A Modal Logic for Coalitional Power in Games

A Modal Logic for Coalitional Power in Games MARC PAULY, Center for Mathematics and Computer Science (CWI), P.O. Box 94079, 1090 GB Amsterdam, The Netherlands. E-mail: 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 in a is associated with a new state, the outcome of the game. A coalition of players is effective for a set of states game if the coalition can guarantee the outcome of the game to lie in . We propose a modal logic (Coalition Logic) to formalize reasoning about effectivity in game frames, where
 expresses that coalition is effective for . 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.

  Keywords: Modal logic, game theory, multiagent systems. 1 Introduction Modelling actions and their effects is a task which has occupied many researchers in computer science, logic, economics and philosophy. In the simplest case, we have one agent (person, process) who can choose between taking different actions which change the state of the world in various ways. A simple model of this scenario will contain an accessibility relation
which associates to every state of the world all those states which the agent can bring about through his actions, i.e. 
 holds if in state  the agent can act so as to bring about state . In modal logic, one introduces a language to talk about such Kripke models:  expresses that the agent can act in such a way that  will be true after his action. This simple one-agent case can easily be extended to many agents by considering a relational structure which contains an accessibility relation

for every agent , where
 expresses that agent  can bring about . The problem with such a multi-agent action logic is that it considers the different agents in isolation. Given a state 
, agent 1 may act to bring about state  and agent 2 may act to bring about state  , but what happens if both of them act simultaneously in 
? Since the actions of the two agents will often not be independent but interact with each other, a more general model of action should associate a resulting state with every pair of actions
    of the two players rather than with actions of the players individually. In this paper, we develop a modal logic based on such more general action models which we shall call game frames. At any state of such a frame, each agent 
 takes an action, and taken together these actions determine the resulting state. This amounts to associating a strategic game form with every state of the frame where the outcomes of the game are states of the frame again. Thus, game frames are essentially extensive game forms with simultaneous ¿ ¿ J. Logic Computat., Vol. 12 No. 1, pp. 149–166 2002 c Oxford University Press 150 A Modal Logic for Coalitional Power in Games actions (see [9]). In Section 2, game frames are introduced together with extensive games without simultaneous moves as well as non-deterministic processes as special cases. Section 3 relates a notion of effectivity to strategic games, formalizing what it means for a coalition of agents to have the ability to force a certain set of outcomes in a strategic game. This notion of effectivity will then be used as the basic semantic notion for the modal logic we develop in Section 4. For a set of agents   , the modal language will contain formulas   which express that the group of agents can bring about , i.e. is effective for . We provide a complete axiomatization of this logic in Section 5, together with some coalitional principles which serve to restrict the power of coalitions enough to yield an axiomatization of extensive games without simultaneous moves. Section 6 discusses the complexity of the satisfiability problem for coalition logic. The possibility for agents to combine strategies when forming a coalition is responsible for making this problem PSPACE-complete rather than NP-complete. Finally, Section 7 provides a unifying game-theoretic view of modal logic where normal as well as non-normal modal logics emerge as restricted versions of Coalition Logic. The logic introduced here can be viewed as a generalization of the modal base logic under[10, 11], an extension of Propositional Dynamic Logic. is lying Parikh’s game logic a logic of determined 2-player games, though a multi-player version is also discussed. The generalization of Coalition Logic consists of dropping the assumption of determinacy and extending the language from individual players to groups of players. While operations on games are not the concern of this paper, such operations could also be added to Coalition Logic, see the remarks in Section 8. 2 A model of interaction: game frames As mentioned in the introduction, we would like an action model where at each state, the actions taken by the agents together determine the resulting state. To obtain such a model, we associate a strategic game with every state of the world. A strategic game 
 

    consists of a non-empty finite set of agents or players  , a non-empty set of strategies or actions 
for every player 
 , a non-empty set of outcome states and an outcome function  

 
 which associates with every tuple of strategies of the players (strategy profile) an outcome state in . In game theory [9, 2], strategic games also come equipped with a preference relation 
  for every player 
 which indicates which outcomes a player prefers. Strictly speaking, our strategic games are only game forms which can be turned into a game by adding these preference relations. For notational convenience, let  



 denote the strategy tuple for coalition   which consists of player  choosing strategy 


. Then given two strategy tuples  and  (where   ),
    denotes the outcome state associated with the strategy profile induced by  and  . Let   be the set of all strategic games among the set of players  over the set of states . Then we define a game frame for players  as a pair
   where is a non-empty set of states and      is a function which associates strategic games to states. In game theoretic terminology, game frames are essentially extensive game forms with simultaneous moves [9], the only difference being that we assume that at every state some game can be played, i.e. there are no (...truncated)