Shapley Values in Weighted Voting Games with Random Weights
  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. 
Yuval Filmus, Joel Oren, Kannan Soundararajan
cs.GT