Generating functions for computing power indices e¢ciently

The Shapley-Shubik power index in a voting situation depends on the number of orderings in which each player is pivotal. The Banzhaf power index depends on the number of ways in which each voter can e¤ect a swing. We introduce a combinatorial method based in generating functions for computing these power indices e¢ciently and we study the time complexity of the algorithms. We also analyze the meet of two weighted voting games. Finally, we compute the voting power in the Council of Ministers of the European Union with the generating functions algorithms and we present its implementation in the system Mathematica. Mathematics Subject Classi...cation 2000: 91A12.