Representation and Complexity in Boolean Games
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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- 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.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.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.Henkin, L.: Some remarks on infinitely long formulas. In: Infinistic Methods, pp. 167–183. Pergamon Press, Oxford (1961)Google Scholar
- 7.Logic and games. Special issue of Journal of Logic, Language and Information (2002)Google Scholar
- 8.Pauly, M.: Logic for Social Software. Ph.D. dissertation, Institute for Logic, Language and Information, Amsterdam (2001)Google Scholar
- 9.Peirce, C.S.: Reasoning and the Logic of Things: The Cambridge Conferences Lectures of 1898. Harvard University Press, Cambridge (1992)Google Scholar
- 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.Zhegalkin, I.I.: The technique of calculation of statements in symbolic logic. Matem. Sbornik 34, 9–28 (1927) (in Russian)Google Scholar