We introduce new power indices to measure the a priori voting power of voters in liquid democracy elections where an underlying network restricts delegations. We argue that our power indices are natural extensions of the standard Penrose-Banzhaf index in simple voting games. We show that computing the criticality of a voter is #P-hard even when voting weights are polynomially-bounded in the size of the instance. However, for specific settings, such as when the underlying network is a bipartite or complete graph, recursive formulas can compute these indices for weighted voting games in pseudo-polynomial time. We highlight their theoretical properties and provide numerical results to illustrate how restricting the possible delegations can alter voters' voting power.

翻译：我们引入新的权力指数，用于衡量流动性民主选举中选民在底层网络限制委托情况下的先验投票权。我们论证这些权力指数是简单投票博弈中标准彭罗斯-班扎夫指数的自然扩展。研究表明，即使投票权重在实例规模上呈多项式有界，计算选民的临界性仍为#P难问题。然而，在特定场景下（例如当底层网络为二分图或完全图时），递归公式可在伪多项式时间内计算加权投票博弈中的这些指数。我们重点阐述了其理论性质，并通过数值结果说明限制委托可能性如何改变选民的投票权。