Abstract
We consider the problem of aggregating votes cast by a society on a fixed set of issues, where each member of the society may vote for one of several positions on each issue, but the combination of votes on the various issues is restricted to a set of feasible voting patterns. We require the aggregation to be supportive, i.e., for every issue, the corresponding component of every aggregator, when applied to a tuple of votes, must take as value one of the votes in that tuple. We prove that, in such a set-up, non-dictatorial aggregation of votes in a society of an arbitrary size is possible if and only if a non-dictatorial binary aggregator exists or a non-dictatorial ternary aggregator exists such that, for each issue, the corresponding component of the aggregator, when restricted to two-element sets of votes, is a majority operation or a minority operation. We then introduce a notion of a uniform non-dictatorial aggregator, which is an aggregator such that on every issue, and when restricted to arbitrary two-element subsets of the votes for that issue, differs from all projection functions. We first give a characterization of sets of feasible voting patterns that admit a uniform non-dictatorial aggregator. After this and by making use of Bulatov’s dichotomy theorem for conservative constraint satisfaction problems, we connect social choice theory with the computational complexity of constraint satisfaction by proving that if a set of feasible voting patterns has a uniform non-dictatorial aggregator of some arity, then the multi-sorted conservative constraint satisfaction problem on that set (with each issue representing a different sort) is solvable in polynomial time; otherwise, it is \(\mathsf {NP}\)-complete.
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- 1.
This came to the attention of the authors only after the work reported here had been essentially completed.
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Acknowledgments
We are grateful to Mario Szegedy for sharing with us an early draft of his work on impossibility theorems and the algebraic toolkit. We are also grateful to Andrei Bulatov for bringing to our attention his “three basic operations” proposition [3, Proposition 3.1], [4, Proposition 2.2]. We sincerely thank the anonymous reviewers of RAMiCS 2017 for their very helpful comments.
Part of this research was carried out while Lefteris Kirousis was visiting the Computer Science Department of UC Santa Cruz during his sabbatical leave from the National and Kapodistrian University of Athens in 2015. Part of the research and the writing of this paper was done while Phokion G. Kolaitis was visiting the Simons Institute of Theory of Computing in the fall of 2016. Lefteris Kirousis’ participation to RAMICS 2017 was funded by the Special Account for Research Grants of the National and Kapodistrian University of Athens.
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Kirousis, L., Kolaitis, P.G., Livieratos, J. (2017). Aggregation of Votes with Multiple Positions on Each Issue. In: Höfner, P., Pous, D., Struth, G. (eds) Relational and Algebraic Methods in Computer Science. RAMICS 2017. Lecture Notes in Computer Science(), vol 10226. Springer, Cham. https://doi.org/10.1007/978-3-319-57418-9_13
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