Optimal Candidate Positioning in Multi-Issue Elections

We investigate voting scenarios (elections) in a multidimensional issue space, represented as R, where voters’ preferences are modelled by points in this space. Voters cast their vote for the candidate whose position is closest to their preferred point, according to a distance measure induced by a norm on R. Candidates strategically position themselves within this space to maximise an objective function, typically aiming to maximise the number of votes received or their ranking among the candidates. We demonstrate that, for an arbitrary number of issues (the dimensionality of the issue space), determining the optimal position for a candidate is NP-hard even when competing against only one candidate. However, when the number of issues is bounded, we prove that this problem becomes polynomial-time solvable for several of its variants. Furthermore, we provide initial approximation algorithms for both single-candidate and multi-candidate versions of our problem, and analyse the applicability of our findings to other electoral models. We find that our model results extend naturally to general scoring systems, including k-approval and Borda voting.