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The binomial system is an electoral system unique in the world. It was used to elect the senators and deputies of Chile during 27 years, from the return of democracy in 1990 until 2017. In this paper we study the real voting power of the different political parties in the Senate of Chile during the whole binomial period. We not only consider the different legislative periods, but also any party changes between one period and the next. The real voting power is measured by considering power indices from cooperative game theory, which are based on the capability of the political parties to form winning coalitions. With this approach, we can do an analysis that goes beyond the simple count of parliamentary seats.

An opinion leader-follower model (OLF) is a two-action collective decision-making model for societies, in which three kinds of actors are considered: "opinion leaders", "followers", and "independent actors". In OLF the initial decision of the followers can be modified by the influence of the leaders. Once the final decision is set, a collective decision is taken applying the simple majority rule. We consider a generalization of \OLF, the gOLF models which allow collective decision taken by rules different from the single majority rule. Inspired in this model we define two new families of collective decision-making models associated with cooperative influence games. We define the "oblivious" and "non-oblivious" influence models. We show that gOLF models are non-oblivious influence models played on a two layered bipartite influence graph. Together with OLF models, the Satisfaction measure was introduced and studied. We analyze the computational complexity of the satisfaction measure for gOLF models and the other collective decision-making models introduced in the paper. We show that computing the satisfaction measure is #P-hard in all the considered models except for the basic OLF inwhich the complexity remains open. On the other hand, we provide two subfamilies of decision models in which the satisfaction measure can be computed in polynomial time. Exploiting the relationship with influence games, we can relate the satisfaction measure with the Rae index of an associated simple game. The Rae index is closely related to the Banzhaf value. Thus, our results also extend the families of simple games for which computing the Rae index and the Banzhaf value is computationally hard., Comment: 29 pages, 8 figures

Centrality is one of the most studied concepts in social network analysis. There is a huge literature regarding centrality measures, as ways to identify the most relevant users in a social network. The challenge is to find measures that can be computed efficiently, and that can be able to classify the users according to relevance criteria as close as possible to reality. We address this problem in the context of the Twitter network, an online social networking service with millions of users and an impressive flow of messages that are published and spread daily by interactions between users. Twitter has different types of users, but the greatest utility lies in finding the most influential ones. The purpose of this article is to collect and classify the different Twitter influence measures that exist so far in literature. These measures are very diverse. Some are based on simple metrics provided by the Twitter API, while others are based on complex mathematical models. Several measures are based on the PageRank algorithm, traditionally used to rank the websites on the Internet. Some others consider the timeline of publication, others the content of the messages, some are focused on specific topics, and others try to make predictions. We consider all these aspects, and some additional ones. Furthermore, we include measures of activity and popularity, the traditional mechanisms to correlate measures, and some important aspects of computational complexity for this particular context., Comment: 33 pages, 3 tables, 3 figures, Information Processing & Management (2016)

In this paper we propose the opinion leader-follower through mediators systems (OLFM systems) a multiple-action collective choice model for societies. In those societies three kind of actors are considered: opinion leaders that can exert certain influence over the decision of other actors, followers that can be convinced to modify their original decisions, and independent actors that neither are influenced nor can influence; mediators are actors that both are influenced and influence other actors. This is a generalization of the opinion leader-follower systems (OLF systems) proposed by van den Brink R, et al. (2011). The satisfaction score is defined on the set of actors. For each actor it measures the number of society initial decisions in which the final collective decision coincides with the one that the actor initially selected. We generalize in OLFM systems some properties that the satisfaction score meets for OLF systems. By using these properties, we provide an axiomatization of the satisfaction score for the case in which followers maintain their own initial decisions unless all their opinion leaders share an opposite inclination. This new axiomatization generalizes the one given by van den Brink R, et al. (2012) for OLF systems under the same restrictions., Comment: 12 pages

In social network analysis, there is a common perception that influence is relevant to determine the global behavior of the society and thus it can be used to enforce cooperation by targeting an adequate initial set of individuals or to analyze global choice processes. Here we propose centrality measures that can be used to analyze the relevance of the actors in process related to spread of influence. In [39] it was considered a multiagent system in which the agents are eager to perform a collective task depending on the perception of the willingness to perform the task of other individuals. The setting is modeled using a notion of simple games called influence games. Those games are defined on graphs were the nodes are labeled by their influence threshold and the spread of influence between its nodes is used to determine whether a coalition is winning or not. Influence games provide tools to measure the importance of the actors of a social network by means of classic power indices and provide a framework to consider new centrality criteria. In this paper we consider two of the most classical power indices, i.e., Banzhaf and Shapley-Shubik indices, as centrality measures for social networks in influence games. Although there is some work related to specific scenarios of game-theoretic networks, here we use such indices as centrality measures in any social network where the spread of influence phenomenon can be applied. Further, we define new centrality measures such as satisfaction and effort that, as far as we know, have not been considered so far. We also perform a comparison of the proposed measures with other three classic centrality measures, degree, closeness and betweenness, considering three social networks. We show that in some cases our measurements provide centrality hierarchies similar to those of other measures, while in other cases provide different hierarchies., Comment: 14 pages, 6 figures, 3 tables

We study the computational complexity of an important property of simple, regular and weighted games, which is decisiveness. We show that this concept can naturally be represented in the context of hypergraph theory, and that decisiveness can be decided for simple games in quasi-polynomial time, and for regular and weighted games in polynomial time. The strongness condition poses the main difficulties, while properness reduces the complexity of the problem, especially if it is amplified by regularity. On the other hand, regularity also allows to specify the problem instances much more economically, implying a reconsideration of the corresponding complexity measure that, as we prove, has important structural as well as algorithmic consequences., Comment: 12 pages

We consider a simple and altruistic multiagent system in which the agents are eager to perform a collective task but where their real engagement depends on the willingness to perform the task of other influential agents. We model this scenario by an influence game, a cooperative simple game in which a team (or coalition) of players succeeds if it is able to convince enough agents to participate in the task (to vote in favor of a decision). We take the linear threshold model as the influence model. We show first the expressiveness of influence games showing that they capture the class of simple games. Then we characterize the computational complexity of various problems on influence games, including measures (length and width), values (Shapley-Shubik and Banzhaf) and properties (of teams and players). Finally, we analyze those problems for some particular extremal cases, with respect to the propagation of influence, showing tighter complexity characterizations., Comment: 23 pages