Skip to main content
SpringerLink
Log in

Weighted Boolean Formula Games

  • Chapter
  • First Online:
Algorithms, Probability, Networks, and Games

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9295))

Abstract

We introduce weighted boolean formula games (WBFG) as a new class of succinct games. Each player has a set of boolean formulas she wants to get satisfied; the formulas involve a ground set of boolean variables each of which is controlled by some player. The payoff of a player is a weighted sum of the values of her formulas. We consider both pure equilibria and their refinement of payoff-dominant equilibria [34], where every player is no worse-off than in any other pure equilibrium. We present both structural and complexity results:

  • We consider mutual weighted boolean formula games (MWBFG), a subclass of WBFG making a natural mutuality assumption on the formulas of players. We present a very simple exact potential for MWBFG. We establish a polynomial monomorphism from certain classes of weighted congestion games to subclasses of WBFG and MWBFG, respectively, indicating their rich structure.

  • We present a collection of complexity results about decision (and search) problems for both pure and payoff-dominant equilibria in WBFG. The precise complexities depend crucially on five parameters: (i) the number of players; (ii) the number of variables per player; (iii) the number of formulas per player; (iv) the weights in the payoff functions (whether identical or not), and (v) the syntax of the formulas. These results imply that, unless the polynomial hierarchy collapses, decision (and search) problems for payoff-dominant equilibria are harder than for pure equilibria.

A preliminary version of this work appeared in the Proceedings of the 3rd International Workshop on Internet and Network Economics, X. Deng and F. Chung Graham eds., pp. 467–481, Vol. 4858, Lecture Notes in Computer Science, Springer-Verlag, December 2007. This work has been partially supported by the German Research Foundation (DFG) within the Collaborative Research Centre “On-the-Fly-Computing” (SFB 901) and by the IST Program of the European Union under contract numbers IST-2004-001907 (DELIS) and 15964 (AEOLUS).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    A boolean formula is the special case of a (boolean) circuit where every boolean gate has fan-out one; so, a boolean formula is a circuit whose underlying graph is a tree.

  2. 2.

    The straightforward depth-preserving conversion of a boolean circuit into an equivalent formula may potentially blow up the size exponentially since pieces of the circuit must be repeated. Nevertheless, the largest shown difference between formula size and boolean circuit size is only \(\mathsf{L}(f) = {\varOmega }(n^{2} \lg ^{-1} n)\) and \(\mathsf{C}(f) = 2n + o(n)\), where f is the storage access function for indirect addressing [54].

  3. 3.

    We note that the work in [9] appeared for the first time in two conference papers published in 2006 [7, 8]; the formulation of, and results about, WBFG in this paper represents independent work.

  4. 4.

    We warn the reader against the formula \({\mathsf {G}}(\mathbf{x}, \mathbf{y}) \equiv 0\) for all \(\mathbf{x}\) and \(\mathbf{y}\). Note that in the constructed game \({\mathsf {\varGamma }}_{{\mathsf {G}}}\), \({\mathsf {f}}_{1} \equiv 0\), \({\mathsf {f}}_{2} \equiv 0\) and \({\mathsf {f}}_{3} \equiv 1\); so, every profile is a pure equilibrium for \({\mathsf {\varGamma }}_{{\mathsf {G}}}\). But this is not a contradiction, since \({\mathsf {G}} \not \in \mathsf{R}\), which implies that \({\mathsf {G}}\) may not be an input for \({\mathsf {\Sigma }}_{2}\)-\({\mathsf {RQBF}}\) (even though \({\mathsf {G}} \not \in {\mathsf {\Sigma }}_{2}\)-\({\mathsf {RQBF}}\)). In fact, we used reduction from \({\mathsf {\Sigma }}_{2}\)-\({\mathsf {RQBF}}\) (as opposed to \({\mathsf {\Sigma }}_{2}\)-\({\mathsf {QBF}}\)) in order to eliminate such degenerate formulas from consideration.

References

  1. Álvarez, C., Gabarró, J., Serna, M.: Equilibria problems on games: complexity versus succinctness. J. Comput. Syst. Sci. 77(6), 1172–1197 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aumann, R.J., Sorin, S.: Cooperation and bounded recall. Game. Econ. Behav. 1(1), 5–39 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bacharach, M., Bernasconi, M.: An experimental study of the variable frame theory of focal points. Game. Econ. Behav. 19(1), 1–45 (1997)

    Article  MATH  Google Scholar 

  4. Bilò, V.: On satisfiability games and the power of congestion games. In: Kao, M.-Y., Li, X.-Y. (eds.) AAIM 2007. LNCS, vol. 4508, pp. 231–240. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  5. Bilò, V., Mavronicolas, M.: The complexity of decision problems about nash equilibria in win-lose games. In: Serna, M. (ed.) SAGT 2012. LNCS, vol. 7615, pp. 37–48. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  6. Blonski, M.: Characterization of pure-strategy equilibria in finite anonymous games. J. Math. Econ. 34(2), 225–233 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bonzon, E., Lagasquie-Schiex, M.-C., Lang, J., Zanuttini, B.: Boolean games revisited. In: Proceedings of the 17th European Conference on Artificial Intelligence, Series Frontiers in Artificial Intelligence and Applications, vol. 141, pp. 265–269, August/September 2006

    Google Scholar 

  8. Bonzon, E., Lagasquie-Schiex, M.-C., Lang, J.: Compact preference representation for boolean games. In: Yang, Q., Webb, G. (eds.) PRICAI 2006. LNCS (LNAI), vol. 4099, pp. 41–50. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  9. Bonzon, E., Lagasquie-Schiex, M.-C., Lang, J., Zanuttini, B.: Compact preference representation and boolean games. Auton. Agents Multi-Agent Syst. 18(1), 1–35 (2009)

    Article  Google Scholar 

  10. Bonzon, E., Lagasquie-Schiex, M.-C., Lang, J.: Dependencies between players in boolean games. Int. J. Approximate Reasoning 50(6), 899–914 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bonzon, E., Lagasquie-Schiex, M.-C., Lang, J.: Effectivity functions and efficient coalitions in boolean games. Synthese 187(1 Supplement), 73–103 (2012)

    Article  MATH  Google Scholar 

  12. Brandt, F., Fischer, F., Holzer, M.: Symmetries and the complexity of pure nash equilibrium. J. Comput. Syst. Sci. 75(3), 163–177 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Case, J.: A class of games having pareto optimal nash equilibria. J. Optim. Theory Appl. 13, 379–385 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, X., Deng, X., Teng, S.-H.: Settling the complexity of two-player nash equilibria. J. ACM 56(3), 14 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Colman, A.M., Bacharach, M.: Payoff dominance and the stackelberg heuristic. Theory Decis. 43(1), 1–19 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Conitzer, V., Sandholm, T.: Complexity results about nash equilibria. Game. Econ. Behav. 63(2), 621–641 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Daskalakis, C., Fabrikant, A., Papadimitriou, C.: The game world is flat: the complexity of nash equilibria in succinct games. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 513–524. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  18. Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a nash equilibrium. SIAM J. Comput. 39(1), 195–259 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Daskalakis, K., Papadimitriou, C.: The complexity of games on highly regular graphs. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 71–82. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  20. Dunkel, J., Schulz, A.S.: On the complexity of pure-strategy nash equilibria in congestion and local-effect games. MatH. Oper. Res. 33(4), 851–868 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dunne, P.E., van der Hoek, W.: Representation and complexity in boolean games. In: Alferes, J.J., Leite, J. (eds.) JELIA 2004. LNCS (LNAI), vol. 3229, pp. 347–359. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  22. Dunne, P.E., Wooldridge, M.: Towards tractable boolean games. In: Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems, vol. 2, pp. 939–946, June 2012

    Google Scholar 

  23. Fabrikant, A., Papadimitriou, C.H., Talwar, K.: The complexity of pure nash equilibria. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pp. 604–612, June 2004

    Google Scholar 

  24. Feigenbaum, J., Koller, D., Shor, P.: A game-theoretic classification of interactive complexity classes. In: Proceedings of the 10th Annual IEEE Conference on Structure in Complexity Theory, pp. 227–237, June 1995

    Google Scholar 

  25. Fischer, F., Holzer, M., Katzenbeisser, S.: The influence of neighbourhood and choice on the complexity of finding pure nash equilibria. Inf. Process. Lett. 99(6), 239–245 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Fortnow, L., Impagliazzo, R., Kabanets, V., Umans, C.: On the complexity of succinct zero-sum games. Comput. Complex. 17(3), 353–376 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Fotakis, D., Kontogiannis, S., Spirakis, P.: Selfish unsplittable flows. Theoret. Comput. Sci. 348(2–3), 226–239 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Gabarró, J., García, A., Serna, M.: The complexity of game isomorphism. Theoret. Comput. Sci. 412(48), 6675–6695 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Gairing, M., Monien, B., Tiemann, K.: Routing (un-)splittable flow in games with player-specific linear latency functions. ACM Trans. Algorithms 7(3), 31 (2011)

    Article  MATH  Google Scholar 

  30. Gale, D., Kuhn, H.W., Tucker, A.W.: On symmetric games. Contributions to the Theory of Games. Annals of Mathematics Studies, vol. 24. Princeton University Press, Princeton (1950)

    MATH  Google Scholar 

  31. Gottlob, G., Greco, G., Scarcello, F.: Pure nash equilibria: hard and easy games. J. Artif. Intell. Res. 24, 357–406 (2005)

    MathSciNet  MATH  Google Scholar 

  32. Harrenstein, P., van der Hoek, W., Meyer, J.-J., Witteveen, C.: Boolean games. In: Proceedings of the 8th Conference on Theoretical Aspects of Rationality and Knowledge, pp. 287–298, July 2001

    Google Scholar 

  33. Harrenstein, P.: Logic in conflict, Ph.D. thesis, Utrecht University (2004)

    Google Scholar 

  34. Harsanyi, J.C., Selten, R.: A General Theory of Equilibrium Selection in Games. The MIT Press, Cambridge (1988)

    MATH  Google Scholar 

  35. Hemaspaandra, E., Hemaspaandra, L.A., Rothe, J.: Exact analysis of dodgson elections: lewis carroll’s 1876 voting system is complete for parallel access to \({\cal {NP}}\). J. ACM 44(6), 806–825 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  36. Howson, J.T.: Equilibria of polymatrix games(Part I). Manag. Sci. 18(5), 312–318 (1972)

    Article  MATH  Google Scholar 

  37. Ianovski, E.: \({ \text{ DValue }}\) for boolean games is \({\cal EXP}\)-Hard. In: CoRR, abs/1403.7428 (2014)

    Google Scholar 

  38. Ianovski, E., Ong, L.: \(\exists { \text{ GuaranteeNash }}\) is \({\cal NEXP}\)-Hard. In: Proceedings of the 14th International Conference on Knowledge Representation and Reasoning, July 2014

    Google Scholar 

  39. Johnson, D.S., Papadimitriou, C.H., Yannakakis, M.: How easy is local search? J. Comput. Syst. Sci. 37(1), 79–100 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  40. Kalai, E., Samet, D.: Unanimity games and pareto optimality. Int. J. Game Theory 14(1), 41–50 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  41. Kearns, M.J., Littman, M.L., Singh, S.P.: Graphical models for game theory. In: Proceedings of the 17th Conference on Uncertainty in Artificial Intelligence, pp. 253–260, August 2001

    Google Scholar 

  42. Krapchenko, V.M.: Complexity of the realization of a linear function in the class of \(\Pi \)-circuits. Math. Notes Acad. Sci. USSR 9(1), 21–23 (1971)

    Google Scholar 

  43. Krentel, M.W.: The complexity of optimization problems. J. Comput. Syst. Sci. 36(3), 490–509 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  44. Leyton-Brown, K., Tennenholtz, M.: Local-effect games. In: Proceedings of the 18th International Joint Conference on Artificial Intelligence, pp. 772–780, August 2003

    Google Scholar 

  45. Mavronicolas, M., Milchtaich, I., Monien, B., Tiemann, K.: Congestion games with player-specific constants. In: Kučera, L., Kučera, A. (eds.) MFCS 2007. LNCS, vol. 4708, pp. 633–644. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  46. Meyer, A.R., Stockmeyer, L.J.: The equivalence problem for regular expressions with squaring requires exponential time. In: Proceedings of the 13th Annual IEEE Symposium on Switching and Automata Theory, pp. 125–129, October 1972

    Google Scholar 

  47. Milchtaich, I.: Congestion games with player-specific payoff functions. Game. Econ. Behav. 13(1), 111–124 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  48. Monderer, D., Shapley, L.S.: Potential games. Game. Econ. Behav. 14(1), 124–143 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  49. Nash, J.F.: Non-cooperative games. Ann. Math. 54, 286–295 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  50. Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Boston (1994)

    MATH  Google Scholar 

  51. Papadimitriou, C.H., Roughgarden, T.: Computing correlated equilibria in multi-player games. J. ACM 55(3), 14 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  52. Papadimitriou, C.H., Zachos, S.: Two remarks on the power of counting. In: Cremers, A.B., Kriegel, H.-P. (eds.) Theoretical Computer Science. LNCS, vol. 145, pp. 269–275. Springer, Heidelberg (1983)

    Chapter  Google Scholar 

  53. Paterson, M., Valiant, L.G.: Circuit size is nonlinear in depth. Theoret. Comput. Sci. 2(3), 397–400 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  54. Paul, W.: A 2.5 lower bound on the combinatorial complexity of boolean functions. SIAM J. Comput. 6(3), 427–443 (1977)

    Article  MathSciNet  Google Scholar 

  55. Rosenthal, R.W.: A class of games possessing pure strategy nash equilibria. Int. J. Game Theory 2(1), 65–67 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  56. Sahni, S.: Computationally related problems. SIAM J. Comput. 3(4), 262–279 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  57. Schoenebeck, G., Vadhan, S.: The computational complexity of nash equilibria in concisely represented games. ACM Trans. Comput. Theory 4(2), 4 (2012)

    Article  MATH  Google Scholar 

  58. Stockmeyer, L.J.: The polynomial time hierarchy. Theoret. Comput. Sci. 3(1), 1–22 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  59. Vollmer, H., Wagner, K.W.: Complexity classes of optimization functions. Inf. Comput. 120(2), 198–218 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  60. Voorneveld, M., Borm, P., van Megan, F., Tijs, S., Facchini, G.: Congestion games and potentials reconsidered. Int. Game Theory Rev. 1(3–4), 283–299 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  61. Wagner, K.W.: More complicated questions about maxima and minima, and some closures of \({\cal NP}\). Theoret. Comput. Sci. 51(1–2), 53–80 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  62. Wagner, K.W.: Bounded query classes. SIAM J. Comput. 19(5), 833–846 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  63. Wegener, I.: The Complexity of Boolean Functions. Wiley, New York (1991)

    MATH  Google Scholar 

  64. Wrathall, C.: Complete sets and the polynomial time hierarchy. Theoret. Comput. Sci. 3(1), 23–33 (1976)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank Paul Spirakis and Karsten Tiemann for many helpful discussions and comments on earlier versions of this work.

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Cyprus, 1678, Nicosia, Cyprus

    Marios Mavronicolas

  2. Faculty of Electrical Engineering, Computer Science and Mathematics, University of Paderborn, 33102, Paderborn, Germany

    Burkhard Monien

  3. Lehrstuhl für Theoretische Informatik, Institut für Informatik, Julius-Maximilians-Universität Würzburg, 97074, Würzburg, Germany

    Klaus W. Wagner

Authors
  1. Marios Mavronicolas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Burkhard Monien
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Klaus W. Wagner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marios Mavronicolas .

Editor information

Editors and Affiliations

  1. University of Patras, Patras, Greece

    Christos Zaroliagis

  2. Technological Educational Institute of Athens, Athens, Greece

    Grammati Pantziou

  3. University of Ioannina, Ioannina, Greece

    Spyros Kontogiannis

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Mavronicolas, M., Monien, B., Wagner, K.W. (2015). Weighted Boolean Formula Games. In: Zaroliagis, C., Pantziou, G., Kontogiannis, S. (eds) Algorithms, Probability, Networks, and Games. Lecture Notes in Computer Science(), vol 9295. Springer, Cham. https://doi.org/10.1007/978-3-319-24024-4_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-24024-4_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-24023-7

  • Online ISBN: 978-3-319-24024-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation