Anonymous and Copy-Robust Delegations for Liquid Democracy

TL;DR

This paper introduces two fractional delegation rules for liquid democracy that overcome previous trade-offs between anonymity and copy-robustness, providing a polynomial-time algorithm for their computation.

Contribution

It demonstrates the equivalence of two fractional delegation rules and develops an efficient algorithm for their implementation, advancing the theoretical understanding of delegation properties.

Findings

The two rules are equivalent and satisfy generalized properties.

A polynomial-time algorithm for computing delegation outcomes is developed.

Applications extend to semi-supervised learning and graph theory.

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.