A three-way decision model based on cumulative prospect theory

Abstract

In three-way decision, the description on the risk attitude of decision-makers is a focus topic. In this paper, we propose a novel three-way decision model based on cumulative prospect theory. First, with the aid of a reference point, the value functions are utilized to describe different risk appetites of decision-makers towards gains and losses. Second, the weight functions incorporate the nonlinear transformation of the conditional probability. The cumulative decision weights are decided by taking into account the increasing order of value functions. Then, with value functions and weight functions, the new decision rules of the proposed model are deduced based on the principle of maximizing the cumulative prospect value rather than minimizing the cost. Further, we analyze and prove the existence and uniqueness of thresholds of our model. Then, the decision rules are simplified based on the conditional probability and numerical solutions of thresholds, and the algorithm for deriving three-way decision rules is constructed. Finally, an illustrative example and a series of relevant comparisons are presented to illustrate and validate the effectiveness and feasibility of our model.

Introduction

Three-way decision (3WD), as an uncertain decision approach, has developed rapidly in the fields of decision-making, granular computing and incomplete data analysis, since it was proposed by Yao in [1], [2]. Three-way decision can be viewed as a generalized theory of traditional rough set, proposed by Pawlak [3], which received much attention in recent literatures [4], [5], [6], [7], [8]. The main concept of three-way decision is to divide the universe into three disjoint regions (positive, boundary and negative regions) by two thresholds α and β. Three types of decision rules, including acceptance, non-commitment and rejection, are generated from three disjoint regions based on the principle of Bayesian risk minimization. In the case that the information is insufficient, the non-commitment is taken as the delayed decision, which is in line with the human decision process. In recent years, three-way decision has been widely applied in diverse territories, such as recommendation systems [9], image recognition [10], cognitive concept learning [11], cost-sensitive learning [12], medical diagnosis [13] and incremental clustering [14].

The theory and development of three-way decision have received a constant attention and achieved many research results [15], [16], [17], [18]. In summary, the study of three-way decision has two main aspects. First, many researchers focused on the extension of the model and notion of three-way decision [19], [20], [21], [22]. Lang et al. [23] presented the concepts of probabilistic conflict, neutral and allied sets of conflicts, and discussed the mechanism for computing the thresholds. Xu and Wang [24] proposed a new aggregation method of interval loss function based on the principle of justifiable granularity. Sun et al. [25] deduced the three-way decision based on the multigranulation fuzzy decision-theoretic rough set over two universes. The second aspect is the development of the sequential three-way decision and granular computing [26], [27], [28]. Li et al. [10] proposed a sequential three-way decision method for cost-sensitive face recognition. Yang et al. [29] proposed a unified dynamic framework of decision-theoretic rough sets for incrementally updating three-way probabilistic regions. Zhang et al. [30] defined a new concept of attribute ratio and established a dynamic three-way decision model. Qian et al. [31] proposed a generalized multigranulation sequential three-way decision model based on multiple different thresholds. In recent years, many scholars have tended to pay more attention to the description on the risk attitude of decision-makers [32], [33], [34], [35]. For example, Liang et al. [36] involved the risk appetite of the decision-maker into three-way decisions. Zhang et al. [37] utilized the utility function as a new risk measurement, and introduced the utility theory into 3WD theory. Thus, how to reflect the risk attitude of decision-makers has become a challenge and attracted increasing attention. However, most of the aforementioned literatures described the risk attitudes of decision-makers from the perspective of loss functions. The risks are mainly measured in terms of losses, while ignoring the impact of gains on decision-makers. In many real-life applications, the risk attitude and behavior of decision-makers are determined by both gains and losses. Some studies have shown that decision-makers may have different risk appetites towards gains and losses [38]. In the actual decision process of decision-makers, the pain of losses can be far greater than the pleasure of gains [39].

Cumulative prospect theory (CPT), proposed by Tversky and Kahneman [40], is an important theory to describe and predict the decision behaviors under risk and uncertainty. It is stated in CPT that the outcomes are represented as gains and losses in comparison with the reference point of decision-makers instead of the final state of wealth [41]. The decision-makers are risk aversion towards gains and risk seeking towards losses [42]. Moreover, two nonlinear transformations of the probability are taken into account towards gains and losses, which overweight small probabilities, while underweight moderate and high probabilities. Instead of transforming each single probability, the cumulative distribution function of the decision weight is also considered. CPT has been verified by a large number of psychological experiments, and it can correctly predict the real decision behavior of decision-makers. In recent years, CPT has attracted growing attention in many diverse decision-making areas. For example, Hu et al. [43] proposed a cumulative prospect theory-based model to describe the charging behavior of battery electric vehicle drivers. Keskin [44] studied cumulative prospect theory preferences in rent-seeking contests. Zhao et al. [39] proposed an optimal execution model with transient price impact and permanent price impact under cumulative prospect theory. Zhou et al. [41] solved the problem of portfolio selection by a group decision-making process and considered the psychological behavior of experts with CPT. Wu et al. [45] established a comprehensive criteria system for the evaluation of renewable power sources based on CPT. Following CPT, Metzger and Rieger [38] investigated a framework for non-cooperative games in normal form where players have behavioral preferences. Li et al. [42] proposed an integrated cumulative prospect theory based hybrid-information multi-criteria decision making approach, which could be utilized to appraise and identify sustainable third party reverse logistics provider. Xu et al. [46] studied psychological behaviors of decision-makers and considered bounded rationality by using CPT.

Motivated by the success of CPT in the researches on decision-making problems, we introduced CPT into 3WD theory to construct a new three-way decision model. A summary of the work completed in this study is shown as follows. First, a 3WD model based on CPT is proposed. Different risk attitudes of decision-makers and appetites towards gains and losses are considered on the basis of the reference point. The conditional probability is generalized into two different weight functions and the decision weight is performed over the cumulative distribution function. With the aid of value functions and weight functions, decision rules are determined based on the principle of maximizing the cumulative prospect value. Then, since the analytic solutions of thresholds of our model cannot be calculated directly, the existence and uniqueness of thresholds are analyzed and proven. The simplified form of decision rules and the algorithm for deriving three-way decision rules are presented. Finally, the proposed model is utilized to support an illustrative emergency decision-making problem. The illustrative example and comparative analysis show the effectiveness of our model.

The remainder of the paper is presented as follows. In Section 2, some basic notions of three-way decision and cumulative prospect theory are presented. Then, in Section 3, the novel three-way decision model based on cumulative prospect theory is proposed. Section 4 provides a further analysis of the thresholds and simplified decision rules. An illustrative example is given to elaborate the effectiveness of our model, and the proposed model is compared with some other 3WD models in Section 5. Section 6 concludes the paper and elaborates on future studies.