Josep Freixas

Weighted voting, abstention, and multiple levels of approval

In this paper we introduce the class of simple games with several ordered levels of approval in the input and in the output – the ( j,k) simple games – and propose a definition for weighted games in this context. Abstention is treated as a level of input approval intermediate to votes of yes and no. Our main theorem provides a combinatorial characterization, in terms of what we call grade trade robustness, of weighted ( j,k) games within the class of all ( j,k) simple games. We also introduce other subclasses of ( j,k) simple games and classify several examples. For example, we show the existence of a weighted representation for the UNSC, seen as a voting system in which abstention is permitted.

Complete simple games

Abstract Completeness is a necessary condition for a simple game to be representable as a weighted voting system. This paper deals with the class of complete simple games and centers on their structure. Using an extension of Isbells desirability relation to coalitions, different from the extension normally used, we associate with any complete simple game a lattice of coalition models based upon the types of indifferent players. We establish the basic properties of a vector with natural components and a matrix with non-negative integer entries, both closely related to the lattice, which are also shown to be characteristic invariants of the game, in the sense that they determine it uniquely up to isomorphisms.

The Shapley-Shubik power index for games with several levels of approval in the input and output

Voting systems with several levels of approval in the input and output are considered in this paper. That means games with n ≥ 2 players, j ≥ 2 ordered qualitative alternatives in the input level and k ≥ 2 possible ordered quantitative alternatives in the output. We introduce the Shapley-Shubik power index notion when passing from ordinary simple games or ternary voting games with abstention to this wider class of voting systems. The pivotal role of players is analysed by means of several examples and an axiomatization in the spirit of Shapley and Dubey is given for the proposed power index.

Banzhaf Measures for Games with Several Levels of Approval in the Input and Output

An axiomatic characterization of ‘a Banzhaf score’ notion is provided for a class of games called (j,k) simple games with a numeric measure associated to the output set, i.e., games with n players, j ordered qualitative alternatives in the input level and k possible ordered quantitative alternatives in the output. Three Banzhaf measures are also introduced which can be used to determine a players ‘a priori’ value in such a game. We illustrate by means of several real world examples how to compute these measures.

Reliability Importance Measures of the Components in a System Based on Semivalues and Probabilistic Values

The main contribution of this paper consists in providing different ways to value importance measures for components in a given reliability system or in an electronic circuit. The main tool used is a certain type of semivalues and probabilistic values. One of the results given here extends the indices given by Birnbaum [3] and Barlow and Proschan [2], which respectively coincide with the Banzhaf [1] and the Shapley and Shubik [15] indices so well-known in game theory.

On ordinal equivalence of the Shapley and Banzhaf values for cooperative games

In this paper I consider the ordinal equivalence of the Shapley and Banzhaf values for TU cooperative games, i.e., cooperative games for which the preorderings on the set of players induced by these two values coincide. To this end I consider several solution concepts within semivalues and introduce three subclasses of games which are called, respectively, weakly complete, semicoherent and coherent cooperative games. A characterization theorem in terms of the ordinal equivalence of some semivalues is given for each of these three classes of cooperative games. In particular, the Shapley and Banzhaf values as well as the segment of semivalues they limit are ordinally equivalent for weakly complete, semicoherent and coherent cooperative games.

Semivalues as power indices

Abstract A restricted notion of semivalue as a power index, i.e. as a value on the lattice of simple games, is axiomatically introduced by using the symmetry, positivity and dummy player standard properties together with the transfer property. The main theorem, that parallels the existing statement for semivalues on general cooperative games, provides a combinatorial definition of each semivalue on simple games in terms of weighting coefficients, and shows the crucial role of the transfer property in this class of games. A similar characterization is also given that refers to unanimity coefficients, which describe the action of the semivalue on unanimity games. We then combine the notion of induced semivalue on lower cardinalities with regularity and obtain a series of characteristic properties of regular semivalues on simple games, that concern null and nonnull players, subgames, quotients, and weighted majority games.

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.

On ordinal equivalence of power measures given by regular semivalues

Tomiyama [Tomiyama, Y., 1987. Simple game, voting representation and ordinal power equivalence. International Journal on Policy and Information 11, 67-75] proved that, for every weighted majority game, the preorderings induced by the classical Shapley-Shubik and Penrose-Banzhaf-Coleman indices coincide. He called this property the ordinal equivalence of these indices for weighted majority games. Diffo Lambo and Moulen [Diffo Lambo, L., Moulen, J., 2002. Ordinal equivalence of power notions in voting games. Theory and Decision 53, 313-325] extended Tomiyamas result to all linear (i.e. swap robust) simple games. Here we extend Diffo Lambo and Moulens result to all the preorderings induced by regular semivalues (which include both classical indices) in a larger class of games that we call weakly linear simple games. We also provide a characterization of weakly linear games and use nonsymmetric transitive games to supplying examples of nonlinear but weakly linear games.

On the ordinal equivalence of the Johnston, Banzhaf and Shapley power indices

In this paper, we characterize the games in which Johnston, Shapley–Shubik and Penrose–Banzhaf–Coleman indices are ordinally equivalent, meaning that they rank players in the same way. We prove that these three indices are ordinally equivalent in semicomplete simple games, which is a newly defined class that contains complete games and includes most of the real–world examples of binary voting systems. This result constitutes a twofold extension of Diffo Lambo and Moulen’s result (Diffo Lambo and Moulen, 2002) in the sense that ordinal equivalence emerges for three power indices (not just for the Shapley–Shubik and Penrose–Banzhaf–Coleman indices), and it holds for a class of games strictly larger than the class of complete games.