Generating functions for computing power indices efficiently

TheShapley-Shubik power index in a voting situation depends on the number of orderings in which each player is pivotal. TheBanzhaf power index depends on the number of ways in which each voter can effect a swing. We introduce a combinatorial method based ingenerating functions for computing these power indices efficiently and we study thetime 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. This work has been partially supported by the Spanish Ministery of Science and Technology under grant SEC2000-1243.