Conquering the Multiverse: The River Voting Method with Efficient Parallel Universe Tiebreaking

TL;DR

This paper introduces an efficient polynomial-time algorithm for the River voting method with Parallel Universe Tiebreaking, ensuring neutrality and computational tractability in elections with ties, and improves its runtime significantly.

Contribution

It presents a polynomial-time algorithm for River with PUT, introduces the semi-River diagram for tiebreaking, and optimizes the algorithm's runtime from O(n^4) to O(n^2 log n).

Findings

River with PUT can be computed in polynomial time.

The semi-River diagram helps in constructing tiebreaks that preserve neutrality.

The algorithm's runtime is improved from O(n^4) to O(n^2 log n).

Abstract

Democracy relies on making collective decisions through voting. In addition, voting procedures have further applications, for example in the training of artificial intelligence. An essential criterion for determining the winner of a fair election is that all alternatives are treated equally: this is called neutrality. The established Ranked Pairs voting method cannot simultaneously guarantee neutrality and be computationally tractable for election with ties. River, the recently introduced voting method, shares desirable properties with Ranked Pairs and has further advantages, such as a new property related to resistance against manipulation. Both Ranked Pairs and River use a weighted margin graph to model the election. Ties in the election can lead to edges of equal margin. To order the edges in such a case, a tiebreaking scheme must be employed. Many tiebreaks violate neutrality or other important properties. A tiebreaking scheme that preserves neutrality is Parallel Universe Tiebreaking (PUT). Ranked Pairs with PUT is NP-hard to compute. The main result of this thesis shows that River with PUT can be computed in polynomial worst-case runtime: We can check whether an alternative is a River PUT winner, by running River with a specially constructed ordering of the edges. To construct this ordering, we introduce the semi-River diagram which contains the edges that can appear in any River diagram for some arbitrary tiebreak. On this diagram we can compute the River winners, by applying a variant of Prims algorithm per alternative. Additionally, we give an algorithm improve the previous naive runtime of River from $\mathcal{O}(n^4)$ to $\mathcal{O}(n^2 \log n)$, where n is the number of alternatives.