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The Nonmanipulative Vote-Deficits of Voting Rules

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Algorithmic Decision Theory (ADT 2021)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 13023))

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Abstract

We introduce a new parameter which we call the nonmanipulative vote-deficit (NMVD) for single-winner voting rules. In particular, the NMVD of a voting rule at an election is the minimum number of votes needed to be added to transform this election into a nonmanipulable one yet without changing the winner. A voting rule has a bounded NMVD if the NMVDs of this rule at all elections are bounded from above by a constant. We show that the prevalent voting rules Borda, Plurality with Runoff, and Maximin have bounded NMVDs. In addition, we show that the NMVD of r-Approval, r-Veto, and Bucklin at every election can be bounded by a function of the number of candidates. For Copeland\(^{\alpha }\), though that in general the NMVDs at elections cannot be bounded by a function of the number of candidates, we show that many special elections are still expected to have small NMVDs. Many of our results are tight.

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Notes

  1. 1.

    Based on a polynomial-time algorithm for a manipulation problem (CCM with exactly one manipulator) presented in [4, 18], \(\mathsf{{C}}_{\varphi }(E)\) can be calculated in polynomial time for all \(\varphi \) considered in this paper.

  2. 2.

    A Condorcet winner is a candidate which beats all the other candidates. A voting rule is Condorcet-consistent if it selects the Condorcet winner as the winner whenever the Condorcet winner exists. Condorcet-consistency is a significant axiomatic property of voting rules (see [22, Figure 9.3] or [21, Table 2] for voting rules and their axiomatic properties).

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Acknowledgements

The author would like to thank the anonymous reviewers of ADT-2021 for their constructive comments.

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Yang, Y. (2021). The Nonmanipulative Vote-Deficits of Voting Rules. In: Fotakis, D., Ríos Insua, D. (eds) Algorithmic Decision Theory. ADT 2021. Lecture Notes in Computer Science(), vol 13023. Springer, Cham. https://doi.org/10.1007/978-3-030-87756-9_15

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  • DOI: https://doi.org/10.1007/978-3-030-87756-9_15

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