Union values for games with coalition structure

•We model situations where decision-makers have several players by games with coalition structure.•We introduce union values for games with coalition structure assigning payoffs to the unions.•We introduce two union values generalizing the Shapley value.•We axiomatize these two solutions differing only in a collusion property.•We apply both solutions to a generalization of airport games.

In a cooperative transferable utility game each decision-making agent is usually represented by one player. We model a situation where a decision-making agent can be represented by more than one player by a game with coalition structure where, besides the game, there is a partition of the player set into several unions. But, whereas usually the decision-making agents are the players in such a game, in this paper the decision-making agents are modeled as the unions in the coalition structure. Consequently, where usually a solution assigns payoffs to the individual players, we introduce the concept of union value being solutions that assign payoffs to the unions in a game with coalition structure. We introduce two such union values, both generalizing the Shapley value for TU-games. The first is the union-Shapley value and considers the unions in the most unified way: when a union enters a coalition then it enters with all its players. The second is the player-Shapley value which takes all players as units, and the payoff of a union is the sum of the payoffs over all its players. We provide axiomatic characterizations of these two union values differing only in a collusion neutrality axiom. After that we apply them to airport games and voting games.