# Shapley Values in Weighted Voting Games with Random Weights

arXiv: https://arxiv.org/abs/1601.06223 · HTML: https://mlsanity.com/paper/1601.06223/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jan 2016.

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

- [Measuring a Priori Voting Power -- Taking Delegations Seriously](https://mlsanity.com/paper/2301.02462/) (2023)

## Abstract (arXiv)

We investigate the distribution of the well-studied Shapley--Shubik values in weighted voting games where the agents are stochastically determined. The Shapley--Shubik value measures the voting power of an agent, in typical collective decision making systems. While easy to estimate empirically given the parameters of a weighted voting game, the Shapley values are notoriously hard to reason about analytically. We propose a probabilistic approach in which the agent weights are drawn i.i.d. from some known exponentially decaying distribution. We provide a general closed-form characterization of the highest and lowest expected Shapley values in such a game, as a function of the parameters of the underlying distribution. To do so, we give a novel reinterpretation of the stochastic process that generates the Shapley variables as a renewal process. We demonstrate the use of our results on the uniform and exponential distributions. Furthermore, we show the strength of our theoretical predictions on several synthetic datasets.