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：具有排序委托的流动民主是一种新颖的投票方案，它将代议制民主的实用性与直接民主的理想主义诉求相结合：每位选民可以选择对当前议题进行投票，或将投票权重委托给其他可信代理人。委托具有传递性，且由于选民可能陷入委托循环，鼓励选民不仅指定单个委托人，而是提供一组潜在委托人及其排序。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。以往研究表明，委托规则的两个属性——匿名性与抗复制性——存在权衡关系。为解决这一问题，我们研究了两种分数委托规则：混合博尔达分支法（推广了满足抗复制性的规则）与随机游走规则（满足匿名性）。通过马尔可夫链树定理，我们证明这两个规则事实上等价，并且同时满足这两个属性的广义版本。结合该定理与Fulkerson算法，我们开发了一种多项式时间算法，用于计算所研究委托规则的输出结果。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。