Generating functions: a useful tool for computing power indices

In the theory of simple games, the study of power indices plays an important role. One of the main di culties with these indices is that computation generally requires the sum of a very large number of terms. The generating functions are e cient tools to make more easy this computation. In this paper, we provide a revision of the main elements of this method when we use it to compute the Shapley-Shubik and the BanzhafColeman power indices. Further, we provide a new method to compute the Banzhaf-Coleman indexS

Evaluación de impacto ambiental de Galicia

Este libro recolle as sesións correspondentes o módulo avaliación do impacto ambiental do Master de Ciencia e Tecnoloxía Ambiental, dirixido polo profesor D. Xosé Luis Armesto Barbeito que se desenvolbe desde 1992 na Facultarle de Ciencia

Power Indices and Minimal Winning Coalitions in Simple Games with Externalities

We propose a generalization of simple games to sit uations with coalitional externalities. The main novelty of our generalization is a monotonicity property that we define for games in partition function form. This property allows us to properly speak about minimal winning embedded coalitions. We propose and characterize two power indices based on these kind of coalitions. We provide methods based on the multilinear extension of the game to compute the indices. Finally, the new indices are used to study the distribution of power in the current Parliament of Andalusia

The Shapley-Shubik Index in the Presence of Externalities

In this note we characterize the restriction of the externality-free value of de Clippel and Serrano (2008) to the class of simple games with externalities introduced in Alonso-Meijide et al. (2015

Two new power indices based on winning coalitions

Deegan and Packel (1979) and Holler (1982) proposed two power indices for simple games: the Deegan–Packel index and the Public Good Index. In the definition of these indices, only minimal winning coalitions are taken into account. Using similar arguments, we define two new power indices. These new indices are defined taking into account only those winning coalitions that do not contain null players. The results obtained with the different power indices are compared by means of two real-world examples taken from the political field

Power indices and minimal winning coalitions for simple games in partition function form

We propose a generalization of simple games to partition function form games based on a monotonicity property that we define in this context. This property allows us to properly speak about minimal winning embedded coalitions. We propose and characterize two power indices based on such coalitions. Finally, the new indices are used to study the distribution of power in the Parliament of Andalusia that emerged after the elections of March 22, 2015

The Least Square Nucleolus is a Normalized Banzhaf Value

In this note we study a truncated additive normalization of the Banzhaf value. We are able to show that it corresponds to the Least Square nucleolus (LS-nucleolus), which was originally introduced as the solution of a constrained optimization problem (Ruiz et al., 1996). Thus, the main result provides an explicit expression that eases the computation and contributes to the understanding of the LS-nucleolus. Lastly, the result is extended to the broader family of Individually Rational Least Square values (Ruiz et al., 1998b)

Mergeable weighted majority games and characterizations of some power indices

In this paper, we introduce a notion of mergeable weighted majority games with the aim of providing the first characterization of the Colomer–Martínez power index (Colomer and Martínez in J Theor Polit 7(1):41–63, 1995). Furthermore, we define and characterize a new power index for the family of weighted majority games that combines ideas of the Public Good (Holler in Polit Stud 30(2):262–271, 1982) and Colomer–Martínez power indices. Finally, we analyze the National Assembly of Ecuador using these and some other well-known power indicesWe would like to thank Balbina V. Casas-Méndez and two anonymous referees for their valuable comments. This work is part of the R+D+I project grants MTM2017-87197-C3-2-P, MTM2017-87197-C3-3-P, PID2021-124030NB-C32, and PID2021-124030NB-C33, that were funded by MCIN/AEI/10.13039/501100011033/ and by “ERDF A way of making Europe”/EU. This research was also funded by Grupos de Referencia Competitiva ED431C-2020/03 and ED431C-2021/24 from the Consellería de Cultura, Educación e Universidades, Xunta de Galicia. Open Access funding provided thanks to the CRUE-CSIC agreement with Springer NatureS

The general elections of 2008 and the game of parliament

En este artículo analizamos el resultado de las elecciones generales desde un punto de vista Teórico de Juegos. Para cuantificar el poder que cada partido tiene en el nuevo Parlamento introducimos tres conceptos, los índices de poder de Shapley-Shubik, de Banzhaf-Penrose y de Deegan-Packel. De esta forma, podemos saber cuál es el partido más decisivo y comparando los resultados obtenidos con los de las elecciones de 2004, qué partidos han aumentado o disminuido más poder. Para terminar se proponen extensiones al modelo propuestoIn this paper we analyze the result of the elections for the Spanish Parliament in a Game Theoretical approach. In order to measure the power each party has in the new Parliament we introduce three concepts, the Shapley-Shubik, Banzhaf-Penrose and DeeganPackel power indices. In this way, we are able to know which the most decisive party is and comparing the result with the elections of 2004, we conclude which party has gain and which has lost more power. At last, we present different extensions of the proposed modelS

On the externality free shapley-shubik index

We address the problem of extending the Shapley-Shubik index to the class of simple games with externalities introduced in Alonso-Meijide et al. (2017). On the one hand, we provide bounds for any efficient, symmetric, and monotonic power index. On the other hand, we characterize the restriction of the externality-free value of de Clippel and Serrano (2008) to the class of games under study by adapting well-known properties