A note on simple games with two equivalence classes of players

Many real-world voting systems consist of voters that occur in just two different types. Indeed, each voting system with a {\lq\lq}House{\rq\rq} and a {\lq\lq}Senat{\rq\rq} is of that type. Here we present structural characterizations and explicit enumeration formulas for these so-called bipartite simple games.Comment: 13 page

The complexity of power indices in voting games with incompatible players

We study the complexity of computing the Banzhaf index in weighted voting games with cooperation restricted by an incompatibility graph. With an existing algorithm as a starting point, we use concepts from complexity theory to show that, for some classes of incompatibility graphs, the problem can be solved efficiently, as long as the players have "small" weights. We also show that for some other class of graphs it is unlikely that we can find efficient algorithms to compute the Banzhaf index in the corresponding restricted game. Finally, we discuss the complexity of deciding whether the index of a player is non-zero

Structural and computational aspects of simple and influence games

Simple games are a fundamental class of cooperative games. They have a huge relevance in several areas of computer science, social sciences and discrete applied mathematics. The algorithmic and computational complexity aspects of simple games have been gaining notoriety in the recent years. In this thesis we review different computational problems related to properties, parameters, and solution concepts of simple games. We consider different forms of representation of simple games, regular games and weighted games, and we analyze the computational complexity required to transform a game from one representation to another. We also analyze the complexity of several open problems under different forms of representation. In this scenario, we prove that the problem of deciding whether a simple game in minimal winning form is decisive (a problem that is associated to the duality problem of hypergraphs and monotone Boolean functions) can be solved in quasi-polynomial time, and that this problem can be polynomially reduced to the same problem but restricted to regular games in shift-minimal winning form. We also prove that the problem of deciding wheter a regular game is strong in shift-minimal winning form is coNP-complete. Further, for the width, one of the parameters of simple games, we prove that for simple games in minimal winning form it can be computed in polynomial time. Regardless of the form of representation, we also analyze counting and enumeration problems for several subfamilies of these games. We also introduce influence games, which are a new approach to study simple games based on a model of spread of influence in a social network, where influence spreads according to the linear threshold model. We show that influence games capture the whole class of simple games. Moreover, we study for influence games the complexity of the problems related to parameters, properties and solution concepts considered for simple games. We consider extremal cases with respect to demand of influence, and we show that, for these subfamilies, several problems become polynomial. We finish with some applications inspired on influence games. The first set of results concerns to the definition of collective choice models. For mediation systems, several of the problems of properties mentioned above are polynomial-time solvable. For influence systems, we prove that computing the satisfaction (a measure equivalent to the Rae index and similar to the Banzhaf value) is hard unless we consider some restrictions in the model. For OLFM systems, a generalization of OLF systems (van den Brink et al. 2011, 2012) we provide an axiomatization of satisfaction. The second set of results concerns to social network analysis. We define new centrality measures of social networks that we compare on real networks with some classical centrality measures.Los juegos simples son una clase fundamental de juegos cooperativos, que tiene una enorme relevancia en diversas áreas de ciencias de la computación, ciencias sociales y matemáticas discretas aplicadas. En los últimos años, los distintos aspectos algorítmicos y de complejidad computacional de los juegos simples ha ido ganando notoriedad. En esta tesis revisamos los distintos problemas computacionales relacionados con propiedades, parámetros y conceptos de solución de juegos simples. Primero consideramos distintas formas de representación de juegos simples, juegos regulares y juegos de mayoría ponderada, y estudiamos la complejidad computacional requerida para transformar un juego desde una representación a otra. También analizamos la complejidad de varios problemas abiertos bajo diferentes formas de representación. En este sentido, demostramos que el problema de decidir si un juego simple en forma ganadora minimal es decisivo (un problema asociado al problema de dualidad de hipergrafos y funciones booleanas monótonas) puede resolverse en tiempo cuasi-polinomial, y que este problema puede reducirse polinomialmente al mismo problema pero restringido a juegos regulares en forma ganadora shift-minimal. También demostramos que el problema de decidir si un juego regular en forma ganadora shift-minimal es fuerte (strong) es coNP-completo. Adicionalmente, para juegos simples en forma ganadora minimal demostramos que el parámetro de anchura (width) puede computarse en tiempo polinomial. Independientemente de la forma de representación, también estudiamos problemas de enumeración y conteo para varias subfamilias de juegos simples. Luego introducimos los juegos de influencia, un nuevo enfoque para estudiar juegos simples basado en un modelo de dispersión de influencia en redes sociales, donde la influencia se dispersa de acuerdo con el modelo de umbral lineal (linear threshold model). Demostramos que los juegos de influencia abarcan la totalidad de la clase de los juegos simples. Para estos juegos también estudiamos la complejidad de los problemas relacionados con parámetros, propiedades y conceptos de solución considerados para los juegos simples. Además consideramos casos extremos con respecto a la demanda de influencia, y probamos que para ciertas subfamilias, varios de estos problemas se vuelven polinomiales. Finalmente estudiamos algunas aplicaciones inspiradas en los juegos de influencia. El primer conjunto de estos resultados tiene que ver con la definición de modelos de decisión colectiva. Para sistemas de mediación, varios de los problemas de propiedades mencionados anteriormente son polinomialmente resolubles. Para los sistemas de influencia, demostramos que computar la satisfacción (una medida equivalente al índice de Rae y similar al valor de Banzhaf) es difícil a menos que consideremos algunas restricciones en el modelo. Para los sistemas OLFM, una generalización de los sistemas OLF (van den Brink et al. 2011, 2012) proporcionamos una axiomatización para la medida de satisfacción. El segundo conjunto de resultados se refiere al análisis de redes sociales, y en particular con la definición de nuevas medidas de centralidad de redes sociales, que comparamos en redes reales con otras medidas de centralidad clásica

Spanning connectivity games

The Banzhaf index, Shapley-Shubik index and other voting power indices measure the importance of a player in a coalitional game. We consider a simple coalitional game called the spanning connectivity game (SCG) based on an undirected, unweighted multigraph, where edges are players. We examine the computational complexity of computing the voting power indices of edges in the SCG. It is shown that computing Banzhaf values and Shapley-Shubik indices is #P-complete for SCGs. Interestingly, Holler indices and Deegan-Packel indices can be computed in polynomial time. Among other results, it is proved that Banzhaf indices can be computed in polynomial time for graphs with bounded treewidth. It is also shown that for any reasonable representation of a simple game, a polynomial time algorithm to compute the Shapley-Shubik indices implies a polynomial time algorithm to compute the Banzhaf indices. As a corollary, computing the Shapley value is #P-complete for simple games represented by the set of minimal winning coalitions, Threshold Network Flow Games, Vertex Connectivity Games and Coalitional Skill Games

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

On Influence, Stable Behavior, and the Most Influential Individuals in Networks: A Game-Theoretic Approach

We introduce a new approach to the study of influence in strategic settings where the action of an individual depends on that of others in a network-structured way. We propose \emph{influence games} as a \emph{game-theoretic} model of the behavior of a large but finite networked population. Influence games allow \emph{both} positive and negative \emph{influence factors}, permitting reversals in behavioral choices. We embrace \emph{pure-strategy Nash equilibrium (PSNE)}, an important solution concept in non-cooperative game theory, to formally define the \emph{stable outcomes} of an influence game and to predict potential outcomes without explicitly considering intricate dynamics. We address an important problem in network influence, the identification of the \emph{most influential individuals}, and approach it algorithmically using PSNE computation. \emph{Computationally}, we provide (a) complexity characterizations of various problems on influence games; (b) efficient algorithms for several special cases and heuristics for hard cases; and (c) approximation algorithms, with provable guarantees, for the problem of identifying the most influential individuals. \emph{Experimentally}, we evaluate our approach using both synthetic influence games as well as several real-world settings of general interest, each corresponding to a separate branch of the U.S. Government. \emph{Mathematically,} we connect influence games to important game-theoretic models: \emph{potential and polymatrix games}.Comment: Accepted to AI Journal, subject to addressing the reviewers' points (which are addressed in this version). An earlier version of the article appeared in AAAI-1