Monge extensions of cooperation and communication structures

Abstract

Cooperation structures without any a priori assumptions on the combinatorial structure of feasible coalitions are studied and a general theory for marginal 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.

The Myerson Value and Superfluous Supports in Union Stable Systems

“…Furthermore, these systems have a close relation with the hypergraph communication situations [4]. Important properties on the class of union stable systems have been studied by Faigle et al [11] who in this framework …nd a meaningful notion of supermodularity that extends Shapley´s convex cooperative model.…”

The core of games on ordered structures and graphs

“…We recall only some basic facts useful for the sequel (see, e.g., Ziegler 1995;Faigle et al 2002 for details). Our exposition mainly follows Fujishige (2005, §1.2).…”

A ranking model for the greedy algorithm and discrete convexity

“…For example, the trivial order on F (with no comparable pairs of elements) is consecutive. Also the set-theoretic containment order (F, ⊆) is consecutive (and yields the combinatorial model investigated in [8]). Often more refined consecutive orderings are "natural".…”

“…We then show that the greedy algorithm is guaranteed to be successful for one type of linear program if and only if it solves the other type as well (Theorem 5.1). Generalizing the approach of Fujishige [13] and the Monge extensions of the game-theoretic model of Faigle et al [8], our model associates with the greedy algorithm a certain functional, which turns out to be concave if and only if the greedy algorithm works. This greedy functional generalizes the so-called Lovász extension of a set function (which coincides with Choquet's [2] discrete integral).…”