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

TL;DR

This work addresses the challenge of neutral, tie-aware voting by focusing on River with Parallel Universe Tiebreaking (PUT). It introduces a polynomial-time framework to decide River PUT winners using a semi-River diagram, a recursively strongest path tree, and a specialized edge-ordering strategy derived from selected edges. The Constructive RV-PUT Check combines these components to determine, in polynomial time, whether a given alternative can be a River PUT winner, and thus to compute the full set of River PUT winners. Compared to Ranked Pairs with PUT (NP-hard), this approach substantially improves computational tractability while preserving key neutrality and immunity properties. The findings have implications for scalable voting in political contexts and applications like AI training, where robust, transparent tie-handling is valuable.

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.

Figures (6)

• Figure 1: A real-world example where the Plurality voting result might have been against the preferences of a majority of voters, according to an opinion article by the economics Nobel laureates maskinOpinionHowMajority2016a. It also illustrates the difference between plurality voting (left) and ranked voting (right).
• Figure 2: Two examples of elections. One with an Condorcet winner (left) and one to illustrate the Condorcet paradox (right)
• Figure 3: The edges $(x,y) \in \mathcal{M}^\textsc{sRV}\xspace\xspace$ with margin $\alpha$ and $(z,y) \in \mathcal{M}^\textsc{sRV}\xspace\xspace$ with margin $\beta$. Note that $\alpha < \beta$. There is the path from $y$ to $z$ in $\mathcal{M}^\textsc{sRV}\xspace$ with strength $\gamma$ and $\beta \geq \gamma$.
• Figure 4: An illustration of $\mathcal{M}^\textsc{sRV}\xspace$ for the proof of \ref{['pps:AlgoRSPT']}. We see that, since $u\in S$ and $v \in V \setminus S \xspace\xspace$, there are crossing edges on the path ${P}_{u, v}$ from $u$ to $v$ with strength $\alpha$ which is $T^{RSP}$T^RSP and the supposedly stronger path ${P}_{u, v}\xspace'$ in $G$ with strength $b$. The edge $e_{\min}$ on ${P}_{u, v}$ has weight $\gamma$ with $\gamma = \alpha$ and the crossing edge on ${P}_{u, v}\xspace'$ has weight $\delta$ with $\delta \geq \beta$. The assumption in the proof is that $\alpha < \beta$ and thus $\gamma<\delta$.
• Figure 5: An illustration of $\mathcal{M}^\textsc{sRV}\xspace$ for Case 1. The path from $w\xspace\xspace$ to $y$ in $T^{RSP}$T^RSP goes through $e=(x,y)$. The edge $e$ has margin $\alpha$. There is the other incoming edge $e_{\geq \text{in}} =(z,y)$ with margin $\beta$. Note that $\alpha < \beta$. Because both edges are in the semi-River diagram, there exists a path ${P}_{y, z}\xspace$ form $y$ to $z$ with strength $\gamma \geq \beta$ in the semi-River diagram.

Theorems & Definitions (46)

• Definition 1
• Corollary 1
• Corollary 2
• Corollary 3
• Definition 2
• Corollary 4
• Lemma 1: Fast River correctness
• proof
• Lemma 2: Fast River runtime
• proof