COMPARATIVE
ANALYSIS OF THE TODIM METHOD ADHERENCE TO PROSPECT THEORY
Alexandre
Leoneti
University
of Sao Paulo, Brazil
E-mail: ableoneti@usp.br
Luiz
Flávio Autran Monteiro Gomes
Ibmec/RJ, Brazil
E-mail: luiz.gomes@professores.ibmec.edu.br
Submission: 10/19/2020
Accept: 11/11/2020
ABSTRACT
Prospect Theory provides a broad and solid frame of
reference for modeling the decision making of rational agents. In the early
1990s, the structure of Prospect Theory was used to propose a method to aid a multicriteria decision based on the process of paired
comparison. The research reported in this article has empirically assessed the
adherence of the mathematical model of the original TODIM method, together with
its variations available in the literature, to Prospect Theory and compared
them with a multicriteria method that does not use
that theory. From a comparative analysis, it was realized that the different
variations of the TODIM method regarding the incorporation of Prospect Theory’s
rationality within the context of Multicriteria
Decision Aid still do not bring the benefits of an already consolidated theory
to the context of decision-making aid. Thus, it is suggested that further studies
be conducted to improve the adherence of Prospect Theory within the structure
of the TODIM method, so that the benefits of a consolidated theory of decision
lead to better results, notably from the perspective of using the method for
the purposes of forecast.
Keywords: TODIM; Prospect Theory; Multicriteria Decision Aid;
Risk Aversion
1.
INTRODUCTION
Having its roots in early writings
of Enlightenment philosophers, Expected Utility Theory was by far the
dominating paradigm for analyzing human decision making until the 70’s (Mongin & Baccelli, 2020; Lengwiler, 2009).
That theory classifies the attitudes of humans facing risky choices as being of
risk aversion, risk proneness, or risk neutrality (Benjamin, Fontana & Kimball, 2020; Breig & Feldman,
2020).
In the 70’s, however, from empirical
analyzes of how agents make decisions, Daniel Kahneman
and Amos Tversky found that the measures of
satisfaction associated with the same physical magnitude are not absolute, but
relative, depending on the situation of gain or loss (Kahneman
& Tversky, 1979).
Specifically, the authors found that
the same magnitude will cause disproportionate displacements depending on their
previous reference point, being: (i) that it will
occur similarly to the modern utility theory when the displacement is
identified as gain; and (ii) that an amplification parameter must be associated
for when the displacement is identified as a loss.
This amplification parameter was
later determined empirically by Tversky and Kahneman (1992) at a value of 2.25. It is noted that the
main contribution of those authors was the proposition of the perspective of
the decision agent regarding the points of reference to the previous state
regarding which gains and losses are evaluated, this being called Prospect
Theory. Prospect Theory is today, notably under the formulation of Cumulative
Prospect Theory, one of the most complete structures for modeling the
decision-making of rational agents (Eisenführ, Weber
& Langer, 2010).
In the early 1990s, Gomes and Lima
(1991, 1992) used the structure of Prospect Theory to propose a multicriteria decision-making method based on the paired
comparison process, which is recurrent in the multicriteria
method structure. The method was named TODIM (acronym for
TOmada de Decisão Interativa e Multicritério, in Portuguese). In TODIM, it was proposed to apply
the concept of Prospect Theory using a conditional function for the pairwise
comparison process between alternatives for each criterion being evaluated.
This conditional function was identified by the Greek letter φ (Phi).
The square root structure used in
the TODIM method can be considered a particular case of the structure proposed
in Tversky and Kahneman
(1992) when considering the value α = β = 0.5. Thus, since its
proposition, variations of the Phi function of the TODIM method have been
suggested in order to optimize the adherence of its mathematical model with
Prospect Theory.
According to Yoon and Hwang (1995)
the adherence of the mathematical model is “how well a method predicts
decisions made without any assistance regardless of the judgments used to
adjust the model”. In this sense, it is possible to find in the literature the
contributions of Gomes and González (2012), Lourenzutti
and Krohling (2013), and Llamazares
(2018).
The present research intends to
empirically evaluate the adherence of the mathematical model of the original
TODIM method, together with its variations available in the literature, to
Prospect Theory and to compare them with a multicriteria
method that does not use this theory. For this, the data from the experiment
presented in Leoneti (2016) were used as input data
for the comparison of these methods.
Furthermore, from the results of the
study, the TOPSIS method (Technique for Order Preference by Similarity to Ideal
Solution) (Yoon & Hwang, 1995), was therefore selected according to Leoneti (2016), because when considering the number of
matches of the method with the surveyed decision makers, the best scores were
achieved by the TOPSIS method, including the 15 times the method correctly
identified the first alternative in the decision makers’ preference, which
meant 79% accuracy.
The next section details
the chosen methods, the results of which are presented and discussed in the
subsequent section.
2.
RESEARCH FRAMEWORK
2.1.
Topsis
The TOPSIS method was proposed in
Yoon and Hwang (1995) and several studies have applied it for aiding
decision-making processes for a quite wide range of situations. In the context
of the COVID-19 pandemic, Majumder, Biswas, and Majumder (2020) applied the TOPSIS method for identifying risk
factors and continuous monitoring of death of COVID-19. Similarly, Mahmudova (2020)
applied the TOPSIS method for improving software efficiency through the
optimization of management process. In its turn, Dehdasht et al.
(2020) used an adapted version of the TOPSIS method to select key drivers for a
successful and sustainable lean construction implementation.
The key highlight of TOPSIS is its
structure based on the comparison of alternatives within the Euclidean space.
The method was developed based on the concept that the best alternative must be
closer to the ideal alternative and have the greatest distance from the
negative solution in a geometric sense. The ideal positive and negative
solutions are basically the solutions composed, respectively, by the best and
worst scores of each criterion for each alternative.
Thus, the order of preference for
the alternatives is produced by comparing Euclidean distances. According to the
creators of TOPSIS, this method defines an index of similarity (or relative
proximity) for the ideal positive solution and dissimilarity (or relative
distance) of the ideal negative solution. The steps of the TOPSIS method are
detailed below.
Step 1: standardize the criteria
using a standardization vector presented in (1):
Step 2: calculate the standardized
values according to (2):
where is the weight value of the jth criterion.
Step 3: identify the ideal positive
and negative solutions according to (3):
where B is the set of benefit
criteria, where the preferable value is greater, and C is the set of
cost criteria, where the preferable value is less.
Step
4: calculate the distance measurements for the ideal positive and negative
alternatives for (3) and (4):
Step 5: calculate the agreement
rates for the alternatives using the formula in (5):
Step 6: order the alternatives
according to the indexes.
2.2.
TODIM
TODIM is a discrete multicriteria method inspired by Prospect Theory (Kahneman & Tversky, 1979) and
was originally presented by Gomes and Lima (1991; 1992). As a decision aiding
method, the number of applications of TODIM as well of some of its extensions
has been significantly increasing in the last ten years. More recently, Lu, Gao
and Zhao (2020) used TODIM for assessing possible environmental impacts of a
pumped hydro energy storage plant. Zindani, Maity and Bhowmik (2020), used
TODIM with fuzzy numbers for providing a version of that method for group
decision making process. Irvanizam et al. (2020) used
an extended version of TODIM with fuzzy numbers for solving the problem under
dual-connection numbers.
While practically all other multicriteria methods start from the premise that the
decision maker always decides seeking the solution corresponding to the maximum
of some global measure of value, the TODIM method makes use of the notion of a
global measure of value calculable by applying the paradigm in which consists
of Prospect Theory, in which there is a relative difference between gains and
losses.
With this, the method is based on a
description, supported by empirical evidence, of how rational agents
effectively decide in the face of risk. In order to be able to apply the
Prospect Theory paradigm to a database derived from calculations and value
judgments, the TODIM method builds the additive difference function (Phi
function) starting from a projection of the differences between the values of
any two alternatives (perceived in relation to each criterion) on a referential
criterion or reference criterion (pairwise comparison). The concept of additive
difference function employed by the TODIM method is based on Tversky (1969) research on the analytical treatment of the
multidimensionality of a value function. The steps of the TODIM method are
further detailed below.
Step 1: standardize the criteria
using the sum of the values per criterion in (6):
Step
2: calculate the Phi function in (7) for each criterion j = 1, …, n
in the pairwise comparisons between the alternatives i
= 1, …, m and k = 1, …, m:
for where wr
is the weight of criterion r = 1, …, n relativized by the
criterion of greatest importance among the n criteria in the form
Step
3: determine the dominance ratio (cost and benefit) expressed in (8):
Step
4: calculate the performance of each alternative based on the sum of the
dominance ratio and standardize the values between zero and 1.0:
Step
5: order the alternatives according to the values .
2.3.
Adaptations in the
Phi Function of TODIM
The original conception of the TODIM
method was adapted by different researchers to better adjust the mathematical
structure of the TODIM method to the principles of Prospect Theory. In this
sense, the works of Gomes and González (2012), Lourenzutti and Krohling (2013),
Lee and Shih (2016), and Llamazares (2018) are
available in the literature. In Gomes and González (2012), the authors proposed
redefining the Phi function of the original TODIM method to allow adherence to
the Cumulative Prospect Theory (CPT), as presented in Tversky
and Kahneman (1992). In this sense, the authors
proposed the changes presented in equation (1), so that it was a substitute for
Equation 7 of the original TODIM method:
having
adopted the same values for parameters λ and α as
suggested in Tversky and Kahneman
(1992), respectively as e
.
Lourenzutti
and Krohling (2013) proposed a generalization of the
TODIM method to deal with diffuse intuitionist numbers in order to make the
method capable of considering random vectors that affect the performance of the
alternatives. To achieve the objective, the authors identified the need for an
adjustment in the Phi function of the original TODIM method, specifically in
the relativization included in the Phi function of
the original method, as can be seen in Equation 7. Thus, the authors proposed
the adaptation of the Phi function of the TODIM method according to equation
(11):
where
the relative value of the criterion is replaced by its original value
.
Finally, Llamazares
(2018) proposed the generalization of the TODIM method with the establishment
of conditions under which the paradoxes related to the relativization
of weights in the original Phi equation could be minimized. The author then
described the Phi equations previously presented in the literature as
particular cases, including those by Gomes and González (2012) and Lourenzutti and Krohling (2013).
In particular, the author also presents the proposals of Lee and Shih (2016),
according to equation (12):
where and
, and also a proposal proclaimed as
the most aligned with the principles of Prospect Theory, from equation (13):
where
the criteria weights are not included in the main structure of the value
function proposed in Kahneman and Tversky
(1979).
3.
METHODOLOGY
To assess the adherence of the
different mathematical models of the Phi function of the TODIM method available
in the literature to Prospect Theory, and to compare them with a multicriteria method that does not use this theory, a
Microsoft Excel® spreadsheet environment was used. In this
spreadsheet, the steps described for the application of the TOPSIS method and
the TODIM method were programmed, with their respective variations, namely: (i) original TODIM (Gomes & Lima, 1991; Gomes & Lima,
1992); (ii) TODIM-CPT (Gomes & González, 2012); (iii) generalized TODIM I (Lourenzutti & Krohling,
2013); (iv) generalized TODIM II (Lee & Shih, 2016); and (v) generalized
TODIM III (Llamazares, 2018). For all models, the
parameter values of the mathematical models were defined as proposed in Tversky and Kahnemann (1992),
where α = β = 0.88 and θ = 2.25.
As an input for this comparison, an
experiment previously presented in Leoneti (2016) was
used, with the participation of twenty students from different years and
courses from the Ribeirão Preto School of Economics,
Administration and Accounting. Those twenty students were selected because they
were taking a course on decision making in organizations and were therefore invited
to evaluate a decision matrix containing five travel destinations (A to E)
based on eight different criteria, namely: (i) rating
of the hotel, (which is the rating of a hotel ranging from 1, worst, to 5, best);
(ii) distance in hours from the destination; (iii) number of days on the trip;
(iv) price of accommodation and airfare (in US $); (v) quantity and diversity
of shopping locations (ranging from 1, worst, to 10, best); (vi) quantity and
diversity of cultural attractions (ranging from 1, worst, to 10, best); (vii)
presence of a natural landscape (ranging from 1, worst, to 10, best); and
(viii) security, if it is safe in terms of health conditions, violence or
terrorism (ranging from 1, most insecure to 10, most secure). The decision
matrix can be seen in Table 1 and the volunteers’ preference vectors, which
were elicited using the Rank Order Centroid (ROC) method, are shown in Table 2.
Table 1: Decision matrix
|
Hotel rating |
Distance in
hours |
Duration of stay |
Cost |
Shopping |
Cultural attractions |
Natural landscape |
Security |
|
|
Destination A |
5 |
2.5 h |
4 |
2839.68 |
5 |
3 |
9 |
8 |
|
Destination B |
3.5 |
12 h |
6 |
3700.00 |
9 |
7 |
3 |
6 |
|
Destination C |
2.5 |
4 h |
5 |
2683.00 |
4 |
5 |
7 |
7.5 |
|
Destination D |
3 |
13 h |
7 |
4150.00 |
6 |
9 |
6 |
7 |
|
Destination E |
4 |
18 h |
9 |
4500.00 |
3 |
8 |
5 |
4 |
Source: adapted from Leoneti (2016)
Table 2: Weight vectors of participants
|
Hotel rating |
Distance in
hours |
Duration of
stay |
Cost |
Shopping |
Cultural attractions |
Natural landscape |
Security |
|
|
1 |
0.079 |
0.016 |
0.054 |
0.152 |
0.111 |
0.340 |
0.033 |
0.215 |
|
2 |
0.079 |
0.033 |
0.054 |
0.215 |
0.152 |
0.111 |
0.016 |
0.340 |
|
3 |
0.152 |
0.340 |
0.079 |
0.215 |
0.054 |
0.033 |
0.111 |
0.016 |
|
4 |
0.111 |
0.033 |
0.215 |
0.054 |
0.152 |
0.340 |
0.079 |
0.016 |
|
5 |
0.054 |
0.016 |
0.340 |
0.215 |
0.079 |
0.152 |
0.111 |
0.033 |
|
6 |
0.054 |
0.016 |
0.033 |
0.111 |
0.079 |
0.340 |
0.215 |
0.152 |
|
7 |
0.215 |
0.079 |
0.054 |
0.152 |
0.016 |
0.033 |
0.340 |
0.111 |
|
8 |
0.152 |
0.016 |
0.215 |
0.111 |
0.054 |
0.340 |
0.079 |
0.033 |
|
9 |
0.016 |
0.111 |
0.079 |
0.054 |
0.152 |
0.340 |
0.215 |
0.033 |
|
10 |
0.152 |
0.054 |
0.111 |
0.215 |
0.033 |
0.079 |
0.340 |
0.016 |
|
11 |
0.152 |
0.054 |
0.215 |
0.340 |
0.016 |
0.033 |
0.111 |
0.079 |
|
12 |
0.054 |
0.033 |
0.111 |
0.152 |
0.016 |
0.079 |
0.340 |
0.215 |
|
13 |
0.111 |
0.054 |
0.152 |
0.079 |
0.016 |
0.340 |
0.033 |
0.215 |
|
14 |
0.079 |
0.016 |
0.215 |
0.152 |
0.054 |
0.340 |
0.033 |
0.111 |
|
15 |
0.079 |
0.016 |
0.215 |
0.152 |
0.033 |
0.340 |
0.111 |
0.054 |
|
16 |
0.079 |
0.016 |
0.152 |
0.111 |
0.033 |
0.340 |
0.215 |
0.054 |
|
17 |
0.111 |
0.016 |
0.054 |
0.340 |
0.079 |
0.152 |
0.033 |
0.215 |
|
18 |
0.215 |
0.016 |
0.054 |
0.152 |
0.033 |
0.079 |
0.340 |
0.111 |
|
19 |
0.152 |
0.016 |
0.079 |
0.215 |
0.340 |
0.054 |
0.033 |
0.111 |
|
20 |
0.340 |
0.016 |
0.215 |
0.152 |
0.111 |
0.079 |
0.033 |
0.054 |
Source: adapted from Leoneti (2016)
The premise for assessing the
adherence of the mathematical models of the Phi function with Prospect Theory
is that suggested by Yoon and Hwang (1995) for which the adherence of the
mathematical model is “how well a method predicts decisions made without any
assistance regardless of judgments used to fit the model”. Consequently,
criteria for evaluating the performance of the different models were
established: (i) the number of correct answers in the
first position of the classification of alternatives; and (ii) the number of
correct answers for any position in the classification of alternatives. The
higher the value in the criteria, the greater the adherence of the models to
the decision maker’s cognitive process, as described by Prospect Theory.
For the calculation of the first
criterion, the count of the times in which the first position in the
classification of the method coincided with that of the decision maker was
used, remembering that in the experiment in Leoneti (2016)
the decision makers evaluated the decision matrix and ordered the alternatives
without the aid of any decision support method. To calculate the second
criterion, how many positions coincided between the two classifications,
whatever they were in the respective classifications, were counted.
4.
RESULTS AND DISCUSSION
The data from the experiment
presented in Leoneti (2016) were inserted in the
electronic spreadsheets containing the mathematical models, which generated the
data that were evaluated comparatively based on the adopted criteria, which can
be seen in Table 3. Note that the differences between the values for the
original TODIM method presented in this table from those in Leoneti
(2016), are due to the fact that here the value for parameter θ was
2.25, for comparison with the other models available in the literature, while
in Leoneti (2016) the value of θ was equal to 10.
Table 3: Values determined for the evaluation criteria
|
|
TOPSIS |
Original TODIM |
TODIM TCP |
Generalized TODIM I |
Generalized TODIM II |
Generalized TODIM III |
|
First position correct answers |
15 |
6 |
4 |
10 |
8 |
8 |
|
Correct answers for some position |
52 |
20 |
17 |
27 |
23 |
21 |
The
highest value found for criteria were identified for the TOPSIS method, which
is not based on Prospect Theory. A possible explanation for this better
performance is that the procedure of pairwise comparison in the TOPSIS method
occurs only between the alternatives and the ideal alternative, reducing the
chance of incorporating irrelevant values into the analysis, for example, the
paired comparison between the worst ranked alternatives.
Regarding the TODIM method, all its
variations that aimed at a greater adherence to Prospect Theory, with the
exception of that proposed in Gomes and González (2012), provided higher
success rates than in the original version of the TODIM method, which
corroborates this approach being a path to be followed for a better use of
Prospect Theory within the context of multicriteria
decision making. It is worth mentioning the variation of the TODIM method
proposed by Lourenzutti and Krohling
(2013), which achieved the highest number of correct answers in the first
position and the highest number of correct answers in any position compared to
the other versions of the TODIM method, including its original version.
However, it was expected that the use of Prospect Theory’s rationality would
provide better rates than those presented by the TOPSIS method, which is not
based on a specific theory of judgment and decision.
In this sense, attention can be
drawn to the fact that the Phi function modeled by Llamazares
(2018), even though being that identified by the author as the most adherent to
the value function proposed by Tversky and Kahneman (1992), has not achieved values higher than TOPSIS
for any of the criteria. This could indicate the need to review the structure
of the Tversky and Kahneman
(1992) own value function for the context of multicriteria
decision making, as their mathematical model did not provide results adherent
to the theory, based on the empirical data found.
In essence, the research presented
in this article has provided new insights on the adherence of Prospect Theory
to the structure of the TODIM method and therefore over previous works of Gomes
and González (2012), Lourenzutti and Krohling (2013), Lee and Shih (2016), and Llamazares (2018).
5.
CONCLUSION
The TODIM method
proposes the use of the most complete theory of rational choice, which is based
on Prospect Theory. It should be noted that the modeling of the method is
relatively simple, which allows its programming and manipulation without many
difficulties in a spreadsheet environment.
However, in its
comparative analysis with a similar method in terms of simplicity, it is noticed that the different variations of the TODIM
method regarding the incorporation of Prospect Theory’s rationality within the
context of multicriteria decision making, do not yet bring the
benefits of a theory already consolidated for the context of decision-making
aid. On the other hand, it is noted that most variations of the TODIM method
presented in the literature seek to provide greater adherence to that theory.
Thus,
it is suggested that further studies be conducted to improve Prospect Theory’s
adherence within the structure of the TODIM method, so that the benefits of a
consolidated theory of decision lead to better results from the perspective of
the method’s prediction, considering the cognition of the decision makers’
regarding the criteria being considered. Due to the increasing importance
of Multicriteria Decision Aid in management and
production decision-making, results that contribute to improve the foundations
of well-established decision aid tools such as the TODIM method are useful to
practitioners as well as researchers.
6.
ACKNOWLEDGEMENTS
The authors would like
to thank FAPESP and CNPq for their support to projects
no. 2016 / 03722-5 and no. 306562 / 2017-0, respectively.
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