Relative Utility Function Research Articles

A utility-based three-way group decision consensus model with overlapping subgroups

This paper introduces a relative utility function-based consensus method for three-way group decision (TWGD) in a social network scenario where overlapping subgroups and non-overlapping subgroups co-exist. Firstly, we construct a three-way decision (TWD) model rooted in utility theory, which proposes a novel objective calculation method for the relative utility functions based on evaluation values. The relative utility matrices of decision makers (DMs) are derived on this basis. Secondly, a utility and trust-driven overlapping clustering algorithm is designed to maximize the overall trust level and the difference between subgroup clustering centers, aiming to divide the group into several manageable subgroups. In addition, we identify contrarian DMs, triggering a delegation-exit mechanism, and determine the weights of subgroups and DMs. Thirdly, a three-way group consensus decision model is formulated to provide targeted adjustment strategies for different types of DMs. To improve group consensus, we address the influence of key DMs on the opinions of overlapping subgroups with a minimum adjustment consensus model. Finally, we demonstrate the feasibility and scientificity of the proposed method through the illustrative example, sensitivity analysis, and comparative analysis.

Regret-based three-way decisions with set pair analysis in incomplete information systems

Set pair analysis is a valuable tool for managing uncertain systems and can effectively handle missing information in the decision-making process. In addition, regret theory can objectively describe the impact of decision-maker psychology on decision-making. However, traditional decision-making methods rarely combine set pair analysis and regret theory to discuss decision problems. Thus, this article aims to integrate regret theory into the three-way decision theory based on set pair analysis, presenting a novel approach to address multi-attribute decision-making problems in real life. Specifically, we first propose a data-driven approach based on set pair analysis to determine attribute weights. Then, a similarity relationship is established based on the weighted distance to calculate conditional probability. Meanwhile, the average value of regret and joy under each attribute is selected as a reference point to determine the relative utility function using regret theory. Subsequently, we develop a new three-way decision method in incomplete information systems and apply it to solve practical problems. Finally, the effectiveness and superiority of our method are demonstrated by case analysis and comparison. We also conduct sensitivity analysis to examine the stability of the new method.

A novel multi-source TWD model based on multi-granularity ball for multiple decision makers

Three-way incomplete multi-attribute decision-making model (TWIMADM) for multiple decision makers is an important topic in data science, which is an effective tool for decision making. However, calculation results of neighborhood blocks are still affected by neighborhood radius. In addition, most of the existing models without considering the information from the evaluations of multiple decision makers separately is difficult to obtain a comprehensive assessment. To address these problems, this paper builds a novel multi-source TWD model based on multi-granularity ball. First, the well-established granular ball is extended to multi-granularity, by which a weighted conditional probability is generated to divide the concerned region into three parts: positive, negative and boundary region. The calculation of the multi-granularity ball is independence of the neighborhood radius parameter. Subsequently, the joint entropy is developed by using the defined multi-granularity to obtain the optimal decision. Furthermore, the relative utility function is utilized to capture the preferences of decision makers, the classification and ranking of each object are determined by calculating the threshold value and expected utility value. Evaluation information from all decision makers are combined and fused. Finally, a novel TWD model is developed to address the challenges of incomplete multi-attribute decision-making problems involving multiple decision makers. The comparative case analyses and experimental evaluations have demonstrated the stability, validity, and feasibility of our model.

BTWM-HF: A behavioral three-way multi-attribute decision-making method with hesitant fuzzy information

Three-way decision (TWD) theory offers a novel paradigm for solving hesitant fuzzy multi-attribute decision-making problems. However, many existing TWD approaches follow the principle of minimum risk, thereby overlooking the influence of the psychological factors of decision-makers (DMs) on decision outcomes when faced with losses and gains. To address this challenge, we propose a behavioral three-way multi-attribute decision-making method with hesitant fuzzy information via hesitant fuzzy TODIM. The proposed method encompasses two key features. One is that the relative utility functions based on the dominance degree are constructed, aiming to portray the psychological behaviors of DMs. The other is the development of an objective calculation of weighted conditional probabilities, achieved without relying on a state set and a binary relation. The feasibility of the proposed method is verified by two numerical examples, one on software selection and the other from the UCI database. The results of the multi-aspect comparative analysis demonstrate that the proposed method exhibits good performance in terms of ranking and classification. Moreover, the experimental results indicate the effectiveness and applicability of the presented method.

Two novel three-way decision models based on fuzzy [formula omitted]-covering rough sets and prospect theory under q-rung orthopair fuzzy environments

Three-way decision (TWD) and prospect theory (PT) are efficacious theories for investigating uncertain problems. TWD can mitigate the risks associated with rapid decision-making in traditional decision-making. PT can describe the psychological behavior of decision-makers. Furthermore, q-rung orthopair fuzzy sets (q-ROFSs) exhibit wider-ranging applications since they generalize intuitive fuzzy sets and Pythagorean fuzzy sets and unify them under one framework. However, the acquisition of q-ROFSs is an important issue in practical problems. In light of these considerations, this paper develops two TWD models under q-rung orthopair fuzzy environments, employing the principles of PT. To begin, we introduce two novel fuzzy β-covering rough sets that meet the inclusion property. Subsequently, utilizing the proposed fuzzy rough set, we provide an objective methodology to acquire q-ROFSs. Following this, the relative utility function is given through a combination of the value function and EDAS method. Additionally, the weighting function is constructed using two distinct approaches: one that continues to incorporate the ideas of EDAS method, and the other that utilizes the probabilistic dominance α-neighborhood classes to calculate conditional probabilities. After that, classification rules and sorting rules are derived. Ultimately, we substantiate the validity and excellence of the proposed TWD models through comparative and experimental analysis.

An attribute fuzzy concept-oriented three-way utility decision model in multi-attribute environments

In recent years, an increasing number of researchers have focused on how to leverage limited information to make more reasonable decisions, thereby reducing the decision-making risk for decision-makers. Three-way decision, as a method for risky decision-making, offers a new approach to address this challenge and has gained significant attention. In view of this, based on the utility theory, this paper proposes a novel three-way utility decision model oriented to attribute fuzzy concept in multi-attribute environments. Firstly, an attribute fuzzy concept is constructed based on the decision-maker’s minimum requirements on each attribute. Subsequently, a new method for calculating the relative utility function is proposed, which is used to calculate the corresponding decision thresholds. Additionally, a generalized conditional probability calculation method is presented based on the attribute fuzzy concept and the basic semantics of conditional probability. Finally, a new three-way decision model is established to obtain the classification and ranking results of all objects. Through application analysis, comparative study and experimental analysis, the proposed method’s rationality and effectiveness are verified.

Three-way multi-attribute decision making under incomplete mixed environments using probabilistic similarity

In recent years, many scholars have explored a variety of methods integrating three-way decision (3WD) and multi-attribute decision making (MADM), which enables the classification and priority ranking of alternatives possible and fully reflects the effectiveness and advantages of 3WD in solving MADM problems. However, few of these methods can effectively deal with the MADM problems with incomplete mixed information that are frequently encountered in real-world situations. This study proposes a three-way MADM method for an incomplete mixed information system (IMIS), where the objective determination of conditional probabilities and utility functions in IMIS without decision label is the pivotal issue. To overcome this issue, we define a probabilistic similarity measure for incomplete mixed information. The probabilistic similarity measure is used to replace the distance measure of classical TOPSIS for estimating the conditional probabilities objectively. The probabilistic similarity class is introduced with arithmetic average method to design a conversion mechanism and obtain the objective relative utility functions of incomplete evaluation values. We then construct a novel 3WD model in IMIS and combine it with two customized ranking principles, to solve the incomplete mixed MADM problems from the perspective of classification and ranking in a more thoughtful and interpretable manner. Our study provides a new perspective for the research on MADM in incomplete mixed information environment. Several examples and experimental comparisons verify the effectiveness and stability of the proposed method. The experiments demonstrate that our method can meet more decision-making requirements and is more accurate and rational in some decision-making scenarios than several prior similar methods.

Uncertain Stochastic Optimal Control with Jump and Its Application in a Portfolio Game

This article describes a class of jump-uncertain stochastic control systems, and derives an Itô–Liu formula with jump. We characterize an optimal control law, that satisfies the Hamilton–Jacobi–Bellman equation with jump. Then, this paper deduces the optimal portfolio game under uncertain stochastic financial markets with jump. The information of players is symmetrical. The financial market is constituted of a risk-free asset and a risky asset whose price process is subjected to the jump-uncertain stochastic Black–Scholes model. The game is formulated by two utility maximization problems, each investor tries to maximize his relative utility, which is the weighted average of terminal wealth difference between his terminal wealth and that of his competitor. Finally, the explicit expressions of equilibrium investment strategies and value functions for the constant absolute risk-averse and constant relative risk-averse utility function are derived by using the dynamic programming principle.

Regret-based three-way decision making with possibility dominance and SPA theory in incomplete information system

In the field of decision making, three-way decision making has been proven more fruitful for providing scopes to make delayed decision along with acceptance and rejection simultaneously. As a result, the decision risk and loss, which could occur due to take rapid decisions in traditional two-way decision making, are effectively reduced. Accordingly, this paper offers a novel three-way multi-attribute decision making model by combining three-way decision making and multi-attribute decision making under an incomplete information system. The incertitude in the information system is illustrated by introducing interval-valued Fermatean connection number based on interval-valued Fermatean fuzzy number and set pair analysis theory. Thereafter, the achievement of this study is five-fold. First, a possibility dominance relation is developed under incomplete information system in the basis of possibility degree measure of interval-valued Fermatean connection number. Second, we put forward a novel procedure to set up the fuzzy state set. Third, the conditional probability is estimated in light of the fuzzy state set and probability dominance relation. Fourth, the relative utility functions are determined in the virtue of regret theory. Lastly, a three-way multi-attribute decision making model is implemented in incomplete information system and exploited to deal with incomplete multi-attribute decision making problems. Eventually, the propriety, stability and superiority of the proposed model is established via conducting the comparative and experimental analysis.

A novel behavioral three-way decision model with application to the treatment of mild symptoms of COVID-19

The Coronavirus Disease 2019 (COVID-19) has popularized since late December 2019. In present, it is still highly transmissible and has severe impact on the public health and global economy. Due to the lack of specific drug and the appearance of different variants, the selection of the antiviral therapy to treat the patients with mild symptom is of vital importance. Hence, in this paper, we propose a novel behavioral Three-Way Decision (3WD) model and apply it to the medicine selection decision. First, a new relative utility function is constructed by considering the risk-aversion behavior and regret-aversion behavior of human beings. Second, based on the relative utility function, some new rules are defined to calculate the thresholds and conditional probabilities in 3WD and some corresponding theorems are explored and proved. Next, a new information fusion mechanism in the framework of evidential reasoning algorithm is developed. Then, the decision results are obtained based on the Bayesian decision procedure and the principle of maximum utility. Finally, an example with large-scale data set and an example about medicine selection for COVID-19 are provided to show the implementation process and effectiveness of the proposed method. Comparative analysis and sensitivity analysis are also performed to illustrate the superiority and the robustness of the current proposal.

TWD-R: A three-way decision approach based on regret theory in multi-scale decision information systems

As a highly prevalent disease, the liver disease is usually diagnosed by testing with several aspects. Due to different accuracies of tests, the obtained results may be different, which may lead to potential errors in diagnosis results. Meanwhile, the psychological behavior and the psychological preference of a decision maker (DM) have a significant influence on decision-making results. Thus, we aim to propose a three-way decision (TWD) approach based on the regret theory (RT) in a multi-scale decision information system (MSDIS), which is named a TWD-R approach. First, by defining a dominating relation and a dominated relation based on the regret-rejoice value of each alternative under each attribute, a novel conditional probability calculation approach is proposed. Second, an objective approach for calculating relative utility functions is proposed to avoid the influence originated from the subjectivity. Third, the key steps and the algorithm of the TWD-R approach are summed up. Finally, the rationality, validity and superiority of the proposed approach are investigated by two cases and some experimental analyses.

A Novel Three-Way Decision Model Based on Utility Theory in Incomplete Fuzzy Decision Systems

The existing three-way decision (TWD)-making methods cannot effectively handle incomplete multiattribute decision-making (MADM) problems in real life; it is necessary to explore an effective three-way MADM model in incomplete fuzzy decision systems (IFDSs). First, we consider the preference of decision makers for each alternative and introduce the concept of predecisions, thus an IFDS is obtained. Then, the weighted conditional probabilities are calculated with the aid of the defined similarity relation. Subsequently, we introduce the notion of relative utility functions and then present an approach to determine the relative utility function values. Afterward, we construct a TWD model in IFDSs and apply it to the modeling of incomplete MADM (IMADM) problems. Our study not only enriches TWD and MADM theories but also provides a new perspective for realistic IMADM problems. At last, the results of comparative and experimental analyses demonstrate the validity, stability, and superiority of our proposed model.

When to Offer Lower Quality or Remanufactured Versions of a Product

ABSTRACTWe investigate the profitability of adding a lower quality or remanufactured product to the product portfolio of a monopoly firm, both in single‐period and steady‐state settings. Consumer behavior is characterized by a deterministic utility function for the original product and a nonlinear relative utility function for the lower quality product. We find a threshold for the cost of the low‐quality product below which it is optimal to add it to the firm's portfolio, and show that while a cost advantage is necessary to make the lower quality offering profitable under linear or convex relative utility functions, market segmentation alone can justify the addition of the lower quality product under concave relative utility functions. In particular, we characterize (i) the new product cost under which it is optimal to offer a lower quality version of the product even if it is as costly to produce as the original product; and (ii) the weighted average of new and remanufactured product costs in the steady state under which it becomes cost effective to offer new products under the remanufactured label. Finally, we also identify the maximum possible profits from customer segmentation and the form of the relative utility function that achieves them. We discuss the implications for the common marketing practices of branding and generics.

Resolving Trans-jurisdictional Water Conflicts by the Nash Bargaining Method: A Case Study in Zhangweinan Canal Basin in North China

Trans-jurisdictional conflict in both water quantity and quality is a general concern in large river basins. In this paper, the relative utility function combined with the asymmetric Nash bargaining method was established to analyze the trans-jurisdictional conflict between water quantity and water quality in the Zhangweinan Canal Basin in China. The basin was divided into four conflict stakeholders, namely, Shanxi, Hebei, Henan, and Shandong Provinces, based on administrative unit. The water usage and pollutant discharge scheme for multiple stakeholders was optimized using the established model to satisfy the environmental flow and water quality objectives at the identified conflict sections. The results indicated that the total water consumption was reduced from 4.38 × 109 m3 in 2007 to 1.97 × 109 m3, and that the allowable COD and NH3-N discharged into the river was less than 3.8 × 104 t and 4.3 × 103 t, respectively. About 81.1 % of COD and 76.1 % of NH3-N should be further reduced compared with the values in 2007.

Multi-Outcome Lotteries: Prospect Theory vs. Relative Utility

This paper discusses two approaches for the analysis of multi-outcome lotteries. The first uses Cumulative Prospect Theory. The second is the Relative Utility Function, which strongly resembles the utility function hypothesized by Markowitz (1952). It is shown that the relative utility model follows Expected Utility Theory with a transformed outcome domain. An illustrative example demonstrates that not only it is a simpler model, but it also provides more sound predictions regarding certainty equivalents of multi-outcome lotteries. The paper discusses estimation procedures for both models. It is noted that Cumulative Prospect Theory has been derived using two-outcome lotteries only, and it is hard to find any evidence in the literature of its parameters ever having been estimated by using lotteries with more than two outcomes. Least squares (mean) and quantile (including median) regression estimations are presented for the relative utility model. It turns out that the estimations for two- and three-outcome lotteries are essentially the same. This confirms the correctness of the model and vindicates the homogeneity of responses given by subjects. An additional advantage of the relative utility model is that it allows multi-outcome lotteries, together with the estimation results, to be presented on a single graph. This is not possible using Cumulative Prospect Theory.

Estimation of Peaked Densities Over the Interval [0,1] Using Two-Sided Power Distribution: Application to Lottery Experiments

This paper deals with estimating peaked densities over the interval [0,1] using two-sided power distribution (Kotz, van Dorp, 2004). Such data were encountered in experiments determining certainty equivalents of lotteries (Kontek, 2010). This paper summarizes the basic properties of the two-sided power distribution (TP) and its generalized form (GTP). The GTP maximum likelihood estimator, a result not derived by Kotz and van Dorp, is presented. The TP and GTP are used to estimate certainty equivalent densities in two data sets from lottery experiments. The obtained results show that even a two-parametric TP distribution provides more accurate estimates than the smooth three-parametric generalized beta distribution GBT (Libby, Novick, 1982) in one of the considered data sets. The three-parametric GTP distribution outperforms GBT for these data. The results are, however, the very opposite for the second data set, in which the data are greatly scattered. The paper demonstrates that the TP and GTP distributions may be extremely useful in estimating peaked densities over the interval [0,1] and in studying the relative utility function.

Lottery Valuation Using the Aspiration/Relative Utility Function

The paper presents a method for lottery valuation using the relative utility function. This function was presented by Kontek (2009) as 'the aspiration function' and resembles the utility curve proposed by Markowitz (1952A). The paper discusses lotteries with discrete and continuous outcome distributions as well as lotteries with positive, negative and mixed outcomes providing analytical formulas for certainty equivalents in each case. The solution is similar to the Expected Utility Theory approach and does not use the probability weighting function – one of the key elements of Prospect Theory. Solutions to several classical behavioral problems, including the Allais paradox, are presented, demonstrating that the method can be used for valuing lotteries even in more complex cases of outcomes described by a combination of Beta distributions. The paper provides strong arguments against Prospect Theory as a model for describing human behavior and lays the foundations for Relative Utility Theory – a new theory of decision making under conditions of risk.

Why Risk is Not Related to Return

This paper argues that in general risk is not empirically correlated with returns in any obvious way. This puzzle is explained as the implication of a utility function in which if people care only about relative wealth, risk is a deviation from what everyone else is doing, and therefore becomes avoidable and unpriced, similar to diversifiable risk in the Capital Asset Pricing Model (CAPM). Using a utility or arbitrage argument, a relative status utility function creates a zero risk-return correlation via a market model that implies a zero risk premium, and has other applications.

On Utility and Individual Choice

A novel relative utility function in cardinal form is put forward in analysing the problem of individual decision-making. Let relative utility denote the ratio of utility which derived from consuming a goods or service to the cost of it. An individual consumer will make her choice to maximize the relative utility function, and thus her desirable utility level will be achieved with the lowest cost. While discarding the assumption of diminishing marginal utility, how an individual consumer will choose among goods and services is effectively analyzed and interpreted in accord with the reality by using the assumption of relative utility maximization. Two instances with unsolved puzzles, lottery and insurance, and Giffen goods, are discussed.

Investigating Excess Returns from Nominal Bonds*

AbstractEstimated real returns on nominal bonds show excess returns of some 200 bp over their index‐linked equivalent. This paper considers two possible explanations for this large difference. First, we assess the likely inflation risk premium by calibrating a model of optimal bond prices under uncertainty. Employing either of CRRA or Abel (1990) relative consumption utility function to derive the stochastic discount factor and covariation risk, we suggest that the inflation risk component of this excess return is unlikely to be much above 50 bp. Secondly, we find little evidence that these excess returns can be ascribed to consistent expectational errors in predicting inflation.