An approach via generating functions to compute power indices of multiple weighted voting games with incompatible players

Abstract

We introduce a new generating function based method to compute the Banzhaf, Deegan–Packel, Public Good (a.k.a. the Holler power index) and Shapley–Shubik power indices in the presence of incompatibility among players. More precisely, given a graph $G=\left( V,E\right)$ with V the set of players and E the edge set, our extension involves multiple weighted voting games (MWVG’s) and incompatible players, i.e., pairs of players belonging to E are not allowed to cooperate. The route to obtain the aforementioned generating functions comprises the use of a key lemma characterizing the set of minimal winning coalitions of the game with incompatibility due to Alonso-Meijide et al. (Appl Math Comput 252(1):377–387, 2015), a tool from combinatorial analysis, namely, the Omega calculus in partition analysis, and basic tools borrowed from commutative algebra involving the computation of certain quotients of polynomial rings module polynomial ideals. Using partition analysis, we obtain new generating functions to compute the Deegan–Packel and Public Good power indices with incompatibility leading to lower time complexity than previous results of Chessa (TOP 22(2):658–673, 2014) and some results of Alonso-Meijide et al. (Appl Math Comput 219(8):3395–3402, 2012). Using a conjunction of partition analysis and commutative algebra, we extend to MWVG’s the generating function approach to compute the Banzhaf and Shapley–Shubik power indices in the presence of incompatibility. Finally, an example taken from the real-world, i.e., the European Union under the Lisbon Treaty, is used to illustrate the usefulness of the Omega package, a symbolic computational package that implements the Omega calculus in Mathematica, due to Andrews et al. (Eur J Comb 22(7):887–904, 2001) in the context of MWVG’s by computing the PG power index of the associated voting game.