http://hdl.handle.net/2117/3429 2013-06-20T11:06:24Z http://hdl.handle.net/2117/17925 Title: Computation of several power indices by generating functions Authors: Alonso Meijide, José María; Freixas Bosch, Josep; Molinero Albareda, Xavier Abstract: In this paper we propose methods to compute the Deegan-Packel, the Public Good, and the Shift power indices by generating functions for the particular case of weighted voting games. Furthermore, we define a new power index which combines the ideas of the Shift and the Deegan-Packel power indices and also propose a method to compute it with generating functions. We conclude by some comments about the complexity to compute these power indices. 2013-02-22T10:25:06Z http://hdl.handle.net/2117/16208 Title: Pure bargaining problems and the Shapley rule Authors: Carreras Escobar, Francisco; Owen Salazar, Guillermo Abstract: Pure bargaining problems with transferable utility are considered. By associating a quasi-additive cooperative game with each one of them, a Shapley rule for this class of problems is derived from the Shapley value for games. The analysis of this new rule includes axiomatic characterizations and a comparison with the proportional rule. 2012-07-09T11:56:32Z http://hdl.handle.net/2117/16206 Title: Accessibility measures to nodes of directed graphs using solutions for generalized cooperative games Authors: Amer Ramon, Rafael; Giménez Pradales, José Miguel; Magaña Nieto, Antonio Abstract: The aim of this paper consists of constructing accessibility measures to the nodes of directed graphs using methods of Game Theory. Since digraphs without a predefined game are considered, the main part of the paper is devoted to establish conditions on cooperative games so that they can be used to measure accessibility. Games that satisfy desirable properties are called test games. Each ranking on the nodes is then obtained according to a pair formed by a test game and a solution defined on cooperative games whose utilities are given on ordered coalitions. The solutions proposed here are extensions of the wide family of semivalues to games in generalized characteristic function form. 2012-07-09T11:34:13Z http://hdl.handle.net/2117/16145 Title: Complete voting systems with two types of voters: weightedness and counting Authors: Freixas Bosch, Josep; Molinero Albareda, Xavier; Roura Ferret, Salvador Abstract: We investigate voting systems with two classes of voters, for which there is a hierarchy giving each member of the stronger class more influence or important than each member of the weaker class. We deduce for voting systems one important counting fact that allows determining how many of them are for a given number of voters. In fact, the number of these systems follows a Fibonacci sequence with a smooth polynomial variation on the number of voters. On the other hand, we classify by means of some parameters which of these systems are weighted. This result allows us to state an asymptotic conjecture which is opposed to what occurs for symmetric games. 2012-06-27T12:59:15Z http://hdl.handle.net/2117/16144 Title: Probabilistic power indices for voting rules with abstention Authors: Freixas Bosch, Josep Abstract: In this paper, we introduce eight power indices that admit a probabilistic interpretation for voting rules with abstention or with three levels of approval in the input, briefly (3, 2) games.Weanalyze the analogies and discrepancies between standard known indices for simple games and the proposed extensions for this more general context. A remarkable difference is that for (3, 2) games the proposed extensions of the Banzhaf index, Coleman index to prevent action and Coleman index to initiate action become non-proportional notions, contrarily to what succeeds for simple games. We conclude the work by providing procedures based on generating functions for weighted (3, 2) games, and extensible to (j,k) games, to efficiently compute them. 2012-06-27T12:46:37Z http://hdl.handle.net/2117/13512 Title: A note on decisive symmetric games Authors: Carreras Escobar, Francisco; Freixas Bosch, Josep; Puente del Campo, María Albina Abstract: Binary voting systems, usually represented by simple games, constitute a main DSS topic. A crucial feature of such a system is the easiness with which a proposal can be collectively accepted, which is measured by the “decisiveness index” of the corresponding game. We study here several functions related to the decisiveness of any simple game. The analysis, including the asymptotic behavior as the number n of players increases, is restricted to decisive symmetric games and their compositions, and it is assumed that all players have a common probability p to vote for the proposal. We show that, for n large enough, a small variation, either positive or negative, in p when p=1/2 takes the decisiveness to quickly approach, respectively, 1 or 0. Moreover, we analyze the speed of the decisiveness convergence. 2011-10-14T16:58:43Z http://hdl.handle.net/2117/12437 Title: Decisiveness of decisive symmetric games Authors: Carreras Escobar, Francisco; Freixas Bosch, Josep; Puente del Campo, María Albina Abstract: Binary voting systems, usually represented by simple games, constitute a main DSS topic. A crucial feature of such a system is the easiness with which a proposal can be collectively accepted, which is measured by the "decisiveness index" of the corresponding game. We study here several functions related to the decisiveness of any simple game. The analysis, including the asymptotic behavior as the number n of players increases, is restricited to decisive symmetric gammes and their compositions, and it is assumed that all players have a common probability p to vote for the proposal. We show that, for n large enough, a small variation, either positive or negaive, in p when p=1/2 takes the decisiveness to quickly approach , respectively, 1 or 0. Moreover, we analyze the speed of the decisiveness convergence. Description: Reseach supported by Grant SGR 2009-01029 of the Catalonia Government (Generalitat de Catalunya) and Frants MTM 2006-06064 and MTM 2009-08037 of the Science and Innovation Spanish Ministry and the European Regional Development Fund. 2011-04-29T15:46:53Z http://hdl.handle.net/2117/12436 Title: Symmetric coalitional binomial semivalues Authors: Carreras Escobar, Francisco; Puente del Campo, María Albina Abstract: We introduce here a family of mixed coalitional values. they extend the binomial semivalues to games endowed with a coalition structure, satisfy the property of symmetry in the quotidient game and the quotient game property, generalize the symmetric coalitional Banzhaf value introduced by Alonso and Fiestras and link and merge the shapley value and the binomial semivalues. A computational procedure in terms of the multilinear extension of the original game is also provided and an application to political science is sketched. Description: Reseach supported by Grant SGR 2009-01029 of the Catalonia Government (Generalitat de Catalunya) and Frants MTM 2006-06064 and MTM 2009-08037 of the Science and Innovation Spanish Ministry and the European Regional Development Fund. 2011-04-29T15:19:06Z http://hdl.handle.net/2117/12035 Title: The proportional coalitional Shapley value Authors: Alonso Meijide, José María; Carreras Escobar, Francisco Abstract: We propose a modification of the Shapley value for monotonic games with a coalition structure. The resulting coalitional value is a twofold extension of the Shapley value in the following sense: (1) the amount obtained by any union coincides with the Shapley value of the union in the quotient game; and (2) the players of the union share this amount proportionally to their Shapley value in the original game (i.e., without unions). We provide axiomatic characterizations of this value close to those existing in the literature for the Owen value and include applications to coalition formation in bankruptcy and voting problems. 2011-03-23T15:15:00Z http://hdl.handle.net/2117/12033 Title: Power and potential maps induced by any semivalue: some algebraic properties and computation by multilinear extensions Authors: Carreras Escobar, Francisco; Giménez Pradales, José Miguel Abstract: The notions of total power and potential, both defined for any semivalue, give rise to two endomorphisms of the vector space of cooperative games on any given player set where the semivalue is defined. Several properties of these linear mappings are stated and the role of unanimity games as eigenvectors is described. We also relate in both cases the multilinear extension of the image game to the multilinear extension of the original game. As a consequence, we derive a method to compute for any semivalue by means of multilinear extensions, in the original game and also in all its subgames, (a) the total power, (b) the potential, and (c) the allocation to each player given by the semivalue. 2011-03-23T14:47:37Z http://hdl.handle.net/2117/11723 Title: Detection of paradoxes of power indices for simple games Authors: Freixas Bosch, Josep; Molinero Albareda, Xavier 2011-03-08T16:06:51Z http://hdl.handle.net/2117/11722 Title: On the notion of dimension and codimension of simple games Authors: Freixas Bosch, Josep; Marciniak, Dorota 2011-03-08T16:04:26Z http://hdl.handle.net/2117/10491 Title: Nodes of directed graphs ranked by solutions defined on cooperative games Authors: Amer Ramon, Rafael; Giménez Pradales, José Miguel; Magaña Nieto, Antonio Abstract: Hierarchical structures, transportation systems, communication networks and even sports competitions can be modeled by means of directed graphs. Since digraphs without a predefined game are considered, the main part of the work is devoted to establish conditions on cooperative games so that they can be used to measure accessibility to the nodes. Games that satisfy desirable properties are called test games. Each ranking on the nodes is then obtained according to a pair formed by a test game and a solution defined on cooperative games whose utilities are given for every ordered coalition. Solutions here proposed are extensions of the wide family of semivalues to games in generalized characteristic function form. 2010-12-02T15:26:30Z http://hdl.handle.net/2117/9550 Title: Maximum tolerance and maximum greatest tolerance Authors: Freixas Bosch, Josep; Molinero Albareda, Xavier Abstract: An important consideration when applying neural networks is the sensitivity to weights and threshold in strict separating systems representing a linearly separable function. Two parameters have been introduced to measure the relative errors in weights and threshold of strict separating systems: the tolerance and the greatest tolerance. Given an arbitrary separating system we study which is the equivalent separating system that provides maximum tolerance or/and maximum greatest tolerance. 2010-10-07T14:19:24Z http://hdl.handle.net/2117/9479 Title: Consorcios y valor de Shapley Authors: Llongueras Arola, Maria Dolors; Magaña Nieto, Antonio Abstract: La coordindación de estrategias en un juego cooperativo, cuando un grupo de jugadores decide actuar en conjunto es la base de la noción de consorcio. Un consorcio en un juego cooperativo es una coalición que posee una estructura interna y, al mismo tiempo, se comporta como un miembro individual. En este trabajo se define el valor de consorcio de Shapley y se determinan expresiones para la diferencia entre el valor de Shapley de un jugador en el juego inicial y su valor de Shapleu en el juego de consorcio. 2010-10-06T18:07:53Z