Anonymous and Copy-Robust Delegations for Liquid Democracy

TL;DR

This paper tackles the problem of distributing voting weight in liquid democracy with ranked delegations while addressing the trade-off between anonymity and copy-robustness. It introduces two fractional delegation rules, Mixed Borda Branching and the Random Walk Rule, and proves their equivalence using the Markov chain tree theorem, while also providing a polynomial-time algorithm built on Fulkerson’s packing framework. The authors extend core axioms to the fractional setting and show that the Random Walk Rule satisfies generalized versions of anonymity, confluence, and copy-robustness, with a non-fractional impossibility result highlighted. The work connects delegation mechanisms to directed power watershed concepts and semi-supervised learning, suggesting broad applicability in graph-based learning and efficient computation of fractional delegations for large-scale systems.

Abstract

Liquid democracy with ranked delegations is a novel voting scheme that unites the practicability of representative democracy with the idealistic appeal of direct democracy: Every voter decides between casting their vote on a question at hand or delegating their voting weight to some other, trusted agent. Delegations are transitive, and since voters may end up in a delegation cycle, they are encouraged to indicate not only a single delegate, but a set of potential delegates and a ranking among them. Based on the delegation preferences of all voters, a delegation rule selects one representative per voter. Previous work has revealed a trade-off between two properties of delegation rules called anonymity and copy-robustness. To overcome this issue we study two fractional delegation rules: Mixed Borda branching, which generalizes a rule satisfying copy-robustness, and the random walk rule, which satisfies anonymity. Using the Markov chain tree theorem, we show that the two rules are in fact equivalent, and simultaneously satisfy generalized versions of the two properties. Combining the same theorem with Fulkerson's algorithm, we develop a polynomial-time algorithm for computing the outcome of the studied delegation rule. This algorithm is of independent interest, having applications in semi-supervised learning and graph theory.

Figures (3)

• Figure 1: Example of a delegation graph. Delegating voters ($v_1$ and $v_2$) are indicated by circles and casting voters ($s_1$ and $s_2$) by squares. Solid edges represent the first-ranked delegations and dashed edges second-ranked delegations.
• Figure 2: Two iterations of \ref{['alg:dual']}. Costs are depicted by edge patterns (solid equals cost $1$, dashed equals cost $2$, and dotted equals cost $3$) and weights are depicted by numbers on the edges.
• Figure 3: Situation in the proof of \ref{['thm:impossibility']}. Solid edges correspond to first-choice delegations, dashed edges to second-choice delegations.

Theorems & Definitions (17)

• Lemma 2.1: Matrix tree theorem Tutt48aDeL20a
• Lemma 2.1
• Lemma 5.0
• Lemma 5.1: Markov chain tree theorem (LeRi86a)
• Theorem 5.2
• Theorem 6.1
• Theorem 7.1
• Theorem 7.2
• Theorem 7.3
• Theorem 7.4