A situation in which a finite set of agents can generate certain payoffs by cooperation can be described by a cooperative game with transferable utility (or simply a TU-game) where each agent is represented by one player in the game. In this paper, we assume that one agent can be represented by more than one player. We introduce two solutions for this multi-player agent game model, both being generalizations of the Shapley value for TU-games. The first is the agent-Shapley value and considers the agents in the most unified way in the sense that when an agent 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 an agent is the sum of the payoffs over all its players. We provide axiomatic characterizations of these two solutions that differ only in a collusion neutrality axiom. The agent-Shapley value satisfies player collusion neutrality stating that collusion of two players belonging to the same agent does not change the payoff of this agent. On the other hand, the player-Shapley value satisfies agent collusion neutrality stating that after a collusion of two agents, the sum of their payoffs does not change. After axiomatizing the player- and agent-Shapley values we apply them to airport games and voting games.
Cooperative TU-game Shapley value multi-player agent collusion neutrality airport games