Marginal Worth Vectors Research Articles

A note on external angles of the core of convex TU games, marginal worth vectors and the Weber set

The paper studies the structure of external angles of the core of convex TU games. It is shown that the external angle at a vertex of the core is proportional to the number of marginal worth vectors defining the vertex. A notion of difference of two convex compact sets is used to define the Weber set of a TU game and to give another proof of the Weber theorem.

Cooperative Games with Size-Truncated Information

We study the marginal worth vectors and their convex hull, the so-called Weber set, from the original coalitional game and the transformed one, which is called the Weber set of level k. We prove that the core of the original game is included in each of the Weber set of level k, for any k, and that the Weber sets of consecutive levels form a chain if and only if the original game is 0-monotone. Even if the game is not 0-monotone, the intersection of the Weber sets for consecutive levels is always not empty, what is not the case for non-consecutive ones. Spanish education system.

A new approach to the core and Weber set of multichoice games

Multichoice games have been introduced by Hsiao and Raghavan as a generalization of classical cooperative games. An important notion in cooperative game theory is the core of the game, as it contains the rational imputations for players. We propose two definitions for the core of a multichoice game, the first one is called the precore and is a direct generalization of the classical definition. We show that the precore coincides with the definition proposed by Faigle, and that the set of imputations may be unbounded, which makes its application questionable. A second definition is proposed, imposing normalization at each level, causing the core to be a convex compact set. We study its properties, introducing balancedness and marginal worth vectors, and defining the Weber set and the pre-Weber set. We show that the classical properties of inclusion of the (pre)core into the (pre)-Weber set as well as their coincidence in the convex case remain valid. A last section makes a comparison with the core defined by van den Nouweland et al.

THE MARGINAL OPERATORS FOR GAMES ON CONVEX GEOMETRIES

In this work we study situations in which communication among the players is not complete and it is represented by a family of subsets of the set of players. Although several models of partial cooperation have been proposed, we shall follow a model derived from the work of Faigle and Kern. We define the games on convex geometries and introduce marginal worth vectors and quasi-supermodular games. Furthermore, we analyze some properties of the marginal operators on the space of games on convex geometries.

Characterization of the extreme core allocations of the assignment game

A characterization of the extreme core allocations of the assignment game is given in terms of the reduced marginal worth vectors. For each ordering in the player set, a payoff vector is defined where each player receives his or her marginal contribution to a certain reduced game played by his or her predecessors. This set of reduced marginal worth vectors, which for convex games coincide with the usual marginal worth vectors, is proved to be the set of extreme points of the core of the assignment game. Therefore, although assignment games are hardly ever convex, the same characterization of extreme core allocations is valid for convex games.

On extreme points of the core and reduced games

Given a balanced cooperative game, we prove that the extreme points of the core have the reduced game property with respect to the Davis and Maschler reduced game. One particular case of this reduction gives when we name marginal games. These games allow us to define the reduced marginal worth vectors, where every player gets his marginal contribution to a successive marginal game. This set of vectors is proved to be the set of extreme points of the core of those balanced games which are almost convex, that is, those balanced games such that all proper subgames are convex.

Even and odd marginal worth vectors, Owen's multilinear extension and convex games

In this paper we characterize convex games by means of Owen's multilinear extension and the marginal worth vectors associated with even or odd permutations.

Solution concepts ofk-convexn-person games

For any positive integersk andn, the subclass ofk-convexn-person games is considered. In casek=n, we are dealing with convexn-person games. Three characterizations ofk-convexn-person games, formulated in terms of the core and certain adapted marginal worth vectors, are given. Further it is shown that fork-convexn-person games the intersection of the (pre)kernel with the core consists of a unique point (namely the nucleolus), but that the (pre)kernel may contain points outside the core.

A note on a core catcher of a cooperative game

In Driessen (1986) it is shown that for games satisfying a certain condition the core of the game is included in the convex hull of the set of certain marginal worth vectors of the game, while it is conjectured that the inclusion holds without any condition on the game. In this note it is proved that the inclusion holds for all games.

K-convexn-person games and their cores

For any natural numbersk andn, the subclass ofk-convexn-person games is introduced. In casek=n, the subclass consists of the convexn-person games. Ak-convexn-person game is characterized in several ways in terms of the core and certain marginal worth vectors. The marginal worth vectors of a game are described in terms of an upper bound for the core and the corresponding gap function. It is shown that thek-convexity of ann-person gamev is equivalent to Examples ofk-convexn-person games are also treated.

Super-modularity: Applications to convex games and to the greedy algorithm for LP

L. S. Shapley ( Internat. J. Game Theory 1 (1971) , 11–26) showed that if a game is convex, then all the marginal worth vectors are members of the core. J. Edmonds ( in “Combinatorial Structures and Their Applications” ( R. Guy et al., Eds.), pp. 69–87, Gordon & Breach, New York, 1970 ) considered a class of linear programming problems for which the greedy algorithm works. The present paper unifies these studies, and establishes the converse of each theorem.