Narrow Results

In this work, we explore solutions for partition function form games, where our basic approach is rooted in a concept of null player, derived directly from the carrier concept of Myerson [(1977) Int. J. Game Theory6(1), 23–31]. We then show that the Myerson value cannot be characterized with the standard translation of Shapley’s axioms of linearity, symmetry, efficiency and null-player to partition function form games. Instead, we provide an axiomatic characterization for the family of solutions for partition function form games satisfying those axioms, and we show that every such solution is a linear combination of specific marginal contributions of players.

The theory of cooperative games with restricted cooperation has been rapidly developing over the last decades. In our study, we present a special game with restricted cooperation — a game with a major player — a modified version of the landlord game presented in Moulin [1988]. Cooperation of players is supposed to be restricted by a communication structure (a star-graph) as well as a coalition structure. We adopt two well-known cooperative allocations — the Myerson value and the ES-value — to the case when there exist restrictions on the cooperation of players and provide their analytical expressions. Additionally, we examine stability of coalition structures using the concept of the Nash equilibrium and formulate conditions guaranteeing such stability for a given coalition structure.

In this paper, we explore network situations where the architecture of the network is described by a hypergraph. In a hypergraph, nodes are the players and — in contrast to bilateral graphs — links between players may involve more than just two players. The value function is defined on the architecture of the network such that different hypergraphs that connect the same individuals can generate different values. We provide axiomatic characterizations for the Myerson value and the position value. Furthermore, we define the notion of the core of a hypergraph situation. By means of an example, we also show that the Myerson value and the position value can lie in the core of a hypergraph situation.