Abstract
This paper studies the anti-manipulation voting method introduced in [8]. We show that the method does not satisfy the consistency condition. The consistency condition characterizes scoring functions. Thus, the method is not a scoring function. Also, the method is not any from a family of not scoring functions comprising Copeland method, instant-runoff voting, majority judgment, minimax, ranked pairs, Schulze method. The paper also shows that the choice of a metric, used by the anti-manipulation method, may imply the winner of the voting.
This research was supported by the National Science Centre, Poland, grant number 2016/21/B/HS4/03016 and SGHS19/07/19.
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Notes
- 1.
In fact, as defined later, the jurors are assigned weights with the most distant jurors given zero weight.
- 2.
The data in Tables 4–10 are taken from https://en.wikipedia.org/wiki/Consistency_criterion.
- 3.
Authors verified calculations.
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A Algorithm Description
A Algorithm Description
As was mentioned in the main text, the set of alternatives is denoted by \( A \), \( V \) is a finite subset of a set of voters \( N \). Any function from \( V \) into a set of linear orders on \( A \) is a profile, denoted by \( w \). Any function \( f \) assigning to a profile \( w \) a nonempty subset of \( A \) is called a social choice function. The task of the algorithm is to check if for a given profile \( w \) the choice function described in the paper (with variants) is consistent. Consistency is defined as the following condition
where \( ( w', w'') \) is a concatenation of profiles leading to a new joint profile. In practice, the data contain a single profile \( ( w', w'') \) and the algorithm checks consistency condition (2) for all possible bi-partitions.
The main algorithm, Algorithm 1, is relatively simple. First, it reads the data that contain a single profile \( ( w', w'') \). Then, it computes the winner for the whole profile. Finally, the consistency condition is checked for all possible bi-partitions. There are two elements of the algorithm that require further explanation. One of these elements is the procedure used to calculate a winner for a given profile \( w \), that is an algorithm used for the social choice function \( f \). The other is a way to efficiently create all possible bi-partitions.
![figure a](http://media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-662-62245-2_2/MediaObjects/504179_1_En_2_Figa_HTML.png)
All bi-partitions are created using Algorithm 2. It takes a number of voters \( k\). There are exactly \( 2^{k- 1} - 1 \) bi-partitions. First, the algorithm creates a list of integers from \( 1 \) to \( 2^{k- 1} - 1 \). Then, each integer is written to the binary and converted into a logical vector, that is used as an index.
![figure b](http://media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-662-62245-2_2/MediaObjects/504179_1_En_2_Figb_HTML.png)
The last element of the main algorithm that requires an explanation is a procedure calculating a winner for a given profile \( w \). This procedure depends on a metric and a type of algorithm used to trim the set of voters. There are two types of metric considered: the standard Euclidean metric and the Manhattan metric. The trimming algorithm has two variants. In all variants 20% of voters are assigned \( 0 \) weight. In one variant of the algorithm, only the whole part of \( k/5 \) is used while in the other also the fraction part is removed. Thus, there are four variants of the procedure used to calculate a winner. Algorithm 3 is used for the procedure calculating a winner.
Algorithm 3 is straightforward, however, one step requires an explanation. The whole algorithm is based around distance groups. If distances between all voters and a mean are all different, then distance groups are just numbers \( 1, 2, \ldots , k\). However, if some distances between a mean and some voters are equal, then all voters with equal distances are assigned the same distance group. As an example consider the following distances \( 5, 5, 4, 4, 1 \). The distance groups for such a vector of distances are \( 1, 1, 2, 2, 3 \).
![figure c](http://media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-662-62245-2_2/MediaObjects/504179_1_En_2_Figc_HTML.png)
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Ramsza, M., Sosnowska, H. (2020). Trials of Characterizations of Anti-manipulation Method. In: Nguyen, N.T., Kowalczyk, R., Mercik, J., Motylska-Kuźma, A. (eds) Transactions on Computational Collective Intelligence XXXV. Lecture Notes in Computer Science(), vol 12330. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-62245-2_2
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