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. Laffond, J. Laine Mathematical Social Sciences 39 2000 35 –53 39
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
N 5 h1,...,i,...,nj of 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 order s 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 PN . 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
i
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G . Laffond, J. Laine Mathematical Social Sciences 39 2000 35 –53
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][PN such that
T 5T 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, xTsy
⇔ uhi [N:xs yju.a.
i
Let D [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 from D to X. The set of S-winners of T [D is the subset ST of X.
Furthermore, if X9,X and T [D, we denote by ST 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 xTx;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 S9 is said to be finer than solution S denoted by S9S if for any tournament T on X, S9T ST . 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 UCT 5 hx[X: there is no
y [X such that y →
x j.
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 A5
hN ,...,N j of N into K
1 k
non-empty subsets called constituencies such that 2;k [ h1,...,Kj, n 5uN u is odd, and
k k
2;k,k9[ h1,...,Kj, n 2n 2. A K-apportionment is perfect if ;k,k9[h1,...,Kj, n 5n .
k k 9
k k 9
Moreover, it is non-degenerated if K ±1,n. The set of all possible apportionments of N is denoted by GN .
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:
6
We are not interested here with the fact that Da may be a proper subset of D for some values of a.
´ G
. Laffond, J. Laine Mathematical Social Sciences 39 2000 35 –53 41
• 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 relation
k
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][PN . Let A5
hN ,...,N j be a K-apportionment of N. The
1 K
tournament among representatives is the complete asymmetric binary relation T s on
A
X defined by: xT s y ⇔
uhk[h1,...,Kj: xT syju.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][PN and A[GN . 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
ment.
´ 42
G . Laffond, J. Laine Mathematical Social Sciences 39 2000 35 –53
Definition 6. Let [s][PN . 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 supports
k
S[T s ] as the set of best candidates. This means that S[T s ] is the set of
k 1k 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][PN and A5
hN ,...,N j[GN. The set of finally el-
1 k
ected candidates
through the
sequential choice
method is
defined by
ST s S[T s ].
A 1k 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][PN , there exists A[GN such that S[T s ]5S[T s ].
A
Furthermore, S is weakly neutral to apportionment if ;N, ;[s][PN , A[GN 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
Note that, since it relates the apportionment to some given profile, this representativeness concept cannot be used as a guideline for real world institutional design: district maps are drawn to endure over diverse issues
and preference profiles.
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. Laffond, J. Laine Mathematical Social Sciences 39 2000 35 –53 43
Definition 9. A Condorcet-consistent solution S is sequentially neutral to appor- tionment if for any electorate N, ;[s][PN , there exists
A[GN such that ST s
S[T s ]5S[T s ]. Furthermore, S is weakly sequentially
A 1k K
k
neutral to
apportionment if
;N, ;[s][PN,
A[GN such
that [ST s
S[T s ]] S[T s ] ± [.
A 1k 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. Whenever
uXu52, 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. Let S,GN . A Condorcet-consistent solution S is non-sensitive to S if
9 9
;[s][PN, ;A5 hN ,...,N j, A95hN ,...,N j[S, ST s
S[T s ]5
1 k
1 K 9
A 1k K
k
ST s S[T s ].
Furthermore, S
is weakly
non-sensitive to
S
A9 1k 9K 9
k
9 9
if ;A5
hN ,...,N j, A95
hN9 ,...,N ,...,N j[S, [ST s
S[T s ]]
1 k
1 1
K 9 A
1k K k
[ST s S[T s ]] ± [.
A9 1k 9K 9
k 9
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 candidates, 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
