# Approval-Based Elections and Distortion of Voting Rules

## TL;DR

This paper analyzes the efficiency loss of approval-based voting rules in metric space elections, introducing acceptability-based distortion and evaluating multiple voting rules' performance in this context.

## Contribution

It extends the concept of distortion to approval preferences and computes acceptability-distortion for various voting rules, providing new insights into their effectiveness.

## Key findings

- Approval Voting distortion computed.
- Acceptability-distortion introduced and analyzed.
- Different rules' acceptability-distortion levels compared.

## Abstract

We consider elections where both voters and candidates can be associated with points in a metric space and voters prefer candidates that are closer to those that are farther away. It is often assumed that the optimal candidate is the one that minimizes the total distance to the voters. Yet, the voting rules often do not have access to the metric space $M$ and only see preference rankings induced by $M$.Consequently, they often are incapable of selecting the optimal candidate. The distortion of a voting rule measures the worst-case loss of the quality being the result of having access only to preference rankings. We extend the idea of distortion to approval-based preferences. First, we compute the distortion of Approval Voting. Second, we introduce the concept of acceptability-based distortion---the main idea behind is that the optimal candidate is the one that is acceptable to most voters. We determine acceptability-distortion for a number of rules, including Plurality, Borda, $k$-Approval, Veto, the Copeland's rule, Ranked Pairs, the Schulze's method, and STV.