  We study the Shapley value in weighted voting games. The Shapley value has
been used as an index for measuring the power of individual agents in
decision-making bodies and political organizations, where decisions are made by
a majority vote process. We characterize the impact of changing the quota
(i.e., the minimum number of seats in the parliament that are required to form
a coalition) on the Shapley values of the agents. Contrary to previous studies,
which assumed that the agent weights (corresponding to the size of a caucus or
a political party) are fixed, we analyze new domains in which the weights are
stochastically generated, modelling, for example, elections processes.
  We examine a natural weight generation process: the Balls and Bins model,
with uniform as well as exponentially decaying probabilities. We also analyze
weights that admit a super-increasing sequence, answering several open
questions pertaining to the Shapley values in such games.
