A simple game is a cooperative game where some coalitions among players or voters became the (monotonic) set of winning coalitions, and the other ones form the set of losing coalitions. It is well-known that weighted voting games form a strict subclass of simple games, where each player has a voting weight so that a coalition wins if and only if the sum of weights of their members exceeds a given quota, otherwise it loses. This work studies how far away a simple game is for being representable as a weighted voting game, which allows for a more compact representation. There are several measures that determine the weightedness of a simple game, such as the dimension, the trade-robustness, the critical threshold value associated with the α -roughly weightedness property, etc. In this work we propose some new weightedness measures, all based on linear programming. In general terms, for a given simple game, a linear program is used to identify its weightedness: (i) the ϵ -roughly value ( μ ϵ ), (ii) the Z + -roughly value ( μ Z + ), (iii) the Δ-roughly value ( μ Δ ), and (iv) the outlier value ( Ψ M ). We show a close relation between the known critical threshold value of weightedness and the new measure μ Δ . Finally, we also present an exhaustive comparison of weightedness measures for simple games with up to six players. • We define new weightedness measures for simple games from inequality systems. • We introduce a new weightedness measure for simple games based on outlier coalitions. • This manuscript provides a comparison of the known α -roughly value with other new weightedness values. • We exhaustively compute some weightedness measures for all simple games up to 7 players.

In this study, our objective is to explore methodologies for the identification of diverse typefaces. Utilizing the gathered data, we conducted a thorough analysis of the outcomes, distinguishing between successes and failures for both uppercase and lowercase letters within each typeface. The analytical framework is anchored in three distinct recognition measures. The initial measure draws upon the relative frequency of accurate responses, providing insights into the overall performance of each typeface. The second measure is constructed around the F-score derived from confusion matrices, offering a comprehensive evaluation of recognition precision and recall. Lastly, the third measure is formulated on the well-established Shapley-Shubik index, extensively scrutinized and endorsed within the realm of classical game theory. This multifaceted approach allows us to comprehensively assess the distinct aspects of typ.eface recognition, contributing to a nuanced under- standing of their effectiveness and characteristics

We describe voting mechanisms to study voting systems. The classical power indices applied to simple games just consider parties, players or voters. Here, we also consider games with a priori unions, i.e., coalitions among parties, players or voters. We measure the power of each party, player or voter when there are coalitions among them. In particular, we study real situations of voting systems using extended Shapley–Shubik and Banzhaf indices, the so-called coalitional power indices. We also introduce a dynamic programming to compute them.

Voting is a commonly applied method for the aggregation of the preferences of multiple agents into a joint decision. If preferences are binary, i.e., “yes” and “no”, every voting system can be described by a (monotone) Boolean function χ:{0,1}n→{0,1}. However, its naive encoding needs 2nbits. The subclass of threshold functions, which is sufficient for homogeneous agents, allows a more succinct representation using n weights and one threshold. For heterogeneous agents, one can represent χ as an intersection of k threshold functions. Taylor and Zwicker have constructed a sequence of examples requiringand provided a construction guaranteeing. The magnitude of the worst-case situation was to be determined by Elkind et al. in 2008, but the analysis unfortunately turned out to be wrong. Here we uncover a relation to coding theory that allows the determination of the minimum number k for a subclass of voting systems. As an application, we give a construction for k≥2n−o(n), i.e., there is no gain from a representation complexity point of view.

An important consideration when applying neural networks is the sensitivity to weights and threshold in strict separating systems representing a linearly separable function. Perturbations may affect weights and threshold so that it is important to estimate the maximal percentage error in weights and threshold, which may be allowed without altering the linearly separable function. In this paper, we provide the greatest allowed bound which can be associated to every strict separating system representing a linearly separable function. The proposed bound improves the tolerance that Hu obtained. Furthermore, it is the greatest bound for any strict separating system. This is the reason why we call it the greatest tolerance.