Abstract
Many bodies around the world make their decisions through voting systems in which voters have several options and the collective result also has several options. Many of these voting systems are anonymous, i.e., all voters have an identical role in voting. Anonymous simple voting games, a binary vote for voters and a binary collective decision, can be represented by an easy weighted game, i.e., by means of a quota and an identical weight for the voters. Widely used voting systems of this type are the majority and the unanimity decision rules. In this article, we analyze the case in which voters have two or more voting options and the collective result of the vote has also two or more options. We prove that anonymity implies being representable through a weighted game if and only if the voting options for voters are binary. As a consequence of this result, several significant enumerations are obtained.
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Notes
The term index covers a broad spectrum of possibilities from physical components to persons or entities.
An apparent more general definition would allow weights and quotas to be real numbers. However, it could be proved that any such representation with real weights and quotas has an equivalent representation in non-negative integer weights and quotas, so there is no reason here to add unnecessary complexity by allowing weights and quotas to be real numbers.
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Acknowledgements
This research was partially supported by funds from the Spanish Ministry of Science and Innovation grant PID2019-I04987GB-I00. We are grateful to the associate editor and two anonymous referees whose interesting comments allowed us to improve the paper.
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Freixas, J., Pons, M. On anonymous and weighted voting systems. Theory Decis 91, 477–491 (2021). https://doi.org/10.1007/s11238-021-09814-3
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DOI: https://doi.org/10.1007/s11238-021-09814-3