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arXiv · 2508.13841

Optimal Candidate Positioning in Multi-Issue Elections

Colin Cleveland·Bart de Keijzer·Maria Polukarov
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Abstract

We study strategic candidate positioning in multidimensional spatial-voting elections. Voters and candidates are represented as points in $\mathbb{R}^d$, and each voter supports the candidate that is closest under a distance induced by an $\ell_p$-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 $O(n^{d+1})$ 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 multidimensional spatial elections and provide practically implementable tools for campaign strategy.

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Colin Cleveland, Bart de Keijzer, Maria Polukarov. 2025-08-19. Optimal Candidate Positioning in Multi-Issue Elections. https://arxiv.org/abs/2508.13841

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