Using the multilinear extension to study some probabilistic power indices

The final publication is available at Springer via http://dx.doi.org/10.1007/s10726-016-9514-6We consider binary voting systems modeled by a simple game, in which voters vote independently of each other, and the probability distribution over coalitions is known. The Owen’s multilinear extension of the simple game is used to improve the use and the computation of three indices defined in this model: the decisiveness index, which is an extension of the Banzhaf index, the success index, which is an extension of the Rae index, and the luckiness index. This approach leads us to prove new properties and inter-relations between these indices. In particular it is proved that the ordinal equivalence between success and decisiveness indices is achieved in any game if and only if the probability distribution is anonymous. In the anonymous case, the egalitarianism of the three indices is compared, and it is also proved that, for these distributions, decisiveness and success indices respect the strength of the seats, whereas luckiness reverses this order.Peer ReviewedPostprint (author's final draft

Matrix representation of TU-games for Linear Efficient and Symmetric values

The aim of this article is to present a new tool for assessing TU-game based on a matrix representation. We focus on TU-games with coalition structures and provide a general matrix form of TU-game. We shed light on some useful properties of the matrix representation of TU-game and the general form obtained is applied to describe the representation for some classical TU-game. The facilities provided by such a representation are used to characterize subclasses of Linear Efficient and Symmetric (LES) values

Matrix representation of TU-games for Linear Efficient and Symmetric values

The aim of this article is to present a new tool for assessing TU-game based on a matrix representation. We focus on TU-games with coalition structures and provide a general matrix form of TU-game. We shed light on some useful properties of the matrix representation of TU-game and the general form obtained is applied to describe the representation for some classical TU-game. The facilities provided by such a representation are used to characterize subclasses of Linear Efficient and Symmetric (LES) values

On ordinal equivalence of the Shapley and Banzhaf values for cooperative games

Cooperative game, Semivalue, Weakly complete game, Semicoherent game, Coherent game, 91A12, 91A40, 91A80, 91B12, C71, D71,

On ordinal equivalence of the Shapley and Banzhaf values for cooperative games

In this paper I consider the ordinal equivalence of the Shapley and Banzhaf values for TU cooperative games, i.e., cooperative games for which the preorderings on the set of players induced by these two values coincide. To this end I consider several solution concepts within semivalues and introduce three subclasses of games which are called respectively: weakly complete, semicoherent and coherent cooperative games. A characterization theorem in terms of the ordinal equivalence of some semivalues is given for each of these three classes of cooperative games. In particular, the Shapley and Banzhaf values as well as the segment of semivalues they limit are ordinally equivalent for weakly complete, semicoherent and coherent cooperative games

Introducció als jocs cooperatius, jocs simples, votació pondederada i poder polític

Al treball s estudia la caracterització d un joc simple, concretament en situacions d una votació sí-no. S analitza matemàticament què vol dir que un sistema de votació sigui ponderat i quines condicions ha de complir un sistema de votació per ser-ho. Seguidament s avalua el concepte de dimensió d una votació. El projecte s ajuda amb exemples de la vida real per fer més visual alguns conceptes que a priori poden resultar una mica abstractes. Finalment s intenta donar una definició matemàtica al concepte de poder basat en l influència que té un individu a l hora d exercir la seva voluntat en una votació. Es defineixen els dos índexs de poder més reconeguts que intenten quantificar aquesta influència i s observen alguns índexs existents. El treball també fa una incursió en l'efecte té sobre aquest poder les incompatibilitats entre dos votants a l hora de formar coalicions