An algebraic framework for the greedy algorithm with applications to the core and Weber set of cooperative games

An algebraic model generalizing submodular polytopes is presented, where modular functions on partially ordered sets take over the role of vectors in ${\mathbb R}^n$. This model unifies various generalizations of combinatorial models in which the greedy algorithm and the Monge algorithm are successful and generalizations of the notions of core and Weber set in cooperative game theory. As a further application, we show that an earlier model of ours as well as the algorithmic model of Queyranne, Spieksma and Tardella for the Monge algorithm can be treated within the framework of usual matroid theory (on unordered ground-sets), which permits also the efficient algorithmic solution of the intersection problem within this model. \u

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.

Capacities and Games on Lattices: A Survey of Result

We provide a survey of recent developments about capacities (or fuzzy measures) and ccoperative games in characteristic form, when they are defined on more general structures than the usual power set of the universal set, namely lattices. In a first part, we give various possible interpretations and applications of these general concepts, and then we elaborate about the possible definitions of usual tools in these theories, such as the Choquet integral, the Möbius transform, and the Shapley value.capacity, fuzzy measure, game, lattice, Choquet integral,Shapley value

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

Solution Concepts for Games with General Coalitional Structure (Replaces CentER DP 2011-025)

We introduce a theory on marginal values and their core stability for cooperative games with arbitrary coalition structure. The theory is based on the notion of nested sets and the complex of nested sets associated to an arbitrary set system and the M-extension of a game for this set. For a set system being a building set or partition system, the corresponding complex is a polyhedral complex, and the vertices of this complex correspond to maximal strictly nested sets. To each maximal strictly nested set is associated a rooted tree. Given characteristic function, to every maximal strictly nested set a marginal value is associated to a corresponding rooted tree as in [9]. We show that the same marginal value is obtained by using the M-extension for every permutation that is associated to the rooted tree. The GC-solution is defined as the average of the marginal values over all maximal strictly nested sets. The solution can be viewed as the gravity center of the image of the vertices of the polyhedral complex. The GC-solution differs from the Myerson-kind value defined in [2] for union stable structures. The HS-solution is defined as the average of marginal values over the subclass of so-called half-space nested sets. The NT-solution is another solution and is defined as the average of marginal values over the subclass of NT-nested sets. For graphical buildings the collection of NT-nested sets corresponds to the set of spanning normal trees on the underlying graph and the NT-solution coincides with the average tree solution. We also study core stability of the solutions and show that both the HS-solution and NT-solution belong to the core under half-space supermodularity, which is a weaker condition than convexity of the game. For an arbitrary set system we show that there exists a unique minimal building set containing the set system. As solutions we take the solutions for this building covering by extending in a natural way the characteristic function to it by using its Möbius inversion.Core;polytope;building set;nested set complex;Möbius inversion;permutations;normal fan;average tree solution;Myerson value

The core of games on k-regular set systems

In the classical setting of cooperative game theory, it is always assumed that all coalitions are feasible. However in many real situations, there are restrictions on the set of coalitions, for example duo to communication, order or hierarchy on the set of players, etc. There are already many works dealing with games on restricted set of coalitions, defining many different structures for the set of feasible coalitions, called set systems. We propose in this paper to consider k-regular set systems, that is, set systems having all maximal chains of the same length k. This is somehow related to communication graphs. We study in this perspective the core of games defined on k-regular set systems. We show that the core may be unbounded and without vertices in some situations.Cooperative game ; feasible coalition ; core

The core of games on distributive lattices : how to share benefits in a hierarchy

Finding a solution concept is one of the central problems in cooperative game theory, and the notion of core is the most popular solution concept since it is based on some rationality condition. In many real situations, not all possible coalitions can form, so that classical TU-games cannot be used. An interesting case is when possible coalitions are defined through a partial ordering of the players (or hierarchy). Then feasible coalitions correspond to teams of players, that is, one or several players with all their subordinates. In these situations, it is not obvious to define a suitable notion of core, reflecting the team structure, and previous attempts are not satisfactory in this respect. We propose a new notion of core, which imposes efficiency of the allocation at each level of the hierarchy, and answers the problem of sharing benefits in a hierarchy. We show that the core we defined has properties very close to the classical case, with respect to marginal vectors, the Weber set, and balancedness.Cooperative game, feasible coalition, core, hierarchy.

Values on regular games under Kirchhoff’s laws

In cooperative game theory, the Shapley value is a central notion defining a rational way to share the total worth of a game among players. In this paper, we address a general framework leading to applications to games with communication graphs, where the set of feasible coalitions forms a poset where all maximal chains have the same length. We first show that previous definitions and axiomatizations of the Shapley value proprosed by Faigle and Kern, and Bilbao and Edelman still work. Our main contribution is then to propose a new axiomatization avoiding the hierarchical strength axiom of Faigle and Kern, and considering a new way to define the symmetry among players. Borrowing ideas from electric networks theory, we show that our symmetry axiom and the classical efficiency axiom correspond actually to the two Kirchhoff’s laws in the resistor circuit associated to the Hasse diagram of feasible coalitions. We finally work out a weak form of the monotonicity axiom which is satisfied by the proposed value.Regular set systems; regular games; Shapley value; probabilistic efficient values; regular values; Kirchhoff’s laws.