Esperanza A. Lebrón

The core of games on convex geometries

Abstract A game on a convex geometry is a real-valued function defined on the family L of the closed sets of a closure operator which satisfies the finite Minkowski–Krein–Milman property. If L is the Boolean algebra 2N then we obtain a n-person cooperative game. We will introduce convex and quasi-convex games on convex geometries and we will study some properties of the core and the Weber set of these games.

The Shapley value for games on matroids: The dynamic model

Abstract. According to the work of Faigle [3] a static Shapley value for games on matroids has been introduced in Bilbao, Driessen, Jiménez-Losada and Lebrón [1]. In this paper we present a dynamic Shapley value by using a dynamic model which is based on a recursive sequence of static models. In this new model for games on matroids, our main result is that there exists a unique value satisfying analogous axioms to the classical Shapley value. Moreover, we obtain a recursive formula to calculate this dynamic Shapley value. Finally, we prove that its components are probabilistic values.

Games with fuzzy authorization structure

A cooperative game consists of a set of players and a characteristic function which determines 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. It is often assumed that the players are free to participate in any coalition, but in some situations there are dependency relationships among the players that restrict their capacity to cooperate within some coalitions. Those relationships must be taken into account if we want to distribute the profits fairly. In this respect, several models have been proposed in literature. In all of them dependency relationships are considered to be complete, in the sense that either a player is allowed to fully cooperate within a coalition or they cannot cooperate at all. Nevertheless, in some situations it is possible to consider another option: that a player has a degree of freedom to cooperate within a coalition. A model for those situations is presented.

On Aitken-Neville Formulae for Multivariate Interpolation

In a paper in J.S.I.A.M., 8, (1960), p.33–42, Thacher and Mil ne showed how the solution of certain polynomial interpolation problems in R s can be constructed from the solutions of s+1 simpler problems. In the present paper we extend the method showing that the number of basic problems to be used depends on the distribution of the points and on the chosen interpolation space, which here is not necessarily a polynomial space.

Games with fuzzy permission structure: A conjunctive approach

Abstract A cooperative game consists of a set of players and a characteristic function which determines 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. A permission structure over the set of players describes a hierarchical organization where there are players who need permission from certain other players before they are allowed to cooperate with others. Various assumptions can be made about how a permission structure affects the cooperation possibilities. In the conjunctive approach it is assumed that each player needs permission from all his superiors. This paper deals with fuzzy permission structures in the conjunctive approach. In this model, players could depend partially on other players, that is, they may have certain degree of autonomy. First, we define a value for games with fuzzy permission structure that only takes into account the direct relations among players and provide a characterization for this value. Finally, we study a value for games with fuzzy permission structure which takes account of the indirect relations among players.

Theτ-value for games on matroids

In the classical model of games with transferable utility one assumes that each subgroup of players can form and cooperate to obtain its value. However, we can think that in some situations this assumption is not realistic, that is, not all coalitions are feasible. This suggests that it is necessary to raise the whole question of generalizing the concept of transferable utility game, and therefore to introduce new solution concepts. In this paper we define games on matroids and extend theτ-value as a compromise value for these games.

Probabilistic Values on Convex Geometries

A game on a convex geometry is a real-valued function defined on the family L of the closed sets of a closure operator which satisfies the finite Minkowski-Krein-Milmanproperty. If L is the Boolean algebra 2N, then we obtain an n-person cooperative game. We will extend the work of Weber on probabilistic values to games on convex geometries. As a result, we obtain a family of axioms that give rise to several probabilistic values and a unique Shapley value for games on convex geometries.

Values for Interior Operator Games

The aim of this paper is to study a new class of cooperative games called interior operator games. These games are additive games restricted by antimatroids. We consider several types of cooperative games as peer group games, big boss games, clan games and information market games and show that all of them are interior operator games. Next, we analyze the properties of these games and compute the Shapley, Banzhaf and Tijs values.

The selectope for games with partial cooperation

Abstract In this paper we define and study some solution concepts for games in which there does not have to be total cooperation between the players. In particular, we show the relations of these solution concepts with the selectope. In this way, we extend the work of Derks, Haller and Peters, METEOR Research Memorandum, Maastricht, RM/97/016 (revised version, 1998).

Convexity properties for interior operator games

Abstract Interior operator games arose by abstracting some properties of several types of cooperative games (for instance: peer group games, big boss games, clan games and information market games). This reason allow us to focus on different problems in the same way. We introduced these games in Bilbao et al. (Ann. Oper. Res. 137:141–160, 2005) by a set system with structure of antimatroid, that determines the feasible coalitions, and a non-negative vector, that represents a payoff distribution over the players. These games, in general, are not convex games. The main goal of this paper is to study under which conditions an interior operator game verifies other convexity properties: 1-convexity, k-convexity (k≥2 ) or semiconvexity. But, we will study these properties over structures more general than antimatroids: the interior operator structures. In every case, several characterizations in terms of the gap function and the initial vector are obtained. We also find the family of interior operator structures (particularly antimatroids) where every interior operator game satisfies one of these properties.