A Survey on Models and Methods for Preference Voting and Aggregation

A Survey on Models and Methods for Preference Voting and Aggregation

Ali Ebrahimnejad (Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshar, Iran) and Farhad Hosseinzadeh Lotfi (Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran)
Copyright: © 2017 |Pages: 26
DOI: 10.4018/978-1-5225-0596-9.ch002
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Abstract

A key issue in the preferential voting framework is how voters express their preferences on a set of candidates. In the existing voting systems, each voter selects a subset of candidates and ranks them from most to least preferred. The obtained preference score of each candidate is the weighted sum of votes receives in different places. Thus, one of the most important issues for aggregating preferences rankings is the determination of the weights associated with the different ranking places. To avoid the subjectivity in determining the weights, various models based on Data Envelopment Analysis (DEA) have been appeared in the literature to determine the most favorable weights for each candidate. This work presents a survey on models and methods to assess the weights in voting systems. The existing voting systems are divided into two areas. In the first area it is assumed that the votes of all the voters to have equal importance and in the second area voters are classifies into different groups and assumed that each group is assigned a different voting power. In this contribution, some of the most common models and procedures for determining the most favorable weights for each candidate are analyzed.
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Introduction

It is often necessary in decision making to rank a group of candidates using a voting system. In a typical ranked voting method, each voter selects a subset of candidates and ranks them from most to least preferred. Among these methods, a popular procedure for obtaining a group consensus ranking is the scoring method where fixed scores are assigned to different places. In this way, the score obtained by each candidate is the weighted sum of the scores he or she receives in different places. The plurality and Borda methods are the most widely used scoring methods. In the plurality method, the selected candidate is the one who obtains the most votes in the first place. In other words, in this method, the first place receives an importance weight of 100% while all other places receive a weight of zero. In Borda’s method, the weight assigned to the first place equals to the number of candidates and each subsequent place receives one unit less than its preceding place. Although Borda’s method has interesting properties, the utilization of a fixed scoring vector implies that the choice of the winner may depend upon which scoring vector is used (Brams & Fishburn, 2002).

To avoid this problem, Cook et al. (Cook & Kress, 1990) suggested evaluating each candidate with the most favorable scoring vector for him/her. With this purpose, they introduced Data Envelopment Analysis (DEA) in this context. DEA determines the most favorable weights for each candidate. Different candidates utilize different sets of weights to calculate their total scores, which are referred to as the best relative total scores and are all restricted to be less than or equal to one. The candidate with the biggest relative total score of unity is said to be efficient candidate and may be considered as a winner. The principal drawback of this method is very often leads to more than one candidate to be efficient candidate. We can judge that the set of efficient candidates is the top group of candidates, but cannot single out only one winner among them. To avoid this weakness, Cook et al. (Cook & Kress, 1990), proposed to maximize the gap between consecutive weights of the scoring vector. However, Green et al. (Green, Doyle, & Cook, 1996) noticed two important drawbacks of the previous procedure. The first one is that the choice of the intensify functions used in their model is not obvious, and that choice determines the winner. The second one is that for an important class of discrimination intensity functions the previous procedure is equivalent to imposing a common set of scores on all candidates. Therefore, when Cook and Kress’s model is used with this class of discrimination intensity functions, the aim pursued by these authors (evaluating each candidate with the most favorable scoring vector for him/her) is not reached.

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