Resource Buying Games

In resource buying games a set of players jointly buys a subset of a finite resource set E (e.g., machines, edges, or nodes in a digraph). The cost of a resource e depends on the number (or load) of players using e, and has to be paid completely by the players before it becomes available. Each player i needs at least one set of a predefined family S_i in 2^E to be available. Thus, resource buying games can be seen as a variant of congestion games in which the load-dependent costs of the resources can be shared arbitrarily among the players. A strategy of player i in resource buying games is a tuple consisting of one of i's desired configurations S_i together with a payment vector p_i in R^E_+ indicating how much i is willing to contribute towards the purchase of the chosen resources. In this paper, we study the existence and computational complexity of pure Nash equilibria (PNE, for short) of resource buying games. In contrast to classical congestion games for which equilibria are guaranteed to exist, the existence of equilibria in resource buying games strongly depends on the underlying structure of the S_i's and the behavior of the cost functions. We show that for marginally non-increasing cost functions, matroids are exactly the right structure to consider, and that resource buying games with marginally non-decreasing cost functions always admit a PNE

Cooperative Games with Lattice Structure

A general model for cooperative games with possibly restricted and hierarchically ordered coalitions is introduced and shown to have lattice structure under quite general assumptions. Moreover, the core of games with lattice structure is investigated. Within a general framework that includes the model of classical cooperative games as a special case, it is proved algorithmically that monotone convex games have a non-empty core. Finally, the solution concept of the Shapley value is extended to the general class of cooperative games with restricted cooperation. It is shown that several generalizations of the Shapley value that have been proposed in the literature are subsumed in this model

On a relation between the domination number and a strongly connected bidirection of an undirected graph

As a generalization of directed and undirected graphs, Edmonds and Johnson introduced bidirected graphs. A bidirected graph is a graph each arc of which has either two positive end-vertices (tails), two negative end-vertices (heads), or one positive end-vertex (tail) and one negative end-vertex (head). We extend the notion of directed paths, distance, diameter and strong connectivity from directed to bidirected graphs and characterize those undirected graphs that allow a strongly connected bidirection. Considering the problem of finding the minimum diameter of all strongly connected bidirections of a given undirected graph, we generalize a result of Fomin et al. about directed graphs and obtain an upper bound for the minimum diameter which depends on the minimum size of a dominating set and the number of bridges in the undirected graph

Lattice polyhedra and submodular flows

Lattice polyhedra, as introduced by Gröflin and Hoffman, form a common framework for various discrete optimization problems. They are specified by a lattice structure on the underlying matrix satisfying certain sub- and supermodularity constraints. Lattice polyhedra provide one of the most general frameworks of total dual integral systems. So far no combinatorial algorithm has been found for the corresponding linear optimization problem. We show that the important class of lattice polyhedra in which the underlying lattice is of modular characteristic can be reduced to the Edmonds–Giles polyhedra. Thus, submodular flow algorithms can be applied to this class of lattice polyhedra. In contrast to a previous result of Schrijver, we do not explicitly require that the lattice is distributive. Moreover, our reduction is very simple in that it only uses an arbitrary maximal chain in the lattice

Note on Representations of Ordered Semirings

The article studies ordered semigroups and semirings with respect to their representations in lattices. Such structures are essentially the pseudolattices of Dietrich and Hoffman. It is shown that a subadditive representation implies the semigroup to be a lattice in its own right. In particular, distributive lattices can be characterized as semirings admitting subadditive supermodular representations. The cover problem asks for a minimal cover of a ground set by representing sets with respect to a semiring. A greedy algorithm is exhibited to solve the cover problem for the class of lattices with weakly subadditive and supermodular representation

Note on Pseudolattices, Lattices and Submodular Linear Programs

A pseudolattice L is a poset with lattice-type binary operations. Assuming that the pseudolattice permits a modular representation as a family of subsets of a set U with certain compatibility properties, we show that L actually is a distributive lattice with the same supremum operation. Given a submodular function r:L o R , we prove that the corresponding unrestricted linear program relative to the representing set family can be solved by a greedy algorithm. This complements the Monge algorithm of Dietrich and Hoffman for the associated dual linear program. We furthermore show that our Monge and greedy algorithm is generally optimal for nonnegative submodular linear programs and their duals (relative to L )