Abstract
A situation in which a finite set of players can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or simply a TU-game. A (single-valued) solution for TU-games assigns a payoff distribution to every TU-game. A well-known solution is the Shapley value. In the literature various models of games with restricted cooperation can be found. So, instead of allowing all subsets of the player set N to form, it is assumed that the set of feasible coalitions is a subset of the power set of N. In this paper we consider such sets of feasible coalitions that are closed under union, i.e. for any two feasible coalitions also their union is feasible. We consider and axiomatize two solutions or rules for these games that generalize the Shapley value: one is obtained as the conjunctive permission value using a corresponding superior graph, the other is defined as the Shapley value of a modified game similar as the Myerson rule for conference structures.
Highlights
A cooperative game with transferable utility, or a TU-game, is a finite set of players and for any subset of players a worth representing the total payoff that the coalition can obtain by cooperating
A situation in which a finite set of players can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or a TU-game
We consider and axiomatize two solutions or rules for these games that generalize the Shapley value: one is obtained as the conjunctive permission value using a corresponding superior graph, the other is defined as the Shapley value of a modified game similar as the Myerson value for games with limited communication
Summary
A cooperative game with transferable utility, or a TU-game, is a finite set of players and for any subset (coalition) of players a worth representing the total payoff that the coalition can obtain by cooperating. We define and axiomatize two solutions for games on union closed systems. The first solution is based on games with a permission structure, the other directly applies the Shapley value to some restricted game. We apply a method similar as Myerson (1977) to define a solution for games on union closed systems which generalizes the Shapley value for games on antimatroids as axiomatized in Algaba et al (2003). A modified or restricted game is defined This game is obtained by assigning to any non-feasible coalition the worth of its largest feasible subset. The union rule for games on union closed systems is defined as the Shapley value of this restricted game.
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