Xavier Molinero

On the existence of a minimum integer representation for weighted voting systems

A basic problem in the theory of simple games and other fields is to study whether a simple game (Boolean function) is weighted (linearly separable). A second related problem consists in studying whether a weighted game has a minimum integer realization. In this paper we simultaneously analyze both problems by using linear programming.For less than 9 voters, we find that there are 154 weighted games without minimum integer realization, but all of them have minimum normalized realization. Isbell in 1958 was the first to find a weighted game without a minimum normalized realization, he needed to consider 12 voters to construct a game with such a property. The main result of this work proves the existence of weighted games with this property with less than 12 voters.

Generic Algorithms for the Generation of Combinatorial Objects

This paper briefly describes our generic approach to the exhaustive generation of unlabelled and labelled combinatorial classes. Our algorithms receive a size n and a finite description of a combinatorial class \(\mathcal{A}\) using combinatorial operators such as union, product, set or sequence, in order to list all objects of size n in \(\mathcal{A}\). The algorithms work in constant amortized time per generated object and thus they are suitable for rapid prototyping or for inclusion in general libraries.

Weighted games without a unique minimal representation in integers

Isbell in 1959 was the first to find a weighted game without a minimum integer realization in which the affected players do not play a symmetric role in the game. His example has 12 players in a weighted decisive game, i.e. a weighted game for which a coalition wins iff its complement loses. The goal of this article is to provide a procedure for weighted games that allows finding out what is the minimum number of players needed to get a weighted game without a minimum integer weighted representation in which the affected players do not play a symmetric role in the game. We prove, by means of an algorithm, that the minimum number of voters required is nine.

Cooperation through social influence

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.

Simple games and weighted games: A theoretical and computational viewpoint

It is a well-known result in the theory of simple games that a game is weighted if and only if it is trade robust. In this paper we propose a variant of trade robustness, that we call invariant-trade robustness, which is enough to determine whether a simple game is weighted. To test whether a simple game is invariant-trade robust we do not need to consider all winning coalitions; a reduced subset of minimal winning coalitions is enough. We make a comparison between the two methods (trade robustness and invariant-trade robustness) to check whether a simple game is weighted. We also provide by means of algorithms a full classification using both methods, for simple games with less than 8 voters according to the maximum level of (invariant-)trade robustness they achieve.

Complete voting systems with two classes of voters: weightedness and counting

We investigate voting systems with two classes of voters, for which there is a hierarchy giving each member of the stronger class more influence or important than each member of the weaker class. We deduce for voting systems one important counting fact that allows determining how many of them are for a given number of voters. In fact, the number of these systems follows a Fibonacci sequence with a smooth polynomial variation on the number of voters. On the other hand, we classify by means of some parameters which of these systems are weighted. This result allows us to state an asymptotic conjecture which is opposed to what occurs for symmetric games.

The Greatest Allowed Relative Error in Weights and Threshold of Strict Separating Systems

An important consideration when applying neural networks is the sensitivity to weights and threshold in strict separating systems representing a linearly separable function. Perturbations may affect weights and threshold so that it is important to estimate the maximal percentage error in weights and threshold, which may be allowed without altering the linearly separable function. In this paper, we provide the greatest allowed bound which can be associated to every strict separating system representing a linearly separable function. The proposed bound improves the tolerance that Hu obtained. Furthermore, it is the greatest bound for any strict separating system. This is the reason why we call it the greatest tolerance.

On the complexity of problems on simple games

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, each of these collections of “yea” voters forms a winning coalition. We are interested in performing a complexity analysis on problems defined on such families of games. This analysis as usual depends on the game representation used as input. We consider four natural explicit representations: winning, losing, minimal winning, and maximal losing. We first analyze the complexity of testing whether a game is simple and testing whether a game is weighted. We show that, for the four types of representations, both problems 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. We analyze strongness, properness, decisiveness and homogeneity, which are desirable properties to be fulfilled for a simple game. We finalize with some considerations on the possibility of representing a game in a more succinct representation showing a natural representation in which the recognition problem is hard.

Computation of several power indices by generating functions

Abstract In this paper we propose methods to compute the Deegan-Packel, the Public Good, and the Shift power indices by generating functions for the particular case of weighted voting games. Furthermore, we define a new power index which combines the ideas of the Shift and the Deegan-Packel power indices and also propose a method to compute it with generating functions. We conclude by some comments about the complexity to compute these power indices.

On the complexity of exchanging

We analyse the computational complexity of the trade robustness problem.We look at the problem of turning winning coalitions into losing coalitions.We consider the trade robustness problem for different representations.We give remaining problems related with trade robustness. We analyze the computational complexity of the problem of deciding whether, for a given simple game, there exists the possibility of rearranging the participants in a set of j given losing coalitions into a set of j winning coalitions. We also look at the problem of turning winning coalitions into losing coalitions. We analyze the problem when the simple game is represented by a list of wining, losing, minimal winning or maximal loosing coalitions.