Mathematical Social Sciences 39 2000 35–53 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. 0165-4896 00 – see front matter
2000 Elsevier Science B.V. All rights reserved.
P I I : S 0 1 6 5 - 4 8 9 6 9 9 0 0 0 0 7 - 4
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G . Laffond, J. Laine Mathematical Social Sciences 39 2000 35 –53
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:
1 1
1 1
Then an overall majority prefers outcome 0 to outcome 1, which means that 0 is the
1
Nurmi 1998 provides a comparative analysis of several voting paradoxes in the case of dichotomous choice; see also Nurmi 1997 on the referendum paradox.
´ G
. Laffond, J. Laine Mathematical Social Sciences 39 2000 35 –53 37
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, ST 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 N lead to a tournament T
on X.
k k
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 ,...,T j of
1 K
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 all uN u are odd integers, this defines a new
k
tournament T on X, called tournament among representatives, which is used to define
A
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
Note that we avoid the fractional problem suggested above.
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G . Laffond, J. Laine Mathematical Social Sciences 39 2000 35 –53
the final set ST of winners, called the representational or indirect outcome. This
A
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 ST ]. Thus, for a given
k k
apportionment A 5 hN ,...,N j, the actual agenda is defined by X 5
ST ; the
1 k
1k K k
set of final winners is then defined by ST X, which is the set of S-winners in the
A
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
5
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
Political parties, such as the French socialist party, are regularly involved in arbitration procedures among several militant tendencies. The actual agenda may be interpreted as the set of all those tendencies which
emerge from local debates. We provide below an alternative interpretation of the sequential choice rule.
´ G
. 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
