Optimal Candidate Positioning in Multi-Issue Elections

We study strategic candidate positioning in multidimensional spatial-voting elections. Voters and candidates are represented as points in Rd and each voter supports the candidate that is closest under a distance induced by an ℓp-norm. We prove that computing an optimal location for a new candidate is NP-hard already against a single opponent, whereas for a constant number of issues the problem is tractable: an O(nd + 1) hyperplane-enumeration algorithm and an O(n log n) radial-sweep routine for d = 2 solve the task exactly. We further derive the first approximation guarantees for the general multi-candidate case and show how our geometric approach extends seamlessly to positional scoring rules such as k-approval and Borda. These results clarify the algorithmic landscape of multi-dimensional spatial elections and provide practically implementable tools for campaign strategy.

Authors

• Colin Cleveland
• Bart de Keijzer
• Maria Polukarov

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