Measuring a Priori Voting Power -- Taking Delegations Seriously
This work addresses the problem of accurately assessing voter influence in liquid democracy systems for researchers and policymakers, though it is incremental as it builds on existing power indices.
The paper tackles the problem of measuring voting power in liquid democracy elections with delegation restrictions, introducing new power indices as extensions of the Penrose-Banzhaf index. It shows that computing criticality is #P-hard in general but provides pseudo-polynomial time algorithms for specific network types, with numerical results illustrating how delegation constraints affect power.
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.