A binary risk decision method based on quantum decision theory

Abstract

In decision process, people are involuntarily influenced by subjective factors. So, when comparing two options, it is significant to consider both objective utility and subjective preference. In this paper, we propose a binary risk decision method based on quantum decision theory. The method defines the prospect probability by combining the objective utility and subjective psychology behavior of the decision maker, which are called utility factor and attraction factor. The decision steps are provided, in which the focus is on defining the attraction factor and the associated sign. The availability of the proposed method is illustrated through an example in investment choice.

Keywords

Risk decision quantum decision making quarter law utility factor attraction factor

1 Introduction

In real world, the decision making problems are so common in every domains. For example, the market must make decisions among the suppliers. And the investor must make decisions among investment products. Even as an ordinary person, he/she sometimes must make decisions to maintain or appreciate the hard-earned wealth. So decision making problems have been researched in depth. Chen et al. [1] research dynamic decision making to meet consumption targets. Saaty et al. [2] provide the practical applications on the modern science of multi-criteria decision making. Pereira et al. [3] discuss multi-criteria decision making under conditions of uncertainty. Zhu et al. [4] propose analytic hierarchy process-hesitant group decision making. Recently, there are more and more researches about decision making [5 –8].

Risk decision making is an important branch in the decision making problems. The risk decision problems can be denoted as comparison on the prospects, in which the possible utilities and the associated probabilities can be obtained. Some classical methods have been provided. Liu et al. [9] discuss risk decision analysis in emergency response based on cumulative prospect theory. Zhong et al. [10] provide a risk decision model of the contract generation for hydropower generation companies in electricity markets. Mukhopadhyay et al. [11] analyze cyber-risk decision models. Hsu et al. [12] provide risk and uncertainty analysis in the planning stages.

Meanwhile, it is well known that the choices of decision makers not only are based on utility, but also are strongly influenced by emotions, prejudices, biases, and other subconscious feelings [13]. So the subjective psychology behavior from the decision maker should be taken into account. Fortunately, as an innovative method, quantum theory has been applied into the decision making problems, which takes the psychological behavior into considerations and explains some paradox and fallacy [14 –16]. Agrawal et al. [17] introduce quantum mechanics and human decision making. Ashtiani et al. [18] give a survey of quantum-like approaches to decision making and cognition. Kocaslan et al. [19] provide quantum interpretation to decision making under risk. Sarris et al. [20] provide quantum models for decision making and opinion dynamic. Feinstein et al. [21] propose quantum multiple-valued decision diagrams containing skipped variables. And some other researches [22 –25].

In some decision problems, there are only two options, while in some other decision problems, there are majority of options, but the decision maker sometimes prefer to compare the options in pair. Therefore, the research on the binary risk decision problems has theoretical and practical significance. Quantum decision theory is a generalization of a quantum theory of measurement [26], which defines the prospect probabilities through taking into account hidden variables as behavioral biases and other subconscious feelings [13]. For the binary risk decision problems, quarter law can be applied to predict the prospect probabilities, in which the key is to define the sign of the attraction factors. In the literature [13], the rule for defining attraction factor signs is provided through combining the influence of the possible gains and the risk.

However, the rule in the literature [13] is not persuasive for some cases, because some necessary and useful information is not taken into consideration. In this paper, we improve the rule defining the attraction factor sign. Not only the maximal gain and the risk associated to the minimal gains, but also the risk associated to the maximal gains and the minimal gains, are taken into account. Then, under the improved rule, we can define the value of the attraction factor based on the quarter law. Ultimately, we propose a binary risk decision method based on quantum decision theory and provide the decision steps. The availability of the proposed method is illustrated through an example in investment choice.

The remainder of the paper is organized as follows. In Section 2, some basic concepts are reviewed, which include quantum decision theory, quarter law and the rule defining the sign of attraction factor. In Section 3, we provide a binary risk decision method based on quantum decision theory and the decision steps. In Section 4, an example is given to illustrate the practicality of the proposed method. Finally, Section 5 draws conclusions and discusses some future researches.

2 Preliminary

In this section, some preliminaries are introduced, which includes quantum decision theory, quarter law and the rule for defining attraction factor signs.

2.1 Quantum decision theory

Based on the quantum decision theory [26], the prospect is measured by both the objective utility terms and the subjective attraction terms.

For the prospects π₁ and π₂: $\begin{matrix} π_{1} & = & {x_{i}, p_{1} (x_{i}) : i = 1, 2, \dots, m} \\ π_{2} & = & {y_{j}, p_{2} (y_{j}) : j = 1, 2, \dots, n}, \end{matrix}$

In π₁ and π₂, the first number (x_i and y_j) of each pair corresponds to the payoff, while the second number (p₁ (x_i) and p₂ (y_j)) is the associated probability.

The prospect probability is defined as $p (π_{k}) = f (π_{k}) + q (π_{k}), k = 1, 2 .$ (1)

f (π_k) and q (π_k) respectively denote the utility factor and attraction factor. And the utility factor f (π_k) is determined by the prospect utilities U (π₁) and U (π₂). $f (π_{1}) = \frac{U (π_{1})}{U (π_{1}) + U (π_{2})},$ (2) $f (π_{2}) = \frac{U (π_{2})}{U (π_{1}) + U (π_{2})} .$ (3)

For the binary risk decision problems, the prospect utilities are defined as $U (π_{1}) = \sum_{i = 1}^{m} x_{i} \times p_{1} (x_{i}),$ (4) $U (π_{2}) = \sum_{j = 1}^{n} y_{j} \times p_{2} (y_{j}) .$ (5)

2.2 Quarter law

The interference term, or the attraction factor [27], q (π_k), is defined by emotions, subconscious feelings and other hidden variable. It is depending on a particular decision maker at a given time. For an ensemble of decision makers the interference term can be treated as a random variable in the interval [- 1, 1], i.e., $- 1 \leq q (π_{k}) \leq 1 .$ (6)

An important property of the attraction factor is the alternation property [13]: $q (π_{1}) + q (π_{2}) = 0 .$ (7)

Corresponding to the non-informative priors, the attraction factor is assumed to be a random quantity that can be characterized by an equiprobable distribution φ (q). The values q₊ and q₋ are defined as $q_{+} = \int_{0}^{1} q φ (q) dq, q_{-} = \int_{- 1}^{0} q φ (q) dq .$ (8)

Thus, the quarter law [27] is determined as $q_{+} = \frac{1}{4} and q_{-} = - \frac{1}{4} .$ (9)

The values can be used for estimating the influence of the attraction factors in the decision making process.

2.3 Rule for defining attraction factor signs

The attraction factor sign is principally important, since it essentially influences the value of the prospect probability. The sign is influenced by the possible gain and the related risk. The rule for defining attraction factor signs is provided in the literature [13].

Two prospects are considered $\begin{matrix} π_{1} & = & {x_{i}, p_{1} (x_{i}) : i = 1, 2, \dots, m} \\ π_{2} & = & {y_{j}, p_{2} (y_{j}) : j = 1, 2, \dots, n} . \end{matrix}$

The related maximal and minimal gains are denoted as $\begin{matrix} x_{max} & = & max_{i} {x_{i}}, x_{min} = min_{i} {x_{i}} . \\ y_{max} & = & max_{j} {y_{j}}, y_{min} = min_{j} {y_{j}} . \end{matrix}$

According to the alternation property (7), the signs of the attraction factors for the binary prospect lattice are opposite $sgn (q (π_{1})) = - sgn (q (π_{2})) .$

So, it is sufficient to analyzed only one of them, say the sign of q (π₁).

The first prospect gain factor is the ratio: $g (π_{1}) \equiv \frac{x_{max}}{y_{max}} .$ (10)

The first prospect gain factor shows how much the maximal gain of the first prospect is larger than that of the second one. On the other side, the smaller the probability of getting the minimal gain in the first prospect, the larger is the ratio $r (π_{2}) \equiv \frac{p_{2} (y_{min})}{p_{1} (x_{min})} .$ (11)

The ratio is called risk factor choosing the second prospect.

The combined influence of possible gain and risk is described by the product g (π₁) × r (π₂). The value defining the sign of attraction factor is

$\begin{matrix} α (π_{1}) & \equiv & g (π_{1}) \times r (π_{2}) - 1 \\ = & \frac{x_{max}}{y_{max}} \times \frac{p_{2} (y_{min})}{p_{1} (x_{min})} - 1 . \end{matrix}$ (12)

Then, the sign of the first attraction factor is defined. $sgn (q (π_{1})) = {\begin{matrix} 1 & α (π_{1}) > 0 \\ - 1 & α (π_{1}) \leq 0 \end{matrix}$ (13)

3 Binary risk decision method based on quantum decision theory

For some risk decision problems, there are only two available options, which can be called binary risk decision. The decision maker must choose the better one between the two options. For some other decision problems, although there are more than two options, people sometimes make pairwise comparison primarily before making decision. Therefore, the binary decision is the basic steps in decision process. In this section, a binary risk decision method based on quantum decision theory is proposed.

3.1 Problem descriptions

A binary risk decision problem is described as follows. Assume that there are two different options, which are denoted by the prospects: π₁ and π₂: $\begin{matrix} π_{1} & = & {x_{i}, p_{1} (x_{i}) : i = 1, 2, \dots, m}, \\ π_{2} & = & {y_{j}, p_{2} (y_{j}) : j = 1, 2, \dots, n} . \end{matrix}$

In the prospects π₁ and π₂, the first number of each pair corresponds to the payoff, while the second number is the associated probability. For instance, in the prospect π₁, for the payoff x_i, the associated probability is p₁ (x_i).

Then, between the two options, which one is the superior?

To answer the question, the two options must be compared in order that the decision maker can choose the preferable one. Obviously, in assessing the options, the objective utility is a primary factor. Additionally, another important factor is uncertainty in the decision process. The uncertainty includes at least two originalities. The first originality is from the risk as well as the uncertainty to the judgment of the gains. The other originality is from subjective behavior of the decision maker, which includes emotions, feelings, experiences, knowledge and so on. Hence, the probability choosing one of the options is simultaneously influenced by objective utility and subjective behavior of the decision makers, which are called utility factor and attraction factor, respectively.

Meanwhile, in quantum decision theory, the prospect is measured by both the objective utility terms and the subjective attraction terms. The objective utility is determined by the payoffs and the associated probabilities, and the attraction factor based on the quarter law includes subjective behavior of the decision makers. So quantum decision theory can be used to such binary risk decision problems.

3.2 Decision method based on quantum decision theory

Based on quantum decision theory, the probability choosing a prospect is combined by utility factor and attraction factor. The utility factor is defined as the Equations (2–5). According to the quarter law [27], the modulus of the attraction factor is $| q (π_{1}) | = \frac{1}{4}$ , then, the focus is how to determine the sign of the attraction factor.

In the literature [13], the sign of the attraction factor is defined as the Equation (13). However, for the case α (π₁) =0, the judgement result is not persuasive. Because, if the prospects π₁ and π₂ are exchanged in order, the sign will be opposite, that is unreasonable. For convenience, we illustrate the contradiction through an example.

Example 1. The prospects are $\begin{matrix} π_{1} & = & {10, 0.2 | 5, 0.6 | 0, 0.2}, \\ π_{2} & = & {10, 0.2 | 9, 0.3 | 1, 0.3 | 0, 0.2} . \end{matrix}$

Here, the first number of each pair corresponds to the payoff, while the second number is the associated probability. For instance, in the prospect π₁, for the payoff 10, the associated probability is 0.2, for the payoff 5, the associated probability is 0.6, and for the payoff 0, the associated probability is 0.2.

The utilities is obtained as: $\begin{matrix} U (π_{1}) & = & 10 \times 0.2 + 5 \times 0.6 = 5, \\ U (π_{2}) & = & 10 \times 0.2 + 9 \times 0.3 + 1 \times 0.3 = 5 . \end{matrix}$

Then the utility factors are respectively computed. $\begin{matrix} f (π_{1}) & = & \frac{U (π_{1})}{U (π_{1}) + U (π_{2})} = 0.5, \\ f (π_{2}) & = & \frac{U (π_{2})}{U (π_{1}) + U (π_{2})} = 0.5 . \end{matrix}$

Obviously, for the both prospects, the maximal and the minimal gains as well as the associated probabilities are exactly same. Specifically, $\begin{matrix} x_{max} & = & y_{max} = 10, x_{min} = y_{min} = 0 . \\ p_{1} (x_{max}) & = & p_{2} (y_{max}) = 0.2, p_{1} (x_{min}) \\ = & p_{2} (y_{min}) = 0.2 \end{matrix}$

Thus, the first prospect gain factor is $g (π_{1}) = \frac{x_{max}}{y_{max}} = \frac{10}{10} = 1 .$

That means the both prospects have the same maximal gains.

The risk factor choosing the second prospect is $r (π_{2}) = \frac{p_{2} (y_{min})}{p_{1} (x_{min})} = \frac{0.2}{0.2} = 1 .$

It means the probabilities of getting the minimal gains in the both prospects are equal.

The combined influence of possible gain and risk is described by the product g (π₁) × r (π₂). The value defining the sign of attraction factor is $α (π_{1}) \equiv g (π_{1}) \times r (π_{2}) - 1 = 0 .$

According to the rule in the literature [13], the sign of the first prospect attraction factor is defined to be −1. p (π₁) = 0.5 − 0.25 = 0.25, p (π₂) =0.5 + 0.25 = 0.75. Consequently, the prospect π₂ is superior to the prospect π₁. However, the result is not persuasive, since the first prospect is more attractive than the second prospect intuitively as well as experimentally.

Therefore, for the case α (π₁) =0, the sign of the first prospect attraction factor is not able to be directly defined as -1. Other factors should be taken into account, which includes the probabilities of the maximal gains as well as the minimal gain values, even includes other gains and the associated probabilities in the prospects. In order to deal with such complex situations, we provide some novel rules to improve the rule determining the attraction factor sigh in the literature [13].

Therefore, the rule defined in the equation (13) can be revised as $sgn (q (π_{1})) = {\begin{matrix} 1 & α (π_{1}) > 0 \\ uncertain & α (π_{1}) = 0 \\ - 1 & α (π_{1}) < 0 \end{matrix} .$ (14)

When α (π₁) =0, there are two cases.

Case 1. The maximal and minimal gains and their probabilities are respectively equal, that is illustrated as the Example 1. To be specific, $\begin{matrix} x_{max} & = & y_{max}, p_{1} (x_{max}) = p_{2} (y_{max}), and \\ x_{min} & = & y_{min}, p_{1} (x_{min}) = p_{2} (y_{min}) . \end{matrix}$

The maximal and minimal gains are temporarily neglected, and the remaining gains and their probabilities are analyzed with the method above.

Case 2. Otherwise, the probabilities of the maximal gains as well as the minimal gain values are taken into consideration. Specifically, we define the risk factor g′ (π₁) and the gain factor r′ (π₁) as: $g^{'} (π_{1}) \equiv \frac{p_{1} (x_{max})}{p_{2} (y_{max})}, r^{'} (π_{1}) \equiv \frac{x_{min}}{y_{min}} .$ (15)

In the risk factor g′ (π₁), the more the probability p₁ (x_max) of getting the maximal gain, the superior the first prospect. Similarly, in the gain factor r′ (π₁), the more the minimal gain x_min, the superior the first prospect. Particularly, if both the minimal gains are 0, respectively, it is defined as r′ (π₁) =1. So, the combined influence of possible gain and risk is described by the product g′ (π₁) × r′ (π₁). The value defining the sign of attraction factor is

$\begin{matrix} α^{'} (π_{1}) & = & g^{'} (π_{1}) \times r^{'} (π_{1}) - 1 \\ = & \frac{p_{1} (x_{max})}{p_{2} (y_{max})} \times \frac{x_{min}}{y_{min}} - 1 . \end{matrix}$ (16)

Then, we have the improved rule: $sgn (q (π_{1})) = {\begin{matrix} 1 & α^{'} (π_{1}) > 0 \\ uncertain & α^{'} (π_{1}) = 0 \\ - 1 & α^{'} (π_{1}) < 0 \end{matrix} .$ (17)

For the case α′ (π₁) =0, except for the maximal and minimal gains, the other gains are necessarily considered.

Example 2. The prospects are $\begin{matrix} π_{1} & = & {10, 0.1 | 6, 0.7 | 3, 0.2}, \\ π_{2} & = & {10, 0.2 | 6, 0.6 | 1, 0.2} . \end{matrix}$

The utilities of the prospects are $\begin{matrix} U (π_{1}) & = & 10 \times 0.1 + 6 \times 0.7 + 3 \times 0.2 = 5.8, \\ U (π_{2}) & = & 10 \times 0.2 + 6 \times 0.6 + 1 \times 0.2 = 5.8 . \end{matrix}$

The utility factors are $f (π_{1}) = 0.5, f (π_{2}) = 0.5 .$ (18)

The first prospect gain factor is $g (π_{1}) = \frac{x_{max}}{y_{max}} = \frac{10}{10} = 1 .$

The second prospect risk factor is $r (π_{2}) = \frac{p_{2} (y_{min})}{p_{1} (x_{min})} = \frac{0.2}{0.2} = 1 .$

The value defining the sign of attraction factor is still $α (π_{1}) \equiv g (π_{1}) \times r (π_{2}) - 1 = 0 .$

According to the rule in the literature [13], the sign of the first prospect attraction factor is defined to be -1. p (π₁) =0.5 − 0.25 = 0.25, p (π₂) =0.5 + 0.25 = 0.75. Consequently, the prospect π₁ is inferior to the prospect π₂, i.e., π₁ ≺ π₂. However, the result is still not persuasive. For most people, the first prospect is more attractive than the second prospect intuitively as well as experimentally.

Based on the rule defined in the Equation (14), it is necessary to do further analysis. $\begin{matrix} p_{1} (x_{max}) & = & 0.1, p_{2} (y_{max}) = 0.2 and \\ x_{min} & = & 3, y_{min} = 1 . \end{matrix}$

The first prospect risk factor associated to the maximal gain is $g^{'} (π_{1}) \equiv \frac{p_{1} (x_{max})}{p_{2} (y_{min})} = \frac{0.1}{0.2} = 0.5 .$

Since α (π₁) =0, we can obtain the further value $\begin{matrix} α^{'} (π_{1}) & = & g^{'} (π_{1}) \times r^{'} (π_{1}) - 1 \\ = & \frac{p_{1} (x_{max})}{p_{2} (y_{max})} \times \frac{x_{min}}{y_{min}} - 1 \\ = & \frac{0.1}{0.2} \times \frac{3}{1} - 1 > 0 . \end{matrix}$

Based on the rule defined in the Equation (18), the sign of the first prospect attraction factor is defined as 1. Consequently, p (π₁) =0.5 + 0.25 = 0.75, p (π₂) =0.5 − 0.25 = 0.25. In other words, the first prospect π₁ is superior to the second prospect π₂, i.e., π₁ ≻ π₂, which is consistent to the empirical analysis.

3.3 Decision steps

Based on quantum decision theory, the binary risk decision problems can be solved by the following steps.

Step 1. Determining the binary decision prospects.

Step 2. Computing the utility factors respect to the Equations (2–5).

Step 3. Defining the sign of the attraction factors respect to the Equations (13–18).

Step 4. Obtaining the probabilities of the prospects through combining the utility and attraction factors according to the Equations (1).

Step 5. Making the optimal choice according to the total probabilities.

Based on the quantum decision theory, the binary risk decision method is provided, which takes into account both the objective utility term and the subjective psychology behavior term, called the utility factor and attraction factor, respectively. The rule defining the attraction factor sign is improved and the decision steps are provided.

4 An illustrative example in investment choice

With the development of the finance market, it is more and more convenience and operational for the general investors. Now, there is an investor from the working class, who has a sum of money to invest. It is well known that the investor should take into account not only the possible gains but also the risk. The main aim of the investor is the maintenance and appreciation of the existing wealth, so he/she is not willing to take the risk for more unreasonable gains. The risk from the investment in stocks and funds is too big for him/her, while the profit from time deposits at the bank is too low. Fortunately, in recent years, the banks and internet financial firms provide some financial products for ordinary customers. After the comprehensive consideration, the investor thinks that financial products are proper for the comparatively satisfied profits as well as a very low risk level.

In this section, the feasibility is illustrated through an example in investment choice about the financial products.

4.1 Analysis to the problem

The investor has a sum of money (for example: 100,000 monetary unit), it is necessary to note that his/her annual income is about 50,000 monetary unit. Now, he/she plans to invest the hard earned money to financial products for the satisfied profits as well as a very low risk level. After primary screening, there are only two options remaining, which are provided by Zhaocaibao (from the internet financial firm: https://zhaocaibao.alipay.com/pf/productList.htm) and fi- nance products from China Construction Bank (http://finance.ccb.com/cn/finance/product.html), re- spectively.

For simplicity, assume that the two options are denoted by the prospects π₁ and π₂: $\begin{matrix} π_{1} & = & {4.3, 0.4 | 3.5, 0.5 | 3, 0.1}, \\ π_{2} & = & {4.3, 0.5 | 3.5, 0.4 | 2, 0.1} . \end{matrix}$

π₁ and π₂ are the financial products provided by Zhaocaibao and China Construction Bank, respectively. The first values in the prospects denote annualized return and the second values denote the associated probability. For example, the first prospect π₁ means that the financial product provides annualized return of 4.3% in probability of 0.4, annualized return of 3.5% in probability of 0.5 and annualized return of 3% in probability of 0.1. Then, for the investor, how to allocate his/her capital between the two financial products?

For such problems, there exists uncertainty in the decision process. The uncertainty includes the risk as well as the uncertainty to judgment of the gains. Meanwhile, the uncertainty is also from subjective behavior of the decision maker, which includes emotions, feelings, experiences, knowledge and so on. Consequently, the probability choosing one of the options is simultaneously influenced by objective utility as well as subjective behavior of the investor.

Therefore, the proposed method based on quantum decision theory can be used to such binary risk decision problem.

4.2 Solutions

Based on the proposed method, we can provide the decision steps as follows.

Step 1. Determining the binary decision prospects.

The prospects have been determined in the above subsection 4.1.

Step 2. Computing the utility factors.

Respect to the Equations (2–5), the utility factors are computed as follows. $\begin{matrix} U (π_{1}) & = & 4.3 \times 0.4 + 3.5 \times 0.5 + 3 \times 0.1 \\ = & 3.77, \\ U (π_{2}) & = & 4.3 \times 0.5 + 3.5 \times 0.4 + 2 \times 0.1 \\ = & 3.75 . \end{matrix}$

Then the utility factors are respectively computed. $\begin{matrix} f (π_{1}) & = & \frac{U (π_{1})}{U (π_{1}) + U (π_{2})} = 0.501, \\ f (π_{2}) & = & \frac{U (π_{2})}{U (π_{1}) + U (π_{2})} = 0.499 . \end{matrix}$

Step 3. Defining the sign of the attraction factors.

Respect to the Equations (13–18), the sign of the attraction factors is defined as follows.

The first prospect gain factor is $g (π_{1}) = \frac{x_{max}}{y_{max}} = \frac{4.3}{4.3} = 1 .$

The second prospect risk factor is $r (π_{2}) = \frac{p_{2} (y_{min})}{p_{1} (x_{min})} = \frac{0.1}{0.1} = 1 .$

The value defining the sign of attraction factor is $α (π_{1}) \equiv g (π_{1}) \times r (π_{2}) - 1 = 0 .$

Based on the rule defined in the Equation (14), it is necessary to do further analysis. $\begin{matrix} p_{1} (x_{max}) & = & 0.4, p_{2} (y_{max}) = 0.5 and \\ x_{min} & = & 3, y_{min} = 2 . \end{matrix}$

The first prospect risk factor associated to the maximal gain is $\begin{array}{l} g' (π_{1}) \equiv \frac{p_{1} (x_{\max})}{p_{2} (y_{\max})} = \frac{0.4}{0.5} = 0.8. \\ r' (π_{1}) \equiv \frac{x_{\min}}{y_{\min}} = \frac{3}{2} = 1.5 \\ α' (π_{1}) = g' (π_{1}) \times r' (π_{1}) - 1 = \frac{p_{1} (x_{\max})}{p_{2} (y_{\max})} \\ \times \frac{x_{\min}}{y_{\min}} - 1 = 0.8 \times 1.5 - 1 = 0.2 > 0. \end{array}$

According to the rule defined in the Equation (18), the sign of the first prospect attraction factor is defined as 1.

Step 4. Obtaining the probabilities of the prospects.

Through combining the utility and attraction factors according to the Equations (1), the probabilities of the prospects are obtained. $\begin{matrix} p (π_{1}) & = & 0.501 + 0.25 = 0.751, \\ p (π_{2}) & = & 0.499 - 0.25 = 0.249 . \end{matrix}$

Step 5. Making the optimal choice respect to the total probabilities.

According to the prospect probabilities, the first prospect π₁ is superior to the second prospect π₂, i.e., π₁ ≻ π₂, which seems to be consistent to the real world.

Ultimately, the investor should distribute all or majority (about 3 quarters) of the money to the first financial product, which is provided by Zhaocaibao, while no more than one quarter of the money is invested into the second financial product, which is provided by China Construction Bank.

Through the example, the feasibility of the proposed method is illustrated, which is much consistent with the actual decision process.

5 Conclusions

As an important branch of the decision problems, risk decision making has drawn much attention. In some risk decision problems, there are only two options to be compared, which are called binary risk decision problems. In this paper, based on the quantum decision theory, we have proposed a binary risk decision method, which takes into account the subjective behavior factor from the decision maker. The prospect probability is combined of objective utility factor and the subjective attraction factor. The utility factor can be computed through the utility function. While, the attraction factor is defined through the quarter law [27] from the quantum decision theory. The focus is to defining the sign of attraction factor, thus we have improved the rule in the literature [13] and provided the decision steps.

There are some novelties and/or characteristics as follows.

The risk decision method is based on quantum decision theory, which takes into account the subjective psychology behavior from the decision maker. The prospect probability is combined by objective utility factor and subjective attraction factor.

The modulus of the attraction factor is defined by the quarter law without any parameters. For the binary risk decision problems, the attraction terms are from the uncertainty, which includes the biases, emotions, knowledge, experience and other subconscious feelings.

The rule defining the attraction factor sign is improved. In the prospects, not only the maximal gain and the risk associated to the minimal gains, but also the risk associated to the maximal gains and the minimal gains, are taken into consideration.

In the future, we shall continue working on decision making problems and the methods based on quantum decision theory, and applied to some other domains such as industrial structure evaluation [28], recommendation systems, behavioral decision making and so on. In real decision making process, there might be some decision makers and multiple attributes to evaluate the alternatives. So the group decision and/or multiple attribute decision problems are common. In the meanwhile, the DM’s subjective psychology behavior should be taken into account. It is practical and interesting to research.

Footnotes

Acknowledgments

This work was supported by the Anhui University Excellent Youth Talent Support Program project (gxyqZD2018010) and the National Natural Science Foundation of China (NSFC) (71771002 and 71503005).

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