Directory UMM :Data Elmu:jurnal:M:Mathematical Social Sciences:Vol39.Issue3.May2000:

Mathematical Social Sciences 39 (2000) 277–301
www.elsevier.nl / locate / econbase

Strategy-proofness, Pareto optimality and strictly convex
norms
Hans van der Stel*
University of Maastricht, Quantitative Economics, P.O. Box 616, 6200 MD Maastricht, The Netherlands
Received 1 December 1996; received in revised form 1 April 1999; accepted 1 April 1999

Abstract
A voting scheme assigns to each profile of alternatives chosen by n individuals a compromise
alternative. Here the set of alternatives is represented by the Euclidean plane. The individual
utilities for the compromise point are equal to the negatives of the distances of this point to the
individually best points. These distances are measured by a given strictly convex norm, common
to all agents. A voting scheme is strategy-proof, if voting for one’s best point is an optimal
strategy for all agents. A characterization is given of all strategy-proof, Pareto optimal voting
schemes. Since the Euclidean norm is strictly convex, this result holds for Euclidean preferences
in particular.  2000 Elsevier Science B.V. All rights reserved.
Keywords: Voting schemes; Strategy-proofness; Pareto optimality; Euclidean preferences; Generalized
medians

1. Introduction
Consider situations in which a group of individuals must decide on the choice of one
alternative from a given set of alternatives. The decision problem is resolved by voting:
each agent votes for one alternative; next, a rule is invoked which selects a unique
compromise alternative. Such a rule will be called a voting scheme (or briefly, a scheme).
Thus, a voting scheme is a rule which prescribes for all possible combinations of
individually reported points what the outcome will be.
In this paper I focus on voting schemes which are strategy-proof (or nonmanipulable)
and Pareto optimal. A voting scheme is strategy-proof if no agent can gain by voting
strategically, no matter what the other agents do. It is Pareto optimal if there is no
*Tel.: 131-43-388-3835; fax: 131-43-325-8535.
E-mail address: h.vanderstel@ke.unimaas.nl (H. van der Stel)
0165-4896 / 00 / $ – see front matter  2000 Elsevier Science B.V. All rights reserved.
PII: S0165-4896( 99 )00031-1

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H. van der Stel / Mathematical Social Sciences 39 (2000) 277 – 301

alternative which is for all agents at least as good as the final outcome, and better for at

least one agent.
A third requirement which might be imposed upon voting schemes is anonymity. A
voting scheme is anonymous if interchanging voters does not affect the outcome. For the
sake of convenience I will use the abbreviation AN. Similarly, SP indicates strategyproofness, and PO Pareto optimality.
The first theoretical results about the existence of voting schemes satisfying SP and
PO were rather negative. The classic result of Gibbard (1973) and Satterthwaite (1975)
implies that only dictatorial schemes (a scheme is dictatorial if there is one voter who
completely determines the outcome) satisfy these requirements, as long as there are at
least three alternatives and no restrictions are made with respect to the preferences an
agent might have.
In many economic and political environments, however, it is reasonable to place some
restrictions on the individual preferences. By this the strategy-proofness condition is
weakened (since there are fewer opportunities for strategic behaviour), which opens the
possibility for the existence of nondictatorial voting rules which satisfy SP and PO.
Among the first authors who followed this approach were Laffond (1980) and Moulin
(1980). At the same time they extended the set of alternatives to a continuum, which
made their theory easier to apply to standard micro-economic models.
In Moulin (1980) the set of alternatives consists of the whole real line. Only
single-peaked preferences are allowed, i.e. preferences with a unique best alternative,
and decreasing when moving away from this alternative in either direction. Moulin

characterizes the class of all voting schemes which satisfy SP, PO and AN. All these
voting schemes turn out to be medians, with as arguments the individually reported
points, plus some extra constants (sometimes called ‘phantom voters’). This result still
holds if only Euclidean preferences are considered, as follows from Proposition 4 in
Border and Jordan (1983). In a Euclidean preference, one point is preferred to another if
it is closer to the best point, measured according to the Euclidean distance. So, a
Euclidean preference is completely characterized by its best point.
Kim and Roush (1984) extend Moulin’s results to higher dimensions. They characterize all continuous, anonymous and strategy-proof voting schemes with the Euclidean
plane as set of alternatives and only Euclidean preferences allowed. Formally, these
voting schemes can be described as follows. Choose an orthogonal pair of axes in the
plane. Coordinates of points are projections on these axes. Choose n 1 1 fixed points in
(R < h 2 `,`j)2 , where n is the number of voters. For each profile of individually
reported points add these fixed points and calculate the two median coordinates; these
will be the coordinates of the compromise point.
Kim and Roush further show that, if Pareto optimality is added as a requirement, then
the number of voters must be odd, and the corresponding voting schemes can be
described in the same way, but without the addition of fixed points. I will call a voting
scheme which can be described in such a way a median scheme [in Peters et al. (1992,
1993) such a scheme is called a generalized median solution].
In the same setting, Peters et al. (1992) show that a voting scheme is continuous if it

satisfies SP and PO. Thus, a voting scheme satisfies SP, PO and AN if and only if it is a

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279

median scheme. They further show that there are no voting schemes satisfying these
requirements if the number of agents is even, or if the set of alternatives is extended to a
Euclidean space with dimension higher than two.
In Peters et al. (1993) these results are generalized to the case where the distance
function which describes the individual preferences is derived from a strictly convex
norm (strict convexity of a norm means that its unit disc is a strictly convex set. Notice
that this is true for the Euclidean norm). Again, if the set of alternatives is twodimensional, a voting scheme satisfies SP, PO and AN if and only if it is a median
scheme with respect to some orthogonal pair of axes. Here, ‘orthogonality’ means
orthogonality in the sense of Birkhoff (1935) (the usual definition of orthogonality does
not apply for an arbitrary strictly convex norm; for norms which are induced by an inner
product, like the Euclidean norm, both concepts are equivalent). For higher dimensions,
the existence of voting schemes satisfying SP, PO and AN depends on the strictly
convex norm by which the preferences are determined.
In this paper I study the same situation as in Peters et al. (1993), but I no longer

require voting schemes to satisfy AN. Thus, voting schemes are assumed to satisfy SP
and PO, where preferences are induced by an a priori fixed strictly convex norm. The
motives for this investigation are two-fold:
(i) Agents may be shareholders, or may represent countries or parties. In such a case
some agents should have more influence in the decision process than others. If so,
anonymity is no longer a reasonable requirement.
(ii) Whenever the number of agents is even, there are no voting schemes satisfying
SP, PO and AN. Therefore, it is interesting to see what class of voting schemes
emerges if the requirement of anonymity is dropped. Next, a member of this class can
be selected such that equality of power (which is the main motive to demand
anonymity) is met, if possible.
In order to be able to describe the main results of this paper I discuss two types of
voting schemes, which are formally defined in Section 3 and satisfy both SP and PO.
Shifted median schemes are defined for the case n 5 3. These are such that the
compromise point is determined by a median scheme for all profiles. However, by which
median scheme the compromise point is determined depends on the position of the
reported points in relation to each other. More precisely, if these points are noncollinear
an angle can be defined by these points, and which median scheme should be used
depends on the fact whether this angle is positive or negative. Of course, if both these
median schemes are the same, the shifted median scheme coincides with this median

scheme.
Coalition dependent median schemes can be defined for an arbitrary number of
agents. These schemes are defined with respect to some orthogonal basis of R 2 and some
class G consisting of coalitions of agents. G is called the class of winning coalitions. If a
coalition belongs to G, its members can enforce the compromise point to be any point in
the Euclidean plane, by voting unanimously for this point. If a coalition dependent
median scheme satisfies SP and PO, its class of winning coalitions must satisfy a

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H. van der Stel / Mathematical Social Sciences 39 (2000) 277 – 301

regularity condition, named distinctiveness. The coordinates of the compromise point
can now be derived by a minmax rule, with as arguments the corresponding coordinates
of the individually reported points.
We are now able to formulate the main results. Theorem 3.4 asserts that, in the case
with three voters, a voting scheme satisfies SP and PO if and only if it it a shifted
median scheme or a dictatorial scheme. This result is used for the proof of Theorem 3.8,
where the number of agents is arbitrary. This theorem states that a voting scheme w
satisfies SP and PO if and only if one of the following assertions is true:

(i) w is a coalition dependent median scheme with respect to some orthogonal basis
and some distinctive class of coalitions.
(ii) There are three agents such that w coincides with a shifted median scheme, with
as arguments the reported points of these agents (so the other agents have no
influence at all).
Since the Euclidean norm is also strictly convex, these results are valid in particular in
the case where all preferences are assumed to be Euclidean. In all other papers on this
class of preferences (Laffond, 1980; Kim and Roush, 1984; Bordes et al., 1990; Peters et
al., 1992, 1993) the main results are on anonymous voting schemes. Luckily, Pareto
optimality is a strong requirement, enabling me to do without anonymity. Due to the fact
that only the strict convexity of the Euclidean norm is essential in the way of reasoning I
follow in this paper, things can be formulated in a more general way.
From the two classes of voting schemes which satisfy SP and PO, the class of shifted
median schemes seems to be very unattractive. I cannot imagine a motive to change to
another coordinate system, when the positions of the expected points change relatively to
one another. Nevertheless, these voting schemes fulfill the requirements. However, when
the number of voters exceeds three, shifted median like voting schemes can easily be
ruled out, by demanding that all agents must have at least some influence. Adding this
requirement, only coalition dependent median schemes turn out to be satisfying (and not
all of them), as long as the number of agents is not equal to three. This result is stated in

Corollary 5.1. Thus, from a practical viewpoint, only coalition dependent median
schemes seem to be interesting.
The class of coalition dependent median schemes is closely related to the class of
generalized median voter schemes studied by Barbera` et al. (1993). Starting with Border
and Jordan (1983), this class appears in many articles on strategy-proofness. The class of
coalition dependent median schemes which are defined with respect to the standard
coordinate system is a subclass of the class of generalized median voter schemes. Since
all generalized median voter schemes are defined with respect to the standard coordinate
system, only these coalition dependent median schemes are generalized median voter
schemes.
The organization of this paper is as follows. Section 2 contains preliminaries. In
Section 3 I define shifted median schemes and coalition dependent median schemes, and
state the main results. Section 4 consists of a proof of Theorem 3.8 and is purely
technical. In Section 5 various issues are considered. First, I discuss briefly which voting
schemes satisfy SP and PO if the set of alternatives is extended to a higher-dimensional

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281

Euclidean space. In the remainder of this section the two-dimensional case is again
considered. Corollary 5.1 represents a way to exclude shifted median schemes in cases
where the number of voters exceeds three. Next, a relaxation of anonymity is proposed,
which is called equality of power (EP). This relaxation allows for the possibility of new
results for cases where the number of voters is even. The first results are promising, but
more research needs to be done. Finally, a special type of coalition dependent median
scheme is discussed, the so-called weighted median scheme. This type of voting scheme
is easy to apply in practical situations, but seems to be neglected in social choice
literature. Section 6 concludes.

2. Preliminaries
Let N 5 h1, . . . ,nj, with n $ 1, denote the set of agents. These agents are assumed to
have preferences on the space R 2 , which have the following form. There is a point
x [ R 2 (the bliss point) and a metric d on R 2 such that y is weakly preferred to z if and
only if d ( y,x) # d (z,x) for all y,z [ R 2 . Moreover, I assume that each preference is
determined by one and the same metric d.
I assume d to be induced by the strictly convex norm i?i (a norm i?i is strictly convex
if for all a,b,c [ R 2 with c [
⁄ convha,bj,1 we have ia 2 ci 1 ib 2 ci . ia 2 bi. This
definition is equivalent to the standard definition in functional analysis; see, for instance,

Kreyszig, 1978, p. 332). An example of such a metric is the Euclidean metric.
A voting scheme is a map w : (R 2 )N → R 2 . The idea is that each individual votes for a
point (for instance, his bliss point), whereupon w determines a compromise point.
Elements of (R 2 )N are called profiles. Given a profile p [ (R 2 )N , p(i) denotes agent i’s
bliss point, and for all M , N, p(M) denotes h p( j) : j [ Mj.
The aim of this paper is to characterize the class of voting schemes which satisfy the
following conditions.
• Strategy-proofness (SP): for all i [ N and all profiles p,q [ (R 2 )N with p( j) 5 q( j)
for all j ± i, we have that d ( p(i),w ( p)) # d ( p(i),w (q)).
• Pareto optimality (PO): for no p [ (R 2 )N is there an x [ R 2 with d (x, p(i)) #
d (w ( p), p(i)) for all i [ N such that at least one of these inequalities is strict.
Strategy-proofness means that it is always optimal to vote for one’s bliss point, no
matter what the other agents do. In game theoretical terms voting for one’s bliss point is
a weakly dominant strategy. Pareto optimality requires that the compromise point is
always optimal in the sense that there is no other point which is for all agents at least as
good as the compromise point and better than the compromise point for at least one
agent. The following conditions are also of interest:

1

‘conv’ denotes ‘the convex hull of’.

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H. van der Stel / Mathematical Social Sciences 39 (2000) 277 – 301

• Weak coalitional strategy-proofness (WSP): for all M , N, x [ R 2 and p,q [ (R 2 )N
with p(M) 5 hxj and p( j) 5 q( j) for all j [ N\M, we have that d (x,w ( p)) # d (x,w (q)).
• Anonymity (AN): w ( p + g ) 5 w ( p) for every p [ (R 2 )N and every permutation
g : N → N.
• Nondictatorship (ND): there is no i [ N with w ( p) 5 p(i) for all p [ (R 2 )N .
Weak coalitional strategy-proofness means that if all agents belonging to some subset
of N have the same bliss point, they cannot gain by simultaneously voting for other
points [in Peters et al. (1993) this condition is called Intermediate Strategy-proofness].
Anonymity states that interchanging agents does not affect the compromise point.
Nondictatorship means that there is no agent who completely determines the compromise point, in the sense that this point always coincides with his reported point (if a
voting scheme does not satisfy ND, it is called dictatorial). It follows by definition that
WSP implies SP. Moreover:
Lemma 2.1 (Peters et al., 1993, Lemma 2.2). Let w be a voting scheme. Then the
following assertions are equivalent:

( i) w satisfies SP,
( ii) w satisfies WSP.
Before I continue to discuss some implications of SP and PO, I introduce some
geometrical notations. All occurring points are elements of R 2 . [x,y]: 5 convhx,yj,
[x,y): 5 [x,y]\hyj, (x,y): 5 [x,y)\hxj, [x,y, → ): 5 hx 1 l( y 2 x) : l [ [0,`)j, (x,y, → ): 5
[x,y, → )\hxj, (←,x,y, → ): 5 [x,y, → ) < [y,x, → ), n(x,y,z): 5 convhx,y,zj. /(x,y,z) denotes the angle between x 2 y and z 2 y whenever x,y,z are noncollinear, and adopts
values between 21808 and 1808. x / /y means that x is parallel to y (here x and y are
considered as vectors).
I am now able to prove the following characterization of Pareto optimality, which I
will use throughout the paper:
Proposition 2.2. Let w be a voting scheme. Then the following assertions are equivalent:

( i) w satisfies PO,
( ii) w ( p) [ convh p(1), . . . , p(n)j for all p [ (R 2 )N .
Proof. ((i) ⇒ (ii)) Assume w
satisfies PO. Let p [ (R 2 )N . Suppose
w ( p) [
⁄ convh p(1), . . . , p(n)j (see Fig. 1). Let ? denote the standard inner product.
Because of a well-known separation theorem from convex analysis (see Rockafellar,
1970, Theorem 11.4) there are t [ R, c [ R 2 such that c ? w ( p) . t and c ? p(i) , t for
all i [ N. Let , 5 hx [ R 2 : c ? x 5 t j. Let q(i) [ [ p(i),w ( p)] > , for i [ N. Then the q(i)
are collinear. Take j,k [ N with q(i) [ [q( j),q(k)] for all i [ N. Since w ( p) [
⁄ [q( j),q(k)],
we have by the strict convexity of i ? i that
iw ( p) 2 q( j)i 1 iw ( p) 2 q(k)i . iq( j) 2 q(k)i.

(1)

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283

Fig. 1. Proof of Proposition 2.2.

Let
iw ( p) 2 q(k)iq( j) 1 iw ( p) 2 q( j)iq(k)
a 5 ]]]]]]]]]]].
iw ( p) 2 q( j)i 1 iw ( p) 2 q(k)i
Then by (1)
iw ( p) 2 q( j)i iq( j) 2 q(k)i
ia 2 q( j)i 5 ]]]]]]]]] , iw ( p) 2 q( j)i.
iw ( p) 2 q( j)i 1 iw ( p) 2 q(k)i
Similarly, ia 2 q(k)i , iw ( p) 2 q(k)i. Furthermore, a [ [q( j),q(k)].
If q(i) [ [q( j),a], then ia 2 q(i)i 5 ia 2 q( j)i 2 iq( j) 2 q(i)i # iw ( p) 2 q( j)i 2
iq( j) 2 q(i)i , iw ( p) 2 q(i)i, because of the strict convexity of i ? i. Similarly, ia 2
q(i)i , iw ( p) 2 q(i)i if q(i) [ [q(k),a]. Hence, ia 2 q(i)i , iw ( p) 2 q(i)i for all i [ N.
Therefore,
ia 2 p(i)i # ia 2 q(i)i 1 iq(i) 2 p(i)i , iw ( p) 2 q(i)i 1 iq(i) 2 p(i)i 5
iw ( p) 2 p(i)i for all i [ N. This contradicts PO. Consequently, w ( p) [
convh p(1), . . . , p(n)j.
(( ii)⇒( i) ). Assume w ( p) [ convh p(1), . . . , p(n)j for all p [ (R 2 )N . Let q [ (R 2 )N and
y [ R 2 with y ± w (q). It suffices to prove that iw (q) 2 q(i)i , iy 2 q(i)i for at least one
i [ N. Because of the assumption we can take j,k [ N and z [ [q( j),q(k)] such that
w (q) [ [z,y]. If y [ [q( j),q(k)], then iz 2 q(i)i # iw (q) 2 q(i)i , iy 2 q(i)i for i 5 j or
i 5 k, and we are done.
Now assume y [
⁄ [q( j),q(k)]. Then by the strict convexity of i?i, iy 2 q( j)i 1 iy 2
q(k)i . iq( j) 2 q(k)i 5 iz 2 q( j)i 1 iz 2 q(k)i. Therefore, it follows that iy 2 q( j)i .
iz 2 q( j)i or iy 2 q(k)i . iz 2 q(k)i.

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H. van der Stel / Mathematical Social Sciences 39 (2000) 277 – 301

Assume w.l.o.g.2 that iy 2 q( j)i . iz 2 q( j)i (otherwise interchange the roles of j and
k). Let l [ [0,1) with w (q) 5 l y 1 (1 2 l)z. Then iw (q) 2 q( j)i 5 i l( y 2 q( j)) 1 (1 2
l)(z 2 q( j))i # liy 2 q( j)i 1 (1 2 l)iz 2 q( j)i , iy 2 q( j)i, and we are done. h
The result stated in Proposition 2.2 does not hold if the set of alternatives is a
Euclidean space with dimension higher than two, unless the norm i?i is induced by an
inner product (cf. van der Stel, 1993). Since I consider a two-dimensional set of
alternatives, this observation does not imply any problems.
Strategy-proofness implies a property called positive association:
Lemma 2.3 (Peters et al., 1993, Lemma 2.3). Let w be a voting scheme which satisfies
SP. Let p,q [ (R 2 )N with q(i) [ [ p(i),w ( p)] for all i [ N. Then w (q) 5 w ( p).
Positive association means that the compromise point will not change if some of the
reported points move into the direction of the compromise point. In particular, replacing
some of the p(i) by w ( p) keeps the compromise point unchanged.
It will often be convenient to write w ( p(1), . . . , p(n)) instead of w ( p). Furthermore, I
will use notations like w (a K ,b L ,c M ), where K,L,M is a partition of N and w (a K ,b L ,c M )
stands for w ( p), where p(K) 5 haj, p(L) 5 hbj and p(M) 5 hcj.
I call two vectors u and v orthogonal (notation: u'v) if iui # iu 1 lvi and
ivi # iv 1 lui for all l [ R. This definition is equivalent to requiring that u and v are
orthogonal in the sense of Birkhoff (1935). It is easy to verify that this definition is
equivalent to the usual definition of orthogonality if i?i is induced by an inner product
(e.g., the Euclidean norm). Since i?i is strictly convex, we have the following result:
Lemma 2.4 (Peters et al., 1993, Lemma 2.6). Let u,v [ R 2 \h0j with u'v. Then
iui , iu 1 lvi and ivi , iv 1 lui for all l [ R\h0j.

3. Main results
In this section I describe three types of voting schemes which satisfy strategyproofness and Pareto optimality. In the subsequent sections I will show that all voting
schemes which satisfy SP and PO must be of one of these types.
Let hu,vj be an orthogonal basis of R 2 . Let n [ N be odd. For all x [ R 2 , x 1 and x 2
denote the coordinates of x with respect to hu,vj, i.e. x 5 x 1 u 1 x 2 v. Then
Med[u,v] : (R 2 )N → R 2 is defined by:
Med[u,v]( p) 5 med( p(1) 1 , . . . , p(n) 1 )u 1 med( p(1) 2 , . . . , p(n) 2 )v,
where ‘med ’ denotes the median of the subsequent real numbers. A voting scheme w

2

Without loss of generality.

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285

for which w 5 Med[u,v] for some orthogonal basis hu,vj of R 2 is called a median
scheme. In Peters et al. (1993) we obtained the following results:
Proposition 3.1 (Peters et al., 1993, Theorem 2.1). Let n be odd. Let w be a voting
scheme. Then w satisfies SP, PO and AN if and only if there is an orthogonal basis hu,vj
of R 2 such that w 5 Med[u,v].
Proposition 3.2 (Peters et al., 1993, Theorem 2.2). Let n be even. Then there are no
voting schemes satisfying SP, PO and AN.
In these propositions the role of anonymity is vital. An example of a voting scheme
which satisfies SP and PO, but not AN (as long as n . 1), is the dictatorial scheme. Note
that this scheme does not satisfy ND by definition. I now define a type of voting scheme
satisfying SP, PO and ND, which is not anonymous in general.
˜ ˜ j be orthogonal bases of R 2 . Then
Let n 5 3. Let hu,vj and hu,v
2 3
2
˜ ˜ ] : (R ) → R is defined by:
Sh[u,v;u,v
˜ ˜ ](x,y,z) 5 Med[u,v](x,y,z) if x,y,z are unequal and /(x,y,z) . 08,
Sh[u,v;u,v
˜ ˜ ](x,y,z) if x,y,z are unequal and /(x,y,z) , 08,
5 Med[u,v
5 Med[u,v](x,y,z) if x,y,z are collinear.
˜ ˜ ] will be called the shifted median scheme with respect to hu,vj and hu,v
˜ ˜ j.
Sh[u,v;u,v
˜ ˜ ](x,y,z). If Med[u,v] 5
Notice that if x,y,z are collinear, Med[u,v](x,y,z) 5 Med[u,v
˜ ˜ ], then Sh[u,v;u,v
˜ ˜ ] 5 Med[u,v] and satisfies AN. In general, however, a shifted
Med[u,v
median scheme is not anonymous.
Lemma 3.3. A shifted median scheme satisfies SP and PO.
˜ ˜ j be orthogonal bases of R 2 . Let w 5 Sh[u,v;u,v
˜ ˜ ]. Proposition
Proof. Let hu,vj and hu,v
˜
˜
3.1 implies that Med[u,v] and Med[u,v ] satisfy PO. So by Proposition 2.2,
˜ ˜ ](x,y,z) [ n(x,y,z) for all x,y,z [ R 2 . Since
Med[u,v](x,y,z) [ n(x,y,z) and Med[u,v
˜ ˜ ](x,y,z)j, this implies that w (x,y,z) [ n(x,y,z) for all
w (x,y,z) [ hMed[u,v](x,y,z), Med[u,v
2
x,y,z [ R . Consequently, because of Proposition 2.2, w satisfies PO.
Now let x,y,z,w [ R 2 . We will prove that
ix 2 w (x,y,z)i # ix 2 w (w,y,z)i.

(2)

ˆ
Let wˆ 5 w (w,y,z). Then by positive association (see Lemma 2.3), wˆ 5 w (w,y,z).
W.l.o.g.
assume that

w (x,y,z) 5 Med[u,v](x,y,z).

(3)

Proposition 3.1 implies that Med[u,v] is strategy-proof. Hence, ix 2 Med[u,v](x,y,z)i #
ˆ
ˆ
ˆ
ix 2 Med[u,v](w,y,z)i.
So, if w (w,y,z)
5 Med[u,v](w,y,z),
we are done. Now assume
˜ ˜ ](w,y,z),
ˆ
ˆ
ˆ
ˆ
ˆ y and z are
that w (w,y,z)
± Med[u,v](w,y,z).
Then w (w,y,z)
5 Med[u,v
w,
ˆ
noncollinear, and 2 1808 , /(w,y,z)
, 08.

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H. van der Stel / Mathematical Social Sciences 39 (2000) 277 – 301

Hence, by (3) we can take xˆ [ [x,wˆ ] > (←,y,z, → ). Then by definition of w,
˜ ˜ ](x,y,z).
˜ ˜]
ˆ
ˆ
ˆ
w (x,y,z)
5 Med[u,v](x,y,z)
5 Med[u,v
Proposition 3.1 implies that Med[u,v
˜ ˜ ](x,y,z)i
˜ ˜ ](w,y,z)i
ˆ
ˆ
ˆ
satisfies SP. So ixˆ 2 Med[u,v
# ixˆ 2 Med[u,v
5 ixˆ 2 w (w,y,z)i
5 ixˆ 2
ŵ i. Since Med[u,v] is also strategy-proof, this implies that:
ix 2 w (x,y,z)i 5 ix 2 Med[u,v](x,y,z)i
ˆ
# ix 2 Med[u,v](x,y,z)i
ˆ
# ix 2 xˆ i 1 ixˆ 2 Med[u,v](x,y,z)i
# ix 2 xˆ i 1 ixˆ 2 wˆ i
5 ixˆ 2 wˆ i.
This proves (2). Similarly, we can prove that:
iy 2 w (x,y,z)i # iy 2 w (x,w,z)i and iz 2 w (x,y,z)i # iz 2 w (x,y,w)i.
Since x,y,z,w [ R 2 can be chosen arbitrarily, this implies that w satisfies SP. h
One might wonder whether there are any other voting schemes for n 5 3 which satisfy
SP and PO. The answer is negative:
Theorem 3.4. Let n 5 3. Let w be a voting scheme. Then w satisfies SP and PO if and
only if one of the following assertions is true:

( i) w is dictatorial,
˜ ˜ j of R 2 such that w 5 Sh[u,v;u,v
˜ ˜ ].
( ii) there are orthogonal bases hu,vj and hu,v
Proof. The if part is implied by Lemma 3.3 and the observation that a dictatorial voting
scheme always satisfies SP and PO. For the only-if part of the proof I refer to van der
Stel (1996), which is an extended version of the present article. h
For an even number of agents, median schemes are nonexisting, whereas shifted
median schemes are only defined for n 5 3. I am now going to discuss a type of voting
scheme which can be defined for all arbitrary n [ N\h0j.
Let G , 2 N with f [
⁄ G and N [ G. Let hu,vj be an orthogonal basis of R 2 . Then
2 N
Cdm[u,v;G] : (R ) → R 2 is defined by:
Cdm[u,v;G]( p) 5min max p(i) 1 u 1min max p(i) 2 v.
S [G

i [S

S [G

i [S

Cdm[u,v;G] will be called the coalition dependent median scheme with respect to hu,vj
and G. Most of the time I impose some specific structure upon G. I call G , 2 N
distinctive, if the following two assertions are true:
(i) For all X,Y , N: if X [ G and X , Y, then Y [ G.
(ii) For all X , N: X [ G if and only if N\X [
⁄ G.

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287

A coalition dependent median scheme corresponds to a simple game, where G plays
here the role of a family of winning coalitions. Condition (i) says that the game is
monotonic, and (ii) says that it is a proper game. See, for example, Peleg (1984).
Lemma 3.5. Let G , 2 N be distinctive. Then:

(a) N [ G and f [
⁄ G.
( b) For all X,Y [ G: X > Y ± f.
Proof.
(a) Since f 5 N\N and f , N, the desired result follows directly from the assumption
that G is distinctive.
(b) Let X,Y [ G. Then by (ii), N\X [
⁄ G. So by (i), Y , N\X. Consequently, X > Y ±
f. h
A coalition dependent median scheme with respect to a distinctive G , 2 N could also
have been defined otherwise. For we have:
Lemma 3.6. Let G , 2 N be distinctive. Let hu,vj be an orthogonal basis of R 2 . Then
Cdm[u,v;G]( p) 5max min p(i) 1 u 1max min p(i) 2 v.
S [G i [S

S [G i [S

Proof. Let t [ h1,2j. Let a (i) 5 p(i) t for i [ N. It suffices to prove that:
max min a (i) 5min max a (i).
S [G

i [S

S [G

i [S

(4)

Let U [ G with max i [U a (i) 5 min S [G max i [S a (i). Since by Lemma 3.5(b) S > U ± f
for all S [ G, we have that min i [S a (i) # max i [U a (i) for all S [ G. Therefore
max min a (i) #min max a (i).
S [G

i [S

S [G

i [S

(5)

Let b 5 min S [G max i [S a (i). Let T 5 hi [ N : a (i) , b j. Then max i [T a (i) , b. Hence,
T[
⁄ G. So N\T [ G. Therefore, maxS [G min i [S a (i) $ mini [N \T a (i) 5 b. Together with
(5) this implies (4). h
Median schemes and dictatorial schemes are special cases of coalition dependent
median schemes. Let hu,vj be an orthogonal basis. Then:
(i) Let n be odd. Let G 5 hX [ 2 N : uXu . ]12 nj. Then Cdm[u,v;G] 5 Med[u,v].
(ii) Let i [ N. Let G 5 hX [ 2 N : i [ Xj. Then Cdm[u,v;G] is the dictatorial voting
scheme with dictator i.
Notice that G is distinctive in both cases. It is easy to verify that a coalition dependent
median scheme satisfies AN if and only if it coincides with a median scheme. Moreover:

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Lemma 3.7. Let G , 2 N be distinctive. Let hu,vj be an orthogonal basis of R 2 . Then
Cdm[u,v;G] satisfies SP and PO.
Proof. Let w 5 Cdm[u,v;G].
(SP) Let i [ N. Let p,q [ (R 2 )N with p( , ) 5 q( , ) for all , [ N\hij. Then u p(i) t 2
min S [G max, [S p( , ) t u # u p(i) t 2 min S [G max, [S q( , ) t u for t 5 1,2. Therefore, u p(i) t 2
w ( p)t u # u p(i)t 2 w (q)t u for t 5 1,2. Since u'v, this implies that i p(i) 2 w ( p)i # i p(i) 2
w (q)i. Consequently, w satisfies SP.
(PO) Suppose w is not Pareto optimal. Then, in view of Proposition 2.2 we take
p [ (R 2 )N such that w ( p) [
⁄ convh p(i) : i [ Nj.
Let w 5 w ( p). By a separation theorem of convex analysis (cf. Rockafellar, 1970,
Theorem 11.4) there are t [ R, c [ R 2 such that:
c ? w . t and c ? x , t for all x [ convh p(i) : i [ Nj.

(6)

˜ 5 t and p(i)
˜ [ ( p(i),w) for i [ N. Since w satisfies
Let p˜ [ (R 2 )N be such that c ? p(i)
SP, we have by positive association (see Lemma 2.3) that w (p˜ ) 5 w. Let K 5 (←,w,w 1
u, → ) < (←,w,w 1 v, → ). Let a,b [ R 2 with
ha,bj 5 K > hx [ R 2 : c ? x 5 t j.
˜
Let q [ (R 2 )N be defined by q(i) 5 a if [p(i),a)
> K 5 f and q(i) 5 b otherwise, for
i [ N. Then by definition of w, w (q) 5 w. Let A 5 hi [ N : q(i) 5 aj. Then w (a A ,
b N \ A) 5 w. Since G is distinctive, A [ G and N\A [
⁄ G, or N\A [ G and A [
⁄ G. In the
first case w 5 a, in the second case w 5 b. Both possibilities contradict (6). Consequently, w satisfies PO. h
Notice that in the proof that Cdm[u,v;G] satisfies SP I do not need the fact that G is
distinctive. The conditions that f [
⁄ G and G ± f would be sufficient (otherwise
Cdm[u,v;G] is not defined).
I now formulate the main result of this paper:
Theorem 3.8. Let w be a voting scheme. Then w satisfies SP and PO if and only if one
of the following assertions is true:

( i) There is an orthogonal basis hu,vj of R 2 and a distinctive set G , 2 N such that
w 5 Cdm[u,v;G].
˜ ˜ j of R 2 and i, j,k [ N such that
( ii) There are orthogonal bases hu,vj and hu,v
˜ ˜ ]( p(i), p( j), p(k)) for all p [ (R 2 )N .
w ( p) 5 Sh[u,v;u,v
The if part follows from Theorem 3.4 and Lemma 3.7. The only-if part is postponed
until the end of Section 4.
In Peters et al. (1993, Lemma 2.5) it is shown that an orthogonal basis of R 2 exists for
all i ? i. So Theorem 3.8 guarantees the existence of nondictatorial voting schemes
satisfying SP and PO.

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289

4. Proofs
This section is devoted to obtain the only-if part of the proof of Theorem 3.8.
Lemma 4.1. Let w : (R 2 )N → R 2 satisfy SP and PO. Let R,S,T , N be disjoint, with
R < S < T 5 N. Let c : (R 2 )3 → R 2 be defined by c (x,y,z) 5 w (x R ,y S ,z T ) for all x,y,z [
R 2 . Then c satisfies SP and PO.
Proof. The Pareto optimality of c follows directly from the definition of Pareto
optimality.
Lemma 2.1 implies that w satisfies WSP. This implies that d (x,c (x,y,z)) 5
d (x,w (x R ,y S ,z T )) # d (x,w (w R ,y S ,z T )) 5 d (x,c (w,y,z)) for all w,x,y,z [ R 2 . Similarly,
d ( y,c (x,y,z)) # d ( y,c (x,w,z)) and d (z,c (x,y,z)) # d (z,c (x,y,w)) for all w,x,y,z [ R 2 .
Therefore, c satisfies SP. h
Throughout the remainder of this section I assume that:
(i) w : (R 2 )N → R 2 satisfies SP and PO.
(ii) a,b,c [ R 2 with ia 2 bi 5 ia 2 ci 5 ib 2 ci ± 0.
(iii) D 5 hS , N : w (a S ,b N \S ) 5 aj.
D will be called the class of dominating coalitions. Lemma 4.2 shows that the
definition of D is independent of the choice of a and b:
Lemma 4.2. Let S , N. Then:

( i) w (a S ,b N \S ) [ ha,bj,
( ii) if S [ D then w (x S ,y N \S ) 5 x for all x,y [ R 2 ,
( iii) if S [
⁄ D then w (x S ,y N \S ) 5 y for all x,y [ R 2 .
Proof. Let c : (R 2 )3 → R 2 be defined by c (x,y,z) 5 w (x S ,y N \S ) for all x,y,z [ R 2 . Then
Lemma 4.1 implies that c satisfies SP and PO. It is easily verified that c cannot be a
shifted median in cases (i), (ii) and (iii). So, c is dictatorial in all these cases, because of
Theorem 3.4. This implies the desired result. h
Lemma 4.3. D is distinctive.
Proof. We have to prove that:
(i) For all X,Y , N with X [ D and X , Y: Y [ D.
(ii) For all X , N: X [ D if and only if N\X [
⁄ D.
(i) Let X,Y , N with X [ D and X , Y. Then by definition of D, w (a X ,b N \X ) 5 a.
Therefore, by positive association (Lemma 2.3), w (a Y ,b N \Y ) 5 a. So Y [ D.

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H. van der Stel / Mathematical Social Sciences 39 (2000) 277 – 301

(ii) (if) Assume N\X [
⁄ D. Then by Lemma 4.2(iii), w (a X ,b N \X ) 5 a. So X [ D.
(only if) Assume X [ D. Then by Lemma 4.2(ii), w (a N \X ,b X ) 5 b. So N\X [
⁄ D. h
If S [ D, and the members of S vote unanimously, their choice determines the
compromise point, as is shown by the following lemma:
Lemma 4.4. Let S [ D. Let x [ R 2 and p [ (R 2 )N with p(i) 5 x for all i [ S. Then
w ( p) 5 x.
Proof. Let w 5 w ( p). Then we have by positive association that w (x S ,w N \S ) 5 w. On the
other hand, Lemma 4.2(ii) implies that w (x S ,w N \S ) 5 x. So w ( p) 5 x. h
What kind of voting scheme w is depends on the structure of D. In the following two
lemmas and in Proposition 4.7 a specific structure of D is assumed, leading to the
conclusion that w can be described in terms of a shifted median scheme. Lemma 4.5
establishes the link with the three voter case, studied in Section 4.
Lemma 4.5. Let disjoint i, j,k [ N with hi, jj,hi,kj,h j,kj [ D. Let d [ R 2 . Let
c : (R 2 )3 → R 2 be defined by c (x,y,z) 5 w (x hi j ,y h j j ,z hkj ,d N \hi, j,kj ) for all x,y,z [ R 2 . Then
c satisfies SP and PO.
Proof. From the definition of SP it follows directly that c satisfies strategy-proofness.
Let x,y,z [ R 2 . In order to show that c satisfies PO, it is sufficient to show that
c (x,y,z) [ ^(x,y,z), because of Proposition 2.2. Let w 5 c (x,y,z). Suppose
w[
⁄ ^(x,y,z). Then a well-known separation theorem of convex analysis (cf. Rockafellar, 1970, Theorem 11.4) implies that there are t [ R, s [ R 2 such that s ? w . t and
s ? u , t for u [ hx,y,zj. Let x˜ [ (x,w) with s ? x˜ 5 t, y˜ [ ( y,w) with s ? y˜ 5 t and
˜ ˜ ˜ ) 5 w. W.l.o.g.
z˜ [ (z,w) with s ? z˜ 5 t. Then we have by positive association that c (x,y,z
˜ ˜ ]. Lemma 4.4 implies that w (y˜ hi, j j ,z˜ hkj ,d N \hi, j,kj ) 5 y.
˜ Hence,
assume that y˜ [ [x,z
˜ ˜ ˜ ) 5 y.
˜ So by SP, ix˜ 2 wi # ix˜ 2 y˜ i. Similarly, we can prove that iz˜ 2 wi # iz˜ 2
c (y,y,z
y˜ i. Therefore, ix˜ 2 wi 1 iz˜ 2 wi # ix˜ 2 y˜ i 1 iz˜ 2 y˜ i 5 ix˜ 2 z˜ i. So, by the strict convexi˜ ˜ ]. Hence, s ? w 5 t. So we have a contradiction. Consequently,
ty of i ? i, w [ [x,z
w [ ^(x,y,z). h
Lemma 4.6. Let disjoint i, j,k [ N with hi, jj,hi,kj,h j,kj [ D. Let p,q [ (R 2 )N with p(i) 5
q(i), p( j) 5 q( j) and p(k) 5 q(k). Then w ( p) 5 w (q).
Proof. Let v 5 w ( p) and w 5 w (q). Let p(i) 5 x, p( j) 5 y and p(k) 5 z. Let
˜ ˜ ˜ ) 5 w (x˜ hi j ,y˜ h j j ,z˜ hk j ,v N \hi, j,k j ) for all x,y,z
˜ ˜ ˜ [ R 2 . Then
c : (R 2 )3 → R 2 be defined by c (x,y,z
by Lemma 4.5, c satisfies SP and PO. Let u 5 w 2 v, xˆ 5 x 1 u, yˆ 5 y 1 u and
ẑ 5 z 1 u. Theorem 3.4 implies that c is a dictatorial scheme, or otherwise a shifted
ˆ ˆ ˆ ) 5 v 1 u 5 w. Hence,
median scheme. In both cases we have that c (x,y,z
w (xˆ hi j ,yˆ h j j ,zˆ hk j ,v N \hi, j,k j ) 5 w. Similarly, we can prove that w (xˆ hi j ,yˆ h j j ,zˆ hkj ,w N \hi, j,kj ) 5
w 1 u. So by WSP (see Lemma 2.1), u 5 0. Therefore, v 5 w. h

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291

Proposition 4.7. Let disjoint i, j,k [ N with hi, jj,hi,kj,h j,kj [ D. Then there are orthogon˜ ˜ j of R 2 such that w ( p) 5 Sh[u,v;u,v
˜ ˜ ]( p(i), p( j), p(k)) for all
al bases hu,vj and hu,v
2 N
p [ (R ) .
Proof. Let w [ R 2 . Let c : (R 2 )3 → R 2 be defined by c (x,y,z) 5 w (x hi j ,y h j j ,z hk j ,w N \hi, j,k j )
for all x,y,z [ R 2 . Then Lemma 4.6 implies that w ( p) 5 c (x,y,z) for all x,y,z [ R 2 and
p [ (R 2 )N with p(i) 5 x, p( j) 5 y and p(k) 5 z. Lemma 4.5 implies that c satisfies SP
and PO. Therefore, the desired result now follows from Theorem 3.4 and the observation
that c is nondictatorial. h
So, if the structure of D is such as assumed in Proposition 4.7, w is a shifted median
scheme, and there is a triple of voters which completely determines the compromise
point. From now on, I will assume that D is of a different structure:
Assumption 4.1. There are no disjoint i, j,k [ N with hi, jj,hi,kj,h j,kj [ D. For all
partitions R,S,T of N we define f(R,S,T ) 5 w (a R ,b S ,c T ).
As we will see, under Assumption 4.1, w must be a coalition dependent median
scheme, which is completely determined by f. For this reason I first analyze f.
Lemma 4.8. Let Assumption 4.1. Let K,L,M be a partition of N. Then the following
assertions are equivalent:

( i) K [ D,
( ii) f(K,L,M) 5 a,
( iii) f(M,K,L) 5 b,
( iv) f(L,M,K) 5 c.
Proof. Let c : (R 2 )3 → R 2 be defined by c (x,y,z) 5 w (x K ,y L ,z M ) for all x,y,z [ R 2 . Then
by Lemma 4.1, c satisfies SP and PO.
(i) ⇒ (ii), (iii), (iv): Then by Lemma 4.4, w (x K ,y L ,z M ) 5 x for all x,y,z [ R 2 . This
implies (ii), (iii) and (iv).
(ii) ⇒ (i): Let f(K,L,M) 5 a. Then c (a,b,c) 5 a. Let d 5 ]12 a 1 ]12 b. Then by strict
convexity of d we have that d (c,d) , d (c,a). So, by SP, c (a,b,d) ± d. Therefore, c is not
a shifted median scheme. Consequently, by Theorem 3.4, c is dictatorial, with
c (a,b,b) 5 a. So K [ D.
Similarly, we can prove that (iii) and (iv) imply (i). h
Lemma 4.9. Let Assumption 4.1. Let f(K,L,M) [ ha,b,cj for all partitions K,L,M of N.
Then w is dictatorial.
Proof. D is distinctive, because of Lemma 4.3. In view of this, take Y [ D such that no
proper subset of Y belongs to D. Suppose uYu . 1. Then let U,V , Y with U >V 5 f,
U ± f, V ± f and U Y 5 f. Then one of the following assertions is
true:

( i) f(K\X,L < X,M) 5 d,

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293

( ii) f(K\Y,L,M < Y) 5 d.
Proof. Suppose neither (i) nor (ii) is true. Then by Lemma 4.11, f(K\X,L < X,M) 5 b
and f(K\Y,L,M < Y) 5 c. So, L < X,M < Y [ D, because of Lemma 4.8. Therefore, since
D is distinctive, Lemma 3.5 implies that (L < X) > (M < Y) ± f. This contradicts the
assumptions. h
Lemma 4.13. Let Assumption 4.1. Let K,L,M be a partition of N, d 5 f(K,L,M) [
⁄ ha,b,cj
and M < h , j [ D for all , [ K < L. Let disjoint U,V , N with U U, Q 5 K >V, R 5 L > U, S 5 L >V. Then we have that f(P <
Q,R < S,M) 5 d. We must prove that f(P < R,Q < S,M) 5 d. Since M < h , j [ D for all
, [ K < L, and D is distinctive, we have that M < X [ D for all nonempty X , K < L.
Therefore, by Lemma 4.8
f(A,B,C) 5 c for all partitions A,B,C of N with C ± M , C.

(11)

We now distinguish between the following cases:
(i) P ± f and Q < S ± f,
(ii) S ± f and P < R ± f,
(iii) all other cases.
Case (i). By (11) and repeated application of Lemma 4.12 we obtain that d 5 f(P <
Q,R < S,M) 5 f(P,R < S < Q,M) 5 f(P < R,Q < S,M).
Case (ii). Similarly, we obtain d 5 f(P < Q,R < S,M) 5 f(P < Q < R,S,M) 5 f(P <
R,Q < S,M).
Case (iii). Since P < R [
⁄ D and M [
⁄ D (because of Lemma 4.8), the distinctiveness
of D implies that Q < S ± f. So, since (i) does not hold, P 5 f. Similarly, S 5 f. So we
have that f(Q,R,M) 5 d. We must prove that f(R,Q,M) 5 d.
Suppose uQu $ 2. Then let i [ Q. Hence by (11) and repeated application of Lemma
4.12, d 5 f(Q,R,M) 5 f(hij,R < Q\hij,M) 5 f(R < hij,Q\hij,M) 5 f(R,Q,M). Similarly, if
uRu $ 2, f(R,Q,M) 5 d.
Now assume uQu 5 uRu # 1. Then by Lemma 4.2(i), uQu 5 uRu 5 1. Let i, j [ N with
Q 5 hij and R 5 h jj. Lemma 4.8 implies that M [
⁄ D. So, since D is distinctive:
hi, jj [ D.

(12)

Lemma 4.2(i) implies that uMu $ 1. So by repeated application of Lemma 4.12 we can
take disjoint I,J , M and m [ M with I < J 5 M\hmj such that f(hij < I,h jj < J,hmj) 5 d.
Suppose hi,mj [
⁄ D. Then by Lemmas 4.8, 4.11 and 4.12, d 5 f(hij < I,h jj < J,hmj) 5
f(I,h jj < J,hm,ij) 5 f(I < J,h jj,hm,ij) 5 f(I < J < hmj,h jj,hij) 5 f(M,R,Q). So Lemma 4.10
implies that f(R,Q,M) 5 d.
Now assume hi,mj [ D. Then because of (12) and Assumption 4.1, h j,mj [
⁄ D. By a

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H. van der Stel / Mathematical Social Sciences 39 (2000) 277 – 301

similar argument as above we obtain that d 5 f(hij < I,h jj < J,hmj) 5 f(hij < I,J,hm, jj) 5
f(hij,I < J,hm, jj) 5 f(hij,I < J < hmj,h jj) 5 f(Q,M,R) 5 f(R,Q,M). h
Lemma 4.14. Let Assumption 4.1. Let K,L,M be a partition of N, and d 5
f(K,L,M) [
⁄ ha,b,cj. Let disjoint U,V , N with U Z, E 5 L > X, F 5 L > Y, G 5 L > Z, H 5
M > X, I 5 M > Y, J 5 M > Z. Then we have that f(A < B < C,E < F < G,H < I < J) 5
d.
Suppose f(A < B < C < I < J,E < F < G,H ) 5 d. Then by Lemma 4.8 and repeated
application of Lemma 4.11, f(B < C < I < J,F < G,A < E < H ) 5 d. So by Lemma
4.14, d 5 f(B < C < I < J,F < G,X) 5 f(B < F < I,C < G < J,X) 5 f(Y,Z,X) 5 d. Therefore, by Lemma 4.10, f(X,Y,Z) 5 d.
Now assume f(A < B < C < I < J,E < F < G,H ) ± d. Then by Lemma 4.12, f(A <
B < C,E < F < G < H,I < J) 5 d.
Suppose f(A < B < C < I,E < F < G < H,J) 5 d. Then by Lemma 4.8 and repeated
application of Lemma 4.11, f(A < B < I,E < F < H,C < G < J) 5 d. So by Lemma 4.14,
d 5 f(A < B < I,E < F < H,Z) 5 f(A < E < H,B < F < I,Z) 5 f(X,Y,Z).
Now assume f(A < B < C < I,E < F < G < H,J) ± d. Then by Lemma 4.12, f(A <
B < C,E < F < G < H < J,I) 5 d. So by Lemma 4.8 and repeated application of Lemma
4.11, f(A < C,E < G < H < J,B < F < I) 5 d. Hence, by Lemma 4.14, d 5 f(A < C,E <
G < H < J,Y) 5 f(C < G < J,A < E < H,Y) 5 f(Z,X,Y). So by Lemma 4.10, f(X,Y,Z) 5
d. h
Having Proposition 4.15, the next lemma makes sure that in the description of w we
can do without the terminology of shifted median schemes.
˜ ˜ j be orthogonal bases of R 2 . Let
Lemma 4.16. Let Assumption 4.1. Let hu,vj and hu,v
˜ ˜ ](a,b,c) [
˜ ˜ ](x,y,z) for
Med[u,v](a,b,c) 5 Med[u,v
⁄ ha,b,cj. Then Med[u,v](x,y,z) 5 Med[u,v
all x,y,z [ R 2 .
˜ → ) < (←,0,v,
˜ → ).
Proof. It suffices to prove that (←,0,u, → ) < (←,0,v, → ) 5 (←,0,u,

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295

W.l.o.g. we assume that Med[u,v](a,b,c) 5 0. Then the definition of the generalized
˜ ˜ [ (←,0,a, → ) < (←,0,b, → ) < (←,0,c, → ). Furmedian scheme implies that u,v,u,v
˜ Hence, u'v and u'v.
˜ Since
thermore, we lose no generality by assuming that u 5 u.
hu,vj is a basis of R 2 , we can take l, m [ R with v˜ 5 lu 1 m v. Suppose l ± 0. Then
because of Lemma 2.4, iv˜ i 5 i lu 1 m vi . i m vi, and i m vi 5 iv˜ 2 lui . iv˜ i. So we
have a contradiction. Therefore, l 5 0. Consequently, v and v˜ are linearly dependent,
implying the desired result. h
Lemma 4.17. Let Assumption 4.1. Then there is an orthogonal basis hu,vj of R 2 such
that for all x,y,z [ R 2 and all partitions K,L,M of N we have that:

w (x K ,y L ,z M ) 5 x if K [ D,
5 y if L [ D,
5 z if M [ D,
5 Med[u,v](x,y,z) if K,L,M [
⁄ D.
Proof. If w (a K ,b L ,c M ) [ ha,b,cj for all partitions K,L,M of N, w is dictatorial, because of
Lemma 4.9. This implies the desired result.
Now assume there is a partition A,B,C of N with w (a A ,b B ,c C ) [
⁄ ha,b,cj. Let d 5
w (a A ,b B ,c C ). Let c : (R 2 )3 → R 2 be defined by c (x,y,z) 5 w (x A ,y B ,z C ) for all x,y,z [ R 2 .
Then c satisfies SP and PO, because of Lemma 4.1. In view of Theorem 3.4 we take an
orthogonal basis hu,vj such that Med[u,v](a,b,c) 5 d.
Let K,L,M be a partition of N. If K [ D, L [ D or M [ D, the desired result follows
straightforwardly from Lemma 4.4. Now assume that K,L,M [
⁄ D. Let c˜ : (R 2 )3 → R 2 be
K
L
M
2
defined by c˜ (x,y,z) 5 w (x ,y ,z ) for all x,y,z [ R . Then c˜ satisfies SP and PO,
because of Lemma 4.1. Proposition 4.15 implies that c˜ (a,b,c) 5 c˜ (b,a,c) 5 d. Hence, by
Lemma 4.16 and Theorem 3.4 we have that c˜ (x,y,z) 5 Med[u,v](x,y,z) for all x,y,z [ R 2 .
Consequently, w (x K ,y L ,z M ) 5 Med[u,v](x,y,z) for all x,y,z [ R 2 . h
Lemma 4.18. Let Assumption 4.1. Then there is an orthogonal basis hu,vj of R 2 such
that for all x,y,z [ R 2 and all partitions K,L,M of N we have that w (x K ,y L ,z M ) 5
Cdm[u,v;D](x K ,y L ,z M ).
Proof. In view of Lemma 4.17 we take an orthogonal basis hu,vj of R 2 such that for all
x,y,z [ R 2 and all partitions K,L,M of N we have:

w (x K ,y L ,z M ) 5 x if K [ D,
5 y if L [ D,
5 z if M [ D,
5 Med[u,v](x,y,z) if K,L,M [
⁄ D.
Let x,y,z [ R 2 and K,L,M be a partition of N. Let p 5 (x K ,y L ,z M ). I distinguish four
cases:

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H. van der Stel / Mathematical Social Sciences 39 (2000) 277 – 301

(i) K [ D,
(ii) L [ D,
(iii) M [ D,
(iv) K,L,M [
⁄ D.
Case (i): Let t [ h1,2j. It suffices to prove that Cdm[u,v;D](x K ,y L ,z M ) t 5 x t . Since
K [ D, Cdm[u,v;D](x K ,y L ,z M ) t # max i [K p(i) t 5 x t . On the other hand, because of
Lemma 3.6 and the fact that D is distinctive, Cdm[u,v;D](x K ,y L ,z M ) t $ min i [K p(i) t 5
x t . So we are done. Cases (ii) and (iii): Similar to case (i). Case (iv): Let t [ h1,2j. It
suffices to prove that Cdm[u,v;D](x K ,y L ,z M ) t 5 med(x t ,y t ,z t ). Since D is distinctive,
K < L,K < M,L < M [ D.
So
Cdm[u,v;D](x K ,y L ,z M ) t # minhmaxhx t ,y t j,maxhx t ,z t j,
maxhy t ,z t jj 5 med(x t ,y t ,z t ). Similarly, because of Lemma 3.6, Cdm[u,v;D](x K ,y L ,z M ) t $
maxhminhx t ,y t j,minhx t ,z t j,minhy t ,z t jj 5 med(x t ,y t ,z t ). Therefore, we are done. h
Proposition 4.19. Let Assumption 4.1. Then there is an orthogonal basis hu,vj of R
such that for all p [ (R 2 )N : w ( p) 5 Cdm[u,v;D]( p).

2

Proof. Let A( j) 5 h p [ (R 2 )N : uh p(i) : i [ Nju # jj for j 5 1, . . . ,n. By Lemma 4.18
there is an orthogonal basis hu,vj of R 2 such that w ( p) 5 Cdm[u,v;D]( p) for all
p [ A(3). Now assume:

w ( p) 5 Cdm[u,v;D]( p