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Trials of Characterizations of Anti-manipulation Method

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Transactions on Computational Collective Intelligence XXXV

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. 1.

    In fact, as defined later, the jurors are assigned weights with the most distant jurors given zero weight.

  2. 2.

    The data in Tables 410 are taken from https://en.wikipedia.org/wiki/Consistency_criterion.

  3. 3.

    Authors verified calculations.

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Correspondence to Michał Ramsza .

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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

$$\begin{aligned} f( w') \cap f( w'') \ne \emptyset \; \Rightarrow f(w') \cap f(w'') = f( (w', w'')), \end{aligned}$$
(2)

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

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

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

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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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