Approval Ratios of Two-Intersecting Double-Interval Societies
Abstract
Consider an election in which the spectrum of political positions is $\mathbb{R}^1$, and each voter approves any platform in some closed interval. If any two voters agree on some platform, then Helly's theorem guarantees there must exist some platform on which they can all agree. I attempt to generalize this result, now assuming instead that each voter has an approval set that is a union of two closed intervals in $\mathbb{R}^1$. There need no longer exist a platform satisfactory to every voter, but I can bound below the proportion of voters who must agree on some position. I apply techniques from graph theory and combinatorics to find upper and lower bounds on the guaranteed approval ratio of the most popular political position. I have shown that the ratio must always be at least $2-\sqrt{3}$, but conjecture that it is always at least $1 / 3$, and that it tends toward this bound asymptotically.
