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 solution for TU-games assigns a set of payoff distributions (possibly empty or consisting of a unique element) to every TU-game. Harsanyi solutions are solutions that are based on distributing dividends. In this paper we consider games with limited communication structure in which the edges or links of an undirected graph on the set of players represent binary communication links between the players such that players can cooperate if and only if they are connected. For such games we discuss Harsanyi solutions whose dividend shares are based on power measures for nodes in corresponding communication graphs. Special attention is given to the Harsanyi degree solution which equals the Shapley value on the class of complete graph games (i.e. the class of TU-games) and equals the position value on the class of cycle-free graph games. Another example is the Harsanyi power solution that is based on the equal power measure, which turns out to be the Myerson value. Various applications of our results are provided.
Highlights
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 TUgame, being a pair (N, v), where N ⊂ IN is a finite set of players and v : 2N → IR is a characteristic function on N such that v(∅) = 0
We show that on the class of cycle-free graph games, the corresponding Harsanyi power solution is equal to the position value, introduced in Borm et al (1992), while it equals the Shapley value on the class of complete graph games
In this paper we studied Harsanyi power solutions for TU-games in which the cooperation possibilities are restricted by a communication graph
Summary
One of the best-known restrictions are the games with limited communication structure in which the members of some coalition S can realize the worth v(S) if and only if they are connected within a given communication graph on the set of players. The novelty of the Harsanyi power solutions for graph restricted games is that we associate sharing systems with some power measure for the underlying communication graph, this measure being a function which assigns a nonnegative real number to every node in the graph. These numbers represent the strength or power of those nodes in the graph.
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