Noname manuscript No.
(will be inserted by the editor)
Preference stability along time: The time cohesiveness measure
T. Gonz´alez-Artega · R. de Andr´es Calle · M. Peral
Received: date / Accepted: date
Abstract This work introduces a non-traditional pers-
pective about the problem of measuring the stability
of agents’ preferences. Specifically, the cohesiveness of
preferences at different moments of time is explored un-
der the assumption of considering dichotomous evalu-
ations. The general concept of time cohesiveness mea-
sure is introduced as well as a particular formulation
based on the consideration of any two successive mo-
ments of time, the sequential time cohesiveness mea-
sure. Moreover, some properties of the novel measure
are also provided. Finally, and in order to emphasize
the adaptability of our proposal to real situations, a
factual case of study about Clinical Decision Making
is presented. Concretely, the study of preference stabil-
ity for life-sustaining treatments of patients with ad-
vanced cancer at end of life is analysed. The research
considers patients who express their opinions on three
life-sustaining treatments at four consecutive periods
of time. The novel measure provides information of pa-
tients preference stability along time and considers the
possibility of cancer metastases.
T. Gonz´alez-Artega
BORDA and PRESAD Research Groups and
Multidisciplinary Institute of Enterprise (IME),
University of Valladolid, E47011 Valladolid, Spain
E-mail: teresag@eio.uva.es
R. de Andr´es Calle
BORDA Research Unit, PRESAD Research Group and
Multidisciplinary Institute of Enterprise (IME),
University of Salamanca, E37007 Salamanca, Spain
E-mail: rocioac@usal.es
M. Peral
Centro de Atenci´on Primaria de Laguna de Duero,
E47140, Valladolid, Spain
E-mail: mperalh@saludcastillayleon.es
Keywords Time cohesiveness measure · Dichotomous
opinions · Preference stability · Patients’ preferences
1 Introduction
Intertemporal decision making is an important scientific
area and it has been obtaining attention from several
research fields such as Economics, Health Economics,
Social Choice, Psychology, Marketing, Decision Analy-
sis, Neuroscience, and so on.
One of the main topics of this area is the study of
preference stability that is often defined like the mea-
surement of the choice consistency among options along
time [8], [20], [28]. Traditionally, preferences have usually
been considered permanent by theory [23], although
there are also different studies to check if they are con-
stant over time [4], [7], [11], [26]. Related to empirical
literature on preference stability, most studies use small
samples in short time periods and they are focused on
a specific type of preferences, the risk preferences [27].
Recently, there has been an increment of works about
time preference [12], [22], [24], while there are few con-
tributions that study the stability of social preference
[10].
From another point of view, a growing number of
studies considers changes in preferences as a result of
shocks such as illness, civil wars, natural disasters, etc.
[9], [15], [21], [25].
The research to date has tend to explore preference
stability by means of statistical approaches: from basic
methods like descriptive analysis and multiple regres-
sion [22], [28] to more elaborate procedures like hierar-
chical generalized linear modelling [8] and others [29].
In order to enhance the preference stability topic,
the aim of this contribution is to develop a new tool
2 T. Gonz´alez-Artega et al.
capable of measuring preference stability from a non-
traditional perspective. For this purpose, the notion of
preference stability is considered in the same vein that
the notion of cohesiveness. This seems natural because
the measurement of preference stability resembles the
notion of measurement of cohesiveness over time, in the
sense that the maximum value captures the notion of
full stability, that is unanimity along time, while the
minimum value captures the notion of total lack of sta-
bility, that is, total disagreement along time.
The cohesiveness or consensus measurement has
been dealt in the Social Choice literature from Bosch’s
seminal work [6]. Subsequently, Alcalde-Unzu and
Vorsatz [1], Alcantud et al. [2] and Garc´ıa-Lapresta and
P´erez-Rom´an [14] introduced several classes of consen-
sus measures based on distances for ordinal informa-
tion. Additionally, several studies related to consensus
problem deal also with cardinal information like the
approaches proposed by Gonz´alez-Arteaga et al. [16],
Gonz´alez-Pach´on and Romero [17], Gonz´alez-Pach´on et
al. [18], Herrera-Viedma et al. [19], and so on. From
another point of view, Alcantud, de Andr´es Calle and
Casc´on [3] introduced a cohesiveness measure when opin-
ions are dichotomous.
Taking into account the previous contributions on
preference stability and cohesiveness measure, this pa-
per is focused on an inter-temporal decision making
problem where a set of agents express their opinions
on an alternative along different moments of time. To
be precise, agents have to approve or disapprove the
alternative under study at diverse point of time. Thus,
the paper objective is to determine how much stability
or cohesiveness agents’ opinions conveys to the group
on the alternative along time. In order to measure such
stability, a new general approach is defined, the time
cohesiveness measure. Following the Social Choice tra-
dition, this measurement takes values in the unit inter-
val considering value 1 full stability and value 0 total
lack of stability. Moreover, an specific formulation of the
time cohesiveness measure is introduced, the sequential
time cohesiveness measure as well as a study of its ana-
lytic properties. Under this approach, the stability of
preferences is understood like the probability that for
a randomly chosen moment of time, two randomly cho-
sen agents have the same opinion at such a time and
its consecutive.
Furthermore, the measurement proposed is put in
practice in a real case of study to emphasise its ap-
plicability. In particular, the stability of preferences for
life-sustaining treatments in terminally cancer patients’
last year of life is analysed.
The paper is structured as follows. It has been di-
vided into three parts. The first part, Section 2, intro-
duces our proposal to measure preference stability: the
time cohesiveness measure. Moreover, an specific type
of this measure, the sequential time cohesiveness mea-
sure, is presented as well as its properties. The second
part, Section 3, includes an application of the novel ap-
proach to a real case of study. Finally, some concluding
remarks are provided.
2 A new tool to measure preference stability:
The time cohesiveness measure
This section is devoted to introduce some notation as
well as our proposal of measurement of preference sta-
bility, namely, the time cohesiveness measure. Then, an
specific formulation, the sequential time cohesiveness
measure, is defined and its properties are examined.
2.1 Notation
Let N = {1, 2, ..., N} a set of agents or experts. Agents
express their opinions on an alternative, x, at different
time moments T = {t
1
, . . . , t
T
} by means of dichoto-
mous opinions.
From now on, the notation used to formalize theses
assessments is the following:
Definition 1 A time preference profile of a set of agents
N on an alternative x at T different time moments is
an N × T matrix
P =
P
1t
1
. . . P
1t
T
.
.
.
.
.
.
.
.
.
P
Nt
1
. . . P
Nt
T
N×T
where P
it
j
is the opinion of the agent i over alternative
x at t
j
moment, in the sense
P
it
j
=
1 if agent i approves x at the t
j
time,
0 otherwise.
Let P
N×T
denote the set of all such N × T matrices.
For simplicity of notation, (1)
N×T
is the N × T matrix
whose cells are universally equal to 1 and (0)
N×T
is the
N × T matrix whose cells are universally equal to 0.
A time preference profile P is unanimous if alter-
native x is approved (resp. disapproved) over T by all
agents. In matrix terms, if the time preference
profile P ∈ P
N×T
is constant, P = (1)
N×T
(resp. P = (0)
N×T
).
Preference stability along time: The time cohesiveness measure 3
Any permutation σ of the agents {1, 2, ..., N } deter-
mines a time preference profile P
σ
by permutation of
the rows of P, that is, row i of the profile P
σ
is row
σ(i) of the profile P.
For each time preference profile P, P
S
is the restric-
tion to a subset of agents, an agent-subprofile on the
agents in S ⊆ N, and it emerges from selecting the rows
of P that are associated with the respective agents in S.
For each time preference profile P, P
I
is the restric-
tion to a subset of consecutive moments of time, time-
subprofile on the moments of time in I ⊆ T, and it
emerges from selecting consecutive columns of P that
are associated with the respective moments of time in
I. Any partition {I
1
, . . . , I
p
} of P generates a decom-
position of P into time-subprofiles P
I
1
, . . . , P
I
p
where
P
I
1
∪ . . . ∪ P
I
p
= P.
An extension of a time preference profile P of a
group of agents N at T = {t
1
, . . . , t
T
} is a time prefe-
rence profile
P at T = {t
1
, . . . , t
T
, t
T +1
, . . . , t
T +q
} such
that the restriction of P to the first T moments of time
of T coincides with P.
A replication of a time preference profile P of a
group of agents N on alternative x is the time prefe-
rence profile P ] P ∈ P
2N×T
obtained by duplicating
each row of P, in the sense that rows r and N + r of
P ] P are row r of P, for each r = 1, ..., N.
For each time preference profile P on alternative x,
n
t
j
0
denotes the number of agents that disapprove x at
the t
j
moment of time, and n
t
j
1
denotes the number of
agents that approve alternative x at the t
j
moment of
time. Therefore, N = n
t
j
0
+ n
t
j
1
for each t
j
∈ T.
In addition, n
t
j
,t
j+1
0,0
denotes the number of agents
that disapprove alternative x at t
j
and keep their opin-
ion at the following point of time t
j+1
. Similarly, n
t
j
,t
j+1
1,1
denotes the number of agents that approve alternative
x at t
j
and keep their opinion at the following point of
time t
j+1
.
In this way, n
t
j
,t
j+1
0,1
is the number of agents that
disapprove alternative x at t
j
but change their opinion
at t
j+1
, and n
t
j
,t
j+1
1,0
is the number of agents that ap-
prove alternative k at t
j
but change their opinion at
t
j+1
. For each t
j
∈ T, n
t
j
0
= n
t
j
,t
j+1
0,0
+ n
t
j
,t
j+1
0,1
and like-
wise n
t
j
1
= n
t
j
,t
j+1
1,1
+ n
t
j
,t
j+1
1,0
. See Table 1 for improving
understanding.
For the purpose of clarifying the use of the pre-
vious notation, the following illustrative example is in-
troduced.
P
P
P
P
P
P
t
j
t
j+1
No Yes
No n
t
j
,t
j +1
0,0
n
t
j
,t
j+1
0,1
n
t
j
0
Yes n
t
j
,t
j +1
1,0
n
t
j
,t
j+1
1,1
n
t
j
1
n
t
j +1
0
n
t
j+1
1
N
Table 1 Notation summary table
Example 1 Let N = {1, 2, . . . , 10} be a set of ten agents
that express their opinions on alternative x along four
consecutive moments of time T = {t
1
, t
2
, t
3
, t
4
}. Their
time preference profile is:
P =
P
1t
1
. . . P
1t
4
.
.
.
.
.
.
.
.
.
P
10t
1
. . . P
10t
4
10×4
=
1 1 1 0
0 1 1 0
0 1 0 0
0 1 0 1
1 0 1 1
1 1 1 1
0 0 0 0
0 0 1 1
0 1 1 0
0 1 0 1
This time preference profile can be summarized in a
table containing the number of agents who approve or
disapprove alternative x at each moment of time t
j
as
well as the number of agents that keep or change their
opinion during consecutive time moments (see Table 2).
2.2 New approach to measure preference stability:
Definition and properties
Influenced by Bosch’s consensus approach [6], our pro-
posal of cohesivenesses measure along time is intro-
duced below.
Definition 2 A time cohesiveness measure for a group
of agents N = {1, ..., N } on an alternative x is a
mapping
τ : P
N×T
→ [0, 1]
that assigns a number τ(P) ∈ [0, 1] to each time preferen-
ce profile P, with the properties:
i) τ(P) = 1 if and only if P is unanimous (full stabi-
lity).
ii) τ(P
σ
) = τ(P) for each permutation σ of the agents
and P ∈ P
N×T
(anonymity).
A time cohesiveness measure is a collection of time
cohesiveness measures for each group of agents N.
4 T. Gonz´alez-Artega et al.
H
H
H
H
t
1
t
2
No Yes
No n
t
1
,t
2
0,0
= 2 n
t
1
,t
2
0,1
= 5 n
t
1
0
= 7
Yes n
t
1
,t
2
1,0
= 1 n
t
1
,t
2
1,1
= 2 n
t
1
1
= 3
n
t
2
0
= 3 n
t
2
1
= 7 N = 10
H
H
H
H
t
2
t
3
No Yes
No n
t
2
,t
3
0,0
= 1 n
t
2
,t
3
0,1
= 2 n
t
2
0
= 3
Yes n
t
2
,t
3
1,0
= 3 n
t
2
,t
3
1,1
= 4 n
t
2
1
= 7
n
t
3
0
= 4 n
t
3
1
= 6 N = 10
H
H
H
H
t
3
t
4
No Yes
No n
t
3
,t
4
0,0
= 2 n
t
3
,t
4
0,1
= 2 n
t
3
0
= 4
Yes n
t
3
,t
4
1,0
= 3 n
t
3
,t
4
1,1
= 3 n
t
3
1
= 6
n
t
4
0
= 5 n
t
4
1
= 5 N = 10
Table 2 Notation summary table for Example 1
Our proposal in contrast to Bosch’s contribution
does not require neutrality property, time moments can
not be exchanged, due to the fact that time order is an
essential aspect to measure the stability of preferences.
Now a particular time cohesiveness measure is in-
troduced. Formally:
Definition 3 The sequential time cohesiveness mea-
sure for a group of agents N = {1, ..., N} on an al-
ternative x is the mapping τ
S
: P
N×T
→ [0, 1] given
by
τ
S
(P) =
=
1
T − 1
·
j=T −1
X
j=1
n
t
j
,t
j+1
0,0
· ( n
t
j
,t
j+1
0,0
− 1)
N(N − 1)
+
1
T − 1
·
j=T −1
X
j=1
n
t
j
,t
j+1
1,1
· ( n
t
j
,t
j+1
1,1
− 1)
N(N − 1)
Intuitively, it measures the probability that for a
randomly chosen moment of time, two randomly cho-
sen agents of a group have the same opinion upon an
alternative at the moment of time selected and its con-
secutive.
It is easy to check that Definition 3 provides a time
cohesiveness measure.
Hereunder, some desirable properties of the sequen-
tial cohesiveness measure are defined and proved.
Properties
Reversal invariance: This property shows that the
main aspect of the time sequential cohesiveness mea-
sure is the stability of agents’ opinions more than an
specific value. If the 0’s are changed for 1’s and vice
verse, then the sequential time cohesiveness measure
reminds equal. Formally:
Let P
c
be the complementary time preference profile
of P defined by P
c
= (1)
N×T
− P. If τ
S
verifies
reversal invariance then τ
S
(P
c
) = τ
S
(P).
Proof Agents’ opinions at t
j
, t
j+1
∈ T do not change
in P and P
c
, then τ
S
does not change. That is, those
agents whose opinions coincide at t
j
and t
j+1
in
P have also coincident opinions at t
j
and t
j+1
in
P
c
although those opinions are different than in P.
Taking into account the Definition 3, τ
S
does not
change.
ut
Time-reducibility: It means that the stability of a
time preference profile is the average of the time
cohesiveness measures of all its consecutive time-
subprofiles of two consecutive moments of time. For-
mally:
Let P ∈ P
N×T
be a time preference profile. We say
that τ
S
verifies time-reducibility if
τ
S
(P) =
1
T − 1
T −1
X
j=1
τ
S
(P
I
j,j+1
)
where P
I
j,j+1
∈ P
N×2
is the time-subprofile of P
containing the columns corresponding to times t
j
and t
j+1
.
Proof It is straightforward from the Definition 3
since
τ
S
(P
I
j,j+1
) =
=
n
t
j
,t
j+1
0,0
(n
t
j
,t
j+1
0,0
− 1)
N(N − 1)
+
n
t
j
,t
j+1
1,1
(n
t
j
,t
j+1
1,1
− 1)
N(N − 1)
ut
Preference stability along time: The time cohesiveness measure 5
Replication monotonicity: When a non-unanimous
time preference profile is replicated, its sequential
time cohesiveness measure increases. Formally:
Let P ∈ P
N×T
be a non unanimous time preference
profile then
τ
S
(P ] P) > τ
S
(P)
Proof Using time-reducibility is enough to prove this
property for only two moments of time. Consider
P
I
j,j+1
∈ P
N×2
a time-subprofile for t
j
and t
j+1
.
τ
S
(P
I
j,j+1
) =
=
n
t
j
,t
j+1
0,0
(n
t
j
,t
j+1
0,0
− 1)
N(N − 1)
+
n
t
j
,t
j+1
1,1
(n
t
j
,t
j+1
1,1
− 1)
N(N − 1)
τ
S
(P
I
j,j+1
] P
I
j,j+1
) =
=
2n
t
j
,t
j+1
0,0
(2n
t
j
,t
j+1
0,0
− 1)
2N(2N − 1)
+
2n
t
j
,t
j+1
1,1
(2n
t
j
,t
j+1
1,1
− 1)
2N(2N − 1)
It is enough that
2z − 1
2N − 1
>
z − 1
N − 1
for each natu-
ral number z ∈ N with z < N . And this is easily
checked. ut
In addition, for an unanimous time preference pro-
file P ∈ P
N×T
, by Definition 3, τ
S
verifies
τ
S
(P ] P) = τ
S
(P) = 1
Minimum time stability: If all agents express their
opinions at a moment of time and change their opin-
ions at the next moment of time, that is, all agents
change their opinions along two successive moments
of time, then the sequential time cohesiveness mea-
sure takes a zero value. It also happens when there
are at most two agents that keep their opinion at
two consecutive moments of time but their opinions
do not coincide each other. Formally:
Let P ∈ P
N×T
be a time preference profile such that
there is at most one agent who has the same opinion
at t
j
and t
j+1
for j ∈ {1, . . . T }, that is, n
t
j
,t
j+1
0,0
≤ 1
and n
t
j
,t
j+1
1,1
≤ 1 for all j ∈ T. Then, τ
S
(P) = 0.
Proof It is immediately from Definition 3. ut
Leaving minimum time stability: In order to leave
the minimum time stability it is needed that at least
the opinions of two agents coincide at the same mo-
ment of time and the next one. Formally:
Let P ∈ P
N×T
be a time preference profile such
that there exists at least a k, k ∈ T, such that
n
t
k
,t
k+1
0,0
> 1 or n
t
k
,t
k+1
1,1
> 1, then τ
S
(P) > 0.
Proof Using Definition 3 is straightforward. ut
Time monotonicity: Consider two time preference
profiles, P and P
0
, that coincide in all their ele-
ments excepting the opinion of an agent m ∈ N,
at t
k
and t
k+1
. Concretely, this agent has different
opinion at t
k
and t
k+1
in P: P
mt
j
6= P
mt
j+1
, and
the agent’s opinion is the same at t
k
and t
k+1
in
P
0
: P
0
mt
j
= P
0
mt
j+1
. In this case, the sequential time
cohesiveness measure verifies τ
S
(P
0
) ≥ τ
S
(P). For-
mally:
Let P, P
0
∈ P
N×T
be time preference profiles such
that:
a) P
it
j
= P
0
it
j
, i ∈ {N \ {m}},
b) P
mt
k
6= P
mt
k+1
, m ∈ N, t
k
, t
k+1
∈ T,
c) P
0
mt
k
= P
0
mt
k+1
, m ∈ N, t
k
, t
k+1
∈ T.
Then, τ
S
(P
0
) ≥ τ
S
(P).
Proof It is enough to prove that τ
S
(P
0
)−τ
S
(P) ≥ 0.
Let n
t
k
,t
k+1
1,1
and n
t
k
,t
k+1
0,0
the number of agents that
approve and disapprove alternative x at t
k
and t
k+1
from P and (n
t
k
,t
k+1
1,1
)
0
and (n
t
k
,t
k+1
0,0
)
0
the number of
agents that approve and disapprove alternative x at
t
k
and t
k+1
from P
0
.
– If P
0
mt
k
= P
0
mt
k+1
= 0, then
(n
t
k
,t
k+1
0,0
)
0
= n
t
k
,t
k+1
0,0
+ 1
and
τ
S
(P
0
) − τ
S
(P) =
=
1
T −1
(n
t
k
,t
k+1
0,0
+1)((n
t
k
,t
k+1
0,0
+1)−1)
N(N −1)
−
1
T −1
n
t
k
,t
k+1
0,0
(n
t
k
,t
k+1
0,0
−1)
N(N −1)
≥ 0
since for all z ∈ N, (z + 1)z − z(z − 1) ≥ 0.
– If P
0
mt
k
= P
0
mt
k+1
= 1, then
(n
t
k
,t
k+1
1,1
)
0
= n
t
k
,t
k+1
1,1
+ 1