Centrality Rewarding Shapley and Myerson Values for Undirected Graph Games

In this paper we introduce two values for cooperative games with communication graph structure. For cooperative games the shapley value distributes the worth of the grand coalition amongst the players by taking into account the worths that can be obtained by any coalition of players, but does not take into account the role of the players when communication between players is restricted. Existing values for communication graph games as the Myerson value and the average tree solution only consider the worths of connected coalitions and respect only in this way the communication restrictions. They do not take into account the position of a player in the graph in the sense that, when the graph is connected, in the unanimity game on the grand coalition all players are treated equally and so players with a more central position in the graph get the same payoff as players that are not central. The two new values take into account the position of a player in the graph. The first one respects centrality, but not the communication abilities of any player. The second value reflects both centrality and the communication ability of each player. That implies that in unanimity games players that do not generate worth but are needed to connect worth generating players are treated as those latter players, and simultaneously players that are more central in the graph get bigger shares in the worth than players that are less central. For both values an axiomatic characterization is given on the class of connected cycle-free graph games