Studia Logica

, Volume 102, Issue 2, pp 339–360 | Cite as

A Modal Logic for Mixed Strategies

  • Joshua Sack
  • Wiebe van der Hoek


Modal logics have proven to be a very successful tool for reasoning about games. However, until now, although logics have been put forward for games in both normal form and games in extensive form, and for games with complete and incomplete information, the focus in the logic community has hitherto been on games with pure strategies. This paper is a first to widen the scope to logics for games that allow mixed strategies. We present a modal logic for games in normal form with mixed strategies, and demonstrate its soundness and strong completeness. Characteristic for our logic is a number of infinite rules.


Modal Logic Logics for Games Mixed Strategies 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aceto, L., W. van der Hoek, A. Ingolfsdottir, and J. Sack, Sigma algebras in probabilistic epistemic dynamics, in TARK XIII, 2011, pp. 191–199.Google Scholar
  2. 2.
    Aumann R.J., Brandenburger A.: Epistemic conditions for Nash equilibrium. Econometrica 63(5), 1161–1180 (1995)CrossRefGoogle Scholar
  3. 3.
    Fagin R., Halpern J.: Reasoning about knowledge and probability. Journal of the ACM 41, 340–367 (1994)CrossRefGoogle Scholar
  4. 4.
    Fagin R., Halpern J.Y., Megiddo N., A logic for reasoning about probabilities. Information and Computation 87, 78–128 (1990)CrossRefGoogle Scholar
  5. 5.
    Kwiatkowska, M., G. Norman, and D. Parker, PRISM 4.0: Verification of probabilistic real-time systems, in G. Gopalakrishnan, and S. Qadeer, (eds.), Proc. CAV’11, vol. 6806 of LNCS, Springer, 2011, pp. 585–591.Google Scholar
  6. 6.
    Osborne M. J., Rubinstein A. A Course in Game Theory. The MIT Press: Cambridge, MA, (1994)Google Scholar
  7. 7.
    Renardel de Lavalette, G., B. Kooi, and R. Verbrugge, Strong completeness for PDL, in P. Balbiani, N. Suzuki, and F. Wolter, (eds.), AiML, 2002, pp. 377–393.Google Scholar
  8. 8.
    van der Hoek, W., and M. Pauly, Modal logic for games and information, in P. Blackburn, J. van Benthem, and F. Wolter, (eds.), Handbook of Modal Logic, Elsevier, Amsterdam, 2006, pp. 1077–1148.Google Scholar
  9. 9.
    van der Hoek W., Walther D., Wooldridge M. Reasoning about the transfer of control. JAIR 37, 437–477 (2010)Google Scholar
  10. 10.
    van der Hoek, W., and M. Wooldridge, Multi-agent systems, in F. van Harmelen, V. Lifschitz, and B. Porter, (eds.), Handbook of Knowledge Representation, Elsevier, 2008, pp. 887–928.Google Scholar
  11. 11.
    Von Stengel, B., Computing equilibria for two-person games, in R.J. Aumann and S. Hart, (eds.), Handbook of Game Theory with Economic Applications, Elsevier, 2002, pp. 1723-1759.Google Scholar
  12. 12.
    Zhou, Ch., Complete deductive systems for probability logic with application to harsanyi type spaces, Ph.D. thesis, Indianapolis, IN, USA, 2007. AAI3278239.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Institute for Logic, Language and ComputationUniversiteit van AmsterdamAmsterdamThe Netherlands
  2. 2.Department of Computer ScienceUniversity of LiverpoolLiverpoolUK

Personalised recommendations