A Proposal for Selecting the Most Value-Aligned Preferences in
Decision-Making Using Agreement Solutions
Aar
´
on L
´
opez-Garc
´
ıa
a
Valencian Research Institute for Artificial Intelligence, Universitat Polit
`
ecnica de Val
`
encia,
Camıde Vera s/n 46022, Valencia, Spain
Keywords:
Decision-Making Modelling, Value-Based Reasoning, Multiple-Criteria Optimization, Weighting Schemes,
TOPSIS, UW-TOPSIS.
Abstract:
Decision-making is mostly subjected to conflict of interest. To solve such a concern, we propose a method-
ology to generate agreement solutions that determine the most value-aligned preference system according to
the stakeholders. These preferences are represented as a weighting scheme that produces a ranking system
through the TOPSIS technique. Such an agreement is obtained utilizing an unweighted multicriteria strategy
and the least-squares approximation. As a result, this weighting vector is an objective data-driven solution,
thus giving empirical evidence and adaptability learning in our proposal. The given solution is also explainable
and scalable per se thanks to the multicriteria technique selected. The agreement weight is used to perform
a ranking system that solves the decision problem considering the value preferences of the stakeholders. We
performed an illustrative example to show the different steps from which the decision problem must be posed
to be resolved. We conclude that our proposal is quite effective for solving value-based decision problems in
which conflicts of interest arise among affective agents. Moreover, we show the interpretation of the agree-
ment solution and its use in decision-making.
1 INTRODUCTION
The conflict of interest is always a major concern in
decision-making (Roy, 1996). When multiple agents
are involved in a decision-making stage, a methodol-
ogy for the selection of the most appropriate strategy
is required. Moreover, there is an important matter to
take into consideration, all stakeholders have to agree
with such a strategy, or at least partially (Ouenniche
et al., 2018). In general, no matter how well thought
out the protocol is. There is always a conflicting sce-
nario caused by human natural factors such as trust
issues, cultural/diversity challenges, and ethical im-
plications among others (Jacquet-Lagreze and Siskos,
1982). Hence, individual biases are substantially
important in real-world decision-making (Samuelson
and Zeckhauser, 1988).
This problem is further exacerbated when it comes
to value-based decision-making (Ormerod and Ulrich,
2013). When human values are considered by affec-
tive agents, their moral reasoning has to move from
“my point of view” to “the approach that is beneficial
for all actors involved” (Ulrich, 2006). Since, in our
a
https://orcid.org/0000-0001-8332-0381
decision context, human values can be understood as
principles, ideals, or beliefs that are important to indi-
viduals and society as a whole; it is crucial to model
the cognitive reasoning of affective agents according
to their values because they highly influence the de-
cision process (Walker and Corporation, 1993). An
additional concern in moral reasoning is the dilemma
related to the trade-off between values, for instance,
conservative values usually conflict with openness to
change (Schwartz, 1992). Then, there is a need for
a cognitive reasoning framework that allows affective
agents to assess the human values involved in the de-
cision scenario as well as a transparent tool for profes-
sional intervention based on critical thinking (Ulrich,
2007).
The Multiple Criteria Decision Analysis (MCDA)
approach is widely utilized for solving decision prob-
lems under the conflict of interests (Watr
´
obski et al.,
2019). There are several kinds of techniques for
decision-making modelling depending on the data in-
formation, complexity, and procedure. In addition,
it has been shown that such techniques are quite
useful for the deployment of autonomous agents in
dynamic environments (Doumpos and Grigoroudis,
López-García, A.
A Proposal for Selecting the Most Value-Aligned Preferences in Decision-Making Using Agreement Solutions.
DOI: 10.5220/0012586300003636
Paper published under CC license (CC BY-NC-ND 4.0)
In Proceedings of the 16th International Conference on Agents and Artificial Intelligence (ICAART 2024) - Volume 1, pages 461-470
ISBN: 978-989-758-680-4; ISSN: 2184-433X
Proceedings Copyright © 2024 by SCITEPRESS – Science and Technology Publications, Lda.
461

2013). However, most of them rely on the selection
of a weighting vector that aggregates the input data.
This concept is crucial in decision analysis because it
has a significant impact on the output. This selection
is generally conducted through an expert panel.
Nonetheless, this step is rather controversial
(Mareschal, 1988) since it triggers once again the con-
flict of interests among stakeholders. Although many
authors have attached this task (Liern and P
´
erez-
Gladish, 2020; Rezaei, 2015), this problem remains
open. For that reason, it is required a decision-making
strategy for selecting the most suitable weighting
scheme for each particular scenario, technique, and
preference system.
This work aims to solve the conflict of interest and
the indeterminacies attached to value-based decision-
making problems. In particular, when the cogni-
tive reasoning of affective agents considers human
values as the evaluation criteria. To this end, we
present a methodology composed of two multiple-
criteria decision-analysis techniques that produce rak-
ing systems and a least-squares method that gener-
ates optimal solutions that represent what we refer
to as agreement weighting schemes. The main con-
sequence of the choice of this agreement solution is
that it shows the underlying representation of the val-
ues of stakeholders. To be more precise, each com-
ponent of the agreement weighting scheme ascertains
the partial relevance of the agents. Then, it can be
subsequently utilized as an objective weighting vec-
tor for solving such decision problems. We have dis-
played an illustrative example indicating how to carry
out the methodology proposed and how to interpret
and extract conclusions from the optimal solutions.
2 DECISION ANALYSIS
The Multiple-Criteria Decision-Analysis (MCDA)
approach is a discipline of operations research which
aims to solve decision-making problems. The meth-
ods and techniques of MCDA are widely utilized for
solving decision problems with conflicting criteria. It
is important to take into account that, in this field, the
term “solving” is interpreted differently than in other
disciplines of operations research. In general, this no-
tion corresponds to the search and selection of the
“best” or “most preferred” alternative from a set of
available options evaluated in some uniform criteria.
2.1 Decision-Making Paradigm
The MCDA paradigm is defined as a decision space
in which all the information given is known by the
stakeholders. On the one hand, it is assumed a set of
finite available choices - known as alternatives - for
which the agents will have to select one or a group of
them as a solution to the problem. On the other hand,
it is considered the multiple attributes in which we as-
sess the qualities of the alternatives. This concept is
known as the criteria and it allows decision-makers to
compare the features of the alternatives. Another el-
ement to consider is the optimal criterion attached to
the attributes, i.e. the behaviour of the variable that in-
dicates whether we have to maximize or minimize it.
With those two concepts, an MCDA problem is stated
with a decision matrix X so that it is composed of N
alternatives (A
1
,...,A
N
) and M criteria (C
1
,...,C
M
).
Finally, we have to determine the decision weights of
the problem as a vector that assigns the relative impor-
tance over the multicriteria model. The formal state-
ment is displayed as follows (Triantaphyllou, 2000):
C
1
C
2
. . . C
M
A
1
x
11
x
12
. . . x
1M
A
2
x
21
x
22
. . . x
2M
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
A
N
x
N1
x
N2
. . . x
NM
w w
1
w
2
. . . w
M
Multiple-criteria optimization strategy has been
shown as a promising approach for addressing value-
based decision problems (Ormerod and Ulrich, 2013).
The MCDA paradigm allows us to perform analyti-
cal models considering human values as the criteria of
our problem. In this manner, affective agents can as-
sess values and ethics by including their own perspec-
tives. The multiple agents can also determine their
attitude towards the decision problem (Ajmeri et al.,
2020), although it leads to totally biased outcomes
and so non-representative results. For that reason, we
must emphasize that mathematical modelling of cog-
nitive reasoning is not an easy task at all (Wittmer,
2019).
2.2 Interpretation of the Weighting
Schemes
A weighting scheme is considered as a decision vector
used when aggregating the multi-attribute utilities of
each alternative to convey the relative importance of
each criterion (Triantaphyllou et al., 1997). As previ-
ously mentioned, the dimension of the weight vector
matches the number of criteria and their values vary
between 0 and 1. Their values are interpreted as bene-
fit criteria because the higher the value, the higher the
impact of the criterion within the aggregation strategy.
To be more specific, when an attribute has attached a
EAA 2024 - Special Session on Emotions and Affective Agents
462

weight of zero, then, it is neglected by the MCDA
model.
In most decision-making problems, the choice of
weighting schemes generates a conflicting scenario
(Hobbs, 1980). The weights determine the result
of MCDA techniques, so the consensus is very con-
troversial because agents inherently add their bias
and personal judgment (Jacquet-Lagreze and Siskos,
1982). In decision-making theory, we can distinguish
between objective (Koksalmis and
¨
Ozg
¨
ur Kabak,
2019) and subjective (Chang et al., 2010) weighting
methods. Nevertheless, the implementation of un-
weighted techniques has attracted the attention of the
researchers (L
´
opez Garc
´
ıa, 2023).
Regarding the interpretation of value-based rea-
soning for affective agents, weights can be under-
stood as the human interpretation of the importance
of a particular value. In a decision scenario com-
posed of human values, there is always a subjective
factor that cannot be neglected because it leads to bi-
ased systems (Serramia et al., 2020) or results that
lack applicability (Wenstøp, 2005). Hence, we must
take into account the impact of ethical dimensions
so that decision-makers’ beliefs are properly assessed
(Rauschmayer, 2001; Kunsch et al., 2009).
The mathematical definition of a weighting
scheme is represented as M positive values - partic-
ularly less than one - whose sum is equal to 1. The set
that determines the domain of all the weight vectors
for an MCDM problem is presented in the following
definition.
Definition 1 (Weighting space Ω). Given an MCDA
problem, we define the domain of weighting schemes
for assessing the relative importance of criteria as:
Ω =
(
(w
1
,...,w
M
) ∈ [0,1]
M



M
∑
j=1
w
j
= 1
)
. (1)
Decision analysis is usually attached to constraints
that represent the conditions and settings of stake-
holders (Nemeth et al., 2019). Although agents aim to
obtain their particular interest, the negotiation about
the weight vector must hold some properties. The
most important condition is that the criteria selected
do not vanish or dominate during the procedure (Fis-
cher, 1995). We can reformulate the weighting space
through the use of bounds and its formulation is pre-
sented as follows.
Definition 2 (Bounded weighting space Ω
lu
). Let l =
{l
j
}
M
j=1
and u = {u
j
}
M
j=1
be two set of bounds so that
0 ≤ l
j
≤ u
j
≤ 1, ∀ j ∈ {1, . . . , M}, we define the Ω
lu
space as the set of weights w ∈ [0,1]
M
that hold l
j
≤
w
j
≤ u
j
for all j ∈ {1, . . . , M}.
A mathematical formalization of the bounded
weighting space is the following:
Ω
lu
=



(w
1
,... , w
M
) ∈ [0,1]
M





∑
M
j=1
w
j
= 1,
l
j
≤ w
j
≤ u
j
,
1 ≤ j ≤ M



. (2)
From this equation, we can see that Ω
lu
is a reduc-
tion of the original weighting space Ω.
3 TECHNIQUE FOR ORDER OF
PREFERENCE BY SIMILARITY
TO IDEAL SOLUTION
The Technique for Order Preference by Similarity
to Ideal Solution, commonly known by its acronym
TOPSIS, defined by (Hwang and Yoon, 1981) is one
of the most known MCDA techniques in the field of
decision-making due to its simplicity and versatility.
A major contribution of TOPSIS is the concept of pos-
itive and negative ideal solutions. They are defined as
synthetic alternatives composed of the best and worst
values of the attributes considered within the problem.
The algorithm calculates the distances between such
ideal solutions as a comparative framework. Then,
the ranking is induced through the notion of relative
proximity regarding the distances computed. This key
concept, also known as the principle of compromise
for TOPSIS, states that the “best” alternative should
have the shortest distance from the positive ideal so-
lution and also the longest distance from the nega-
tive ideal solution (Lai et al., 1994). It is an easy-to-
understand idea and it offers explainable and trans-
parent results. Finally, the alternatives are sorted in
descending order indicating the preference order for
the stakeholders.
3.1 Original TOPSIS Version
The classic TOPSIS technique was originally pre-
sented by (Hwang and Yoon, 1981) meant a signifi-
cant breakthrough in the field of decision analysis. Its
algorithmic implementation is quite straightforward
and the results are data-driven using the concept of
relative distance to the ideal solutions.
The implementation of the original TOPSIS algo-
rithm is described in the following steps:
Step 1 Given a decision matrix X = [x
i j
], we normal-
ize the decision matrix computing the ℓ
2
vector
normalization per each criteria:
r
i j
=
x
i j
v
u
u
t
M
∑
j=1
x
2
i j
∈ [0,1], (3)
A Proposal for Selecting the Most Value-Aligned Preferences in Decision-Making Using Agreement Solutions
463

for all i ∈ {1,...,N} and j ∈ {1, . . . , M}.
Step 2 Given a weighting scheme (w
1
,...,w
M
) ∈ Ω,
we obtain the weighted normalized decision ma-
trix by computing:
v
i j
= w
j
r
i j
, (4)
per each i ∈ {1,...,N} and j ∈ {1, . . . , M}.
Step 3 Determine the positive ideal and negative
ideal solutions as PIS = (v
+
1
,...,v
+
M
) and NIS =
(v
−
1
,...,v
−
M
) so that:
v
+
j
=



max
1≤i≤n
{v
i j
} if j ∈ J
max
min
1≤i≤n
{v
i j
} if j ∈ J
min
1 ≤ j ≤ M,
v
−
j
=



min
1≤i≤n
{v
i j
} if j ∈ J
max
max
1≤i≤n
{v
i j
} if j ∈ J
min
1 ≤ j ≤ M,
where J
max
is the set of criteria to be maximized
and J
min
is the set of criteria to be minimized.
Step 4 Calculate the separation measures with regard
to PIS and NIS per each i ∈ {1 . . . N} as:
D
+
i
=
v
u
u
t
M
∑
j=1
(v
i j
− v
+
j
)
2
,
D
−
i
=
v
u
u
t
M
∑
j=1
(v
i j
− v
−
j
)
2
.
(5)
Step 5 Calculate the relative proximity index to the
ideal solutions using the following quotient:
R
i
=
D
−
i
D
+
i
+ D
−
i
, 1 ≤ i ≤ N. (6)
Step 6 Make a ranking of the alternatives in descend-
ing order of the values of the relative proximity
index {R
i
}
N
i=1
.
In essence, note that TOPSIS transforms a feature
matrix X ∈ R
N×M
into a R
N
vector making full use of
the selected attributes. Such a transformation gives us
decisive information about the global situation of the
alternatives involved since it produces the final rank-
ing.
It has been shown that the use of distances be-
tween such ideal solutions accurately mimics the psy-
chological characteristics of loss aversion and regret
aversion that occur in real-world scenarios (Yoon and
Kim, 2017; Liu et al., 2023). In the field of human
behaviour analysis, it can be understood as the eco-
nomic interpretation of the endowment effect (Thaler,
1980) or the loss aversion (Kahneman and Tversky,
1984), which are well-known concepts in economic
theory.
For further methodological and extensions see
(N
˘
ad
˘
aban et al., 2016) or (Papathanasiou and Ploskas,
2018). For applicability purposes or case studies ar-
eas see (Behzadian et al., 2012).
3.2 Unweighted TOPSIS Version
The UnWeighted TOPSIS technique (UW-TOPSIS) is
a generalization of the TOPSIS approach for ranking
decision alternatives considering the relative impor-
tance index as a function instead of a real-valued vec-
tor. The main advantage is that UW-TOPSIS does not
require the use of a fixed weighting scheme. The algo-
rithm presented by (Liern and P
´
erez-Gladish, 2020)
utilizes instead a set of bounds in which the relative
importance of each criterion varies. Thus, this tool
gives major flexibility to decision-makers when im-
plementing it. Unlike the classical TOPSIS technique,
this method solves two non-linear optimization prob-
lems considering the relative proximity index (6) as
the objective function. There is also a constraint in
which weights are contained in the selected bounds.
As a result, the output gives us information about both
minimal and maximal possible rank values per each
alternative.
For a formal implementation of the unweighted
TOPSIS technique, we have to consider the bounded
weighting space of Definition 2. In this manner,
the Ω
lu
establishes the constraint of the optimization
problem previously mentioned. The steps to carry out
the implementation of the UW-TOPSIS technique are
presented as follows:
Step 1 Given a decision matrix X = [x
i j
], w normal-
ize it like in TOPSIS-Step 1 to get [r
i j
].
Step 2 Determine the positive and negative ideal so-
lutions according to the normalized matrix as
PIS = (u
+
1
,...,u
+
M
) and NIS = (u
−
1
,...,u
−
M
) so
that:
u
+
j
=



max
1≤i≤n
{r
i j
} if j ∈ J
max
min
1≤i≤n
{r
i j
} if j ∈ J
min
1 ≤ j ≤ M,
u
−
j
=



min
1≤i≤n
{r
i j
} if j ∈ J
max
max
1≤i≤n
{r
i j
} if j ∈ J
min
1 ≤ j ≤ M,
where J
max
is the set of criteria to be maximized
and J
min
is the set of criteria to be minimized.
Step 3 Given two set of bounds l = {l
j
}
M
j=1
and u =
{u
j
}
M
j=1
so that 0 ≤ l
j
≤ u
j
≤ 1 per each j ∈
{1,...,M}, we defined the bounded set of weight-
ing schemes Ω
lu
.
EAA 2024 - Special Session on Emotions and Affective Agents
464

Step 4 We consider the separating functions D
+
i
,D
−
i
:
Ω
lu
→ [0, 1] regarding the ideal solutions per each
i ∈ {1,...,N}. So, given w ∈ Ω
lu
, we have:
D
+
i
(w) =
v
u
u
t
M
∑
j=1
w
2
j
(r
i j
− u
+
j
)
2
,
D
−
i
(w) =
v
u
u
t
M
∑
j=1
w
2
j
(r
i j
− u
−
j
)
2
.
(7)
Step 5 The relative proximity function to the ideal so-
lutions is defined as the function R
i
: Ω
lu
→ [0,1],
so that per each w ∈ Ω
lu
:
R
i
(w) =
D
−
i
(w)
D
+
i
(w) + D
−
i
(w)
, 1 ≤ i ≤ N. (8)
Step 6 For each alternative i ∈ {1,...,N}, we calcu-
late the score values R
L
i
and R
U
i
by solving the
non-linear mathematical programming problems
considering R
i
(8) as objective function and the
set of weights in Ω
lu
as the variables of the prob-
lem:
R
L
i
= min
{
R
i
(w) | w ∈ Ω
lu
}
,
R
U
i
= max
{
R
i
(w) | w ∈ Ω
lu
}
,
(9)
where the l
j
lower bound and u
j
upper bound
are subjected as restriction per each w
j
, ∀ j ∈
{1,...,M}.
Step 7 Considering the score intervals [R
L
i
,R
U
i
] and a
utility parameter λ ∈ [0, 1], we compute the aggre-
gated UW-TOPSIS score as:
R
UW
i
= (1 − λ)R
L
i
+ λR
U
i
, 1 ≤ i ≤ N. (10)
Step 8 Make a ranking of the alternatives in descend-
ing order of the values of the relative proximity
index {R
UW
i
}
N
i=1
.
It is noteworthy to mention that the output of UW-
TOPSIS is not only a decision interval [R
L
i
,R
U
i
] that
shows the range of possible scores over Ω
lu
, but also a
set of optimal weights {w
∗L
i
,w
∗U
i
} attached to the op-
timization problem of Step 6. Such optimal weights
give us meaningful information about the behaviour
of the proximity index and specifically about the rela-
tive importance of the criteria.
In terms of applicability, UW-TOPSIS has shown
multiple advantages with respect to the original TOP-
SIS in several areas and/or tasks. For instance, see
(Blasco-Blasco et al., 2021; L
´
opez-Garc
´
ıa et al.,
2023; L
´
opez-Garc
´
ıa et al., 2023). For a multi-agent
strategy or a multi-phase approach, there exists the
MUW-TOPSIS variant, which analyzes multiple de-
cision matrices simultaneously (Bouslah et al., 2023).
The source code for implementing the UW-TOPSIS
technique with the Python programming language can
be found in (L
´
opez-Garc
´
ıa, 2021).
4 EXTRACTION OF THE
AGREEMENT SOLUTIONS
The main objective of this paper is the extraction of
an agreement solution w
∗
∈ Ω
lu
so that it ascertains
the most suitable weighting scheme for the decision-
makers. Since conflict of interest is always present
when multiple agents participate in a decision prob-
lem, the use of a weighting agreement vector solves
such a concern because it establishes an optimal
choice that meets the needs of the stakeholders. Fur-
thermore, w
∗
determines the underlying value-aligned
preferences stated in the decision scenario. In this
section, we explain the strategy for achieving such an
agreement.
Once we perform the UW-TOPSIS technique,
we have a ranking system originated by the R
UW
score vector. We want to emphasize that each score
has attached the relative importance of their optimal
weighting schemes {w
∗L
i
,w
∗U
i
}. Hence, the impact
of the final solution does not follow a homogeneous
strategy. For that reason, the key is to generate a uni-
form weighting scheme that somehow replicates the
R
UW
ranking through a single TOPSIS implementa-
tion.
We aim to extract agreement solutions for giv-
ing uniform evaluation criteria in the decision-making
stage. If we have the values of a weighting scheme
with no conflict of interest, we just have to apply the
TOPSIS technique because, in this way, we gener-
ate a ranking system that reduces the concern caused
by the agents involved. In case the bounded weight-
ing space Ω
lu
established by decision-makers is com-
posed of a single element w
lu
- i.e. l
j
= w
lu
j
= u
j
per
each j ∈ {1,...,M} - the problem is solved by ap-
plying classic TOPSIS with w
lu
. If not, we have to
extract an agreement solution - known as w
∗
- that
satisfies the needs of the stakeholders using approxi-
mation methods. Given a score vector R
UW
obtained
by UW-TOPSIS, the problem described is equivalent
to the existence of solutions over Ω
lu
for the following
system of equations:
R
i
(w) =
D
−
i
(w)
D
+
i
(w) + D
−
i
(w)
= R
UW
i
, ∀i ∈ {1, . . . , N},
so that the relative proximity function R
i
(w) is the
function presented in (8). In case there exists a weight
in Ω
lu
that solves it, this weight will be considered as
the agreement solution. When we cannot guarantee
the existence of such an element, we have to imple-
ment a regression task that approximates the weight-
ing scheme, thus generating the feasible agreement
solution over Ω
lu
.
A Proposal for Selecting the Most Value-Aligned Preferences in Decision-Making Using Agreement Solutions
465

In this section, we show our proposal for generat-
ing such an agreement solution. The point is to per-
form a regression model that fits the R
UW
scores em-
ploying a single weight vector for the TOPSIS tech-
nique. Similar to the UW-TOPSIS optimization pro-
cess, we add the constraint that the domain is the Ω
lu
set. In this manner, we can generate a final ranking
TOPSIS-based utilizing w
∗
as the weighting scheme.
When our approach is applied by affective agents,
the objective is to extract the partial relevance of the
human values involved in the problem. In this man-
ner, the cognitive evaluation is directly attached to
each element of the agreement solution, thus guid-
ing the underlying preferences in the decision. As the
solution is obtained by employing a regression tech-
nique, the agreement solution fits the cognitive repre-
sentation of every agent. Hence, it justifies the final
TOPSIS ranking that will determine the most appro-
priate decision.
4.1 Regression Analysis of TOPSIS
Score Vectors
The agreement solution is obtained by means of a
regression model that fits the R
UW
i
score vector em-
ploying the image of TOPSIS for w ∈ Ω
lu
. For the
sake of simplicity, we consider TOPSIS(w) as the rel-
ative proximity vector R
i
that results after applying
the TOPSIS technique described in 3.
We can formulate the scenario optimization ap-
proach for a given loss function L as follows:
min L

R
UW
i
, TOPSIS(w)

s.t.
M
∑
j=1
w
j
= 1,
l
j
≤ w
j
≤ u
j
,
1 ≤ j ≤ M.
(11)
With this optimization problem, we implicitly
seek the weights that return the same score for both
TOPSIS and UW-TOPSIS over Ω
lu
. If we repre-
sent the solution as w
∗
, we can say that w
∗
contains
the relative importance that generates an agreement
for the different alternatives regarding the values se-
lected. It is worth mentioning that the existence of w
∗
does not necessarily guarantee the same ranking as
the UW-TOPSIS technique. Then, we must note that
we are approximating the importance that determines
the preferences of the agents.
As a result of the optimization scenario, the so-
lution to the problem is the w
∗
weighting scheme,
which has attached the optimal relative proximity in-
dex R
∗
i
= TOPSIS(w
∗
). We can generate a final that
meets the needs of decision-makers in terms of an
agreement solution.
4.2 Least-Squares Approximation
Approach
The regression task previously mentioned generates
a TOPSIS-based parameter estimation in which the
weights are the selected decision variables. For that
reason, we have selected a least-squares approxima-
tion approach for solving such a fitting task. The idea
is to minimize the residual error produced in the ap-
proximation of the R
UW
score because it is a theo-
retical score vector that has been obtained employing
UW-TOPSIS instead of the original TOPSIS. The ob-
jective function that measures the loss of the problem
is the residual sum of squares (RSS). If we consider
R
i
(w) as the image of the TOPSIS method for the w
weight, the RSS loss function is defined as follows:
RSS : Ω
lu
→ R
+
w 7→
N
∑
i=1

R
UW
i
− R
i
(w)

2
.
(12)
For our particular case, we have removed the num-
ber of combinations attached to the permutations. It
is justified because it does not have any impact on the
optimization problem. We want to remark that the
least squares problem is usually defined over the pa-
rameter space. Nonetheless, in this paper, we have
added the constraints attached to the Ω
lu
domain.
4.3 Procedure for Generating
Agreement Solutions
We have already explained how to extract the agree-
ment weight through the least-squares approximation
approach and how to perform the fitting strategy uti-
lizing (12). In this section, we show the procedure
that decision-makers have to carry out for generating
the ranking system induced by w
∗
. Depending on the
conditions stated, the work routine passes over differ-
ent conditional expressions.
Following the notation previously described, the
algorithmic procedure for extracting the agreement
weighting scheme and the ranking system is presented
in the Algorithm 1.
The cognitive representation transmitted to the af-
fective agents is the agreement solution that leads
to the final TOPSIS ranking. Since the equilibrium
between the preference systems of the agents is de-
termined through the agreement solution, the result
also follows an equilibrium among values in the de-
cision problem. Hence, agents have achieved the
most value-aligned solution considering their individ-
ual preferences.
EAA 2024 - Special Session on Emotions and Affective Agents
466

Input : Decision matrix X ∈ R
N×M
.
Bounded weighting space Ω
lu
.
begin
if Is Ω
lu
composed of a single weighting
scheme?
then
Apply TOPSIS.
else
Apply UW-TOPSIS.
if Is there a known w ∈ Ω
lu
that
generates the R
UW
ranking?
then
Apply TOPSIS with w.
else
Apply the regression task that
extracts w
∗
.
Apply TOPSIS with w
∗
.
end
end
end
Output: Agreement weighting scheme w
∗
.
Ranking system ≼ induced by the
R
∗
score vector.
Algorithm 1: Extraction of the agreement weighting
scheme in a decision problem.
5 ILLUSTRATIVE EXAMPLE
This section presents an illustrative example based on
selecting the best service in which conflicting criteria
apply to the problem and both human and economic
values are considered. Here we show the advantage
of the unweighted technique for solving the conflict
of interest and the extraction of an agreement solution
that illustrates the underlying values of the agents and
generates the final ranking.
Let us introduce a decision-making problem in
which we have to pick the best service and deci-
sion support is required. Once the stakeholders have
shown their position and the conditions that need to
be met, three services (S
1
,S
2
,S
3
) have been taken
as alternatives to the problem. The assessment of
the services has been conducted using five criteria.
These criteria are divided into social (accountabil-
ity Acc, sustainability Sus, and environmental factors
Env) and economic (total costs Cost and associated
times Time) values. The valuations of each service
are displayed in Table 1 where the elements of the de-
cision matrix stand for the agents’ assessments on a
scale of 1 to 5 following a Likert scale. This aspect is
justified because each service has its market volume
and scope, so this rating scale is utilized to estimate
the cognitive representation of the different affective
agents.
The affective agents in charge of the evaluation of
the decision process decide to carry out a subjective
strategy to highlight as much as possible the advan-
tages of their assigned service. The rules for assess-
ing the criteria of the problem state that no criterion
(i.e. no human value) should have a value of less than
0.10 or more than 0.35 in a [0,1] scale. Bearing this
in mind, the agents present their particular scenario
in which different weighting schemes are selected for
computing the decision task. The weight vectors of
the three different scenarios stated (A, B and C) are
shown in Table 2. It is worth mentioning that no ele-
ment of the weighing scheme has a value equal to the
bounds. As a result, we find an indeterminate solution
- since there is no preferred alternative - caused by the
agent biases introduced to the problem.
As a manner to solve such concerns, we carry
out the methodology proposed in this paper. First,
we implement the UW-TOPSIS technique with the
same settings considering the lower and upper bounds
stated by the stakeholders. That is to say:

l = (0.10, . . . , 0.10)
u = (0.35, ..., 0.35)
∈ R
5
.
The resultant score rankings (min, max and aggre-
gated) obtained per each service are displayed in Ta-
ble 4. With this information, the UW-TOPSIS output
returns the following order:
UW-TOPSIS order: S
2
≽ S
1
≽ S
3
.
In order to show additional information about the
optimization problem conducted (i.e. the problem
that we carry out in Step 6), we have shown the opti-
mal weighting schemes {w
∗L
,w
∗U
} in Table 3. Since
weights determine the relative importance of each cri-
terion, we have also counted the number of times that
certain attribute matches the values of its lower and/or
upper bound. It has been indicated as a l or u match.
Second, we continue with the extraction of the
agreement weighting scheme. As we can see in Ta-
ble 3, there is no consensus in the selection of a
weighting vector. Then, we performed the least-
square approximation strategy for the UW-TOPSIS-
based R
UW
score. The fitting technique gives us a
residual error of magnitude of 10
−7
, thus yielding the
following results:
R
∗
= (0.4591, 0.5853, 0.4198),
w
∗
= (0.2106, 0.1854, 0.1644, 0.2179, 0.2216),
RSS = 1.0774 · 10
−7
,
Rank = S
2
≽ S
1
≽ S
3
.
For a better understanding of the accuracy of the least-
squares weighting estimation, we have visually repre-
sented the different TOPSIS-induced scores in Fig-
ure 1.
A Proposal for Selecting the Most Value-Aligned Preferences in Decision-Making Using Agreement Solutions
467

Table 1: Decision matrix for our example with three services and four criteria.
Social Values Economic Values
Acc Sus Env Cost Time
Service 1 (S
1
) 5 2 3 4 1
Service 2 (S
2
) 4 2 5 1 3
Service 3 (S
3
) 3 5 1 2 4
Optimality max max max min min
Table 2: Ranking results for the different TOPSIS scenarios associated with the selected weights.
Weighting schemes
Acc Sus Env Cost Time Ranking
Scenario A 0.1849 0.1178 0.1735 0.1899 0.3339 S
1
≽ S
2
≽ S
3
Scenario B 0.1831 0.1167 0.3306 0.1880 0.1815 S
2
≽ S
1
≽ S
3
Scenario C 0.1082 0.3443 0.1738 0.1902 0.1836 S
3
≽ S
2
≽ S
1
Table 3: Weighting schemes {w
∗L
,w
∗U
} obtained for the UW-TOPSIS associated with minimum and maximum scores to-
gether with the number of matches from with weights have the same value as lower or upper bounds.
Weights min

w
∗L

Weights max

w
∗U

Acc Sus Env Cost Time Acc Sus Env Cost Time
S
1
0.10 0.35 0.10 0.35 0.10 0.35 0.10 0.10 0.10 0.35
S
2
0.10 0.35 0.10 0.10 0.35 0.10 0.10 0.35 0.35 0.10
S
3
0.10 0.10 0.35 0.10 0.35 0.10 0.35 0.10 0.35 0.10
l match 3 1 2 2 1 2 2 2 2 2
u match 0 2 1 1 2 1 1 1 1 1
Figure 1: Ranking systems obtained with the different TOPSIS versions

R
L
, R
U
, R
UW
, R
∗

.
Table 4: Results obtained with UW-TOPSIS using the
bounds given by the stakeholders.
R
L
R
U
R
UW
Service 1 0.1994 0.7189 0.4592
Service 2 0.3387 0.8317 0.5851
Service 3 0.1773 0.6630 0.4201
From the solution of the least-squares approxima-
tion problem, we can see that the most-value-aligned
preferences in the decision model mean an impact of
0.5604 on the social values and 0.4396 on the eco-
nomic values. Then, the underlying values of the
stakeholders show a greater impact on the social fac-
tors, although economic criteria have a higher im-
pact when analyzing it individually. As expected, the
agreement weighting scheme satisfies the same rank-
ing order as the one obtained with the UW-TOPSIS
technique. Regarding the values of w
∗
, we can see
a balanced weighting vector because the equally dis-
tributed weight would have all values as
1
5
. Then, it is
less biased than the weights obtained in both the indi-
vidual agents’ choice (Table 2) and the optimization
stage (Table 3).
EAA 2024 - Special Session on Emotions and Affective Agents
468

6 CONCLUSIONS
In this paper, we have proposed a procedure to reduce
the conflict of interest in value-based decision-making
when affective agents evaluate the decision scenario.
To this end, we have used multiple-criteria techniques
and a multi-agent strategy. Our approach is based on
the extraction of an agreement solution that fulfils the
requirements of stakeholders. Such an agreement so-
lution is a weighting scheme and it is understood as
the preference system that most closely matches with
the values of decision-makers. Then, the agreement
weights can be applied to decision problems as an ob-
jective, explainable and transparent scheme.
With the use of the agreement solutions, we can-
not only apply or scale them to further decision-
making problems but also know and evaluate the in-
herent values of the decision scenario. Thus, we can
offer an assessment of biases and conflicting patterns
which are present in the problem. Therefore, we can
carry out our methodology as a knowledge-based sys-
tem that leads to a consensus among stakeholders.
We have also remarked on the limitations attached
to TOPSIS and showed how to tackle them using its
unweighted version. Even though UW-TOPSIS avoid
the usual shortcomings in decision indeterminacy, the
computational costs associated with the optimization
problems can pose a problem over large data sets.
Hence, future work on the stability of this technique
is required.
Although the extraction of agreement solutions
has been conducted utilizing a constrained least-
squares problem, it would be interesting to consider
alternative fitting strategies that generate more accu-
rate results. Further regression methods and/or alter-
native loss functions could lead to solutions adapted
to the requirements of the decision scenario.
Finally, the application of our proposal on datasets
with a larger number of alternatives or with different
values should be studied. As future lines of work, it
would also be interesting to study the trade-off that
arises when directly opposite human values are taken.
ACKNOWLEDGEMENTS
This work has been supported for the Consel-
leria de Innovaci
´
on Universidades, Ciencia y
Sociedad Digital “Programa/Ayudas PROM-
ETEO” (Ref. CIPROM/2021/077) and VAE-
VADEM TED2021-131295B-C32, funded by
MCIN/AEI/10.13039/501100011033 and the Euro-
pean Union NextGenerationEU/PRTR.
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