# Constructing a Measurement Method of Differences in Group Preferences Based on Relative Entropy

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

_{i}(i = 1, 2,..., m), and the information entropy of the system can be defined as follows:

_{i}= Q

_{i}, then D (P:Q) = 0. According to the principle of relative entropy, to obtain optimal results, the result of gathering should be the closest to the probability (a priori probability) of a choice distribution among all the probability distributions satisfying a given constraint.

## 3. Materials and Methods

#### 3.1. Questions

- First: to measure the respective attitude tendency degrees of different groups;
- Second: to sort the variable based on the attitudes of different groups; and
- Third: to measure the differences in attitude among the different groups.

#### 3.2. Problem Model and Solution Target

#### 3.3. Model Structure

#### 3.3.1. Data Gathering Based on the Relative Entropy Method

_{j}, j = 1, 2, …, n}; policymakers set group E = {e

_{i}, i = 1, 2, …, m}, and x

_{ij}represents the evaluation by policy-makers e

_{i}of project a

_{j}. If we assume that a larger value means the project is more certain, supposing that the group preference g can be measured, and its measure value is x

_{gj}, then x

_{gj}is the mapping of evaluation value a

_{j}. Thus, if the preference amount of the group preference is Xg = (x

_{g1}, x

_{g2}, ..., x

_{gn})

^{T}, when x

_{gj}is obtained, the decision scheme can be sorted and the group preferences compared. The main application in the decision-making field is to compare the decision-making scheme, and as this paper is for group decision-makers—that is, comparing groups who are making evaluations—then the scheme set W = {w

_{i}, i = 1, 2, …, m} is a set of decision-makers weights and is combined to 1. Since the continuous variables are too complex, it must be assumed here that the variables used to evaluate the scheme are discrete variables and that different groups are making independent evaluations. The programming model (p) is formulated below as Equation (4):

^{*}

_{g}= (X

^{*}

_{g1}, X

^{*}

_{g2}, …, X

^{*}

_{gN})

^{T}of scheme problem P, which is called the preference amount, can be solved:

#### 3.3.2. Distance of Data Measurement Based on the Relative Entropy Method

- Measurement of the demand degree. Since the preference amount is a relative index, the total preference amount of line i in the table is valued at 1, which is in line with the conditions of the information entropy of Equation (1). Therefore, information entropy can be used to reflect the discrete degree of each row of data from the preference amount of each indicator in each line. Define the preference entropy of any line i as H
_{g}, based on Equation (1):$${\mathrm{H}}_{\mathrm{g}}=-{\displaystyle \sum _{\mathrm{j}=1}^{\mathrm{n}}}{\mathrm{x}}_{\text{gj}}{\text{lnx}}_{\text{gj}},\mathrm{s}.\mathrm{t}.{\displaystyle \sum _{\mathrm{j}=1}^{\mathrm{n}}}{\mathrm{x}}_{\text{gj}}=1,{\mathrm{x}}_{\text{gj}}0$$

_{g}means that the data are discrete and contain a large amount of information. As reflected in the reality, indicating that there are substantial differences among the survey respondents in the different indicators of the degree of preference, the “love and hate” degree is large, and more information is demanded. Conversely, a larger value of H

_{g}represents an even distribution of data. As reflected in the reality, the differences in the degree of preference indicators between the different respondents is small; they give nearly the same score to the evaluation of each indictor, and less information is demanded. Thus, the reciprocal of preference entropy H

_{g}, i.e., 1/H

_{g}, can be used to measure the degree of “love and hate”, which is also known as the intensity of demand.

- 2.
- Measurement of the distance between the components and the total. In contrast to the mean value, the preference amount is a relative quantity. Thus, its separate data make little sense; in a meaningful amount of preference array, the total array value is 1. A comparison of the different preference amount array proximities is actually a comparison of the distance between the two groups of data distributions. The overall amount of the preference data array is X
^{*}_{g}= (X^{*}_{g1}, X^{*}_{g2}, …, X^{*}_{gj})^{T}; when setting a certain amount of the preference component array as X^{*}_{i}= (X^{*}_{i1}, X^{*}_{i2}, …, X^{*}_{ij})^{T}, the distance can be measured by Equation (2), which belongs to the K-L measure in mathematics. However, using the K-L measure requires accordance with a condition, namely, for any i, there must be P_{i}≥ Q_{i}to guarantee a non-negative conclusion. To solve this problem, one can simply take the absolute value of the method, but people often do not use this method in mathematics, instead preferring the method of extraction of a root after squaring. Based on relative entropy theory and mathematical practice, we define two distributions, and the distance D_{i}of the component X^{*}_{i}to the total X^{*}_{g}is:$${\mathrm{D}}_{\mathrm{i}}=\sqrt{{\displaystyle \sum _{\mathrm{j}=1}^{\mathrm{m}}}{\left({\mathrm{x}}_{\text{ij}}^{*}\text{ln}\frac{{\mathrm{x}}_{\text{ij}}^{*}}{{\mathrm{x}}_{\text{gj}}^{*}}\right)}^{2}}$$ - 3.
- Measurement of the components’ centrifugal force or centripetal force. Regarding a component i, w
_{i}is the measurement of the component in the total amount of weight; multiplying the weight by the distance forms a tendency of the component to deviate from the total indictor, called the centrifugal force. Define the centrifugal force of a component, i = w_{i}D_{i}. Since the distance variable belongs to the fixed distance variables, addition and subtraction can be used instead of multiplication and division; thus, the corresponding centripetal force = w_{i}(1 − D_{i}) is defined. The centrifugal force and the centripetal force have two mathematical characteristics after being defined.

_{i}D

_{i}+ w

_{i}(1 − D

_{i}) = w

_{i}.

## 4. Discussion and Results

#### 4.1. The Formation of the Measured Variables

- ①
- Do you think the regulatory authorities’ past regulatory policy for the stock market has been effective?
- ②
- What do you think of the effects of a series of policies and measures that were taken when the stock market crashed?
- ③
- Do you think the reform of non-tradable shares has proven successful?
- ④
- On 20 April 2008, the China Securities Regulatory Commission (CSRC) issued the “Guidance Opinions on Releasing the Transfer of Restricted Stocks of Listed Companies”. What do you think of its effects in practice?
- ⑤
- Do you think the regulatory policy of the CSRC on market manipulation and insider trading and its implementation have been successful?

#### 4.2. Comparison of the Mean Value and Preference Amount Sorting

_{ij}. The b

_{11}of the first person (i = 1) to question 1 (j = 1) is:

_{ij}is aggregated to the ratio of the group evaluation values. For example, to calculate the ratio of the evaluation value of the regulator group (g = 1) to question 1 (j = 1), given the total number of regulators is 19 and the weight of each regulator to the whole regulator group is w

_{1}= 1/19, the evaluation value ratio of the whole regulator group (g = 1) to question 1 (j = 1) is:

- Average score conclusion: question 2 > question 4 > question 6 > question 5 > question 1
- Preference conclusion: question 2 > question 4 > question 6 > question 5 > question 1
- In most cases, the order of each indicator sorted by average score and by preference are the same.

#### 4.3. Main Contradiction Found through a Significance Test

#### 4.4. Characteristic Analysis of Different Groups

## 5. Conclusions

#### 5.1. Relative Entropy Theory Solves the Quantitative Measure of Group Preference

#### 5.2. Preference Entropy and Center Distance are the Specific Methods for Measurements

#### 5.3. The Empirical Research Has Been Successful

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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No. | Individual | Question 1 | Question 2 | Question 3 | Question 4 | Question 5 | Total |
---|---|---|---|---|---|---|---|

1 | Regulator | 1 | 5 | 3 | 3 | 3 | 15 |

… | … | … | … | … | … | … | … |

Total | Regulator | 63 | 103 | 89 | 83 | 77 | 415 |

No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Value | 1 | 1 | 5 | 5 | 5 | 1 | 5 | 5 | 1 | 1 |

No. | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | — |

Value | 5 | 5 | 5 | 1 | 1 | 1 | 5 | 5 | 5 | — |

Group(g) | Question 1 | Question 2 | Question 3 | Question 4 | Question 5 | Total |
---|---|---|---|---|---|---|

Regulator (g_{1}) | 0.1294 | 0.2572 | 0.2180 | 0.2022 | 0.1871 | 0.9939 |

General investors (g_{2}) | 0.1297 | 0.2165 | 0.2116 | 0.1708 | 0.1663 | 0.8949 |

Listed companies (g_{3}) | 0.1240 | 0.2431 | 0.1919 | 0.1683 | 0.1565 | 0.8838 |

Fund companies (g_{4}) | 0.1156 | 0.2594 | 0.1837 | 0.1817 | 0.1861 | 0.9265 |

Securities traders (g_{5}) | 0.1311 | 0.2380 | 0.1868 | 0.1303 | 0.1791 | 0.8654 |

**Table 4.**Evaluation indicators of the five types of market participants for the securities regulatory problems.

Q1 | Q2 | Q3 | Q4 | Q5 | ||
---|---|---|---|---|---|---|

Regulators (19 persons) | Preference amount | 0.130 | 0.259 | 0.219 | 0.203 | 0.188 |

Mean value | 3.32 | 5.42 | 4.68 | 4.37 | 4.05 | |

General investors (40 persons) | Preference amount | 0.145 | 0.242 | 0.236 | 0.191 | 0.186 |

Mean value | 3.25 | 4.50 | 4.15 | 3.55 | 3.65 | |

Listed Companies (32 persons) | Preference amount | 0.140 | 0.275 | 0.217 | 0.190 | 0.177 |

Mean value | 3.06 | 4.88 | 3.88 | 3.44 | 3.44 | |

Fund companies (26 persons) | Preference amount | 0.125 | 0.280 | 0.198 | 0.196 | 0.200 |

Mean value | 3.23 | 5.69 | 4.15 | 4.15 | 4.15 | |

Securities traders (22 persons) | Preference amount | 0.152 | 0.275 | 0.216 | 0.151 | 0.207 |

Mean value | 3.64 | 5.73 | 4.36 | 3.18 | 4.36 | |

Total (139 persons) | Preference amount | 0.139 | 0.264 | 0.219 | 0.187 | 0.190 |

Mean value | 3.27 | 5.13 | 4.19 | 3.69 | 3.86 |

**Table 5.**The difference test between the mean value and the difference test of the preference amount for all of the questions.

Question Combinations | Mean Value | Preference Value | ||||
---|---|---|---|---|---|---|

Difference b/w the Two | Sig. (Two-Tailed) | Differences | Difference b/w the Two | Sig. (Two-Tailed) | Differences | |

Q1–Q2 | −1.856 | 0.000 | Significant | −0.125 | 0.000 | Significant |

Q1–Q3 | −0.921 | 0.000 | Significant | −0.08 | 0.005 | Significant |

Q1–Q4 | −0.417 | 0.078 | Not significant | −0.048 | 0.095 | Not significant |

Q1–Q5 | −0.590 | 0.018 | Significant | −0.051 | 0.075 | Not significant |

Q2–Q3 | 0.935 | 0.000 | Significant | 0.045 | 0.245 | Not significant |

Q2–Q4 | 1.439 | 0.000 | Significant | 0.077 | 0.052 | Significant |

Q2–Q5 | 1.266 | 0.000 | Significant | 0.074 | 0.061 | Not significant |

Q3–Q4 | 0.504 | 0.003 | Significant | 0.032 | 0.373 | Not significant |

Q3–Q5 | 0.331 | 0.075 | Not significant | 0.029 | 0.418 | Not significant |

Q4–Q5 | −0.173 | 0.312 | Not significant | −0.003 | 0.928 | Not significant |

Groups | Ratio (w_{i}) | Preference Entropy (H_{g}) | Center Distance |
---|---|---|---|

Regulators | 0.137 | 1.59 | 0.020 |

General investors | 0.288 | 1.59 | 0.029 |

Listed companies | 0.23 | 1.58 | 0.017 |

Fund companies | 0.187 | 1.58 | 0.032 |

Securities traders | 0.158 | 1.58 | 0.041 |

Total | 1 | 1.59 | 0.000 |

© 2017 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Zhang, S.; Liu, W.; He, Q.; Hao, X. Constructing a Measurement Method of Differences in Group Preferences Based on Relative Entropy. *Entropy* **2017**, *19*, 24.
https://doi.org/10.3390/e19010024

**AMA Style**

Zhang S, Liu W, He Q, Hao X. Constructing a Measurement Method of Differences in Group Preferences Based on Relative Entropy. *Entropy*. 2017; 19(1):24.
https://doi.org/10.3390/e19010024

**Chicago/Turabian Style**

Zhang, Shiyu, Wenzhi Liu, Qin He, and Xuguang Hao. 2017. "Constructing a Measurement Method of Differences in Group Preferences Based on Relative Entropy" *Entropy* 19, no. 1: 24.
https://doi.org/10.3390/e19010024