Clustering methods group individuals or objects based on information about their similarity or proximity. When the raw information to generate clusters cannot be easily observed or verified, the cluster designer must rely on information reported by individuals behind the observations. When these individuals receive utility from a public decision taken with aggregated data within each own's cluster and have single-peaked preferences, we prove that there do not exist clustering methods such that truth-revealing behavior is always a dominant strategyclustering methods

Javier Perote

Juan Perote-PeÃ±a

English

Research Papers in Economics

The impossibility of strategy−proof clustering 
Juan Perote−Peña Javier Perote
Universidad Pablo de Olavide de Sevilla Universidad de Salamanca
Abstract
Clustering methods group individuals or objects based on information about their similarity
or proximity. When the raw information to generate clusters cannot be easily observed or
verified, the cluster designer must rely on information reported by individuals behind the
observations. When these individuals receive utility from a public decision taken with
aggregated data within each own's cluster and have single−peaked preferences, we prove that
there do not exist clustering methods such that truth−revealing behavior is always a dominant
strategy
The authors wish to acknowledge to the Fundación Centro de Estudios Andaluces (CentrA) for financial support.
Citation: Perote−Peña, Juan and Javier Perote, (2003) "The impossibility of strategy−proof clustering." Economics
Bulletin, Vol. 4, No. 23 pp. 1−9
Submitted: October 1, 2002.  Accepted: May 15, 2003.
URL: http://www.economicsbulletin.com/2003/volume4/EB−02D70012A.pdf1 Introduction
Let us imagine a society in which the agents or individuals live in some locations
in a given region. A social planner must eﬃciently locate some public facilities -
say, health services, police stations, schools, etc.- in the region based on reported
information about each individual location. The individuals would like the
public facility to be located on their own location -to minimize the transport
costs, for example- and have preferences on the set of feasible locations that are
continuous and single-peaked: the further the location is along a straight line
starting from the single most preferred alternative -top location or peak, i.e.,
the agent’s own location-, the worse it is for the agent. When individuals are
asked to report their locations, they might have an incentive to lie and report
false information in order to achieve a better location for themselves. When just
one public facility must be chosen, the allocation problem can be solved in some
cases. For example, in the one-dimensional case when the public facility must
be located on a street, it is possible to design Pareto-optimal allocation methods
such that no individual will ever have an incentive to report false information
about her own location -see Moulin (1980)-.
In the real world usually more than just one public facility can be located and
in this case, there are diﬀerent possibilities open. For example, individuals might
be allowed to choose the service point they like the most from a ﬁxed amount
of points to be chosen by society -see, for example, Miyagawa (1998, 2001)-.
Other possibility consists in dividing society into diﬀerent areas that group the
set of individuals associated to each service point. In this note, we assume
that the social planner has no clear constraint in the number of public facilities
to locate, but would like to group individuals that live ”close” to each other
to allocate service points with a minimum of transport costs. Since the only
information the planner has to build the groups or ”neighborhoods” is about
reported locations, she applies a clustering technique to those points generating
the diﬀerent areas that will deserve a service point of their own. For the problem
to be meaningful we shall need to impose a minimal constraint on the admissible
clustering techniques to avoid trivial results. Therefore, the extreme cases of
always considering a unique cluster or as many clusters as individuals cannot be
proper cluster techniques. After excluding these two extreme and trivial cases,
we ﬁnd that every other admissible clustering technique cannot rule out the
possibility of individuals lying about their locations.
Notice that our model is not restricted to a spatial allocation problem, but
admits a much wider range of economic problems. The information to be re-
ported can be any measures of diﬀerent socioeconomic variables of agents to be
taken into account to create clusters. The next section is devoted to both the
model and the main impossibility result.
12 The model and the result
Let N = {1,...,n} be a society composed by a ﬁnite number n ≥ 3o fagents
or individuals denoted by i,j,h,w ∈ N. Let π,π0 denote diﬀerent partitions
of society, i.e., π =
©
S1,...,Skª
,w h e r ek denotes the number of sets in the
partition. We use indexes t,l ∈ {1,...,k} to denote speciﬁcs e t si n t oag i v e n
partition π =
©
S1,...,Skª
. Let Π be the set of all possible partitions of society
N. Therefore, for all π =
©
S1,...,Skª
∈ Π, it must hold that St ∩Sl = ∅∀ t,l ∈
{1,...,k},t6= l and ∪k
t=1St = N. Notice that given any partition π,i fw ed e n o t e
as πi = {St ∈ π | i ∈ St} ∀i ∈ N, it must hold that ∀i,j ∈ N, either πi ∩πj = ∅
or πi = πj and πi 6= ∅∀ i ∈ N (since it must hold that i ∈ Si ∀i ∈ N). Let
E be the real line and let x,y,z ∈ E2 denote arbitrary elements -locations-
in E2. Let d(x,y) ∈ E+ denote the euclidean distance between the points or
locations x and y. Each individual i ∈ N has a preference relation deﬁned on
the set E2 denoted as Ri, where Pi and Ii are the asymmetric and symmetric
parts of Ri respectively. A utility function u is a function of the kind: u :
E2 → E. A preference relation Ri is represented by a utility function ui if
∀x,y ∈ E2,x R iy ←→ ui(x) ≥ ui(y). We require admissible preferences to be
continuous in E2 and such that (1) and (2) hold.
(1). For every individual i ∈ N,t h e r ee x i s t sau n i q u el o c a t i o np(Ri) ∈ E2
such that p(Ri)Pix ∀x ∈ E2\p(Ri).
(2). For every individual i ∈ N,∀x ∈ E2\p(Ri), ∀λ ∈ (0,1),
[λp(Ri)+( 1− λ)x]Pix.
Let < be the set of all continuous preferences on E2 such that (1) and (2)
hold1.A neconomy R =(R1,...,Rn) ∈ <n is a list of one admissible preference
relation for every agent, and for any economy R ∈<n there will be a vector of
associated peaks p(R)=( p(R1),...,p(Rn)) ∈ E2n. We shall also make use of
the following notation, given any agent i ∈ N: R =(Ri,R−i) ∈ <n.
A clustering technique is a function C : E2n → Π. A clustering technique is
intended to put together as members of the same group or cluster the agents
that are ”similar” among themselves and separate them from those that are
less similar, the deﬁnition of the similarity degree being implicit in function C.
Therefore, ∀x =( x,y,...,z) ∈ E2n,C (x) is a possible partition of the set N,
so whenever i ∈ Cj(x), it must be true that j ∈ Ci(x)a n dCj(x)=Ci(x). In
this case we say that agents i ∈ N and j ∈ N belong to the same cluster, that
is, Cj(x)=Ci(x) ∈ C(x) ∈ Π. The elements in E2n used by function C to
generate clusters are interpreted as the reported ”peaks” of the agents, i.e., any
agent i ∈ N report a peak xi ∈ E2 that can be either the true peak p(Ri)
(agent i’s most preferred location in the plane E2)o rad i ﬀerent one2.T h e r e
1Conditions (1) and (2) together with the assumption of continuity of preferences is a
possible deﬁnition of the concept of ”single-peakedness” when extended to two dimensions.
2Notice that the declared or reported vector of peaks of the agents x =( x1,...,x n)d o
not need to coincide with the true vector of peaks p(R)=( p(Ri),...,p(Rn)) given the (true)
economy R =( R1,...,Rn) ∈ <n, but they are the only information that the planner has to
make the clustering decision.
2are hundreds of clustering techniques proposed in the literature -see Everitt
(1993) for an excellent survey, for example-, each one characterized by diﬀerent
concepts of the ”distance” between groups of individuals and the way they are
formed, either using algorithms or optimization techniques. We only need a
very loose characterization of admissible clustering techniques that is fulﬁlled
by most of them.
Deﬁnition 1 Ac l u s t e r i n gt e c h n i q u eC is admissible if the following two con-
ditions hold:
(i). ∃b d ∈ E++, such that ∀x =( xi,x j,x −i−j) ∈ E2n such that ∃i ∈ N with
d(xi,x j) ≥ b d ∀j ∈ N → i/ ∈ Cj(x) ∀j ∈ N.
(ii). ∃e d ∈ E++, such that ∀x =( xi,x j,x −i−j) ∈ E2n such that ∃i,j ∈ N
with d(xi,x j) ≤ e d → i ∈ Cj(x).
Condition (i) implies that when an agent’s reported peak is far enough from
any other’s peak, it deserves to form an independent cluster. Analogously,
condition (ii) means that when two agents’ reported peaks are close enough
they must belong to the same cluster. Notice that, in particular, (ii) guarantees
that any two agents with preferences such that their peaks are located in the
same place must always be in the same cluster. Now, we provide a very general
model for the public choice taken in each cluster. We only assume that the choice
must be eﬃcient when considering the members of the cluster alone -they might
vote or reach agreements to locate the public service.
Deﬁnition 2 Given any clustering technique C, a public location function
is a function LC : <n → E2n such that ∀R ∈ <n with associated vector of peaks
p(R)=( p(R1),...,p(Rn)) ∈ E2n,L C(R)=( LC
1 (R),...,LC
n(R)) ∈ E2n is such
that the following two conditions hold:
(i’). ∀i,j ∈ N, i ∈ Cj(p(R)) → LC
i (R)=LC
j (R) ∈E2.
(ii’). ∀i ∈ N, there does not exist z ∈ E2 such that zPjLC
i (R) ∀j ∈ N such
that j ∈ Ci(p(R)).
A public location function establishes a common Pareto-eﬃcient location for
the public service for all members within each cluster. Condition (i’) forces a
single location for all the agents belonging to the same cluster and condition
(ii’) imposes that any location chosen for a given cluster cannot be Pareto-
dominated by some other location. Notice that for any economy and for any
cluster there always exist eﬃcient locations and well-deﬁned public location
functions (consider, for example, choosing ∀R ∈ <n, ∀i ∈ N, LC
i (R)=p(Rh),
where h ∈ N is deﬁned as h =m i n{j ∈ Ci(R)}).
Deﬁnition 3 An allocation method in our model is a pair (C,L) where C is
an admissible clustering technique and L is a public location function.
Now, we deﬁne the strategic property we are interested in:
3Deﬁnition 4 An allocation method (C,L) is manipulable at economy R =
(Ri,R−i) ∈ <n by individual i ∈ N, if there exists a preference relation R0
i ∈ <
such that LC
i (R0
i,R−i)PiLC
i (R).
Deﬁnition 5 An allocation method (C,L) is strategy-proof if it is not ma-
nipulable at any economy R ∈ <n by any individual i ∈ N.
We are now prepared to state our result.
Theorem 1 There are no strategy-proof allocation methods under our assump-
tions
Proof. We prove it by contradiction. Let (C,L) be a strategy-proof allo-
cation method. Let us consider any two individuals i,j ∈ N and the following
economy R =( Ri,R j,R−i−j) ∈<n : Ri ∈ <, with peak p(Ri)a t( 0 ,0) is any
preference relation that can be represented by the following utility function:
∀x =( x1,x 2) ∈ E2,
ui(x1,x 2)=
½
−sign(x2)x2 + bx1 if x1 ≤ 0
−sign(x2)x2 − cx1 if x1 ≥ 0 (1)
where b =
e δ
2
³
b δ + ε
´ and c =1 ( ε > 0i sa n yp o s i t i v en u m b e r ) 3.I ti se a s yt o
check that preferences Ri ∈ < above have indiﬀerence curves on the plane that
correspond to a rhomboid with center in agent i0s peak - the origin (0,0)- like
the one depicted in Figure 1. The axis of the rhombus coincide with both x1
and x2. When we have b = c =1 ,w eo b t a i nas y m m e t r i cr h o m b u s .N o t i c et h a t
parameter b determines the slopes of the rhomboid left sides (for negative values
of x1) and parameter c determines the slopes of the rhomboid for positive values
of x2.T h en u m b e r−a (the intercept of the rhomboid for negative values of x2)
is used as the measure of the utility that generates utility function ui(x1,x 2),
which is in fact a pyramid generating the rhomboids as level curves.
Agent j0s preferences Rj ∈ <, with peak p(Rj)a t(e d,0) is any preference re-
lation that can be represented by the following utility function: ∀x =( x1,x 2) ∈
E2,
uj(x1,x 2)=
(
−sign(x2)x2 + b(x1 − e d)i f x1 ≤ e d
−sign(x2)x2 − c(x1 − e d)i f x1 ≥ e d
(2)
where b =1a n dc =
e δ
2
³
b δ + ε
´ (ε > 0 is any positive number). This
preference relation is analogous to the former one, the indiﬀerent curves being
rhomboids centered at (x1,x 2)=( e d,0). The remaining agents h ∈ N\{i,j}
3Note that b δ and e δ have already been deﬁned inside Deﬁnition 1.
4 
    x
2 
     
       
    a  
     
   (b)    (-c) 
 
   
      -a/b              a/c           x
1 
   
   (-b)    (c) 
 
        -a      
Figure 1:
h a v et h es a m ep r e f e r e n c e sRh ∈ <, with peak p(Rh)a t( e d + b d,0), that can be
represented by the following utility function: ∀x =( x1,x 2) ∈ E2,
uh(x1,x 2)=



−sign(x2)x2 + b(x1 −
h
e d + b d
i
)i f x1 ≤ e d + b d
−sign(x2)x2 − c(x1 −
h
e d + b d
i
)i f x1 ≥ e d + b d
(3)
Again, with b = c =1 . L e tp(R) ∈ E2n be the vector of peaks of economy
R ∈ <n, i.e., p(R)=( p(Ri),p(Rj),p(R)−i−j). By construction, (i) and (ii), it
holds for R =( Ri,R j,R−i−j) ∈<n that i ∈ Cj(p(R)) and ∀h,w ∈ N\{i,j},
it holds that h ∈ Cw(p(R)) by (ii), while i/ ∈ Ch(x) ∀h ∈ N\{i,j}, so
since C(p(R)) is a partition of N, it must be the case of j/ ∈ Ch(x) ∀h ∈
N\{i,j}.N o w , w e p r o v e t h a t LC must be such that LC
i (R)=LC
j (R) ∈ h
(0,0),(e d,0)
i
by (i’) and (ii’). Given preferences Ri,R j ∈ <, the only Pareto-
optimal locations are on the straight line connecting both peaks -a segment lying
on the x axis-: any other location (x1,x 2) ∈ E2 such that x1 < 0o rx1 > e d
are Pareto-dominated when considering only members of the cluster i,j ∈ N by
(x1,x 2)=( 0 ,0) and (x1,x 2)=(e d,0) respectively and any other location such
that 0 ≤ x1 ≤ e d and x2 6= 0 is dominated by location (x1,0) -both agents i
and j attain a higher utility level given their preferences in (1) and (2) at the
beginning of the proof-. Now, there are only two possibilities to consider: Case
51: LC
i (R)=LC
j (R)=( x1,0) with
e d
2
≤ x1 ≤ e d. Then, we can imagine the
admissible economy R0 =( R0
i,R−i) ∈ <n where all preferences are the same
as those in R ∈ < with the exception of individual i0s new preferences R0
i ∈ <,
which can be represented by the following utility function u0
i: ∀x =( x1,x 2) ∈
E2,
u0
i(x1,x 2)=
(
−sign(x2)x2 + b(x1 +b δ)i f x1 ≤ 0
−sign(x2)x2 − c(x1 +b δ)i f x1 ≥ 0
(4)
where c = b = 1. This preference relation is a rhomboid centered at (−b δ,0),
so p(R0
i)=( −b δ,0). First of all, notice that given preferences Ri ∈ <, it holds
that ui(−b δ,0) >u i(x1,0), or, in other words,
(−b δ,0)Pi(x1,0) (5)
Now, let us consider economy R0 =( R0
i,R−i) ∈ <n. Let p(R0) ∈ E2n be
the vector of peaks associated to economy R0 ∈ <n. Therefore, p(R0
i)=( −b δ,0),
p(R0
j)=p(Rj)=( e δ,0) and p(R0
h)=p(Rh)=( e δ + b δ,0) ∀h ∈ N\{i,j}. Now,
notice that by (i), j,h / ∈ Ci(p(R0)) ∀h ∈ N\{i,j}, so agent i ∈ N form her own
cluster in economy R0 ∈ <n and by (i’) and (ii’) above, it must be the case of
LC
i (R0)=p(R0
i)=( −b δ,0), while the public location for agent i ∈ N associated
to economy R ∈ <
n is by assumption LC
i (R)=( x1,0). Now, (5) above can be
written as: LC
i (R0)=LC
i (R0
i,R−i)PiLC
i (R0), so the allocation method (C,L)i s
manipulable at economy R ∈ <
n with vector of peaks p(R) ∈ E2n by individual
i ∈ N by means of reporting false preferences R0
i ∈ <. Hence, (C,L)i sn o t
strategy-proof, entering into contradiction with our initial assumption. The
only case left is Case 2: LC
i (R)=LC
j (R)=( x1,0) with
e d
2
≥ x1 ≥ 0. If this
is the case, let us consider the following economy: R00 =( R0
j,R−j) ∈ <n, with
associated vector of peaks p(R00) ∈ E2n where all agents with the exception of
j ∈ N have the same preferences they had in economy R ∈ <
n, and agent j’s
new preferences are: R0
j = Rh ∀h ∈ N\{i,j} (i.e., those represented by utility
function (3) above). The vector of peaks p(R00) ∈ E2n will then be: p(R00)=
(p(Ri),p(R0
j),p(R)−i−j)=( ( 0 ,0),(e δ+b δ,0),...,(e δ+b δ,0)). B y( i )a n d( i i ) ,i th o l d s
for economy R00 ∈ <n that j,h / ∈ Ci(p(R00)) ∀h ∈ N\{i,j} and ∀h,w 6= i, h ∈
Cw(p(R00)). By (i’) and (ii’), Lj(R00)=p(R0
j)=p(Rh)=( e δ +b δ,0). Now, notice
that given preferences Rj ∈ <, it holds that uj(e δ +b δ,0) >u i(x1,0), -see Figure
2- or, in other words,
(e δ +b δ,0)Pi(x1,0) (6)
Now, notice that expression (6) above can be written as:
LC
j (R00)=LC
j (R0
j,R−j)PiLC
j (R). (7)
6 
              x
2 
 
 
 
 
                               (0,0) 
 
                           x
1 
                            
 
 
 
                                                        δ
~
              δˆ  
 
Figure 2:
Hence, allocation method (C,L) is manipulable at economy R ∈ <
n with vector
of peaks p(R) ∈ E2n by individual j ∈ N by means of reporting false preferences
R0
j ∈ <. Thus, (C,L) is not strategy-proof in this case either, so we cannot avoid
contradictions and the theorem is proved.
7References
[1] Barber` a, S., & M. Jackson (1994): ”A characterization of Strategy- Proof
Social Choice Functions for Economies with Pure Public Goods”. Social
Choice and Welfare 11, 241-252.
[2] Brams, S.J., Jones, M.A. & Kilgour, D.M. (2001): ”Single-Peakedness and
Disconnected Coalitions”. Journal of Theoretical Politics, forthcoming.
[3] Everitt, B.S.: Cluster Analysis, London: Arnold cop., 1993.
[4] Miyagawa, E (1998): ”Strategy-proof provision of a menu”. Mimeo,
Columbia University.
[5] Miyagawa, E (2001): ”Locating libraries on a street”. Social Choice and
Welfare 18, 527-541.
[6] Moulin, H. (1980): ”On Strategy- proofness and Single- peakedness”. Public
Choice 35, 437-455.
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The impossibility of strategy-proof clustering

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