Representation and Complexity in Boolean Games

  • Paul E. Dunne
  • Wiebe van der Hoek
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3229)


Boolean games are a class of two-player games which may be defined via a Boolean form over a set of atomic actions. A particular game on some form is instantiated by partitioning these actions between the players – player 0 and player 1 – each of whom has the object of employing its available actions in such a way that the game’s outcome is that sought by the player concerned, i.e. player i tries to bring about the outcome i. In this paper our aim is to consider a number of issues concerning how such forms are represented within an algorithmic setting. We introduce a concept of concise form representation and compare its properties in relation to the more frequently used “extensive form” descriptions. Among other results we present a “normal form” theorem that gives a characterisation of winning strategies for each player. Our main interest, however, lies in classifying the computational complexity of various decision problems when the game instance is presented as a concise form. Among the problems we consider are: deciding existence of a winning strategy given control of a particular set of actions; determining whether two games are “equivalent”.


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  1. 1.
    Dunne, P.E.: The Complexity of Boolean Networks. Academic Press, London (1988)MATHGoogle Scholar
  2. 2.
    Goranko, V.: The Basic Algebra of Game Equivalences. In: Pauly, M., Sandhu, G. (eds.) ESSLLI Workshop on Logic and Games (2001)Google Scholar
  3. 3.
    Harrenstein, P.: Logic in Conflict – Logical Exploration in Strategic Equilibrium. Ph.D. dissertation, Dept. of Computer Science, Univ. of Utrecht (2004) (submitted)Google Scholar
  4. 4.
    Harrenstein, B.P., van der Hoek, W., Meyer, J.-J., Witteveen, C.: Boolean Games. In: van Benthem, J. (ed.) Proc. 8th Conf. on Theoretical Aspects of Rationality and Knowledge (TARK 2001), pp. 287–298. Morgan-Kaufmann, San Francisco (2001)Google Scholar
  5. 5.
    Henkin, L.: Some remarks on infinitely long formulas. In: Infinistic Methods, pp. 167–183. Pergamon Press, Oxford (1961)Google Scholar
  6. 6.
    Hyafil, L., Rivest, R.: Constructing Optimal Binary Decision Trees is np–complete. Inf. Proc. Letters 5, 15–17 (1976)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Logic and games. Special issue of Journal of Logic, Language and Information (2002)Google Scholar
  8. 8.
    Pauly, M.: Logic for Social Software. Ph.D. dissertation, Institute for Logic, Language and Information, Amsterdam (2001)Google Scholar
  9. 9.
    Peirce, C.S.: Reasoning and the Logic of Things: The Cambridge Conferences Lectures of 1898. Harvard University Press, Cambridge (1992)Google Scholar
  10. 10.
    Wooldridge, M., Dunne, P.E.: On the Computational Complexity of Qualitative Coalitional Games. Technical Report, ULCS-04-007, Dept. of Computer Science, Univ. of Liverpool (2004) (to appear: Artificial Intelligence)Google Scholar
  11. 11.
    Zhegalkin, I.I.: The technique of calculation of statements in symbolic logic. Matem. Sbornik 34, 9–28 (1927) (in Russian)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Paul E. Dunne
    • 1
  • Wiebe van der Hoek
    • 1
  1. 1.Dept. of Computer ScienceUniversity of LiverpoolLiverpoolUnited Kingdom

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