Geometric construction of voting methods that protect voters' first choices

TL;DR

This paper explores the geometric design of voting methods that incentivize voters to honestly rank their favorite candidate first, analyzing their properties and limitations through a geometric framework.

Contribution

It classifies voting methods satisfying the Strong Favorite Betrayal Criterion into four geometric categories and analyzes their restrictions and potential.

Findings

Two categories closely relate to positional point systems.

Most SFBC methods only weakly distinguish between first and second choices.

Relaxing conditions allows for more meaningful voting methods.

Abstract

We consider the possibility of designing an election method that eliminates the incentives for a voter to rank any other candidate equal to or ahead of his or her sincere favorite. We refer to these methods as satisfying the ``Strong Favorite Betrayal Criterion" (SFBC). Methods satisfying our strategic criteria can be classified into four categories, according to their geometrical properties. We prove that two categories of methods are highly restricted and closely related to positional methods (point systems) that give equal points to a voter's first and second choices. The third category is tightly restricted, but if criteria are relaxed slightly a variety of interesting methods can be identified. Finally, we show that methods in the fourth category are largely irrelevant to public elections. Interestingly, most of these methods for satisfying the SFBC do so only ``weakly," in that these methods make no meaningful distinction between the first and second place on the ballot. However, when we relax our conditions and allow (but do not require) equal rankings for first place, a wider range of voting methods are possible, and these methods do indeed make meaningful distinctions between first and second place.