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.