Bounds for the Nakamura number

The Nakamura number is an appropriate invariant of a simple game to study the existence of social equilibria and the possibility of cycles. For symmetric quota games its number can be obtained by an easy formula. For some subclasses of simple games the corresponding Nakamura number has also been characterized. However, in general, not much is known about lower and upper bounds depending of invariants on simple, complete or weighted games. Here, we survey such results and highlight connections with other game theoretic concepts.Comment: 23 pages, 3 tables; a few more references adde

Decisiveness indices are semiindices: addendum

In the paper Decisiveness indices are semiindices (Freixas and Pons, 2016) it was shown that any decisiveness index obtained from an anonymous probability distribution is a semiindex, and that the converse is not true. In this note we characterize the semiindices which are indices of decisiveness.Peer ReviewedPostprint (author's final draft

A Fibonacci sequence for linear structures with two types of components

We investigate binary voting systems with two types of voters and a hierarchy among the members in each type, so that members in one class have more influence or importance than members in the other class. The purpose of this paper is to count, up to isomorphism, the number of these voting systems for an arbitrary number of voters. We obtain a closed formula for the number of these systems, this formula follows a Fibonacci sequence with a smooth polynomial variation on the number of voters.Comment: All the results contained in this file are included in a paper submitted to Annals of Operations Research in October, 2008 on ocasion of the Conference on Applied Mathematical Programming and Modelling, that held in Bratislava in May, 200

Achievable hierarchies in voting games with abstention

It is well known that he influence relation orders the voters the same way as the classical Banzhaf and Shapley-Shubik indices do when they are extended to the voting games with abstention (VGA) in the class of complete games. Moreover, all hierarchies for the influence relation are achievable in the class of complete VGA. The aim of this paper is twofold. Firstly, we show that all hierarchies are achievable in a subclass of weighted VGA, the class of weighted games for which a single weight is assigned to voters. Secondly, we conduct a partial study of achievable hierarchies within the subclass of H-complete games, that is, complete games under stronger versions of influence relation. (C) 2013 Elsevier B.V. All rights reserved.Peer ReviewedPostprint (author’s final draft

An aggregation rule based on the binomial distribution

Many decision-making situations require the evaluation of several voters or agents. In a situation where voters evaluate candidates, the question arises of how best to aggregate evaluations so as to compare the candidates. The aim of this work is to propose a method of aggregating the evaluations of the voters, which has outstanding properties and serve as a potential evaluative tool in many contexts. Ordered weighted averages is a family of rules appropriate for studying this problem. In this paper, I propose as a solution an ordered weighted average that satisfies compelling properties and whose weights are derived from the binomial distribution.This research is part of the I+D+i project PID2019-104987GB-I00 supported by MCIN/AEI/10.13039/501100011033/.Peer ReviewedPostprint (published version

The Banzhaf value for cooperative and simple multichoice games

This is a post-peer-review, pre-copyedit version of an article published in Group Decision and Negotiation. The final authenticated version is available online at: https://doi.org/10.1007/s10726-019-09651-4.This article proposes a value which can be considered an extension of the Banzhaf value for cooperative games. The proposed value is defined on the class of j-cooperative games, i.e., games in which players choose among a finite set of ordered actions and the result depends only on these elections. If the output is binary, only two options are available, then j-cooperative games become j-simple games. The restriction of the value to j-simple games leads to a power index that can be considered an extension of the Banzhaf power index for simple games. The paper provides an axiomatic characterization for the value and the index which is closely related to the first axiomatization of the Banzhaf value and Banzhaf power index in the respective contexts of cooperative and simple games.Peer ReviewedPostprint (author's final draft

A value for j-cooperative games: some theoretical aspects and applications

This is an Accepted Manuscript of a book chapter published by Routledge/CRC Press in Handbook of the Shapley value on December 6, 2019, available online: https://www.crcpress.com/Handbook-of-the-Shapley-Value/Algaba-Fragnelli-Sanchez-Soriano/p/book/9780815374688A value that has all the ingredients to be a generalization of the Shapley value is proposed for a large class of games called j-cooperative games which are closely related to multi-choice games. When it is restricted to cooperative games, i.e. when j equals 2, it coincides with the Shapley value. An explicit formula in terms of some marginal contributions of the characteristic function is provided for the proposed value. Different arguments support it: (1) The value can be inferred from a natural probabilistic model. (2) An axiomatic characterization uniquely determines it. (3) The value is consistent in its particularization from j-cooperative games to j-simple games. This chapter also proposes various ways of calculating the value by giving an alternative expression that does not depend on the marginal contributions. This chapter shows how the technique of generating functions can be applied to determine such a value when the game is a weighted j-simple game. The chapter concludes by presenting several applications, among them the computation of the value for a proposed reform of the UNSC voting system.Peer ReviewedPostprint (author's final draft

On minimum integer representations of weighted games

We study minimum integer representations of weighted games, i.e., representations where the weights are integers and every other integer representation is at least as large in each component. Those minimum integer representations, if the exist at all, are linked with some solution concepts in game theory. Closing existing gaps in the literature, we prove that each weighted game with two types of voters admits a (unique) minimum integer representation, and give new examples for more than two types of voters without a minimum integer representation. We characterize the possible weights in minimum integer representations and give examples for $t\ge 4$ types of voters without a minimum integer representation preserving types, i.e., where we additionally require that the weights are equal within equivalence classes of voters.Comment: 29 page

Effect of a science communication event on students’ attitudes towards science and technology

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