, Volume 193, Issue 3, pp 781–811 | Cite as

Partial-order Boolean games: informational independence in a logic-based model of strategic interaction

  • Julian Bradfield
  • Julian Gutierrez
  • Michael WooldridgeEmail author
S.I. : Logic and the Foundations of Decision and Game Theory (LOFT)


As they are conventionally formulated, Boolean games assume that players make their choices in ignorance of the choices being made by other players – they are games of simultaneous moves. For many settings, this is clearly unrealistic. In this paper, we show how Boolean games can be enriched by dependency graphs which explicitly represent the informational dependencies between variables in a game. More precisely, dependency graphs play two roles. First, when we say that variable x depends on variable y, then we mean that when a strategy assigns a value to variable x, it can be informed by the value that has been assigned to y. Second, and as a consequence of the first property, they capture a richer and more plausible model of concurrency than the simultaneous-action model implicit in conventional Boolean games. Dependency graphs implicitly define a partial ordering of the run-time events in a game: if x is dependent on y, then the assignment of a value to y must precede the assignment of a value to x; if x and y are independent, however, then we can say nothing about the ordering of assignments to these variables—the assignments may occur concurrently. We refer to Boolean games with dependency graphs as partial-order Boolean games. After motivating and presenting the partial-order Boolean games model, we explore its properties. We show that while some problems associated with our new games have the same complexity as in conventional Boolean games, for others the complexity blows up dramatically. We also show that the concurrency in partial-order Boolean games can be modelled using a closure-operator semantics, and conclude by considering the relationship of our model to Independence-Friendly (IF) logic.


Boolean games Foundations of games Concurrency theory Logic 


  1. Abramsky, S. (2003). Sequentiality vs. concurrency in games and logic. Mathematical Structures in Computer Science, 13(4), 531–565.CrossRefGoogle Scholar
  2. Abramsky, S. (2006). Socially responsive, environmentally friendly logic. Acta Philosophica Fennica 78. Truth and Games: Essays in Honour of Gabriel SanduGoogle Scholar
  3. Abramsky, S., & Melliès, P.A. (1999). Concurrent games and full completeness. In LICS (pp. 431–442). IEEE Computer SocietyGoogle Scholar
  4. Balabanov, V., Chiang, H. J., & Jiang, J. H. (2014). Henkin quantifers and Boolean formulae: A certification perspective of DQBF. Theoretical Computer Science, 523, 86–100.CrossRefGoogle Scholar
  5. Bonzon, E., Lagasquie, M., Lang, J., & Zanuttini, B. (2006) Boolean games revisited. In ECAI Google Scholar
  6. Bonzon, E., Lagasquie-Schiex, M. C., & Lang, J. (2009). Dependencies between players in Boolean games. International Journal of Approximate Reasoning, 50(6), 899–914.CrossRefGoogle Scholar
  7. Boppana, R. B., & Sipser, M. (1990). The complexity of finite functions. In A. V. Aho & M. J. Corasick (Eds.), Handbook of theoretical computer science volume A: Algorithms and complexity (pp. 757–804). Amsterdam: Elsevier Science Publishers B.V.Google Scholar
  8. Bradfield, J. C. (2006). Independence: Logics and concurrency. Acta Philosophica Fennica, 78, 47–70. Truth and Games: Essays in Honour of Gabriel Sandu.Google Scholar
  9. Clairambault, P., Gutierrez, J., & Winskel, G. (2012). The winning ways of concurrent games. In LICS (pp. 235–244) IEEE Computer SocietyGoogle Scholar
  10. Clairambault, P., Gutierrez, J., & Winskel, G. (2013) Imperfect information in logic and concurrent games. In Computation, Logic, Games, and Quantum Foundations, LNCS, vol. 7860, (pp. 7–20). SpringerGoogle Scholar
  11. Cook, S., & Soltys, M. (1999). Boolean programs and quantified propositional proof systems. Bulletin of the Section of Logic, 28(3), 119–129.Google Scholar
  12. Dunne, P.E., Kraus, S., van der Hoek, W., & Wooldridge, M. (2008) Cooperative Boolean games. In AAMAS Google Scholar
  13. Emerson, E. A. (1990). Temporal and modal logic. In J. van Leeuwen (Ed.), Handbook of theoretical computer science volume B: Formal models and semantics (pp. 996–1072). Amsterdam, The Netherlands: Elsevier Science Publishers B.V.Google Scholar
  14. Fagin, R., Halpern, J. Y., Moses, Y., & Vardi, M. Y. (1995). Reasoning about knowledge. Cambridge, MA: The MIT Press.Google Scholar
  15. Ghallab, M., Nau, D., & Traverso, P. (2004). Automated planning: Theory and practice. San Mateo, CA: Morgan Kaufmann Publishers.Google Scholar
  16. Godefroid, P. (1996). Partial-order methods for the verification of concurrent systems., Lecture Notes in Computer Science New York: Springer.CrossRefGoogle Scholar
  17. Grant, J., Kraus, S., Wooldridge, M., Zuckerman, I. (2011) Manipulating Boolean games through communication. In IJCAI Google Scholar
  18. Gutierrez, J. (2011). Concurrent logic games on partial orders. In WoLLIC, LNCS, vol. 6642, (pp. 146–160). SpringerGoogle Scholar
  19. Gutierrez, J., & Wooldridge, M. (2014). Equilibria of concurrent games on event structures. In LNCS, ACM PressGoogle Scholar
  20. Harrenstein, P., van der Hoek, W., Meyer, J.J., & Witteveen, C. (2001) Boolean games. In TARK, (pp. 287–298)Google Scholar
  21. Hearn, R. A., & Demaine, E. D. (2009). Games, puzzles, & computation. Wellesley, MA: A. K. Peters Ltd.Google Scholar
  22. Henkin, L. (1961). Some remarks on infinitely long formulas. Journal of Symbolic Logic, 30(1), 167–183.Google Scholar
  23. Jurdziński, M., Nielsen, M., & Srba, J. (2003). Undecidability of domino games and hhp-bisimilarity. Information and Computation, 184(2), 343–368.CrossRefGoogle Scholar
  24. Koller, D., & Milch, B. (2003). Multi-agent influence diagrams for representing and solving games. Games and Economic Behavior, 45(1), 181–221.CrossRefGoogle Scholar
  25. Mann, A. L., Sandu, G., & Sevenster, M. (2011). IIndependence-friendly logic. A game-theoretic approach., LMS Lecture Note Series Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  26. Manna, Z., & Pnueli, A. (1992). The temporal logic of reactive and concurrent systems. Berlin: Springer.CrossRefGoogle Scholar
  27. Mavronicolas, M., Monien, B., & Wagner, K.W. (2007). Weighted Boolean formula games. In WINE (pp. 469–481)Google Scholar
  28. Melliès, P. A., & Mimram, S. (2007). Asynchronous games: Innocence without alternation (pp. 395–411)., CONCUR, LNCS Heidelberg: Springer.Google Scholar
  29. Nielsen, M., & Winskel, G. (1995). Models for concurrency. In Handbook of logic in computer science. Oxford University Press: Oxford, EnglandGoogle Scholar
  30. Osborne, M. J., & Rubinstein, A. (1994). A course in game theory. Cambridge, MA: The MIT Press.Google Scholar
  31. Pauly, M. (2001). Logic for social software. Ph.D. thesis, University of AmsterdamGoogle Scholar
  32. Peterson, G. L., Reif, J. H., & Azhar, S. (2001). Lower bounds for multiplayer noncooperative games of incomplete information. Computers and Mathematics with Applications, 41, 957–992.CrossRefGoogle Scholar
  33. Saraswat, V.A., Rinard, M.C., & Panangaden, P. (1991) Semantic foundations of concurrent constraint programming. In POPL (pp. 333–352). ACM PressGoogle Scholar
  34. Winskel, G. (2012). Deterministic concurrent strategies. Formal Aspects of Computing, 24(4–6), 647–660.CrossRefGoogle Scholar
  35. Wooldridge, M., Endriss, U., Kraus, S., & Lang, J. (2013). Incentive engineering for Boolean games. Artificial Intelligence, 195, 418–439.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Julian Bradfield
    • 1
  • Julian Gutierrez
    • 2
  • Michael Wooldridge
    • 2
    Email author
  1. 1.University of EdinburghEdinburghUK
  2. 2.University of OxfordOxfordUK

Personalised recommendations