The Communication Tree Value for TU-Games with Graph Communication

Abstract

A new solution is presented for transferable utility games with graph communication where the cooperation possibilities are represented by a graph. Players are only able to cooperate and obtain some worth in a coalition if they form a connected set in the given graph. To determine the payoff for each player, a single-valued solution, the communication tree value, is proposed for this class of games. The idea is that to form the grand coalition of all players a player can join a set of players only if this player is connected in the graph to at least one of the players in the set. To a set of players, starting with an arbitrary single player, from each maximally connected subset of remaining players one player joins who is connected to one or more of the players in that set. In this way a (rooted) tree on the set of players is obtained, called a communication tree. For a given game each communication tree of the graph induces a marginal contribution vector, in which any player receives a payoff equal to what he contributes in worth when he joins his subordinates in the tree. The average payoff over all communication trees of the graph determines the value. In case the underlying graph is cycle-complete the value coincides with the average tree solution. When there is complete communication between all players, which is the special case of cycle-completeness, players join one by one, yielding the Shapley value. A weak form of convexity is introduced, under which the value is guaranteed to be an element of the core. For games with complete graph communication the condition coincides with convexity and in case the underlying graph is cycle-free it is weaker than super-additivity.