# Practical Algorithms for Multi-Stage Voting Rules with Parallel Universes Tiebreaking

## TL;DR

This paper introduces new algorithms and ILP formulations for computing winners in multi-stage voting rules like STV and RP under parallel universes tiebreaking, addressing computational complexity issues.

## Contribution

It provides the first algorithms for PUT-winners under STV and RP, including DFS-based methods with heuristics and novel ILP models, improving computational efficiency.

## Key findings

- Algorithms outperform ILP in speed
- Heuristics and sampling improve search efficiency
- Experimental results validate practical applicability

## Abstract

STV and ranked pairs (RP) are two well-studied voting rules for group decision-making. They proceed in multiple rounds, and are affected by how ties are broken in each round. However, the literature is surprisingly vague about how ties should be broken. We propose the first algorithms for computing the set of alternatives that are winners under some tiebreaking mechanism under STV and RP, which is also known as parallel-universes tiebreaking (PUT). Unfortunately, PUT-winners are NP-complete to compute under STV and RP, and standard search algorithms from AI do not apply. We propose multiple DFS-based algorithms along with pruning strategies, heuristics, sampling and machine learning to prioritize search direction to significantly improve the performance. We also propose novel ILP formulations for PUT-winners under STV and RP, respectively. Experiments on synthetic and real-world data show that our algorithms are overall faster than ILP.