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.

翻译：液态民主结合排名委托是一种新颖的投票机制，融合了代议制民主的实用性与直接民主的理想主义诉求：每位选民既可选择直接对当前议题投票，也可将其投票权重委托给其信任的代理人。委托具有传递性，且由于选民可能形成委托循环，系统鼓励选民不仅指定单一代理人，而是提供一组潜在代理人及其排序偏好。基于所有选民的委托偏好，委托规则为每位选民选择一位代表。此前研究表明，委托规则的匿名性与抗拷贝性这两个属性之间存在权衡。为解决此问题，我们研究了两种分数委托规则：混合博达分支法（推广了一种满足抗拷贝性的规则）和随机游走规则（满足匿名性）。运用马尔可夫链树定理，我们证明这两种规则实质上等价，并同时满足这两类属性的广义版本。通过将该定理与Fulkerson算法相结合，我们开发了一种多项式时间算法来计算所研究委托规则的结果。该算法具有独立价值，在半监督学习与图论中均有应用前景。