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Parrondo's paradox was introduced by Juan Parrondo in 1996. In game theory, this paradox is described as: A combination of losing strategies becomes a winning strategy. At first glance, this paradox is quite surprising, but we can easily explain it by using simulations and mathematical arguments. Indeed, we first consider some examples with the Parrondo's paradox and, using the software R, we simulate one of them, the coin tossing. Actually, we see that specific combinations of losing games become a winning game. Moreover, even a random combination of these two losing games leads to a winning game. Later, we introduce the major definitions and theorems over Markov chains to study our Parrondo's paradox applied to the coin tossing problem. In particular, we represent our Parrondo's game as a Markov chain and we find its stationary distribution. In that way, we exhibit that our combination of two losing games is truly a winning combination. We also deliberate possible applications of the paradox in some fields such as ecology, biology, finance or reliability theory.

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

This paper studies the complexity of computing a representation of a simple game as the intersection (union) of weighted majority games, as well as, the dimension or the codimension. We also present some examples with linear dimension and exponential codimension with respect to the number of players., Comment: 5 pages

Voting is a commonly applied method for the aggregation of the preferences of multiple agents into a joint decision. If preferences are binary, i.e., "yes" and "no", every voting system can be described by a (monotone) Boolean function $\chi\colon\{0,1\}^n\rightarrow \{0,1\}$. However, its naive encoding needs $2^n$ bits. The subclass of threshold functions, which is sufficient for homogeneous agents, allows a more succinct representation using $n$ weights and one threshold. For heterogeneous agents, one can represent $\chi$ as an intersection of $k$ threshold functions. Taylor and Zwicker have constructed a sequence of examples requiring $k\ge 2^{\frac{n}{2}-1}$ and provided a construction guaranteeing $k\le {n\choose {\lfloor n/2\rfloor}}\in 2^{n-o(n)}$. The magnitude of the worst-case situation was thought to be determined by Elkind et al.~in 2008, but the analysis unfortunately turned out to be wrong. Here we uncover a relation to coding theory that allows the determination of the minimum number $k$ for a subclass of voting systems. As an application, we give a construction for $k\ge 2^{n-o(n)}$, i.e., there is no gain from a representation complexity point of view., Comment: 13 pages, 1 table

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 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

We investigate binary voting systems with two types of voters and a hierarchy among the members in each type, so that members in one class have more influence or importance than members in the other class. The purpose of this paper is to count, up to isomorphism, the number of these voting systems for an arbitrary number of voters. We obtain a closed formula for the number of these systems, this formula follows a Fibonacci sequence with a smooth polynomial variation on the number of voters., Comment: All the results contained in this file are included in a paper submitted to Annals of Operations Research in October, 2008 on ocasion of the Conference on Applied Mathematical Programming and Modelling, that held in Bratislava in May, 2008

Simple games cover voting systems in which a single alternative, such as a bill or an amendment, is pitted against the status quo. A simple game or a yes-no voting system is a set of rules that specifies exactly which collections of ``yea'' votes yield passage of the issue at hand. A collection of ``yea'' voters forms a winning coalition. We are interested on performing a complexity analysis of problems on such games depending on the game representation. We consider four natural explicit representations, winning, loosing, minimal winning, and maximal loosing. We first analyze the computational complexity of obtaining a particular representation of a simple game from a different one. We show that some cases this transformation can be done in polynomial time while the others require exponential time. The second question is classifying the complexity for testing whether a game is simple or weighted. We show that for the four types of representation both problem can be solved in polynomial time. Finally, we provide results on the complexity of testing whether a simple game or a weighted game is of a special type. In this way, we analyze strongness, properness, decisiveness and homogeneity, which are desirable properties to be fulfilled for a simple game., Comment: 18 pages, LaTex file