Manuel Ordóñez

Axiomatizations of the Shapley value for games on augmenting systems

This paper deals with cooperative games in which only certain coalitions are allowed to form. There have been previous models developed to confront the problem of unallowable coalitions. Games restricted by a communication graph were introduced by Myerson and Owen. In their model, the feasible coalitions are those that induce connected subgraphs. Another type of model is introduced in Gilles, Owen and van den Brink. In their model, the possibilities of coalition formation are determined by the positions of the players in a so-called permission structure. Faigle proposed another model for cooperative games defined on lattice structures. We introduce a combinatorial structure called augmenting system which is a generalization of the antimatroid structure and the system of connected subgraphs of a graph. In this framework, the Shapley value of games on augmenting systems is introduced and two axiomatizations of this value are showed.

Clifford Theory: A Geometrical Interpretation of Multivectorial Apparent Power

In this paper, a generalization of the concept of electrical power for periodic current and voltage waveforms based on a new generalized complex geometric algebra (GCGA) is proposed. This powerful tool permits, in n-sinusoidal/nonlinear situations, representing and calculating the voltage, current, and apparent power in a single-port electrical network in terms of multivectors. The new expressions result in a novel representation of the apparent power, similar to the Steinmetzs phasor model, based on complex numbers, but limited to the purely sinusoidal case. The multivectorial approach presented is based on the frequency-domain decomposition of the apparent power into three components: the real part and the imaginary part of the complex-scalar associated to active and reactive power respectively, and distortion power, associated to the complex-bivector. A geometrical interpretation of the multivectorial components of apparent power is discussed. Numerical examples illustrate the clear advantages of the suggested approach.

Games on fuzzy communication structures with Choquet players

Myerson (1977) used graph-theoretic ideas to analyze cooperation structures in games. In his model, he considered the players in a cooperative game as vertices of a graph, which undirected edges defined their communication possibilities. He modified the initial games taking into account the graph and he established a fair allocation rule based on applying the Shapley value to the modified game. Now, we consider a fuzzy graph to introduce leveled communications. In this paper players play in a particular cooperative way: they are always interested first in the biggest feasible coalition and second in the greatest level (Choquet players). We propose a modified game for this situation and a rule of the Myerson kind.

Geometric algebra: a multivectorial proof of Tellegen's theorem in multiterminal networks

A generalised and multivectorial proof of Tellegens theorem in multiterminal systems is presented using a new power multivector concept defined in the frequency domain. This approach permits in nonsinusoidal/linear and nonlinear situations formulating Tellegens theorem in a novel complex-multivector representation, similar to Steinmetzs phasor model, based on complex numbers and limited to the purely sinusoidal case. In this sense, a suitable notation of voltage and current complex-vectors, associated to the elements and nodes of the network, is defined for easy development to Kirchhoffs laws in this environment. A numerical example illustrates the clear advantages of the suggested proof.

Myerson values for games with fuzzy communication structure

In 1977, Myerson considered cooperative games with communication structure. A communication structure is an undirected graph describing the bilateral relationships among the players. He introduced the concept of allocation rule for a game as a function obtaining an outcome for each communication structure among the players of the game. The Myerson value is a specific allocation rule extending the Shapley value of the game. More recently, the authors studied games with fuzzy communication structures using fuzzy graph-theoretic ideas. Now we propose a general framework in order to define fuzzy Myerson values. Players in a coalition need to measure their profit using their real individual and communication capacities at every moment because these attributes are fuzzy when the game is proposed. So, they look for forming connected coalitions working at the same level. The different ways to obtain these partitions by levels determine different Myerson values for the game. Several interesting examples of these ways are studied in the paper, following known models in games with fuzzy coalitions: the proportional model and the Choquet model.

The core and the Weber set of games on augmenting systems

This paper deals with cooperative games in which only certain coalitions are allowed to form. There have been previous models developed to confront the problem of unallowable coalitions. Games restricted by a communication graph were introduced by Myerson and Owen. In their model, the feasible coalitions are those that induce connected subgraphs. Another type of model is introduced in Gilles, Owen and van den Brink. In their model, the possibilities of coalition formation are determined by the positions of the players in a so-called permission structure. Faigle proposed a general model for cooperative games defined on lattice structures. In this paper, the restrictions to the cooperation are given by a combinatorial structure called augmenting system which generalizes antimatroid structure and the system of connected subgraphs of a graph. In this framework, the core and the Weber set of games on augmenting systems are introduced and it is proved that monotone convex games have a non-empty core. Moreover, we obtain a characterization of the convexity of these games in terms of the core of the game and the Weber set of the extended game.

Cooperation among agents with a proximity relation

A cooperative game consists of a set of players and a characteristic function determining the maximal gain or minimal cost that every subset of players can achieve when they decide to cooperate, regardless of the actions that the other players take. The relationships of closeness among the players should modify the bargaining among them and therefore their payoffs. The first models that have studied this closeness used a priori unions or undirected graphs. In the a priori union model a partition of the big coalition is supposed. Each element of the partition represents a group of players with the same interests. The groups negotiate among them to form the grand coalition and later, inside each one, players bargain among them. Now we propose to use proximity relations to represent leveled closeness of the interests among the players and extending the a priori unions model.

The geometric algebra as a power theory analysis tool

In this paper, a multivectorial decomposition of power equation in single-phase circuits for periodic n-sinusoidal /linear and nonlinear conditions is presented. It is based on a frequency-domain Clifford vector space approach. By using a new generalized complex geometric algebra (GCGA), we define the voltage and current complex-vector and apparent power multivector concepts. First, the apparent power multivector is defined as geometric product of vector-phasors (complex-vectors). This new expression result in a novel representation and generalization of the apparent power similar to complex-power in single-frequency sinusoidal conditions. Second, in order to obtain a multivectorial representation of any proposed power equation, the current vector-phasor is decomposed into orthogonal components. The power multivector concept, consisting of complex-scalar and complex-bivector parts with magnitude, direction and sense, obeys the apparent power conservation law and it handles different practical electric problems where direction and sense are necessary. The results of numerical examples are presented to illustrate the proposed approach to power theory analysis.

A Banzhaf value for games with fuzzy communication structure: Computing the power of the political groups in the European Parliament

Abstract In 2013, Jimenez-Losada et al. introduced several extensions of the Myerson value for games with fuzzy communication structure. In a fuzzy communication structure the membership of the players and the relations among them are leveled. Now we study a Banzhaf value for these situations. The Myerson model is followed to define the fuzzy graph Banzhaf value taking as base point the Choquet integral. We propose an axiomatization for this value introducing leveled amalgamation of players. An algorithm to calculate this value is provided and its complexity is studied. Finally we show an applied example computing by this fuzzy value the power of the groups in the European Parliament.

Duality on combinatorial structures. An application to cooperative games

Abstract This paper deals with cooperative games in which only certain coalitions are feasible. Several combinatorial structures have been used to represent feasible coalition systems for different situations: poset of ideals, connected sets in graphs, convex geometries or antimatroids. Augmenting systems are combinatorial structures generalizing antimatroids defined in the framework of the game theory. In this paper, we introduced decreasing systems as dual structures of the augmenting systems. We connect the classical concept of dual game with the duality between these structures. The Shapley and the Banzhaf values are studied and compared for augmenting and decreasing systems.