José Miguel Giménez

The modified Banzhaf value for games with coalition structure: an axiomatic characterization

Abstract Different axiomatic systems for the Shapley and Banzhaf values can be found in the literature. For games with a coalition structure, Owen’s modification of the Shapley value (the Owen coalition value) has also been axiomatized in several ways; in this paper we provide an axiomatic characterization for Owen’s modification of the Banzhaf value.

A connectivity game for graphs

Abstract.The introduction of a {0,1}-valued game associated to a connected graph allows us to assign to each vertex a value of weighted connectivity according to the different solutions that for cooperative games are obtained by means of semivalues.The marginal contributions of each vertex to the coalitions differentiate an active connectivity from another reactive connectivity, according to whether the vertex is essential to obtain the connection or is the obstacle for the connection between the vertices in the coalition. We offer general properties of the connectivity, as well as the behaviour of different families of graphs with regard to this concept. We also analyse the effect on different vertices due to the addition of an edge to the initial graph.

Accessibility in oriented networks

The aim of this work consists of allocating a value that allows us to emphasize the importance of each player in a cooperative game when the cooperation possibilities are limited according to the links of an oriented network. The proposed concept of accessibility tries to conjugate the marginal contributions of each node as a game player with the cooperation geometry imposed by the digraph that models the network. We study general properties of this concept and particularly with respect to oriented paths. Concrete applications are proposed.

Accessibility measures to nodes of directed graphs using solutions for generalized cooperative games

The aim of this paper consists of constructing accessibility measures to the nodes of directed graphs using methods of Game Theory. Since digraphs without a predefined game are considered, the main part of the paper is devoted to establish conditions on cooperative games so that they can be used to measure accessibility. Games that satisfy desirable properties are called test games. Each ranking on the nodes is then obtained according to a pair formed by a test game and a solution defined on cooperative games whose utilities are given on ordered coalitions. The solutions proposed here are extensions of the wide family of semivalues to games in generalized characteristic function form.

Partnership formation and multinomial values

We use multinomial values to study the effects of the partnership formation in cooperative games, comparing the joint effect on the involved players with the alternative alliance formation. The simple game case is especially considered and the application to the Catalonia Parliament (Legislature 2003-2007) is also studied.

Power and potential maps induced by any semivalue: Some algebraic properties and computation by multilinear extensions

The notions of total power and potential, both defined for any semivalue, give rise to two endomorphisms of the vector space of cooperative games on any given player set where the semivalue is defined. Several properties of these linear mappings are stated and the role of unanimity games as eigenvectors is described. We also relate in both cases the multilinear extension of the image game to the multilinear extension of the original game. As a consequence, we derive a method to compute for any semivalue by means of multilinear extensions, in the original game and also in all its subgames, (a) the total power, (b) the potential, and (c) the allocation to each player given by the semivalue.

Some properties for probabilistic and multinomial (probabilistic) values on cooperative games

We investigate the conditions for the coefficients of probabilistic and multinomial values of cooperative games necessary and/or sufficient in order to satisfy some properties, including marginal contributions, balanced contributions, desirability relation and null player exclusion property. Moreover, a similar analysis is conducted for transfer property of probabilistic power indices on the domain of simple games.

An axiomatic characterization for regular semivalues

Abstract Semivalues form a wide family of solutions for cooperative games. The payoff that a semivalue allocates to each player in a given game is a weighted sum of the marginal contributions of the player in the game, and the weighting coefficients depend only on the coalition size. In this paper we provide an axiomatic characterization for each semivalue the weighting coefficients of which are all positive (regular semivalue).

On cooperative games, inseparable by semivalues

Abstract.Semivalues like the Shapley value and the Banzhaf value may assign the same payoff vector to different games. It is even possible that two games attain the same outcome for all semivalues. Due to the linearity of the semivalues, this exactly occurs in case the difference of the two games is an element of the kernel of each semivalue. The intersection of these kernels is called the shared kernel, and its game theoretic importance is that two games can be evaluated differently by semivalues if and only if their difference is not a shared kernel element. The shared kernel is a linear subspace of games. The corresponding linear equality system is provided so that one is able to check membership. The shared kernel is spanned by specific {−1,0,1}-valued games, referred to as shuffle games. We provide a basis with shuffle games, based on an a-priori given ordering of the players.

A general procedure to compute mixed modified semivalues for cooperative games with structure of coalition blocks

Semivalues are solution concepts for cooperative games that assign to each player a weighted sum of his/her marginal contributions to the coalitions, where the weights only depend on the coalition size. The Shapley value and the Banzhaf value are semivalues. Mixed modified semivalues are solutions for cooperative games when we consider a priori coalition blocks in the player set. For all these solutions, a computational procedure is offered in this paper.