# Measuring a Priori Voting Power -- Taking Delegations Seriously

arXiv: https://arxiv.org/abs/2301.02462 · HTML: https://mlsanity.com/paper/2301.02462/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 59 references that are in the map.
- First posted on arXiv 6 Jan 2023.

## Builds on (most cited first, 14 of 14)

- [On Rational Delegations in Liquid Democracy](https://mlsanity.com/paper/1802.08020/) (2018)
- [Binary Voting with Delegable Proxy: An Analysis of Liquid Democracy](https://mlsanity.com/paper/1707.08741/) (2017)
- [Power in Liquid Democracy](https://mlsanity.com/paper/2010.07070/) (2020)
- [Incentivising Participation in Liquid Democracy with Breadth-First Delegation](https://mlsanity.com/paper/1811.03710/) (2018)
- [Iterative Delegations in Liquid Democracy with Restricted Preferences](https://mlsanity.com/paper/1809.04362/) (2018)
- [Liquid Democracy with Ranked Delegations](https://mlsanity.com/paper/2112.07509/) (2021)
- [Proxy Voting for Better Outcomes](https://mlsanity.com/paper/1611.08308/) (2016)
- [Flexible Representative Democracy: An Introduction with Binary Issues](https://mlsanity.com/paper/1811.02921/) (2018)
- [Unravelling multi-agent ranked delegations](https://mlsanity.com/paper/2111.13145/) (2021)
- [Measuring voting power in convex policy spaces](https://mlsanity.com/paper/1312.5936/) (2013)
- [Shapley Values in Weighted Voting Games with Random Weights](https://mlsanity.com/paper/1601.06223/) (2016)
- [Average Weights and Power in Weighted Voting Games](https://mlsanity.com/paper/1905.04261/) (2019)
- [Pirates in Wonderland: Liquid Democracy has Bicriteria Guarantees](https://mlsanity.com/paper/2105.05142/) (2021)
- [Tracking Truth with Liquid Democracy](https://mlsanity.com/paper/2107.11868/) (2021)

## Built on by (most cited first, 1 of 1)

- [As Time Goes By: Adding a Temporal Dimension Towards Resolving Delegations in Liquid Democracy](https://mlsanity.com/paper/2307.12898/) (2023)

## Abstract (arXiv)

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.