We study the distribution of Shapley values in weighted voting games. The Shapley values measure the voting power collective decision making systems. While easy to estimate empirically given the parameters of a weighted voting game, the Shapley values are 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.