81 research outputs found
Cooperative Games on Antimatroids
AMS classification: 90D12;game theory;cooperative games;antimatroids
Entropy of capacities on lattices and set systems
We propose a definition for the entropy of capacities defined on lattices.
Classical capacities are monotone set functions and can be seen as a
generalization of probability measures. Capacities on lattices address the
general case where the family of subsets is not necessarily the Boolean lattice
of all subsets. Our definition encompasses the classical definition of Shannon
for probability measures, as well as the entropy of Marichal defined for
classical capacities. Some properties and examples are given
On Hierarchies and Communication
Many economic organizations have some relational structure, meaning that economic agents do not only differ with respect to certain individual characteristics such as wealth and preferences, but also belong to some relational structure in which they usually take different positions. Two examples of such structures are communication networks and hierarchies. In the literature the distinction between these two types of relational structures is not always clear. In models of restricted cooperation this distinction should be defined by properties of the set of feasible coalitions. We characterize the feasible sets in communication networks and compare them with feasible sets arising from hierarchies
Cooperative games under augmenting systems
The goal of this paper is to develop a theoretical framework inorder to analyze cooperative
games inwhic h only certaincoalition s are allowed to form. We will axiomatize the structure
of such allowable coalitions using the theory of antimatroids, a notion developed for combinatorially
abstract sets. 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.
We introduce a new combinatorial structure called augmenting system, which is a generalization of
the antimatroid structure and the system of connected subgraphs of a graph. The main result of
the paper is a direct formula of Shapley and Banzhaf values for games under augmenting systems
restrictions
Monge extensions of cooperation and communication structures
Cooperation structures without any {\it a priori} assumptions on the combinatorial structure of feasible coalitions are studied and a general theory for mar\-ginal values, cores and convexity is established. The theory is based on the notion of a Monge extension of a general characteristic function, which is equivalent to the Lovász extension in the special situation of a classical cooperative game. It is shown that convexity of a cooperation structure is tantamount to the equality of the associated core and Weber set. Extending Myerson's graph model for game theoretic communication, general communication structures are introduced and it is shown that a notion of supermodularity exists for this class that characterizes convexity and properly extends Shapley's convexity model for classical cooperative games.
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