Measuring a Priori Voting Power -- Taking Delegations Seriously

TL;DR

This paper develops new power indices for liquid democracy, accounting for delegation networks, and analyzes their computational complexity and properties through theoretical and numerical methods.

Contribution

It introduces novel power indices extending the Penrose-Banzhaf index to liquid democracy with delegation networks, and explores their computational aspects and implications.

Findings

Computing voter criticality is #P-hard in general.

Recursive formulas enable pseudo-polynomial time computation in specific network structures.

Delegation restrictions significantly impact voters' voting power.

Abstract

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.