Maximum Group Perceived Utility Consensus Models Considering Regret Aversion

XUE Yuqin(薛雨沁)，CHENG Dong(程 棟)

Glorious Sun School of Business and Management, Donghua University, Shanghai 200051, China

Abstract: In the consensus-reaching process (CRP), the actual utility of decision-makers (DMs) is often influenced by the psychological behavior of regret aversion. However, the influence of regret aversion on DMs’ utilities is rarely taken into account in the existing consensus models. The consensus-reaching problem of DMs with regret aversion is explored to maximize their perceived utilities under a limited budget. Firstly, three basic types of perceived utility functions are constructed based on the regret theory to describe the perceived utility of DMs with various preferences. Then, considering the limited budget and individual regret aversion, the maximum perceived utility consensus models based on types of left-skewed, right-skewed, middle-skewed, and heterogeneous utility preferences are proposed to achieve the consensus that maximizes the group perceived utility. After that, an example of land-transfer price negotiation in China is given to illustrate the validity of the proposed models. Finally, the model comparison and the sensitivity analysis are presented to reveal the influence of DMs’ regret aversion on the CRP. The results suggest that the DMs’ regret aversion will not affect left-skewed and right-skewed groups, but will affect the consensus results of middle-skewed and heterogeneous groups.

Key words: regret aversion; consensus model; interval preference; maximum perceived utility

Introduction

The consensus-reaching process (CRP) aims to draw on the opinions of all decision-makers (DMs) in a group so as to agglomerate agreement. It generally requires the assistance of a moderator, who uses cost compensation and other means to convince DMs to modify their opinions[1-2]. For the moderator’s interest, he/she always hopes to reach a consensus at the lowest cost, so the minimum cost consensus model (MCCM) is proposed[3-6]. Since the MCCM can improve consensus efficiency and reduce costs, it has been widely applied to various important group decision-making (GDM) issues, such as commercial negotiations, demolition compensation, and pollution control consultations[1]. However, the MCCM only considers the feelings of the moderator and ignores the effect of DMs’ utilities on the CRP[7-8]. Fully preserving the utility preferences of all DMs is the key to ensure the effectiveness of consensus opinion and its successful implementation in the GDM[9-10]. As the main body of the CRP, each DM hopes that his/her opinion will be given enough attention. In other words, the moderator needs to pay more attention to the utility of each DM, rather than just considering the compensation cost in the CRP. Therefore, the utility consensus models are required to be constructed in order to obtain the consensus that can be supported and satisfied by the DMs.

Regret aversion, or the feeling of avoiding regret, is a common psychological phenomenon among DMs and may affect their utilities in the CRP[11-12]. Each DM will compare the consensus opinion with his/her own opinion, which may result in two kinds of psychology behavior: one is that the DM will feel regret if he/she finds that the individual opinion can better satisfy his/her preference than the consensus opinion; otherwise, he/she will feel joyful. When the DM anticipates the potential regret after the decision, his/her regret aversion will affect his/her decision to avoid making decisions that make him/her feel regretful. This expectation of regret or rejoicing about the decision results will affect both their utilities and opinions. The regret theory, proposed by Bell[13], and Loomes and Sugden[14], is used to measure the influence of the regret aversion on individual utility. In recent years, it has been successfully applied to GDM problems. For example, Zhouetal.[15]obtained the regret value of the alternatives by constructing a regret function to select the best alternative. Peng and Yang[16]took the DMs’ regret aversion and prospect preference into consideration, and proposed two algorithms to solve stochastic multi-criteria decision-making problem. These models work well to solve the GDM problem of alternative selection, but they pay little attention to the CRP, which usually reaches the consensus by means of an optimization method. Yang and Sen[17]built a multi-objective optimization model considering utility functions to obtain the best solution. Hoetal.[18]considered DMs’ utilities and risk attitudes, and proposed a fuzzy goal programming model in order to improve DMs’ satisfaction. Researchers[8-9]constructed the maximum utility consensus model based on linear and non-linear utility functions under limited budget constraints. Subsequently, based on the risk preference of DMs, Gongetal.[19]adopted piecewise linear utility functions to establish a utility consensus model so as to describe the dynamic preference and the consensus level.

Maximum group perceived utility consensus models consider regret aversion of DMs. However, the psychology of DMs’ regret aversion is often neglected in the current utility consensus model, which uses the expected utility function to measure the DMs’ utilities. This may lead to the consensus result deviating from the solution recommendation obtained by applying the extant utility consensus theory. Therefore, it is necessary to consider the impact of DMs’ regret aversion when constructing the utility consensus model.

This paper aims to explore how to obtain the optimal consensus that maximizes the group perceived utility considering DMs’ regret aversion. To portray the perceived utility of DMs with different psychological preferences, we first define three kinds of perceived utility functions based on regret theory, namely, left-skewed, right-skewed, and middle-skewed types. Then, the maximum group perceived utility consensus models under different preference structures are constructed, which improve and extend the current utility consensus theory. The main contributions of the study are as follows.

(1) Considering the influence of DMs’ regret aversion, a consensus model of maximum group perceived utility under limited cost is proposed. Although Gongetal.[8]explored the problem of maximum group epxected utility consensus models for the first time, they did not consider DMs’ regret aversion.

(2) Three types of perceived utility functions incorporating DMs’ regret aversion are defined. Compared with the expected utility function, this article is based on the regret theory to construct the perceived utility function to reflect the actual utility of the DM as much as possible.

(3) The influence of different individual preference types of DMs on the CRP is explored. Based on the case of land transfer, this paper presents future work of solving the GDM problem with different DMs’ preference types by constructing a heterogeneous maximum perceived utility model. The results show that group members with different preference types tend to set a neutral land-transfer price, which is beneficial to the long-term cooperation between farmers and companies.

The rest of this article is organized as follows. Section 1 describes in detail the research problem and explains the parameters to be used in the subsequent modeling. Section 2 focuses on the construction of three types of perceived utility functions based on regret theory. Then, four types of maximum group perceived utility consensus models are constructed in section 3. Section 4 gives an application example to verify the validity of the proposed models. To further illustrate the influence of regret aversion on the proposed consensus models, a comparative analysis and a sensitivity analysis are included in section 5. Finally, section 6 concludes the whole paper and presents the future work.

1 Problem Description

2 Construction of Perceived Utility Function Based on Regret Theory

This section briefly introduces the knowledge related to regret theory, and then constructs three types of perceived utility functions accordingly: left-skewed, right-skewed, and middle-skewed regret utility functions. According to the individual perceived utility measurement formula, this section provides a basis for the following construction of the maximum group perceived utility consensus model.

2.1 Regret theory

The basic idea of regret theory is that the DM not only pays attention to the results that can be obtained from the selected option, but also compares the possible results under various options. When the DM finds that the results of choosing other options are better, he/she will feel regret. Otherwise, he/she will feel delighted. In regret theory, the perceived utility of the DM is composed of the expected utility function of the current result and the regret-rejoice function. Suppose that there are two decision resolutionsxandy, anduxanduyrepresent the expected utility value obtained by the decision resolutionsxandy, respectively. Then, the perceived utilityUxyobtained by the user for the decision resolutionxcan be expressed as

Uxy=ux+R(ux-uy),

(1)

whereR(ux-uy) represents the regret-rejoice value generated by the DM who abandons the resolutionybecause of the choice of the resolutionx. Regret-rejoice functionR(ux-uy)is a monotonically increasing concave function, satisfyingR(0)=0,R′(ux-uy)>0, andR″(ux-uy)<0[21], which usually is presented as a negative exponential function formgR(ux-uy)[22]:

gR(ux-uy)=1-exp[-δ(ux-uy)],

(2)

where the parameterδ∈[0, +∞] is the regret coefficient of the DM. The larger the value ofδis, the more impact of regret aversion on the DM will have.

2.2 Construction of perceived utility function

Considering the complexity of GDM, individuals prefer interval numbers to a crisp number to express their views more comprehensively. Unlike the crisp number, DMs often have different preferences for the values in the interval. In the CRP, the utility function is usually used to reflect the individual preference structure of DMs. According to different types of interval preferences, these functions are usually divided into left-skewed, right-skewed, and middle-skewed types[8]. The traditional utility functions only demonstrate the expected utility of DMs, which ignore the impact of DMs’ psychology on themselves and cannot reflect the real DMs’ utilities in the CRP. Therefore, the perceived utility functions are constructed to not only measure the impact of DMs’ regret aversion on utility, but also reflect the real utility as much as possible.

We use the utility function in Gongetal.[8]to obtain the expected utility of three types of individual preference. A DM of a left-skewed type prefers the lower bound of his/her opinion interval, and his/her expected utility value will decrease with the increase of the opinion value.

Definition1Letoibe the individual opinion value of the left-skewed DMdi, and then the perceived utility function ofdiis defined as

(3)

(4)

Similarly, a DM belongs to the right-skewed type if he/she is inclined to the upper bound of the opinion interval and gains more utility when increasing the value of his/her opinion.

Definition2Letoibe the individual opinion value of the right-skewed DMdi, and then his/her perceived utility function is defined as

(5)

(6)

If the DM gains the largest utility when his/her individual opinion is equal to the median value of the opinion interval, his/her personal preference is called the middle-skewed type.
Definition3Letoibe the individual opinion value of the middle-skewed DMdi, andΔκibe the utility difference between the consensus opinion and his/her opinion, and then the perceived utility function ofdiis defined as

(7)

Since the expected utility function of middle-skewed DMs is piecewise linear, we will discuss the utility difference caused by the comparison of consensus opinions and individual opinions in the following four situations.

(8)

(9)

(10)

(11)

The utility difference reflects the closeness covering the individual opinion, the consensus opinion, and the median of the interval. When the utility difference is positive, the consensus opinion is closer to the median of the interval than the individual opinion. Otherwise, the individual opinion is closer than the consensus opinion to the median of the interval.

3 Construction of Maximum Group Perceived Utility Consensus Models

After constructing the perceived utility function based on the regret theory, three kinds of DMs’ perceived utilities can be obtained. On the base of section 2, this section will discuss the construction of homogeneous and heterogeneous maximum group perceived utility consensus models. Homogeneity means that DMs in the group have the same type of individual preference, while heterogeneity means that their preferences are not exactly the same. Considering DMs’ regret psychology and limited consensus budget, the maximum group perceived utility consensus models can be constructed based on homogeneous (left-skewed, right-skewed, and middle-skewed types) and heterogeneous utility preferences.

3.1 Homogeneous maximum group perceived utility consensus model

3.1.1 Left-skewedmaximumgroupperceivedutilityconsensusmodel

When the DMs’ preferences are all left-skewed, the group can be defined as a left-skewed group. Based on the left-skewed regret perceived utility function defined in section 2.2, a left-skewed maximum group perceived utility consensus model under the constraints of limited cost and interval opinions is constructed.

(12)

Lemma1If the feasible region of a convex optimization problem is a continuous non-empty set, then there must be an optimal solution.

ProofPlease refer to Ref. [23].

Theorem1Left-skewed regret perceived utility model (12) is convex optimization and has an optimal solution.

ProofThe standard form of the model is

(13)

(14)

3.1.2 Right-skewedmaximumgroupperceivedutilityconsensusmodel

If all DMs in the group belong to the same preference type with the right-skewed type, we call them a right-skewed group. Similar to section 3.1.1, the construction of right-skewed maximum group perceived utility consensus model is

(15)

Theorem2Right-skewed regret perceived utility model (15) is convex optimization and has an optimal solution.

ProofThe standard form of the model is

(16)

which is similar to model (13). The proof that the inequality constraint is a convex function is the same as Theorem 1. Thus by proving that the objective function is convex, it is possible to prove that model (16) is convex optimization.

(17)

3.1.3 Middle-skewedmaximumgroupperceivedutilityconsensusmodel

The middle-skewed group means that all the DM’s preference tends to the median of the interval. The construction of middle-skewed maximum group perceived utility consensus model is similar to section 3.1.1, which is constructed as follows.

(18)

Theorem3Middle-skewed maximum group perceived utility consensus model (18) is convex optimization and has an optimal solution.

ProofThe standard form of the model is

(19)

which is similar to model (13). The proof that the inequality constraint is a convex function is the same as Theorem 1. Thus by proving that the objective function is convex, it is possible to prove that model (19) is convex optimization.

(20)

(21)

(22)

(23)

3.2 Heterogeneous maximum group perceived utility consensus model

In many practical decision-making issues, the goals of DMs are not always the same, which may result in their individual preference being different and often contradictory. The above three models only consider the situation of the single personal preference in the decision group, which is relatively simple and it is difficult to fully reflect the actual complex decision-making behavior. Therefore, the construction of maximum group perceived utility model in the situation of the various individual preferences is discussed in this section.

The definition of a heterogeneous group is that the types of DMs’ personal preferences are not exactly the same and may be any of the left-skewed, the right-skewed and the middle-skewed types. Based on the three perceived utility functions, under the constraints of limited cost, a heterogeneous maximum group perceived utility consensus model is constructed.

(24)

4 Case Study of Land-Transfer Price Negotiation

Since reform and opening up in 1978, China has experienced great economic growth and rapid urbanization. The ongoing economic development, and urbanization have been found to lead to land fragmentation in rural areas[25-26]. The fragmentation of rural land management not only affects mechanized operations, but also reduces production efficiency and increases production costs. To solve this problem, farmers are encouraged to transfer their land management rights, which is called land transfer, hoping to increase the utilization rate of land[27]. Many farmers are willing to circulate the land contract management right. To improve circulation efficiency and maintain sustainable land transfer, a proper land-transfer price is the core issue. Too low land-transfer prices may cause farmers to be unwilling to transfer their lands, while too high prices may cause contractors to be unwilling to contract the farmers’ lands. Therefore, the land-transfer price negotiation is an indispensable part during the transfer process. The land-transfer price negotiation usually involves the psychological expectations of government departments, contractors, farmers, and village committee members whose interests are closely related to the price. They have their own expected range for land-transfer price and their utility preferences are different. Contractors are left-skewed DMs, and they hope that the transfer price is as low as possible. Farmers are right-skewed DMs and hope that the transfer price is as high as possible. In order to promote rural land transfer, the village committee member always hopes that the transfer price is as moderate as possible, who is a middle-skewed DM. The government can be regarded as a moderator. If they fail to reach a consensus on the land-transfer price during the negotiation, it will be difficult to carry out land-transfer activities effectively. Therefore, how to coordinate the different utility preference types of contractors, farmers and village committee members on land-transfer price is a key issue related to the success or failure of rural land transfer. So, in this section four types of regret perceived utility consensus models proposed in the previous section are used to deal with rural land-transfer price problem, aiming to achieve the most suitable price for all of DMs.

Suppose there are four DMs， {d1,d2,d3,d4}, whered1is a contractor,d2andd3are farmers, andd4is a village committee member. Liu and Han[28]used the rural land transfer price model to calculate the land-transfer price in Liaoning Province, Jiangsu Province, Hubei Province, and Sichuan Province reasonably. In 2016, among four regions, the highest circulation price was 150 RMB/(m2·d) and the lowest circulation price was 40 RMB/(m2·d). Based on this interval[40, 150], the initial opinion intervals of four DMs about the land-transfer price are generated randomly, which areo1=[54, 67],o2=[75, 90],o3=[60, 88], ando4=[40, 98].o′is the final land-transfer price. The unit subsidies that the government gives to four DMs arec1=15,c2=5,c3=5, andc4=2, respectively. In this instance, it is assumed that the weights of four DMs are equal, that is,w1=w2=w3=w4=0.25. The regret coefficient represents the degree of each DM’s regret aversion, of which the value range is [0, +∞]. For the convenience of calculation, the regret coefficient of four DMs is unified to be equal to random numbers within the interval, which isδ=1.5. Using the MCCM[8]for reference, the minimum cost is 40. Therefore, the limited cost budget in the model should be larger than the minimum consensus cost, and we takeB=50.

In the case of land-transfer price negotiation, the decision group is a heterogeneous group, so the heterogeneous consensus model needs to be constructed. According to section 3.2, the maximum group perceived utility consensus model is as follows.

(25)

In order to demonstrate the feasibility of homogeneous consensus models, the values in the case are substituted into three types of proposed homogeneous models as a numerical example in the CRP.

The left-skewed maximum group perceived utility consensus model is

(26)

The right-skewed maximum group perceived utility consensus model is

(27)

The middle-skewed maximum group perceived utility consensus model is

(28)

Using the Matlab optimization toolbox, the optimal solutions to the above models can be obtained (shown in Table 1). For a homogeneous group, the consensus opinion is determined by the group’s common utility preference. The left-skewed group has the smallest consensus opinion, while the right-skewed group has the largest one. For a heterogeneous group, the consensus opinion tends to the utility preference of the most similar DMs in the group. The heterogeneous group is composed of two farmers, a contractor and a village committee member. Among them, the number of farmers is the largest, and the land-transfer price is more in line with farmers’ left-skewed preference. The group perceived utility of the heterogeneous group is negative, representing the uncomfortable or painful feelings that the consensus opinion brings to the group.

Table 1 Results of four types of maximum perceived utility consensus models

5 Model Discussion

5.1 Comparative analysis

This section further analyzes the impact of DMs’ regret aversion on consensus. First, we construct the maximum utility consensus model without considering regret aversion in the same way as the maximum group perceived utility consensus model (see Appendix A). The only difference between these two models is that the utility constraint in the model not considering regret aversion obtains the expected utility rather than the perceived utility. Then the data in the example of land-transfer price negotiation are substituted into these models, and the results are shown in Table 2.

Table 2 Results of the consensus model without considering regret aversion

By comparing Table 1 and Table 2, we could obtain the following conclusions.

(1) Regret avoidance psychology may make one DM feel regretful as well as joyful. For the left-skewed and the middle-skewed groups, the group perceived utility with considering regret (0.950 and 2.146, respectively) is greater than the expected utility without considering regret (0.791 and 1.929, respectively). At this time, most of DMs in the group are joyful. For the right-skewed and the heterogeneous groups, the group perceived utility with considering regret (0.129 and -0.080, respectively) is lower than the expected utility without considering regret (0.355 and 0.197, respectively) and most of DMs in the group feel regret for the consensus result.

(2) The consensus of the middle-skewed and the heterogeneous groups affected by regret aversion will be slight, while the consensus of the left-skewed and the right-skewed groups will not be influenced. For a heterogeneous group composed of farmers, contractors, and village committee members, the regret aversion will make DMs pay attention to the type of preference of the minority in the group. In this instance, the contractor with the left-skewed preference is the minority in the group and prefers the price to be as low as possible. Therefore, the land-transfer price of the heterogeneous group considering regret aversion is lower than that without considering regret.

5.2 Sensitivity analysis

In this section, the cost and the regret coefficient of the DM are taken as variables, respectively. With other factors remaining unchanged, we will explore the impact of different costs and regret coefficients on the consensus results of four consensus models. The specific results of the impact on group perceived utility and consensus opinion are shown in Figs. 1 and 2, and the specific results of different regret coefficients are shown in Figs. 3 and 4.

Fig. 1 Influence of budget on group perceived utility

Fig. 2 Influence of budget on consensus opinion

Fig. 3 Influence of regret coefficient on group perceived utility

Fig. 4 Influence of regret coefficient on consensus opinion

It can be seen from Fig. 2 that the movement direction of consensus opinions under the change of budget is related to the type of utility preference. For the left-skewed and the middle-skewed groups, the consensus opinion decreases with the increase of budget. While, for the right-skewed and the heterogeneous groups, the consensus opinion gradually increases. From Fig. 1, as budget changes, the increase or decrease of the group perceived utility level is independent of the type of utility preference. For four types of regret-perceived utility models, the group perceived utility increases with the increase of the budget. The larger budget the moderator gives, the more DMs can be compensated, and the group perceived utility will be greater.

From Figs. 3 and 4, it can be found that the negative effect of regret has greater impact on the group perceived utility than the positive effect of rejoicing. With the increase of the regret coefficient, the impact of regret aversion will be greater. The perceived utilities of the left-skewed and the middle-skewed groups under the impact of rejoicing gradually increase, while the perceived utilities of the right-skewed and the heterogeneous groups gradually decrease. Also, the increase in perceived utility is significantly less than the decrease in utility. Moreover, the influence of the DMs’ regret aversion on the CRP is limited. When the regret coefficient increases to a certain level, the consensus opinion will no longer change.

6 Conclusions

Since the regret aversion of DMs has an important impact on the utility, the purpose of this study is to determine how to obtain the consensus opinion that maximizes the group perceived utility when considering regret aversion. Based on regret theory, three perceived utility functions are defined to obtain the perceived utility of DMs with different individual preference types. Then, under the constraint of limited cost, homogeneous and heterogeneous maximum group perceived utility consensus models are constructed based on perceived utility functions. The main conclusions are as follows.

(1) For a homogeneous group, the consensus opinion is determined by common utility preference of the group. The left-skewed group has the smallest consensus opinion, while the right-skewed group has the largest one. For a heterogeneous group, the consensus opinion tends to the utility preference of the most similar DMs in the group.

(2) The regret aversion brings both regret and rejoicing to the DMs. When most of the DMs in the group feel regret, the group perceived utility will be less than the expected utility of the group without considering the regret. Otherwise, the group perceived utility will be larger than the expected utility. Also, the negative effect produced by regret will have a greater impact on the group perceived utility than the positive effect produced by rejoicing.

(3) The consensus of the middle-skewed and the heterogeneous groups affected by regret aversion will be more neutral, while the consensus of the left-skewed and the right-skewed groups will not be influenced. For a middle-skewed group, the consensus opinion and the individual opinion will be larger under the influence of regret aversion. For a heterogeneous group, the regret aversion will make DMs pay attention to the preference needs of the minority in the group. In the case of land-transfer price negotiation, the impact of regret aversion caused by different kinds of individual preferences will lead to a neutral land-transfer price, which helps to the long-term cooperation between farmers and companies.

(4) As the compensation cost changes, the movement direction of consensus opinions is related to the type of group preferences, while the increase or decrease of the group utility level has nothing to do with the type of individual preferences. The consensus opinions of the left-skewed and the middle-skewed maximum group perceived utility consensus models decrease with the increase of the cost, while the consensus opinions of the right-skewed and the heterogeneous models increase. The group perceived utility of the above four types of models all gradually increase with the increase of the cost.

Although this article provides a consensus method considering the bounded rationality of DMs, we assume that the regret coefficients of DMs are consistent, ignoring the impact of different perceptions of regret among individuals. Therefore, the maximum group perceived utility consensus model under different circumstances of regret coefficients is an interesting direction in future study.