and corresponds to the actual practice of the parliamentary type.
The second method, called the sequential choice method, endows each constituency with the power to influence the final agenda. We suppose that the actual choice set X* open to representatives (or actual agenda) is a subset of X (which is interpreted as the set of all ex ante candidates). A potential candidate belongs to the actual agenda if it is supported by at least one constituency N [i.e. belongs to S(T )]. Thus, for a givenk k
apportionment A5hN ,...,N1 kj, the actual agenda is defined by X*5 <1#k#KS(T ); thek
set of final winners is then defined by S(T /X*), which is the set of S-winners in theA
tournament among representatives restricted to the actual agenda. Such a choice method obviously coincides with the previous one in the dichotomous case. It may be conceived as a model of internal democratic organization within political parties: viewing constituencies as local committees having charge of the definition of the party policy platform, the final step would correspond to the party congress where the assembly of all
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local representatives choose among all proposals made by local committees.
Having specified what is the final outcome (for some district map) of a representation-al system, we can now study the referendum paradox with at least three candidates. The paper mainly addresses the following question: is it always possible, given some preference profile, to design an apportionment such that direct and representational outcomes both coincide (or at least intersect)? The answer is trivially positive in the dichotomous choice case. We show below that this is no longer true in a higher dimensional setting.
When dealing with the one-shot rule, a sufficient condition for avoiding the referendum paradox is to choose an apportionment A such that T and T are identical.A
We call such an apportionment representative. Hence the problem becomes the following: does at least one representative apportionment exist for any preference profile? We exhibit (Theorem 2) a profile for which this is not the case. Moreover, we show (Theorems 5 and 6) that, when the candidate set is larger than three, direct and indirect outcomes may be disjoint for any district map (with a fixed number of constituencies), and for any Condorcet-consistent solution.
These general inconsistency results are obtained in the case where no restriction bears upon the individual preferences. One may look for domain restrictions which ensure identical direct and indirect outcomes. For instance, it is easy to check that, in the dichotomous case, one cannot have one candidate supported by more than 2 / 3 of the electorate and the other supported by more than 2 / 3 of the representatives. More generally, a widespread social agreement allows avoidance of the referendum paradox for any district map. It is shown in Wagner (1983), (1984); Nurmi (1998) that such agreement corresponds to a qualified majority of 3 / 4 of the electorate. We prove below that this ‘75% rule’ still applies in the general case (Theorem 3). More precisely, we
5
show that, for any majority margin lower than 75%, there exists a profile allowing for no representative apportionment (Theorem 4).
Turning to the sequential choice rule, it appears that an even larger discrepancy exists between direct and representational system. Indeed, we show that direct and indirect outcomes may not overlap even for profiles allowing for any apportionment to be representative (Theorem 7). Thus the sufficient condition for no paradox in the one-shot rule case does not apply to the sequential rule one. Consistency between direct and representational outcomes may be conceived as a normative criterion for collective choice. A related separability axiom has been introduced by Smith (1973) for the dichotomous choice case: it states that, whenever the electorate is divided into two constituencies in such a way that both choose the same candidate, then the society should choose accordingly. This axiom is easily extended to more than two candidates: the society should choose the intersection between the constituency choice outcomes. It is already known (see Young, 1975) that no Condorcet-consistent solution verifies this generalized version of Smith’s axiom. Our consistency requirement between direct and representational outcomes goes in the reverse direction: whenever the overall society chooses some set of candidates, there is a way to design local bodies such that this set coincides (or at least intersects) with a specific subset of all candidates chosen in at least one local body. We then prove that this consistency axiom is incompatible with Condorcet-consistency.
Finally, one may argue that, despite all distortions just listed, most polities does rest on a representational system. Hence, the main criterion is not the consistency between direct and indirect outcomes, but instead the sensitivity of the indirect outcome to some change in the district map. We prove (Theorem 9) that any Condorcet-consistent solution is sensitive to such a change. Indeed, we exhibit a profile for which any apportionment is representative, whereas the outcomes of any two different apportionments are mutually disjoint.
The paper is organized as follows: Section 2 formalizes all concepts described in the introduction. Section 3 is devoted to the existence of representative apportionments. The study of the referendum paradox, as well as the analysis of sensitivity of representational outcomes to the district map, appear in Section 4. The paper ends with comments on directions for further research.
2. Representativeness concepts
We now introduce several concepts dealing with representation in a majority voting system. Let us begin with simple notions about tournaments. Consider a population
N5h1,...,i,...,njof n voters (where n is odd). Each voter i has preferences over a finite set X of social alternatives (or candidates), which are represented by a complete linear orders on X. A profile [s]5(s ) is a vector of n preference orders on X. The set
i i i[N
of all possible preference profiles is denoted by P(N ).
Now with each profile [s][P may be associated the complete asymmetric binary relation T(s) on X (called the majority tournament for [s]) defined by: ;x, y[X, xT(s)y⇔uhi[N:xs yju.0.5n. Inversely, it is well-known that, as long as the number n
of voters can be chosen large enough, every tournament T on X corresponds to some preference profile [s], i.e. ' a finite set of integers N and [s][P(N ) such that
T5T(s) (see MacGarvey, 1953).
The first definition introduces a restriction on profiles which is useful for the sequel:
Definition 1. Let a .0.5n. A preference profile [s] is said to have strength a if
;x, y[X, xT(s)y⇔uhi[N:xs yju.a. i
LetD[resp.D(a)] denote the set of all tournaments on X (resp. of all tournaments on
6
X associated with profiles of strength at least a). A tournament solution is a multi-valued application S fromDto X. The set of S-winners of T[Dis the subset S(T ) of X. Furthermore, if X9,X and T[D, we denote by S(T /X9) the set of S-winners of T once restricted to the subset X9 of candidates. A Condorcet-winner of X for T[D is a candidate x*[X such that x*Tx;x±x*. A tournament solution S is
Condorcet-con-sistent if it always selects the Condorcet-winner as unique winner whenever it exists. Furthermore, a solution S9is said to be finer than solution S (denoted by S9#S ) if for
any tournament T on X, S9(T )#S(T ). A large and growing literature is devoted to the
search for a ‘satisfactory’ Condorcet-consistent solution concept, satisfactory meaning based on relevant and desirable axioms for collective choice. Special attention will be paid below to a specific solution concept, the Uncovered Set (see Miller, 1977, 1980;
´
Shepsle and Weingast, 1982; Moulin, 1986; Laffond and Laine, 1994), which is defined as follows:
2
Definition 2. Let T[D. The covering relation → is defined on X by: x→y⇔xTy and
[;z[X, yTz⇒xTz]. The Uncovered Set of X for T is the set UC(T )5hx[X: there is no y[X such that y→xj.
Almost all Condorcet-consistent solutions proposed in the literature are finer than the Uncovered Set (the only exception is the top-cycle).
Let us turn now to several concepts related to the design of a representative democratic system. Many alternative ways to divide the electorate N into mutually disjoint subsets (or constituencies) may be a priori conceived. The following one involves rather weak requirements:
Definition 3. A K-apportionment of N is a partition A5hN ,...,N1 kj of N into K non-empty subsets called constituencies such that 2;k[h1,...,Kj, nk5uNku is odd, and
2;k,k9[h1,...,Kj, nk2nk9#2. A K-apportionment is perfect if;k,k9[h1,...,Kj, nk5n .k9
Moreover, it is non-degenerated if K±1,n. The set of all possible apportionments of N is
denoted by G(N ).
Such a definition can be motivated as follows: in order to choose one or several candidates within X, the society implements a two-step voting process where:
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• in the first step, voters, once allocated in constituencies, ask a representative to promote their opinions (i.e. their preferences over X ) in parliamentary debates;
• in the second step, the final choice is made from all representatives’ opinions about candidates.
We obviously restrict our analysis to non-degenerated apportionments. Majority voting within each constituency N will generally leads to a non-transitive binary relationk
on X. The first condition in Definition 3 implies that it will lead to a tournament T (s) k
defined as the restriction of T(s) to N (i.e. xT (s)y⇔uhi[N : xs yju.0.5n ). Hence,
k k k i k
a representative’s opinion is a tournament on X. Since we suppose that n is odd, it follows that K is also odd. A perfect apportionment implies that each representative represents the same number of voters. This ‘equal weight’ requirement for representa-tives ensures that each individual preference is given ex ante the same potential influence on final decisions. However, perfectness implies some restriction on the electorate size. This explains the second condition in Definition 3, which formalizes an ‘almost equal representation principle’. It is worth noting that none of the inconsistency results obtained below rests on the necessity to design constituencies of unequal sizes: indeed, all proofs involve perfect apportionments.
In order to make precise the actual choice process considered here, we define the notion of tournament among representatives as follows:
Definition 4. Let [s][P(N ). Let A5hN ,...,N1 Kj be a K-apportionment of N. The tournament among representatives is the complete asymmetric binary relation T (s) on
A
X defined by: xT (s)y⇔uhk[h1,...,Kj: xT (s)yju.0.5K.
A k
This means that an outcome x will be socially preferred to an outcome y if x defeats y in more than one half of constituencies under majority rule.
Now suppose that S has been chosen by the society as the relevant choice concept. We now formalize two alternative choice methods which may prevail in a representational system:
2.1. The one-shot method
It consists in choosing the set of winners from the tournament among representatives (instead of from the initial tournament in a direct democracy). This leads to the following definition:
Definition 5. Let [s][P(N ) and A[G(N ). The set of S-winners in the one-shot method
is defined by S[T (s)]. A
The initial majority tournament T(s) generally differs from the tournament among
representatives T (s). This suggests an additional restriction on the design of apportion-A
Definition 6. Let [s][P(N ). An apportionment A is said to be representative if
7
T(s)5T (s). G(N,s) denotes the set of all representative apportionments of N. A R
This representativeness concept states that voters are allocated among constituencies in such a way that majority opinions are not biased through representation, in the sense that pairwise majority comparisons are identical in both representational and referendum systems. Note that, in the dichotomous case, the referendum paradox amounts to the existence of at least one non-representative apportionment.
2.2. The sequential choice method
Another way to design the choice process is to select within each constituency a set of winners, and then, at the final level, to choose the set of finally elected candidates among those previously chosen at the constituency level. Therefore, for a given profile [s],
S[T (s)] represents the set of elected candidates in constituency N . As long as there is
k k
no strategic bias in the representation process, each N -representative then supportsk
S[T (s)] as the set of best candidates. This means that < S[T (s)] is the set of
k 1#k#K k
candidates among which the final choice will prevail. These candidates are then compared according to T (s): indeed, xT (s)y means that more than one half of
A A
representatives (or constituencies) prefer x to y. This is formalized in the next definition:
Definition 7. Let [s][P(N ) and A5hN ,...,N1 kj[G(N ). The set of finally
el-ected candidates through the sequential choice method is defined by
S(T (s) /< S[T (s)]). A 1#k#K k
Note that in the dichotomous case, the sequential and the one-shot methods coincide. It is straight-forward to exhibit a profile for which this is no longer the case for more than two outcomes and some Condorcet-consistent solution.
It is now possible to formalize the way to compare referendum and representational systems, and more generally the sensitivity of final choices to the apportionment procedure. The next definition deals with the one-shot method:
Definition 8. A Condorcet-consistent solution S is neutral to apportionment if for any
electorate N, ;[s][P(N ), there exists A[G(N ) such that S[T (s)]5S[T(s)]. A
Furthermore, S is weakly neutral to apportionment if;N,;[s][P(N ),'A[G(N ) such that S[T (s)]>S[T(s)]±[.
A
When the one-shot method is used, both systems allow for the same set of final outcomes as long as the apportionment is representative. Moreover, such a set is not sensitive to the choice of a specific representative apportionment. In other words, an electoral reform does not matter as long as the representativeness property is not violated.
The next two definitions deal with the sequential choice method:
7
Definition 9. A Condorcet-consistent solution S is sequentially neutral to
appor-tionment if for any electorate N, ;[s][P(N ), there exists A[G(N ) such that S(T (s) /< S[T (s)])5S[T(s)]. Furthermore, S is weakly sequentially
A 1#k#K k
neutral to apportionment if ;N, ;[s][P(N ), 'A[G(N ) such that [S(T (s) /< S[T (s)])]>S[T(s)]±[.
A 1#k#K k
A solution is neutral (or sequentially neutral under the sequential rule) to an apportionment if it selects the same set of winners in both referendum and representa-tional systems. It is weakly neutral (or sequentially neutral) to an apportionment if the two sets of final outcomes intersect. WheneveruXu52, any Condorcet-consistent solution
S is neutral to any representative apportionment. We will see below that this is no longer
true for larger sets of candidates.
Another sensitivity concept is defined as follows:
Definition 10. LetS,G(N ). A Condorcet-consistent solution S is non-sensitive to Sif
9 9
;[s][P(N ), ;A5hN ,...,Nj, A95hN ,...,N j[S, S(T (s) /< S[T (s)])5
1 k 1 K9 A 1#k#K k
S(T (s) /< S[T (s)]). Furthermore, S is weakly non-sensitive to S A9 1#k9#K9 k
9 9
if ;A5hN ,...,N1 kj, A95hN91,...,N ,...,N1 K9j[S, [S(T (A s) /<1#k#KS[T (k s)])]>
[S(T (s) /< S[T (s)])]±[. A9 1#k9#K9 k9
A solution S is non-sensitive to a set of apportionments if it provides the same set of winners whatever the actual apportionment that prevails among admissible ones. It is weakly non-sensitive if any pair of apportionments share at least one winning outcome. The informal idea which underlines this definition is that, even if the choice between a direct and an indirect democratic system does influence the finally elected candidate(s), the latter should not change in case of a reform of the apportionment method. Such a requirement allows for restricting the set of potentially admissible electoral reforms.
Our approach to the representative voting systems naturally suggests the following questions, which are studied in the next two parts:
• Is it true that for any preference profile, there exists a representative apportionment? If not, is it possible to define a non-obvious restriction on the set of profiles which ensures the existence of such a representative apportionment?
• Does the representativeness requirement for apportionment matter? In particular, is it possible, for both collective choice methods defined above, to exhibit a profile for which a Condorcet-consistent solution is non-neutral to any non-representative apportionment?
• Can we find a Condorcet-consistent solution which is neutral to the set of all representative apportionments?
3. Existence of representative apportionments
Theorem 1. There exists a preference profile[s][P(N) such that G (N,s)5[. R
However, the proof presented in the reference above involves few voters and a very large set of candidates. The next theorem states that the same inexistence problem may arise even if the ratio candidates / voters is less than one:
Theorem 2. There exists a profile[s][P(N) such thatuXu/uNu#7 /9 andG(N,s)5[. R
Proof. Let N5h1,...,9jand X5h1,...,7j and let [s] be such that:
• ;i[N, 1si2⇔i[N95h1,...,5j
• ;i[N9,;m[X2h1,2j, ms 2⇔i5m22 i
• ;i[(N2N9), ;m[X2h1,2j, ms2 i
(All other majority pairwise comparisons do not matter.)
It is obvious to check that T(s) is such that 2 is a Condorcet loser. Now let
F 5hT ,T ,T1 2 3jbe the set of sub-tournaments on X associated with any 3-apportionment
A5hN ,N ,N1 2 3jof N, where each constituency involves 3 voters. We claim that A is not representative. Indeed, suppose that A[G(N,s) and w.l.g. that 1T 2 and 1T 2. Since
R 1 2
;m±1,2, mT(s)2, it follows that;m±1,2, m22[(N1<N ) (otherwise 2 would get a2
majority of votes against m in N and N , which would imply that 2 T (s)m, a
1 2 A
contradiction). Thus one can suppose w.l.g. that N1,N9 and uN2>N9u52. Let i0[N1
and m05i012. Since i is the only individual in N who prefers m to 2, then 2T m .0 1 0 1 0
Furthermore, every individual in N2>N9prefers m to 2, hence 2T m .Thus0 2 0 'm0[A2
h1,2jsuch that 2T m and 2T m , hence 2 T (s)m , which contradicts the
representa-1 0 2 0 A 0
tiveness of A. h
Theorem 2 points out a significant difference between the dichotomous and the general case: in the former case, there always exist a district map which yields agreement with direct democracy, whereas in the latter case, maybe no district map yields a social preference relation that coincides with overall majority preferences.
An essential ingredient in the proof above is the existence of majority dominations with tiny margins. What about the possibility to design representative apportionments when the overall tournament involves strong majorities, i.e. widespread social agree-ments? Suppose that n5Km, where K[h3,n21j denotes the (odd) number of con-stituencies (hence allowing for a perfect apportionment). Moreover, suppose that a majority of voters prefers some candidate y to some other one x. Then, in order for x to defeat y in a representational system, it must be the case that x is preferred to y by at least 0.25(K11)(m11) voters (this is a necessary requirement for x to get a majority of votes against y in a majority of constituencies). It follows that the minimal proportion q
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Theorem 3. ;a$0.75, ;[s][P(N) such that T(s)[D(a), G(N,s)5G(N).
This result provides an interesting restriction on the set of profiles which ensures that majority preferences will not be biased through the representation process. It simply states that ‘popular willingness’ has to be strong enough. An immediate corollary of Theorem 3 is the existence of at least one non-representative apportionment when at least one majority margin is less than 75% (as long as the number of constituencies as well as the size of the electorate are large enough). In fact, things may become even worse: indeed, we now prove that, as long as some pairwise majority margin is lower than 0.75, there exists at least one profile for which no representative apportionment
8
Step 1. Let h[h1,...,2k11jand w,w9[X. Let w (resp. wh 9h)5(w>N ) [resp. (wh 9>N )].h
We claim that the preference profile ensures the following four simple statements:
9
1. If uw u$( p11), uw9 u$( p11) and wPw9, then wT w9: indeed, i[w ⇒wsw9 (if
h h h h i
9 9
i[wh>w , this follows from Point 1 and wPwh 9, whereas if i[(wh2w ), this followsh
from Point 3) and thus it follows that at least ( p11) individuals in N prefer w to wh 9.
9
2. Ifuw u$( p11),uw u,( p11) and w9Pw, then w9T w: note first that i[w ⇒w9sw:
h h h h i
9 9
indeed, i[(w 2w )⇒w9s w from Point 2, and i[w >w ⇒w9sw from Point 1
h h i h h i
and w9Pw. Thenuwhu$( p11) implies that more than ( p11) individuals in N preferh
w9 to w.
9 9
3. Ifuwhu,( p11) and wPw9, then w9T w: one has ih [⁄ wh⇒w9siw: indeed, i[(wh2
9 9
w )⇒w9sw from Point 2, and i[[N 2(w <w )]⇒w9s w from Point 4 and
h i h h h i
9
wPw9. ThusuNh2whu$( p11) implies that more than ( p11) individuals in N preferh
w9 to w.
Step 2. m#(2p11)k1p(k11)⇒G(N,s)5[. R
Proof. Let w* be the Condorcet-winner of T(s)5P. We can deduce from Step 1 that if
A is representative, it must be true that ;w[X2hw*j, uhh[h1,...,2k11j: uwhu$( p1
1)ju$(k11): indeed, this ensures that w*T (s)w;w±w*. Now suppose that m#(2p1 A
1)k1p(k11). Let wk9 be the alternative defined by w5hi , 1x #x#mj, where hi ,...,i1 ( 2p11 )kjj5
<
h51N , and whereh V 5hi( 2p11 )k11,...,imj is allocated among the remaining constituencies of A in such a way that;h[hk11,...,2k11j,uNh>Vu#p. It is9
obvious to check that uhh[h1,...,2k11j: uwhu$( p11)ju,(k11), which implies that
w9T (s)w*. Hence there is no representative apportionment. A
Step 3. ;e .0, N can be chosen such that T(s)[D(0.752e) and G(N,s)5[. R
Proof. Let m5(2p11)k1p(k11). It is straightforward to check that ;k, p$0, m /n,
0.75. The fact that limk5p→` [m /n]50.75 ends the proof of Theorem 4. h
4. Sensitivity of tournament solutions to representation
4.1. Representative systems under one-shot method
Theorem 5. There exists a profile [s][P(N) such that uXu/uNu#7 /9, and ;A[G(N),
UC[T(s)]±UC[T (s)]. A
Proof. Let N5h1,...,9jand X5h1,...,7jand let [s] be defined by (voters’ names appear
in the first row, and alternatives are listed in decreasing preference order):
1 2 3 4 5 6 7 8 9
1 2 a a a 3 4 1 2
a a 3 4 5 5 2 5 4
b b b b b 1 5 3 3
2 4 1 5 1 4 1 2 1
3 5 5 2 3 2 3 4 5
4 3 4 3 2 b b b b
5 1 2 1 4 a a a a
It is easily checked that T(s) (hereafter denoted by T ) is such that 1 T 2,4,a,b, 2 T
3,4,a,b, 3 T 1,4,a,b, 4 T 5,b, 5 T 1,2,3,b, and a T 3,4,5,b. Thus b is a Condorcet loser, and it is easily checked that UC(T )5ha,1,2,3,4,5j.
Let A5(N ) be a 3-apportionment of N and let S 5[T (s)] be the
p hp51,2,3j p p51,2,3
associated set of tournaments on X. It follows from Theorem 2 thatG (N,s)5[. The R
proof is organized according to the following strategy: since no representative
apportion-2 9
ment exists,'(x, y)[X such that xTy and yT (s)x; each step of the proof is devoted to A
a specific set of (x, y)-type pairs of candidates.
Step 1. y5b, x[h1,...,5j.
Case 1. x51.
It follows that one may assume w.l.g. that voters 2,3,4 and 5 form majorities in both
N and N , and that either a is a Condorcet winner in T or 2 T a T 1,3,4,5,b. In the1 2 A A A
former case, UC(T )5haj±UC(T ). In the later case, UC(T )5UC(T )⇒1,3,4,5 T 2.
A A A
Since hi[N: 4s 2j5h3,4,6,7j, andhi[N: 3s 2j5h3,5,6,8j, it is obviously seen that 4
i i
T 2A ⇔2 T 3, hence either 3A [⁄ UC(T ), or 4A [⁄ UC(T ).A
Case 2. x52.
Voters 1,3,4 and 5 form a majority in N and A in N , which implies that either a is a1 2
Condorcet winner in T or 1 T a T 2,3,4,5,b. As above, UC(T )A A A 5UC(T )A ⇒the later case prevails, and 2,3,4,5 T 1 [condition (*)]. Sincehi[N: 4s1j5h2,4,7,9j, one has to
A i
consider the following possible sub-cases:
9
Case 2.1.h2,4j,N and1 h7,9j,N : it must be the case that N is either2 1 h1,2,4jorh2,3,4j, or h2,4,5j. N15h1,2,4j andh1,3,4,5j have a majority in N and N imply with (*) that1 2
A5hN ,N ,N1 2 3j[h(1,2,4),(6,7,9),(3,5,8)j<h(1,2,4),(6,7,8),(3,5,9)j⇒2 TA
3,4,5⇒h3,4,5j>UC(T )A 5[. Moreover, N15h2,3,4j, (*), 3 TA 1 and b TA 2⇒A5
h(2,3,4),(7,6,9),(1,5,8)j⇒4 TA 2,3,5⇒h2,3,5j>UC(T )A 5[. Finally, the reader may easily check that N15h2,4,5j, 5 T 1 and b T 2A A ⇒A5h(2,4,5),(7,6,9),(1,3,8)j⇒4 T 3,5,A
whereas 2 T 3 T 5 T 2 and 2 T 4⇒3[⁄ UC(T ), hence UC(T )±UC(T ).
A A A A A A
Case 2.2. h2,7j,N and1 h4,9j,N : b T 22 A ⇒N2[h1,4,9j<h3,4,9j<h4,5,9j. In the first
case, b T 2 and 5 T 1A A ⇒A5h(2,7,8),(1,4,9),(3,5,6)j⇒2 T 3,4,5A ⇒h3,4,5j>UC(T )A 5 [. Moreover, if N25h3,4,9j, then b T 2, 5 T 1 and 3 T 1 are mutually incompatible.A A A
Finally, if N25h4,5,9j, then b T 2 and (*)A ⇒A5hN ,N ,N1 2 3j[h(2,6,7),(4,5,9),(1,3,8)j<
h(2,7,8),(4,5,9),(1,3,6)j. In both cases, one get 2,4,5 T 3, hence 3A [⁄ UC(T ).A
Case 2.3. h2,9j,N and1 h4,7j,N : b T 22 A ⇒N2[h1,4,7j<h3,4,7j<h4,5,7j. In the first
case, (*) and b TA 2⇒A5h(2,8,9),(1,4,7),(3,5,6)j⇒2,4,5 TA 3⇒3[⁄ UC(T ). In theA
second case, (*) and b TA 2⇒A[h(2,8,9),(1,4,7),(3,5,6)j<h(2,8,9),(3,4,7),(1,5,6)j,
which leads to the same conclusion. Finally, if N25h4,5,7j, b T 2A ⇒ either 1 T 5 or 1A
TA 3, hence either 5[⁄ UC(T ) or 3A [⁄ UC(T ). This proves that b TA A 2 implies that
UC(T )±UC(T ). A
Case 3. x[h3,4,5j.
This means that voters 1,2,4 and 5 form a majority in both N and N . It follows from1 2
[s] that a T 3,4,5,b. Suppose thatA h1,2j,N and1 h4,5j,N . Then if N2 15h1,2,3j, a is a Condorcet winner in T , hence UC(T )±UC(T ). Thus UC(T )5UC(T )⇒N 5h1,2,zj,
A A A 1
where z[h6,...,9j, which implies that 1,2 T a T 3,4,5,b. It is straightforward to checkA A
that the same conclusion prevails for all other way to allocate 1,2,4 and 5 among N and1
N . Now assume that 1 T 2; UC(T )2 A 5UC(T )A ⇒3,4,5 T 1. Moreover, 1 T 2 and 4 T 1A A A
implies the existence of at least one constituency k[h1,2,3jwhere 4 T 1 T 2. But thisk k
is impossible sincehi[N: 4s 1j5h1,3,5,6,8j>hi[N: 1s2j5h2,4,7,9j5[. Thus 1 T
i i A
4 and thus 4[⁄ UC(T ). Finally, suppose that 2 T 1. Similarly, it must be the case thatA A
3,4,5 T 2. But this is impossible since hi[N: 2s1j5h2,4,7,9j>hi[N: 3s2j5
A i i
h3,5,6,8j5[. Thus, 3[⁄ UC(T ). This proves that if x53, UC(T )±UC(T ). The reader
A A
may check that if x[h4,5j, the same argument applies. This concludes the proof of Step
1.
Step 2. y5b, x5a.
This implies that voters 6,7,8 and 9 form a majority in two constituencies. It follows that a is a Condorcet loser in T , hence a[⁄ UC(T ), thus UC(T )±UC(T ).
A A A
Step 3. x5a, y[h3,4,5j.
Step 4. y5a, x[h1,2j.
Suppose first that a T 1. It follows from [s] that voters 2,3,4 and 5 form a majority A
in two constituencies. But this implies that b T 1, and thus UC(T )±UC(T ) from Step
A A
1. Finally, if a T 2, voters 1,3,4 and (form a majority in two constituencies, whichA
implies that b T 2, hence again UC(T )±UC(T ) from Step 1. This concludes the proof.
A A
h
When no restriction bear upon the ratio between the number of voters and the number of candidates, things may become even worse: it is possible to exhibit a preference profile such that, for any Condorcet-consistent tournament solution, direct and repre-sentative voting systems always leads to mutually disjoint sets of winners.
Theorem 6. No Condorcet-consistent tournament solution S is weakly neutral to
apportionment.
Proof. Consider the preference profile [s] defined in the proof of Theorem 4. Let
1 2 1
Let us examine the tournament among representatives T (A s). For w,w9[X, define a, b, c and resp. by:
Case 4. (c2b).ua2du. We get w9T (A s)w. Moreover (c2b).(a2d )⇔uhh[h1,...,2k1
Finally, for any apportionment A, any Condorcet-consistent solution S will select from
1 2
T (s) as unique winner the top-element of X for P , whereas S[T(s)] is clearly
A A
reduced to the top-element of X for P. The theorem follows from the fact that
1
;A[G(N ), X ±[. h A
4.2. Representative system under the sequential choice method
We turn now to the comparison between the direct system and the representative system based on the sequential choice method. The next result points out that the same negative result prevails as for the one-shot method:
Theorem 7. No Condorcet-consistent solution S is weakly sequentially neutral to
apportionment.
Proof. Consider again the profile introduced in the proof of Theorem 4. It is shown in
the proof of Theorem 4 that ;A5hN , 1#h#2k11j, T (s) is a linear order on X
Theorem 8. There exists a profile [s][P(N) such that G(N)5G (N,s) and, R
for any Condorcet consistent solution S finer than UC,
[S(T(s))]>[S(T (s) /< S(T (s)))]5[ ;AhN,1#k#Kj. A 1#k#K k k
´
Proof. See Laffond and Laine (1996). We just sketch the proof here. It involves a slight
modification of the profile introduced in the proof of Theorem 4: let X5hw,N:
uwu52p21jand P be a linear order on X; voter i’s preferences over a pair hw,w9j are now defined by:
• if i[[w>w9], wsw9⇔w9Pw i
• if i[[w2w9], wsw9 i
• if i[N2[w<w9], wsw9⇔wPw9. i
Let w* be the maximal element of X for P. It is rather easy to check that P5T(s),
hence that w* is the Condorcet-winner of T(s). Let A5hN ,...,N1 Kj[G(N ) where uNku[h2p21,2p11j, and let w,w9[X be such that wPw9. Suppose that w*[⁄ A. Consider
constituency N . It follows from [k s] that w9T (k s)w⇔uNk>w9u.p. Thus uhk: w9T (s)wju#1 (since constituencies are mutually disjoint). This implies that A is
k
representative. Now suppose that w*[⁄ A. Let w[X be s.t. w*T (s)w. We have from 1
[s] that uN12wu.0.5uN1u. Moreover,;i[(N12w), N1siw. Since N1siw*;i[N , we1
get that N1→w* in T (k s), hence w*[⁄ UC[T (1 s)]. In the case where w*[A, it is left to
the reader to prove that w**→w* in all T (s), where w** is the second top element of k
X for P. Thus w*[⁄ <1#k#KUC[T (k s)]. Hence w*[⁄
<
1#k#KS[T (k s)] for any S,UC. h
This result may be given the following statistical-type interpretation, dealing with the accuracy of predictions about the election results: suppose that the prevailing system in a society N with (2m11)(2k11) voters is direct democracy, and that the set of candidates is X5hw,N: uwu52p21j; therefore, the actual voting outcome is given by S[T(s)];
moreover, suppose that T(s) has a Condorcet winner w*. Now, consider an opinion
research institute which tries to forecast the election outcome, using a sample N of sizeh
2k11. The survey outcome is then given by S[T (s)]. The above proof defines a profile k
for which this outcome announce that, for any possible sampling design, w* will be defeated.
Our last result deals with the sensitivity of Condorcet-consistent solutions to the choice of a specific apportionment. It states that any such solution is sensitive to the set of representative apportionments. More precisely, the profile described in the proof of the next theorem allows for two representative apportionments leading to mutually disjoint sets of winners:
Theorem 9. No Condorcet consistent solution S is weakly non-sensitive to G(N,s). R
Proof. Consider the profile defined in the proof of Theorem 8 and assume that
1 2 2 2 2
for k51,2,3 the sets Xk and Xk by Xk 5hw[X: uw>Nku$2j and Xk 5(X2Xk ).
*
Using the same arguments as in the proof of Theorem 6, one get that T has wk k as
2
*
Condorcet winner, where wk is the worst element of Xk for P. Now suppose that P is
2
such that w15h1,2,3jPw25h4,5,6jPw35h7,8,9jPw45h1,4,7j Pw55h2,5,8j P w65
2
h3,6,9j P w ;w±w , h51,...,6. Let A and A9 be the apportionments, respectively h
9 9 9
defined by A5hN ,N ,N1 2 3j5hw ,w ,w1 2 3j and A95hN ,N ,N1 2 3j5hw ,w ,w4 5 6j. It
fol-9
lows that
<
h51,2,3S(T )h 5hw ,w ,w1 2 3j, whereas<
h51,2,3S(T )h 5hw ,w ,w4 5 6j. ThusS(T /A
<
k51,2,3S(T )k 5[. It is left to the reader to check that the proof is easily generalized to any number n of voters. h5. Concluding comments
This paper may be conceived as an attempt to enlarge the set of axioms for tournament solution concepts with an additional one having a positive flavor, for most collective decision procedures are actually of sequential nature. It appears that, unless when there is a widespread agreement on pairwise comparisons between candidates, there is no hope for a sincere political body to design apportionments of the electorate without bearing the risk to be accused of gerrymandering. Indeed, sensitivity to the drawing of electoral districts appears to be an intrinsic feature of a representational system based on the majority rule.
Many routes for further research may be followed. One is to plug a representative democratic setting into already studied models of political competition. We guess that a better understanding of the way political parties can strategically use the apportionment mechanism is still to be achieved, and would be of great interest to improve the positive
´ theory of political institutions. A first set of results is given in Laffond and Laine (1996). More generally, all results obtained in voting game theory could be reexamined within our setting. This is the topic of an ongoing research.
Acknowledgements
The helpful comments of two anonymous referees are gratefully acknowledged. A first version of this paper was written while the second author was visiting the Economics Department of Keele University. The authors thank the staff of this department, and especially Richard Cornes, Peter Lawrence and Gauthier Lanot for valuable discussions.
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