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The River Method

Michelle Döring, Markus Brill, Jobst Heitzig

TL;DR

River is a Condorcet-consistent refinement of Split Cycle that constructs a tree-shaped margin diagram by adding edges in decreasing margin while forbidding cycles and branching, yielding a unique winner with easily interpretable rebutting paths. It achieves independence from Pareto-dominated alternatives (IPDA) and independence from Smith-dominated alternatives (ISDA), and, with impartial or Pareto-consistent tiebreakers, independence of clones (IoC) and quasi-Pareto independence (IQDA). The method combines simplicity (hand-calcable procedure and tree diagram) with strong axiomatic guarantees, addressing agenda-manipulation concerns that affect related methods like RP, BP, and SV. Computationally, River-PUT is polynomial-time, making River competitive for practical use and robust against extensive tie-breaking scenarios, with supplementary material illustrating real-world-scale examples.

Abstract

We introduce River, a novel Condorcet-consistent voting method that is based on pairwise majority margins and can be seen as a simplified variation of Tideman's Ranked Pairs method. River is simple to explain, simple to compute even 'by hand', and gives rise to an easy-to-interpret certificate in the form of a directed tree. Like Ranked Pairs and Schulze's Beat Path method, River is a refinement of the Split Cycle method and shares with those many desirable properties, including independence of clones. Unlike the other three methods, River satisfies a strong form of resistance to agenda-manipulation that is known as independence of Pareto-dominated alternatives.

The River Method

TL;DR

River is a Condorcet-consistent refinement of Split Cycle that constructs a tree-shaped margin diagram by adding edges in decreasing margin while forbidding cycles and branching, yielding a unique winner with easily interpretable rebutting paths. It achieves independence from Pareto-dominated alternatives (IPDA) and independence from Smith-dominated alternatives (ISDA), and, with impartial or Pareto-consistent tiebreakers, independence of clones (IoC) and quasi-Pareto independence (IQDA). The method combines simplicity (hand-calcable procedure and tree diagram) with strong axiomatic guarantees, addressing agenda-manipulation concerns that affect related methods like RP, BP, and SV. Computationally, River-PUT is polynomial-time, making River competitive for practical use and robust against extensive tie-breaking scenarios, with supplementary material illustrating real-world-scale examples.

Abstract

We introduce River, a novel Condorcet-consistent voting method that is based on pairwise majority margins and can be seen as a simplified variation of Tideman's Ranked Pairs method. River is simple to explain, simple to compute even 'by hand', and gives rise to an easy-to-interpret certificate in the form of a directed tree. Like Ranked Pairs and Schulze's Beat Path method, River is a refinement of the Split Cycle method and shares with those many desirable properties, including independence of clones. Unlike the other three methods, River satisfies a strong form of resistance to agenda-manipulation that is known as independence of Pareto-dominated alternatives.

Paper Structure

This paper contains 29 sections, 19 theorems, 9 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Preliminaries
  4. Immunity against Majority Complaints
  5. Social Choice Functions
  6. Ranked Pairs ($RP$)
  7. Beat Path ($BP$)
  8. Stable Voting ($SV$)
  9. Tiebreaking and Uniquely Weighted Profiles
  10. Parallel Universe Tiebreaking
  11. The River Method
  12. Axiomatic Properties
  13. Independence of Smith-Dominated Alternatives
  14. Independence of Pareto-Dominated Alternatives
  15. Independence of Clones
  16. ...and 14 more sections

Key Result

Proposition 3.2

River is a refinement of Split Cycle, i. e., $x\in RV\xspace(\mathbf{P}\xspace)$ implies $x\in SC\xspace(\mathbf{P}\xspace)$.

Figures (8)

  • Figure 1: An election with 6 alternatives showing the behaviour of the different social choice functions. From left to right are the margin graph $\mathcal{M}\xspace$, the Split Cycle graph $\mathcal{M}\xspace^{SC\xspace}$ with winning set $\{a,b,c\}$, the Stable Voting winner $a$ with the deciding election without $e$, the Ranked Pairs graph $\mathcal{M}\xspace^{RP\xspace}$ with winner $c$, and a table of all beatpaths with Beat Path winner $b$.
  • Figure 2: The River diagram $\mathcal{M}\xspace^{RV}$ for the margin graph of \ref{['fig:example_SC_functions']} with $RV\xspace(\mathbf{P}\xspace)=\{a\}$. On the right, it is arranged as a tree with the root $a$ at the top.
  • Figure 3: Margin graphs for the preference profiles used in the proof of \ref{['thm:IPDA_otherSC']}.
  • Figure 4: Illustration of state $(2)$ for clone set $D$ with $D[1]=d_1$.
  • Figure 5: Large real-life example with 14 alternatives. From left to right are the margin graph, the Ranked Pairs diagram and the River diagram. Margin-zero edges are drawn in gray and all other margins are given on the edges. Lines dashed or dotted of the same kind illustrate that this set of edges would have to be decided by a tiebreaker and that only one of those edges can be in the diagram.
  • ...and 3 more figures

Theorems & Definitions (55)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 45 more