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The Condorcet Dimension of Metric Spaces
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abstract
A Condorcet winning set is a set of candidates such that no other candidate is preferred by at least half the voters over all members of the set. The Condorcet dimension, which is the minimum cardinality of a Condorcet winning set, is known to be at most logarithmic in the number of candidates. We study the case of elections where voters and candidates are located in a $2$-dimensional space with preferences based upon proximity voting. Our main result is that the Condorcet dimension is at most $4$, under both the Manhattan norm and the infinity norm, which are natural measures in electoral systems. We also prove that any set of voter preferences can be embedded into a metric space of sufficiently high dimension for any $p$-norm, including the Manhattan and infinity norms.
Forward citations
Cited by 2 Pith papers
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Bi-Criteria Metric Distortion
In line metrics, a constant-size committee can achieve the same cost as the optimal single winner, bypassing the factor-3 barrier that applies to any single winner.
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