Francesc Carreras

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.

A comparative axiomatic characterization of the Banzhaf-Owen coalitional value

A compact axiomatic characterization of the modified Banzhaf value for games with a coalition structure (Banzhaf-Owen value, for short) is provided. The axiomatic system used here can be compared with parallel axiomatizations of other coalitional values such as the Owen value or the Alonso-Fiestras value, thus giving arguments to defend the use of one of them that will depend on the context where they are to be applied.

Evaluation of the Catalonian Parliament, 1980-1984

Abstract We analyze here the process of coalition in the Catalonian Parliament. Using Owens modification of the Shapley value (for games with a priori unions), we show that the actual result was not unreasonable, given the underlying political structure.

The multilinear extension and the modified Banzhaf-Coleman index

Abstract Owen suggests a modification of the Banzhaf-Coleman index for games where a coalition structure is given. We provide a procedure to obtain this new index by using the multilinear extension of the original game and we apply it to real-world examples.

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 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.

A decisiveness index for simple games

Abstract The decisiveness index introduced in this paper is designed to provide a normalized measure of the agility of all simple games, primarily viewed as collective decision-making mechanisms. We study the mathematical properties of the index and derive different axiomatic characterizations for it. Moreover, a close relationship is shown to the Banzhaf index of power––for which twice the decisiveness index plays the role of potential function––that gives rise to an effective computational procedure. Some real-world examples illustrate the usefulness of the decisiveness index, together with the Banzhaf power index, in applications to political science.

Games and cooperation indices

We provide a model for quantitatively restricted cooperation. It leads to a modification of both the game and the Shapley value whose properties are studied, giving rise to an axiomatic characterization. An individually rational extension of the Myerson value is included, as well as a description of several former restriction methods that are shown to be generalized by our model.

Semivalue Versatility and Applications

Semivalues are shown to exhibit a capability of modification that enables us to introduce additional information in the evaluation of games. After using a general method to design modified versions of the Shapley and Banzhaf values, we apply them to some political, economic and sociological problems.

Cooperation indices and coalitional value

SummaryWe introduce an allocation rule for situations defined by a TUgame, a cooperation index and a coalition structure, and characterize it axiomatically. This rule is an extension of Owens coalitional value; in fact, a variety of previously studied game situations is shown to be embodied and unified by our model.