Computing power indices for weighted voting games via dynamic programming

Abstract

We study the efficient computation of power indices for weighted voting games using the paradigm of dynamic programming. We survey the state-of-the-art algorithms for computing the Banzhaf and Shapley-Shubik indices and point out how these approaches carry over to related power indices. Within a unified framework, we present new efficient algorithms for the Public Good index and a recently proposed power index based on minimal winning coalitions of smallest size, as well as a very first method for computing Johnston indices for weighted voting games efficiently. We introduce a software package providing fast C++ implementations of all the power indices mentioned in this article, discuss computing times, as well as storage requirements.

Dynamic Programming for Computing Power Indices for Weighted Voting Games with Precoalitions

“…In this section we introduce the technique of dynamic programming for counting coalitions in weighted voting games efficiently, discuss the state-of-the-art algorithms for the Banzhaf and Shapley-Shubik indices [7,27,28] and present generalizations of existing algorithms that will come out to be useful when we deal with precoalitions and internal games within precoalitions in Section 4. In other words: We are not computing any power indices with precoalitions in this section, but lay the groundwork for doing so in Section 4.…”

“…We will use dynamic programming to count losing and winning coalitions and present algorithms following the work by Kurz [7]. For a more general introduction to counting coalitions via dynamic programming (including examples) we refer to the recent article [28] or the textbook by Chakravarty, Mitra and Sarkar [35], chapter 12.…”

“…for x from q + w(i) to w do 5: return T number of winning coalitions of weight x ∈ [q, w] 10: end procedure Algorithms 1 and 2 can be generalized for the distribution of cardinalities of coalitions in a straightforward manner [7,28]. Given that we normally assume q > w 2 (and hence w − q + 1 ≤ q), we prefer to work with the winning coalitions from above.…”

“…If the subsets of players reach only very few different weight sums, this method is favoured [7] and fast-access data structures for polynomials with few coefficients in computer algebra systems like Mathematica [25] can be used. In this paper we use the strongly related, though mathematically less sophisticated, paradigm of dynamic programming for power index computation [26][27][28]. The recent article [28] presents a new algorithm for the Johnston index [29], discusses algorithms for power indices based on minimal winning coalitions [30][31][32], surveys the state-of-the-art and introduces publicly available software.…”

“…In this paper we use the strongly related, though mathematically less sophisticated, paradigm of dynamic programming for power index computation [26][27][28]. The recent article [28] presents a new algorithm for the Johnston index [29], discusses algorithms for power indices based on minimal winning coalitions [30][31][32], surveys the state-of-the-art and introduces publicly available software. The work by Molinero and Blasco [33] studies algorithms for the Banzhaf-Owen and Owen indices and, to our knowledge, it is the only paper on dynamic programming algorithms for weighted voting games with precoalitions.…”