An improved three-way decision model based on prospect theory

Abstract

In three-way decision, how to describe the risk attitudes of decision-makers is an important issue. The prospect theory is widely used to reflect decision-makers' risk attitudes and the decision rules are based on the maximum prospect value. However, the previous three-way decision models based on prospect theory neglected time outcome or they included time outcome in monetary outcome in decision process. In fact, in some cases, time outcome is a more important element in decision making, and without considering it directly may weaken the rationality of decision results. To address these problems, we construct an improved three-way decision model based on prospect theory, which simultaneously and directly considers time outcome and monetary outcome. Specially, this model is a multi-objective optimization model. Firstly, prospect theory is used to describe decision-makers' risk attitudes toward monetary gains and losses as well as time gains and losses. Secondly, we construct a multi-objective optimization model and introduce preference coefficients to transform it into a single objective optimization model, which is based on the maximum comprehensive prospect value. Further, the existence and uniqueness of thresholds are proven, and the decision rules are given. Finally, an illustrative example and some comparative analyses are presented, which validate the rationality and superiority of our improved model.

Introduction

Three-way decision (3WD), proposed by Yao [1], [2], is an important method dealing with uncertain decision problems. Its main idea is to divide the universe into three disjoint regions (positive region, negative region and boundary region) by the thresholds α, β and γ. The three regions can be interpreted as three decision results: acceptance, rejection and uncertain, respectively [3]. Compared with traditional two-way decision, three-way decision has one more delayed decision when the information is insufficient to make a decision to reject or accept. This decision making process is consistent with human cognitive process. In the last decade, three-way decision has been extensively applied in many research fields [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. The researches of three-way decision can be divided into two main directions [16]. One of the research directions is the extension of the model and the concept of three-way decision [17], [18], [19], [20], [21], [22], [23], [24], [25]. And the development of the granular computing and sequential three-way decision is another important research direction [26], [27], [28], [29], [30]. In addition, how to express decision-makers' risk attitudes in three-way decision has become an important research issue in recent years [31], [32], [33], [34]. Classical three-way decision (C3WD) model utilizes the loss function to measure the risk attitudes of decision-makers [1], [2]. However, C3WD model does not consider the risk attitudes of different decision-makers. Then, Zhang et al. [35] proposed a utility theory-based three-way decision (U3WD) model, which uses utility function to measure risk attitudes and takes decision-makers' risk preferences for gains and losses into account, the model makes decision based on the maximum utility. However, in some decision environments, utility theory cannot reflect the real preferences and risk attitudes of decision-makers [36]. For this problem, prospect theory provides a good solution.

Prospect theory, proposed by Kahneman and Tversky [37], describes the performances of decision-makers under uncertainty and risk. It is based on psychological behaviors. Many psychological experiments [36], [38] indicated that people tend to prospect value maximization rather than utility maximization in decision making process. According to prospect theory, people are not completely rational in decision making, and decision-makers show risk aversion to gains and risk chasing to losses. For instance, an experiment showed that when faced with “win $80 for sure or 85% to win $100”, the majority of participants selected a win $80 for sure. However, when faced with “lose $80 for sure or 85% to loss $100”, the majority of participants selected an 85% to loss $100 [39]. In prospect theory, the outcomes of taking an action are expressed as gains and losses compared with the decision-makers' reference points [40], and the value function and weight function are used to describe the risk attitudes of decision-makers. Prospect theory is applicable to the situation that the decision-makers are bounded rationality. For example, when facing an investment project, the decision-makers usually have psychological expectations for its return, when the actual return exceeds their psychological expectations, decision-makers will show conservative risk attitudes toward the project; otherwise, decision-makers will show risk chasing attitudes toward the project. Prospect theory has been widely used in diverse decision making areas [41], [42], [43], [44], [45], [46]. For instance, Zhu et al. [45] proposed multi-reference points risk decision method under static and dynamic conditions. Wang et al. [46] proposed a group emergency decision method based on prospect theory. In particular, prospect theory is also well applied for multi-criteria decision making problems, for example, Gomes and Lima [47] took into account decision-makers' psychological behaviors, and proposed TODIM (an acronym in Portuguese “TOmada de Decisão Iterativa Multicritério”) method based on prospect theory, which has been improved and extended [48], [49].

As prospect theory performed well on decision making under the condition of risk and uncertainty, Wang et al. [16] first introduced prospect theory into three-way decision and constructed a prospect theory-based three-way decision (P3WD) model to describe decision-makers' risk attitudes. Further, some improved prospect theories have also been introduced into three-way decision, such as cumulative prospect theory [33] and third-generation prospect theory [50], which have achieved good results. These studies greatly promoted the development of three-way decision and prospect theory. However, these studies only consider monetary outcome without time outcome or include time outcome in monetary outcome in decision process, which may fail to reflect the impact of time outcome on the decision making adequately. Generally, “time outcome” represents the time spent of performing an action, which is a negative indicator, and we always want to spend less time. In fact, in some decision processes, time outcome is important, and even more important than monetary outcome. For example, in the selection of a route from point A to point B, the time outcome may have a priority. And time outcome is also a priority in the emergency rescue. Motivated by these observations, we construct an improved P3WD model by directly taking into account time outcome in addition to monetary outcome in the decision process. Specially, this model is also a multi-objective optimization model based on the maximum monetary prospect value and maximum time prospect value. Further, because decision-makers may have different preferences for monetary prospect and time prospect in different situations, we introduce monetary preference coefficient and time preference coefficient in the improved P3WD model.

The novelty of this paper can be shown as follows:

• The improved P3WD model takes into account monetary outcome and time outcome simultaneously and directly. That is, this model is a multi-objective optimization model that expects both the maximum monetary prospect value and maximum time prospect value, which makes the decision results more reasonable.

• The improved P3WD model describes decision-makers' different preferences for monetary prospect and time prospect. In different circumstances, decision-makers can adjust the monetary preference coefficient and time preference coefficient to make decision. Hence, this model can be flexibly applied to various decision making environments.

The structure of this paper is presented as follows. Section 2 introduces the basic concepts of three-way decision and prospect theory. Section 3 constructs the improved three-way decision model based on prospect theory. Section 4 gives an illustrative example to apply the improved P3WD model and analyze the sensitivity of thresholds and decision results. In Section 5, some comparative analyses are presented, which validate the rationality and superiority of the improved P3WD model. Section 6 concludes this paper and presents the future work.