The properties of crescent preference vectors and their utility in decision making with risk and preferences

doi:10.1016/j.fss.2020.06.008

Fuzzy Sets and Systems

Volume 409, 15 April 2021, Pages 114-127

https://doi.org/10.1016/j.fss.2020.06.008 Get rights and content

Abstract

The Crescent Method is a recently proposed decision method that can consider problems involving both risk and preferences. In this work, we elaborately discuss why and how to use this interesting method in decision making. We present its advantages in accurately merging both types of decisions. However, not all preferences are suitable to use with the Crescent Method and for melting with probability information. This study systematically proposes and analyzes those subclasses of preference vectors that are suitable for the Crescent Method. Unimodal preferences are shown to be suitable for the Crescent Method, but they are not closed under convex combination. Pure crescent preferences are shown to be suitable for the Crescent Method and to have the property of convexity. The interrelations and inclusions of certain different subclasses of preference vectors along with some examples are presented in detail.

Introduction

Evaluation for certain alternatives is the underlying problem in various decision-making situations. Aggregation functions (also known as aggregation operators) based information fusion techniques play a crucial role in numerous quantitative evaluation problems [3], [5], [6], [15], [16], [31]. The development of aggregation functions has been an active area of research for decades [7], [8], [9], [10], [11], [12], [14], [17], [22], [23], [24], [25], [28], [30], [31]. We consider an important type of aggregation function, namely the averaging function, and discuss two of its applications in two well-known uncertain decision problems, namely risk decisions and preference decisions.

A standard decision problem is characterized by the following three factors:

•
m decision alternatives $d_{r}$ ( $r = 1, . . ., m$ );
•
n states of nature $s_{i}$ ( $i = 1, . . ., n$ ); and
•
mn corresponding payoffs (with respect to each decision alternative and each state of nature) $x_{r i}$ ( $r = 1, . . ., m$ , $i = 1, . . ., n$ ).

The decision aim is to select the best decision alternative after comprehensively considering all the payoffs under the different states of nature for every alternative. Decision making under uncertainty generally includes two situations: probability information of the states of nature being available and probability information of the states of nature being unavailable.

When the required probability information regarding the likelihood of the states of nature is available, the situation is referred to as a decision under risk (or a risk decision). In this situation, each state of nature $s_{i}$ ( $i = 1, . . ., n$ ) is assigned a probability $P (s_{i})$ . Then, the decision makers consider the expected value of the decision alternative $d_{r}$ to be an overall payoff of it, i.e., the weighted payoff $E_{P} (d_{r}) = \sum_{i = 1}^{n} P (s_{i}) x_{r i}$ . In the language of aggregation functions, $E_{P} (d_{r})$ is a weighted averaging with the weight vector being $p = {(P (s_{i}))}_{i = 1}^{n}$ .

When the probability information regarding the states of nature is unavailable, the scarcity of information renders the whole decision problem more uncertain. To practically help decision makers work through the difficult alternatives and choose among them, some decision scenarios have to be effectively simplified wherein the emotions, conceptions, experiences, and preferences of decision makers are considered and measured. An effective method to circumvent the uncertainty is to consider and model a type of bipolar optimism/pessimism preference. The following three decision preferences of decision makers are commonly used:

(i)
The optimistic/sanguine approach;
(ii)
The pessimistic/conservative approach; and
(iii)
The Laplace decision approach.

The decision makers who use the optimistic approach are inclined to believe the state of nature with best payoff will definitely occur, i.e.,

E^{⁎} (d_{r}) = \max_{i \in {1, . . ., n}} {x_{r i}}

. Those with the pessimistic approach tend to believe the state of nature with the worse payoff will occur, i.e.,

E_{⁎} (d_{r}) = \min_{i \in {1, . . ., n}} {x_{r i}}

. As a type of equilibrium preference, the Laplace decision approach evenly considers all the states of nature and regards the uncertain situation as a risk situation with a uniform probability distribution. Hence, the approach considers that each state of nature has an equal probability of occurring, i.e.,

P (s_{i}) = 1 / n

for each

i \in {1, . . ., n}

and accordingly

E_{L} (d_{r}) = (1 / n) \sum_{i = 1}^{n} x_{r i}

Ordered weighted averaging (OWA) operators [29] provided a generalized model that builds a continuum of preferences from the optimistic approach via the Laplace approach to the pessimistic approach. Liu [15] applied this theory to the decision problem without probability information. When the decision problem is associated with probability information, decision makers generally use that risk information and the expected values/payoffs to measure alternatives. When the decision problem is without probability information, the bipolar optimism/pessimism preferences of the decision makers, which often embody their experiences or intuitions, exclusively decide the corresponding aggregation of all related payoffs and consequently affect the decision results. Therefore, the decision made without probability information but with decision makers' preferences is referred to as a preference decision.

Both risk decisions and preference decisions are important in decision theories. Because both involve uncertainties, we subsume them into a larger class of decision problems called uncertain decisions. In this study, the term “uncertain” does not refer to data types such as interval numbers and fuzzy numbers that themselves contain uncertainties.

Decision makers and researchers have generally not considered the important situation where both probability information and bipolar optimism/pessimism preferences are involved. Hence, there is a paucity of suitable models and methods to address this problem in decision analysis. Actually, this situation is natural and pervasive in real decision-making practices because even if probability information about the states of nature is provided, it still has uncertainty. Then, it is reasonable for decision makers to apply their bipolar optimism/pessimism to the problem and properly melt it with the given probability information.

A feasible, reasonable, and accurate melting method that melts preferences with probability information has significant utility for the aforementioned decision situations and can provide reasonable choices to decision makers in real scenarios. The recently introduced Crescent Method [11] is suitable to address this problem. Because of its feasibility and certain features, namely orness/andness affinity and consistency, it can quantitatively incorporate the preferences of the decision makers.

In this study, we consider preference vectors as the embodiments of bipolar optimism/pessimism preferences. We systematically analyze the classifications of all feasible preference vectors in the Crescent Method and provide methods to generate some classes of those feasible vectors and to classify some relations and differentiations of those different classes of preference vectors.

The rest of this paper is organized as follows. Section 2 elaborates the procedures and advantages of the Crescent Method for decision problems involving both risk and preferences. Section 3 analyzes unimodal preference vectors and shows their meaningfulness and certain properties. In Section 4, we propose the class of pure crescent preference vectors, analyze their related properties, and derive certain overall interrelations and inclusions among all subspaces of the preference vectors discussed in this study. Section 5 concludes the paper.

Section snippets

Using the Crescent Method to melt preference vectors with probability information and generate fuzzy measures

Throughout this paper, some involved inputs/payoffs, preferences, and probability information will be expressed by functions defined on finite sets. For convenience, as we concentrate on aggregation techniques involving preferences rather than empirical studies, we only consider one certain alternative d and its types of evaluations $E (d)$ . To specify,

•
denotes the states of nature by $N = {1, . . ., n}$ ;
•
a vector/function $p : N \to [0, 1]$ indicates the probability of each state of nature;
•
a payoff/input vector

Crescent Method with unimodal preference vectors and its properties

There are many similar well-known definitions of unimodal distribution and unimodal function. We note the following simple definition. A unimodal function f on [0,1] is defined in such a way that there exists at least one point $x_{0} \in [0, 1]$ such that f is non-decreasing when $x < x_{0}$ and f is non-increasing when $x > x_{0}$ . As for the unimodal vector defined on a linearly ordered set ( $N; \leq$ ), we adopt the following definition that has the same mathematical structure but a different purpose.

Definition 11

[21] A preference $w = ($

Crescent Method with pure crescent preference vectors and its properties

In this section, we first discuss a new subspace of preference vectors that is convex and suitable for the Crescent Method. Then, we discuss the relations between certain different subspaces and certain special preference vectors.

Next, we define pure crescent non-decreasing, pure crescent non-increasing, and pure crescent preference vectors, respectively.

Definition 12

Given any preference vector $w \in W^{< n >}$ , its dual preference is defined to be a preference vector $w^{'} \in W^{< n >}$ such that $w (i) = w^{'} (n - i + 1)$ ( $i \in N$ ).

Remark

For any

Conclusions

When decision makers need to simultaneously consider probability information and their preferences in decision making, the Crescent Method is an effective and accurate method. We reviewed that the decision problem without probability can be perfectly generalized by Yager's preference decision approach. Our study showed that miscellaneous crescent preference vectors can be perfectly blended with traditional risk decisions.

We discussed two important subclasses of crescent preference vectors,

Declaration of Competing Interest

There is no interests' conflict with any form related to this paper.

Acknowledgements

The authors would like to thank three anonymous referees, the Associate Editor, and the Editor-in-Chief for their many valuable comments, suggestions, and help, which significantly improved the contents and representations in this paper. This work was supported under: Scientific Research Start-up Foundation with Grant 184080H202B165. The work on this paper was supported by grants APVV-18-0052 and by the project of Grant Agency of the Czech Republic (GAČR) no. 18-06915S. The work was also partly

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The properties of crescent preference vectors and their utility in decision making with risk and preferences

Abstract

Introduction

Section snippets

Using the Crescent Method to melt preference vectors with probability information and generate fuzzy measures

Crescent Method with unimodal preference vectors and its properties

Crescent Method with pure crescent preference vectors and its properties

Conclusions

Declaration of Competing Interest

Acknowledgements

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Inf. Sci.

Fuzzy Sets Syst.

Inf. Sci.

Inf. Fusion

Fuzzy Sets Syst.

Inf. Sci.

Inf. Sci.

A method of introducing weights into OWA operators and other symmetric functions

Comparing apples and oranges: the weighted OWA function

Int. J. Intell. Syst.

Theory of capacities

Ann. Inst. Fourier

Aggregation Functions

The metric space of ordered weighted average operators with distance based on accumulated entries

Int. J. Intell. Syst.