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.
Highlights
A situation in which a finite set of players can generate certain payoffs by cooperation can be described by a cooperative game with transferable utility
Considering payoff allocation we focus on the Shapley value, but other solutions such as the nucleolus, Banzhaf value or Core are considered in the literature
From the many applications of games with a permission structure, we will briefly discuss two: (i) auction games which are an application of peer group games, and (ii) hierarchically structured firms where the permission structure is a rooted tree and the game is convex such that the only nonnull players are those at the lowest level of the hierarchy, i.e. those players that have no successors
Summary
A situation in which a finite set of players can generate certain payoffs by cooperation can be described by a cooperative game with transferable utility (or a TUgame). Since antimatroids are union closed (i.e. the union of any two feasible coalitions is feasible), a similar approach as for games with a permission structure can be followed by defining a restricted game that assigns to every coalition the worth of its largest feasible subset in the original game, and applying the Shapley value (or any other TU-game solution) to this restricted game. From the many applications of games with a permission structure, we will briefly discuss two: (i) auction games which are an application of peer group games, and (ii) hierarchically structured firms where the permission structure is a rooted tree and the game is convex such that the only nonnull players are those at the lowest level of the hierarchy, i.e. those players that have no successors (the other players are supposed to be managers who coordinate the production process but do not produce value themselves)
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