Games with a permission structure - A survey on generalizations and applications

Abstract

In the field of cooperative games with restricted cooperation, various restrictions on coalition formation are studied. The most studied restrictions are those that arise from restricted communication and hierarchies. This survey discusses several models of hierarchy restrictions and their relation with communication restrictions. In the literature, there are results on game properties, Harsanyi dividends, core stability, and various solutions that generalize existing solutions for TU-games. In this survey, we mainly focus on axiomatizations of the Shapley value in different models of games with a hierarchically structured player set, and their applications. Not only do these axiomatizations provide insight in the Shapley value for these models, but also by considering the types of axioms that characterize the Shapley value, we learn more about different network structures. A central model of games with hierarchies is that of games with a permission structure where players in a cooperative transferable utility game are part of a permission structure in the sense that there are players that need permission from other players before they are allowed to cooperate. This permission structure is represented by a directed graph. Generalizations of this model are, for example, games on antimatroids, and games with a local permission structure. Besides discussing these generalizations, we briefly discuss some applications, in particular auction games and hierarchically structured firms.This invited paper is discussed in the comments available at

The priority value for cooperative games with a priority structure

“…From a mathematical point of view, our model is identical to the acyclic permission structures introduced in Gilles et al (1992) and the structures modeling precedence constraints introduced in Faigle and Kern (1992), even if these structures are sometimes defined by means of digraphs. These two models have been extensively studied in the literature (van den Brink, 2017;Algaba et al, 2017, survey some results) and have in common that the associated structure imposes some restriction on the formation of coalitions or on the subset of coalitions to which an allocation rule is sensitive. In van den Brink and Gilles (1996), van den Brink (1997) and van den Brink and Dietz (2014), the so-called conjunctive, disjunctive and local permission values are computed as the Shapley value of a restricted game in which only the "feasible" part of a coalition is productive, where the "feasible" part is the largest subset of the coalition that contains all the hierarchical superiors deemed necessary for the worth generation (these vary depending on the model: conjunctive, disjunctive or local).…”

Comments on: Games with a permission structure - A survey on generalizations and applications

“…Comments on: Games with a Permission Structure * Juan Vidal-Puga † Published in Top ‡ Van den Brink (2017) presents an excellent survey on how a permission structure affects the sharing of benefits from cooperation among a finite set of players. The survey covers from the simplest and most intuitive permission structure, given by a directed graph, to a more general one, given by the abstract concept of normal antimatroid (Dilworth, 1940).…”