Piercing Numbers in Approval Voting

TL;DR

This paper surveys how discrete geometry results, especially piercing numbers, inform the analysis of approval voting models where voter preferences are represented geometrically.

Contribution

It provides a comprehensive overview of geometric results relevant to approval voting, connecting discrete geometry with voting theory.

Findings

Piercing numbers relate to voter approval set intersections.

Geometric constraints influence approval voting outcomes.

Survey of implications for various voter preference models.

Abstract

We survey a host of results from discrete geometry that have bearing on the analysis of geometric models of approval voting. Such models view the political spectrum as a geometric space, with geometric constraints on voter preferences. Results on piercing numbers then have a natural interpretation in voting theory, and we survey their implications for various classes of geometric constraints on voter approval sets.