A modal logic for coalitional power in games

Abstract

We present a modal logic for reasoning about what groups of agents can bring about by collective action. Given a set of states, we introduce game frames which associate with every state a strategic game among the agents. Game frames are essentially extensive games of perfect information with simultaneous actions, where every action profile is associated with a new state, the outcome of the game. A coalition of players is effective for a set of states X in a game if the coalition can guarantee the outcome of the game to lie in X. We propose a modal logic (Coalition Logic) to formalize reasoning about effectivity in game frames, where [C]φ expresses that coalition C is effective for φ. An axiomatization is presented and completeness proved. Coalition Logic provides a unifying game-theoretic view of modal logic: Since nondeterministic processes and extensive games without parallel moves emerge as particular instances of game frames, normal and non-normal modal logics correspond to 1-and 2-player versions of Coalition Logic. The satisfiability problem for Coalition Logic is shown to be PSPACE-complete.