# Comparing Election Methods Where Each Voter Ranks Only Few Candidates

## TL;DR

This paper investigates how well election rules like positional scoring and Minimax can be approximated using partial preferences collected through randomized or deterministic methods, providing theoretical bounds and simulation results.

## Contribution

It introduces bounds on approximation ratios for election rules based on partial preferences and compares randomized versus deterministic collection methods.

## Key findings

- Randomized approach generally yields better approximations.
- Borda rule approximations are effective with just two candidates per voter.
- Minimax rule requires larger candidate subsets for good approximation.

## Abstract

Election rules are formal processes that aggregate voters preferences, typically to select a single candidate, called the winner. Most of the election rules studied in the literature require the voters to rank the candidates from the most to the least preferred one. This method of eliciting preferences is impractical when the number of candidates to be ranked is large. We ask how well certain election rules (focusing on positional scoring rules and the Minimax rule) can be approximated from partial preferences collected through one of the following procedures: (i) randomized-we ask each voter to rank a random subset of $\ell$ candidates, and (ii) deterministic-we ask each voter to provide a ranking of her $\ell$ most preferred candidates (the $\ell$-truncated ballot). We establish theoretical bounds on the approximation ratios and we complement our theoretical analysis with computer simulations. We find that mostly (apart from the cases when the preferences have no or very little structure) it is better to use the randomized approach. While we obtain fairly good approximation guarantees for the Borda rule already for $\ell = 2$, for approximating the Minimax rule one needs to ask each voter to compare a larger set of candidates in order to obtain good guarantees.