Related papers: Influence and interaction indexes for pseudo-Boole…
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…
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 $f$.…
We introduce an index for measuring the influence of the k-th smallest variable on a pseudo-Boolean function. This index is defined from a weighted least squares approximation of the function by linear combinations of order statistic…
The Lovasz extension of a pseudo-Boolean function $f : \{0,1\}^n \to R$ is defined on each simplex of the standard triangulation of $[0,1]^n$ as the unique affine function $\hat f : [0,1]^n \to R$ that interpolates $f$ at the $n+1$ vertices…
In many applications of cooperative game theory -- from corporate governance and cartel formation to parliamentary voting -- not all winning coalitions are feasible. Ideological distances, institutional constraints, or pre-electoral…
Weighted voting games are frequently used in decision making. Each voter has a weight and a proposal is accepted if the weight sum of the supporting voters exceeds a quota. One line of research is the efficient computation of so-called…
Weighted voting games are ubiquitous mathematical models which are used in economics, political science, neuroscience, threshold logic, reliability theory and distributed systems. They model situations where agents with variable voting…
Weighted voting games are a family of cooperative games, typically used to model voting situations where a number of agents (players) vote against or for a proposal. In such games, a proposal is accepted if an appropriately weighted sum of…
In this paper, we propose an improved version of the power index related to the Banzhaf power index for weighted voting systems. This index now takes into account the mutual persuasion power matrix(PPM) existing among the voters. This…
Feature attribution methods have become essential for explaining machine learning models. Many popular approaches, such as SHAP and Banzhaf values, are grounded in power indices from cooperative game theory, which measure the contribution…
This paper deals with a new measure of the influence of each feature on the response variable in classification problems, accounting for potential dependencies among certain feature subsets. Within this framework, we consider a sample of…
We explore the switching-algebraic computation of the Banzhaf indices for general and monotone or unrestricted systems. This computation is achieved via (a) two Boolean-quotient formulas that are valid when the voting system is not…
This paper provides a serious attempt towards constructing a switching-algebraic theory for weighted monotone voting systems, whether they are scalar-weighted or vector-weighted. The paper concentrates on the computation of a prominent…
We examine a hierarchy of equivalence classes of quasi-random properties of Boolean Functions. In particular, we prove an equivalence between a number of properties including balanced influences, spectral discrepancy, local strong…
This paper employs switching-algebraic techniques for the calculation of a fundamental index of voting powers, namely, the total Banzhaf power. This calculation involves two distinct operations: (a) Boolean differencing or differentiation,…
We investigate the influences of variables on a Boolean function $f$ based on the quantum Bernstein-Vazirani algorithm. A previous paper (Floess et al. in Math. Struct. in Comp. Science 23: 386, 2013) has proved that if a $n$-variable…
In simple games, larger coalitions typically wield more power, but do all players align their efforts effectively? Consider a voting scenario where a coalition forms, but needs more voters to pass a bill. The cohesion of the new group of…
Weighted voting is a classic model of cooperation among agents in decision-making domains. In such games, each player has a weight, and a coalition of players wins the game if its total weight meets or exceeds a given quota. A players power…
Following Zhang and Grossi~(AAAI 2021), we study in more depth a variant of weighted voting games in which agents' weights are induced by a transitive support structure. This class of simple games is notably well suited to study the…
The influence of a variable is an important concept in the analysis of Boolean functions. The more general notion of influence of a set of variables on a Boolean function has four separate definitions in the literature. In the present work,…