Games on lattices, multichoice games and the shapley value: a new approach

Abstract

Multichoice games, as well as many other recent attempts to generalize the notion of classical cooperative game, can be casted into the framework of lattices. We propose a general definition for games on lattices, together with an interpretation. Several definitions of the Shapley value of a multichoice games have already been given, among them the original one due to Hsiao and Raghavan, and the one given by Faigle and Kern. We propose a new approach together with its axiomatization, more in the spirit of the original axiomatization of Shapley, and avoiding a high computational complexity.

A model of influence in a social network

“…An interesting generalization would be also to assume that the influence function B is a probabilistic function. Another improvement could be to enlarge the set of possible decisions to games on product lattices (Grabisch and Lange, 2007). Moreover, we would like to provide an axiomatic characterization of the influence indices.…”

Axiomatization of an Importance Index for Generalized Additive Independence Models

“…Hence, we are facing here a double difficulty: to propose a "value" both valid for k ≥ 1 and for nonincreasing models. There have been many proposed values for multichoice games, e.g., Hsiao and Raghavan [11], van den Nouweland et al [17], Klijn et al [12], Peters and Zank [14] and Grabisch and Lange [10], etc. All of them satisfy the classical efficiency axiom.…”

Monge extensions of cooperation and communication structures

“…In Myerson's [25] communication graph model, for example, only those sets of agents are feasible for communication that induce connected subgraphs. Other examples arise from models where N is (partially) ordered by some dominance or preference relation (e.g., Derks and Gilles [8], Faigle and Kern [14,15], Gilles et al [17], Grabisch and Lange [18], Hsiao and Raghavan [19]). The latter model was further relaxed and studied by Algaba et al [3], Bilbao et al [2,5] to combinatorial coalition structures of so-called antimatroids, convex geometries and augmenting systems, and by Lange and Grabisch to regular set systems [22].…”