The Complexity of Power-Index Comparison

We study the complexity of the following problem: Given two weighted voting games G' and G'' that each contain a player p, in which of these games is p's power index value higher? We study this problem with respect to both the Shapley-Shubik power index [SS54] and the Banzhaf power index [Ban65,DS79]. Our main result is that for both of these power indices the problem is complete for probabilistic polynomial time (i.e., is PP-complete). We apply our results to partially resolve some recently proposed problems regarding the complexity of weighted voting games. We also study the complexity of the raw Shapley-Shubik power index. Deng and Papadimitriou [DP94] showed that the raw Shapley-Shubik power index is #P-metric-complete. We strengthen this by showing that the raw Shapley-Shubik power index is many-one complete for #P. And our strengthening cannot possibly be further improved to parsimonious completeness, since we observe that, in contrast with the raw Banzhaf power index, the raw Shapley-Shubik power index is not #P-parsimonious-complete.Comment: 12 page

Weighted Banzhaf power and interaction indexes through weighted approximations of games

The Banzhaf power index was introduced in cooperative game theory to measure the real power of players in a game. The Banzhaf interaction index was then proposed to measure the interaction degree inside coalitions of players. It was shown that the power and interaction indexes can be obtained as solutions of a standard least squares approximation problem for pseudo-Boolean functions. Considering certain weighted versions of this approximation problem, we define a class of weighted interaction indexes that generalize the Banzhaf interaction index. We show that these indexes define a subclass of the family of probabilistic interaction indexes and study their most important properties. Finally, we give an interpretation of the Banzhaf and Shapley interaction indexes as centers of mass of this subclass of interaction indexes

Influence in Classification via Cooperative Game Theory

A dataset has been classified by some unknown classifier into two types of points. What were the most important factors in determining the classification outcome? In this work, we employ an axiomatic approach in order to uniquely characterize an influence measure: a function that, given a set of classified points, outputs a value for each feature corresponding to its influence in determining the classification outcome. We show that our influence measure takes on an intuitive form when the unknown classifier is linear. Finally, we employ our influence measure in order to analyze the effects of user profiling on Google's online display advertising.Comment: accepted to IJCAI 201

The men who weren't even there: Legislative voting with absentees

Voting power in voting situations is measured by the probability of changing decisions by altering the cast 'yes' or 'no' votes. Recently this analysis has been extended by strategic abstention. Abstention, just as 'yes' or 'no' votes can change decisions. This theory is often applied to weighted voting situations, where voters can cast multiple votes. Measuring the power of a party in a national assembly seems to fit this model, but in fact its power comprises of votes of individual representatives each having a single vote. These representatives may vote yes or no, or may abstain, but in some cases they are not even there to vote. We look at absentees not due to a conscious decision, but due to illness, for instance. Formally voters will be absent, say, ill, with a certain probability and only present otherwise. As in general not all voters will be present, a thin majority may quickly melt away making a coalition that is winning in theory a losing one in practice. A simple model allows us to differentiate between winning and more winning and losing and less losing coalitions reected by a voting game that is not any more simple. We use data from Scotland, Hungary and a number of other countries both to illustrate the relation of theoretical and effective power and show our results working in the practice.a priori voting power; power index; being absent from voting; minority; Shapley-Shubik index; Shapley value

Measuring the interactions among variables of functions over the unit hypercube

By considering a least squares approximation of a given square integrable function $f\colon[0,1]^n\to\R$ by a multilinear polynomial of a specified degree, we define an index which measures the overall interaction among variables of $f$. This definition extends the concept of Banzhaf interaction index introduced in cooperative game theory. Our approach is partly inspired from multilinear regression analysis, where interactions among the independent variables are taken into consideration. We show that this interaction index has appealing properties which naturally generalize the properties of the Banzhaf interaction index. In particular, we interpret this index as an expected value of the difference quotients of $f$ or, under certain natural conditions on $f$, as an expected value of the derivatives of $f$. These interpretations show a strong analogy between the introduced interaction index and the overall importance index defined by Grabisch and Labreuche [7]. Finally, we discuss a few applications of the interaction index

BARGAINING, VOTING, AND VALUE

This paper addresses the following issue: If a set of agents bargain on a set of feasible alternatives 'in the shadow' of a voting rule, that is, any agreement can be enforced if a 'winning coalition' supports it, what general agreements are likely to arise? In other words: What influence can the voting rule used to settle (possibly non-unanimous) agreements have on the outcome of negotiations? To give an answer we model the situation as an extension of the Nash bargaining problem in which an arbitrary voting rule replaces unanimity to settle agreements by n players. This provides a setting in which a natural extension of Nash's solution is obtained axiomatically. Two extensions admitting randomization on voting rules based on two informational scenarios are considered.Bargaining, voting, value, bargaining in committees.

Computation of power indices

Power indices are a useful tool for studying voting systems in which different members have different numbers of votes. Many international organisations are organised in this way in order to accommodate differences in size of countries, including the system of QMV in the European Union Council of Ministers. A power index is a measure of power based on the idea that a member’s power is his ability to swing a decision by changing the way his vote is cast. This paper addresses the problem of the computation of the two most widely used power indices, the so-called classical power indices, the Banzhaf index (and also the related Coleman indices) and the Shapley-Shubik index. It discusses the various methods that have been proposed in the literature: Direct Enumeration, Monte Carlo Simulation, Generating Functions, Multilinear Extensions Approximation, the Modified MLE Approximation Method. The advantages and disadvantages of the algorithms are discussed including computational complexity. It also describes methods for so called “oceanic games”. The paper also discusses the so-called “inverse problem” of finding what the weights should be given the desired power indices. The method is potentially useful as providing a basis for designing a voting system with a given desired distribution of power among the members, for example, to reflect differences in population or financial contributions. Examples are given from the International Monetary Fund, shareholder voting and the European Union Council

Computation of Power Indices

Lecture Notes prepared for Summer School, “EU Decision Making : Assessment and Design of Procedures”, San Sebastian, Spain, July 8-11, 2002.

Computational implementation of indices of power

In this paper we present algorithms and computational implementation of the Shapley Value and Banzhaf– Coleman Index of Power. Both indices describe the real power of the coalitions involved in strategic interactions. The system allows the study of complex Electoral Applications. The data input can be done in two different ways: by considering all the possible coalitions, or only the basic coalitions (political parties, sectorial groups, etc.). The system also allows to introduce restrictions (incompatibilities) among some coalitions. We present some applications for computing the Electoral Power in the election of authorities in Universidad Nacional de San Luis. We describe the client–server design and the implementation of these tools, using the languages C and Tcl/Tk. The server program is written in C and requires Linux. The Client Program is written in Tcl/Tk with namespace mechanism and it supports Linux and Windows.Área: Informática Teórica - Inteligencia Artificial - Lenguajes - CompiladoresRed de Universidades con Carreras en Informática (RedUNCI

Mutual information-based group explainers with coalition structure for machine learning model explanations

In this article, we propose and investigate ML group explainers in a general game-theoretic setting with the focus on coalitional game values and games based on the conditional and marginal expectation of an ML model. The conditional game takes into account the joint distribution of the predictors, while the marginal game depends on the structure of the model. The objective of the article is to unify the two points of view under predictor dependencies and to reduce the complexity of group explanations. To achieve this, we propose a feature grouping technique that employs an information-theoretic measure of dependence and design appropriate groups explainers. Furthermore, in the context of coalitional game values with a two-step formulation, we introduce a theoretical scheme that generates recursive coalitional game values under a partition tree structure and investigate the properties of the corresponding group explainers.Comment: 46 pages, 69 figure