The Mathematics and Statistics of Voting Power
In an election, voting power—the probability that a single vote is 
decisive—is affected by the rule for aggregating votes into a single outcome. 
Voting power is important for studying political representation, fairness and 
strategy, and has been much discussed in political science. Although power 
indexes are often considered as mathematical definitions, they ultimately 
depend on statistical models of voting. Mathematical calculations of voting 
power usually have been performed under the model that votes are decided 
by coin flips. This simple model has interesting implications for weighted 
elections, two-stage elections (such as the U.S. Electoral College) and 
coalition structures. We discuss empirical failings of the coin-flip model of 
voting and consider, first, the implications for voting power and, second, 
ways in which votes could be modeled more realistically. Under the random 
voting model, the standard deviation of the average of n votes is proportional 
to 1/√n, but under more general models, this variance can have the form cn^(−α) or √a−b log n. Voting power calculations undermore realistic models 
present research challenges in modeling and computation.