Multistage voting is a common voting form through which the winners are selected. By virtue of weighted multistage voting rules, in this paper, we establish a weighted voting model by analyzing the correlation between individual preference and group preference. The weights of voters in each voting stage are adjusted through preference deviation degrees between individual preferences and group preference, and the ranking among candidates in each stage is determined according to weighted Borda function value. Examples are given to verify our model, which shows that weighted information aggregation model can mine more useful information from different individual preferences of voters to quicken the aggregation of group preference.
Many social choice activities usually adopt multistage voting form. Multistage voting is a dynamic group decisionmaking procedure, in which the final winner is selected through roundbyround elimination. In multistage voting system, the final voting result is generated via multiple interrelated exhaustive processes. A voter can take part in multiple times of voting; one or more candidates are eliminated till only the qualified candidate is left. Ranking for candidates is a representation form of voters' individual preferences, which means a voter expresses both his favorite candidate and the ranking of other candidates in accordance with his preference. During multistage voting system, voters need to express their preference ranking in each voting stage, so it is a major issue to fairly and effectively aggregate the individual preferences into group preference.
Weighted multistage voting is a voting system embodying a voter's power on final result, which can embody the influences of different voters on the final result, so it is necessary to measure the power of voters. Common power indices include ShapleyShubik power index [
Therefore this paper mainly discusses how to determine the weight of each voter in accordance with the relation between individual preferences and group preference in multistage voting system and puts forward relevant voting model. The less the deviation degree between individual preference of a voter and group preference is, the larger the weight of a voter will be, and vice versa. In fact, a voting process is a value judgment process not fussed about right or wrong, so we view the result (group preference) supported by most of the voters as right standard in certain period. With the changes of individual preferences and consequent changes of group preference in every round, determining weights of voters becomes a dynamic changing process referring to initial weight.
To facilitate our discussion, the paper is organized as follows. In Section
A binary relation
In individual preference, for two candidates
The order relation
Let
There are many social choice functions, such as Condorcet function, Borda function, and Kemeney function to aggregate individual preferences into group preference. Different functions can generate different results. Borda function is “the best tool for acquiring all acceptance when an effective approach is sought for achieving a common objective and the fairest method when divergent opinions must be unified” [
If there are
In essence, Borda number is the times of each candidate’s superiority to other candidates in the preference rankings given by voters. In this paper, Borda numbers are used to express the utility value of each candidate in a given individual preference ranking.
Let
Let a voter’s individual preference ranking for candidate set
Let
Given the voter set
Borda function is defined as
According to voters' preference given in Example
Deviation degree of individual preference, which means the departure of the individual preference from group preference, is defined as
The amendatory value of weight for the voter
The new weight of the voter
After weights of voters are adjusted, voters closer to group preference can get larger weights, which will adequately indicate voter's individual value and facilitate concentration of group opinions.
A multistage voting process includes voting and votes counting. In this subsection, we express individual preferences by ranking and aggregate individual preferences into group preference by weighted Borda function.
Weighted Borda function is defined as
Suppose weights vector of the voter set
In weighted multistage voting, each voter has an initial weight at first, but the voters more quickly getting close to group opinions will get more power with the increase of voting times; that is, voter weight will increase. On the contrary, the weight of voters deviating more from group opinions will decrease.
Basic principle of weighted multistage voting method is described as follows. At every voting stage, all voters give respective preference ranking for candidates firstly; then weighted Borda function values of candidates will be calculated based on adjusted weights and individual preference ranking of voters. If any candidate meets specified condition for win, the voting will end; otherwise the candidate with the lowest weighted Borda function value in this stage will be eliminated. Meanwhile, weights of voters will be readjusted according to the relation between individual preference and group preference at this stage, and then the next voting stage starts.
The steps are as follows.
Let the voter set be
Obtain the voter preference matrix
Calculate the weighted Borda function value
If any candidate meets specified winning condition, the voting will end.
Adjust voter's weights. According to (
Eliminate the unique candidate with minimal weighted Borda function value. If there are several candidates with the same minimal weighted Borda function values, an additional voting will be held to select only one to eliminate.
Let the voter set be
According to (
Apparently,
Next voter weights will be adjusted. Suppose weight correction coefficient and then individual preference deviation, weight correction, and postcorrection voter weight in voting stage 1 are calculated according to (
Weights correction results of the first stage voting.
 











 

0.000  0.000  0.559  0.433  1.601  0.000  0.000  0.612  0.250  0.000  0.559 

0.129  0.129  0.061  0.077  0.000  0.129  0.129  0.055  0.099  0.129  0.061 

0.102  0.102  0.082  0.087  0.064  0.102  0.102  0.080  0.093  0.102  0.082 
From Table
Next weighted Borda function values of candidates are calculated as follows:
Obviously,
Weights correction results of the second stage voting.
 











 

0.471  0.471  0.816  0.471  0.816  0.000  0.000  0.816  0.471  0.000  0.471 

0.083  0.083  0.000  0.083  0.000  0.196  0.196  0.000  0.083  0.196  0.083 

0.097  0.097  0.057  0.085  0.045  0.130  0.130  0.056  0.090  0.130  0.082 
From Table
Because
The relation of the value of

0.99  0.97  0.95  0.9  0.89  0.85  0.7  0.5 

0.459  0.465  0.476  0.497  0.501  0.517  0.573  0.639 

0.541  0.535  0.503  0.503  0.499  0.482  0.427  0.361 
The following conclusions can be drawn from above analysis.
The paper focuses on multistage voting model based on ordinal individual preferences through dynamically adjusting weights of voters. Because performance of candidates and cognition of voters differ, preference rankings of voters may be inconsistent. Weighted information aggregation mode can mine more useful information from different individual preference rankings of voters and quicken the aggregation of group preference. Weights of voters at each voting stage are determined mainly through the proximity between individual preferences and group preference, so weights of voters in every voting stage are adjusted, and then weighted Borda function values are calculated to decide the ranking of this stage. We know that
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is partially supported by the National Science Foundation of China (no. 11001289), Ministry of Education in China Project of Humanities and Social Sciences (13YJC630252), Chongqing Natural Science Foundation, China (CSTC2010BB9321), and the fund of Chongqing Normal University, China (11XWB016).