Towards the computation problems in multi-stage and multi-winner voting rules

Abstract

Multi-stage and multi-winner voting rules are playing an increasingly important role in society. The former consists of a large number of various procedures of multiple rounds based on repeated ballots and/or sequential elimination, such as Single Transferable Vote (STV), Baldwin rule, Coombs rule, Ranked Pairs (RP), etc., while the latter can select multiple candidates to win election to an office. For a long time, they have been well studied for their properties in different axioms, their manipulability, and the computational complexity of calculating a manipulation. As for the multi-stage voting rules, however, the literature is surprisingly vague about the impact of tiebreaking. Unlike previous default handling methods like alphabetical tiebreaking, our goal focuses on computing the set of all possible winners under any tiebreaking mechanism, henceforth known as parallel-universes tiebreaking (PUT). In this research, multiple algorithms are proposed based on depth-first search together with pruning strategies, heuristics, sampling, and machine learning which prioritizes search direction and substantially improves the performance. Also, novel integer linear programming (ILP) formulations are proposed for PUT-winners under STV and Ranked Pairs as a comparison, and the experiments' results show that the search algorithms are overall faster than ILP. Besides, much of this dissertation focuses on the margin of victory (MOV) problem. Margin of victory, concisely defined as the minimum number of ballots needed to be changed to alter the outcome, is another widely studied issue in voting theory. It is a crucial measurement for robustness of election outcome, which is very difficult and time-consuming to compute for multi-stage voting rules. To address this, this dissertation proposes an efficient algorithm for the exact MOV of STV computation, which has a significant time reduction compared to the current best-known Blom et al.'s approach. And this research first provides an efficient search algorithm for exact MOV of ranked pairs with the help of linear programming as a basic tool and machine learning as an accelerator technique. In addition, this dissertation also investigates multi-winner voting rules experimentally and exploratively. Contrary to the approximation algorithms discussed in most of the literature so far, this dissertation proposes the first search algorithms for solving the exact committee of winners for Chamberlain-Courant rule, where the machine learning aided prioritization proves to be valid again in early discovery. Also, this dissertation proposes an integer linear programming formulation which is the first for computing the exact MOV under the Chamberlain-Courant rule.