Network Structures with Hierarchy and Communication

Abstract

Agents, participating in different kind of organizations, usually take different positions in some network structure. Two well-known network structures are hierarchies and communication networks. This paper aims at introducing a new type of network structure having both communication and hierarchical features. We describe a network by a collection of feasible sets, being the sets of network positions (nodes), that can organize themselves and act as a group. We introduce a new type of network structure, that has communication as well as hierarchical features. We compare these new network structures with other structures from the literature, and study its basis, i.e. the 'smallest' representation of the network. Finally, we apply these new network structures to cooperative games, where cooperation is restricted by some network structure, and provide an axiomatization of an extension of the Shapley value to this class of games.

Harsanyi power solutions for cooperative games on voting structures

“…For this special subclass of union stable systems the characteristic of power is contained in the implicit interpretation of the winning coalitions in a voting game. Moreover, the interpretation of power in these systems is quite different from the one given by accessible union stable systems (Algaba et al, 2018) which combines communication and hierarchical properties. Namely, these set systems could be also studied taking into account only the feature of power which is in contrast with the system of feasible coalitions derived from an acyclic directed graph by the conjunctive approach (Gilles et al, 1992) or under precedence constraints (Faigle and Kern, 1992).…”

“…On GU S, Algaba et al (2015) provide axiomatic characterizations of the Harsanyi power solutions and of two well-known Harsanyi power solutions, namely the Myerson value (Myerson, 1977) and the position value (Meessen, 1988). As the latter two Harsanyi power solutions have been initially defined for communication graph games, union stable systems are mainly viewed as generalizations of communication graphs or networks (see also Algaba et al, 2018). In the same way, the proposed power measures are generalizations of power measures for graphs (the degree measure, the equal power measure, etc).…”

The Shapley Value and Games with Hierarchies

“…The conjunctive permission value satisfies efficiency and linearity 11 . Although the conjunctive permission value does not satisfy the weak hierarchical strength axiom, it satisfies a version of the p-strength axiom, where in the unanimity game of the 'grand coalition', the payoffs are allocated equally over the players, i.e.…”

Cooperative games on intersection closed systems and the Shapley value

“…It remains to prove that Ω S = sup Q∈max Ω S Q in equality in (4). It suffices to note that, for each T ∈ 2 N \ ∅, c(T ) ⊆ S if and only if there is Q ∈ max Ω S such that Q ⊇ T .…”

“…Since then, many other set systems have been used to model restricted cooperation: Algaba et al (2000,2001) study cooperative games on union stable systems, Algaba et al (2004) consider TU-games on antimatroids, Bilbao (1998) and Bilbao and Edelman (2000) introduce TU-games on convex geometries, Bilbao (2003) studies TUgames on augmenting set systems, and Lange and Grabisch (2009) consider TU-games on regular set systems. More recently, van den Brink et al (2011) consider TU-games on union closed systems and Algaba et al (2018) introduce TU-games on accessible union stable systems.…”