Theory of Computing Systems

, Volume 54, Issue 1, pp 83–110 | Cite as

Computational Aspects of Uncertainty Profiles and Angel-Daemon Games

  • Joaquim Gabarro
  • Alina Garcia
  • Maria Serna


We analyze the complexity of equilibria problems for a class of strategic zero-sum games, called angel-daemon games. Those games were introduced to asses the performance of the execution of a web orchestration on a moderate faulty or under stress environment. Angel-daemon games are a natural example of zero-sum games whose representation is naturally succinct. We show that the problems of deciding the existence of a pure Nash equilibrium or of a dominant strategy for a given player are \({\Sigma}^{p}_{2}\)-complete. Furthermore, computing the value of an angel-daemon game is EXP-complete. Thus, our results match the already known classification of the corresponding problems for the generic families of succinctly represented games with exponential number of actions.


Nash equilibrium Zero-sum games Succinct games Angel-daemon games Algorithmic game theory Orchestrations Fault tolerance Computational complexity 



The authors want to thank the anonymous referees for a careful reading and useful suggestions that help us to correct some inaccuracies and improve the presentation and readability of the paper.


  1. 1.
    Akerlof, G., Schiller, R.: Animal Spirits. Princeton University Press, Princeton (2009) Google Scholar
  2. 2.
    Akkarajitsakul, K., Hossain, E., Niyato, D.: Coalition-based cooperative packet delivery under uncertainty: a dynamic Bayesian coalitional game. IEEE Trans. Mob. Comput. 12(2), 371–385 (2013) CrossRefGoogle Scholar
  3. 3.
    Àlvarez, C., Gabarro, J., Serna, M.: Equilibria problems on games: complexity versus succinctness. J. Comput. Syst. Sci. 77(6), 1172–1197 (2011) CrossRefzbMATHGoogle Scholar
  4. 4.
    Ardagna, D., Panicucci, B., Passacantando, M.: A game theoretic formulation of the service provisioning problem in cloud systems. In: Proceedings of the 20th International Conference on World Wide Web, WWW’11, pp. 177–186. ACM, New York (2011) CrossRefGoogle Scholar
  5. 5.
    Babaioff, M., Kleinberg, R., Papadimitriou, C.H.: Congestion games with malicious players. Games Econ. Behav. 67(1), 22–35 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Bonzon, E., Lagasquie-Schiex, M.C., Lang, J., Zanuttini, B.: Boolean games revisited. In: Brewka, G., Coradeschi, S., Perini, A., Traverso, P. (eds.) Proceedings of the 2006 Conference on ECAI 2006: 17th European Conference on Artificial Intelligence, Riva del Garda, Italy, August 29–September 1, 2006. Including Prestigious Applications of Intelligent Systems (PAIS 2006), Frontiers in Artificial Intelligence and Applications, vol. 141, pp. 265–269. IOS Press, Amsterdam (2006) Google Scholar
  7. 7.
    Bougé, L.: Le mòdele de programmation à parallélisme de donés: une perspective sémantique. Tech. Sci. Inform. 12(5), 541–562 (1993) zbMATHGoogle Scholar
  8. 8.
    Castro, M., Liskov, B.: Practical byzantine fault tolerance. In: Seltzer, M.I., Leach, P.J. (eds.) Proceedings of the Third USENIX Symposium on Operating Systems Design and Implementation (OSDI), February 22–25, 1999, pp. 173–186. USENIX Association, New Orleans (1999) Google Scholar
  9. 9.
    Chakrabarty, D., Karande, C., Sangwan, A.: The effect of malice on the social optimum in linear load balancing games. CoRR (2009). arXiv:0910.2655 [cs.GT]
  10. 10.
    Chen, X., Deng, X., Teng, S.H.: Settling the complexity of computing two-player Nash equilibria. J. ACM 56(3), 14 (2009) MathSciNetGoogle Scholar
  11. 11.
    Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a Nash equilibrium. SIAM J. Comput. 39(1), 195–259 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Díaz, J., Mitsche, D., Rustagi, N., Saia, J.: On the power of mediators. In: Leonardi, S. (ed.) Proceedings of the 5th International Workshop on Internet and Network Economics, WINE 2009, Rome, Italy, December 14–18, 2009. Lecture Notes in Computer Science, vol. 5929, pp. 455–462. Springer, Berlin (2009) Google Scholar
  13. 13.
    Dolev, D.: The byzantine generals strike again. J. Algorithms 3(1), 14–30 (1982) CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Eliaz, K.: Fault tolerant implementation. Rev. Econ. Stud. 69(3), 589–610 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Fabrikant, A., Papadimitriou, C.H., Talwar, K.: The complexity of pure Nash equilibria. In: Babai, L. (ed.) Proceedings of the 36th Annual ACM Symposium on Theory of Computing, June 13–16, 2004, pp. 604–612. ACM, Chicago (2004) Google Scholar
  16. 16.
    Feigenbaum, J., Koller, D., Shor, P.: A game-theoretic classification of interactive complexity classes. In: SCT’95: Proceedings of the 10th Annual Structure in Complexity Theory Conference (SCT’95), p. 227. IEEE Comput. Soc., Washington (1995) CrossRefGoogle Scholar
  17. 17.
    Fortnow, L., Impagliazzo, R., Kabanets, V., Umans, C.: On the complexity of succinct zero-sum games. In: IEEE Conference on Computational Complexity, pp. 323–337 (2005) Google Scholar
  18. 18.
    Gabarro, J., Garcia, A., Clint, M., Kilpatrick, P., Stewart, A.: Bounded site failures: an approach to unreliable grid environments. In: Danelutto, M., Fragopoulou, P., Getov, V. (eds.) Making Grids Work, pp. 175–187. Springer, Berlin (2008) CrossRefGoogle Scholar
  19. 19.
    Gabarró, J., García, A., Serna, M.: The complexity of game isomorphism. Theor. Comput. Sci. 412(48), 6675–6695 (2011) CrossRefzbMATHGoogle Scholar
  20. 20.
    Gabarro, J., Garcia, A., Serna, M., Kilpatrick, P., Stewart, A.: Analysing orchestrations with risk profiles and angel-daemon games. In: Gorlatch, S., Fragopoulou, P., Priol, T. (eds.) Grid Computing Achievements and Propects, pp. 121–132. Springer, Berlin (2008) CrossRefGoogle Scholar
  21. 21.
    Gabarro, J., Serna, M., Stewart, A.: Web services and incerta spiriti: a game theoretic approach to uncertainty. In: Liu, W. (ed.) 11th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty, ECSQARU 2011. Lecture Notes in Computer Science, vol. 6717, pp. 651–662. Springer, Berlin (2011) Google Scholar
  22. 22.
    Gabarro, J., Serna, M., Stewart, A.: Analysing web orchestrations under stress using uncertainty profiles (2012). Submitted Google Scholar
  23. 23.
    Gabarro, J., Serna, M., Stewart, A.: Orchestrating unreliable services: strategic and probabilistic approaches to reliability. In: Bruni, R., Sassone, V. (eds.) 6th International Symposium on Trustworthy Global Computing, TGC 2011, Aachen, Germany, June 9–10, 2011. Lecture Notes in Computer Science, vol. 7173, pp. 197–211. Springer, Berlin (2012). Revised selected papers Google Scholar
  24. 24.
    Gao, A., Yang, D., Tang, S., Zhang, M.: QoS-driven web service composition with inter service conflicts. In: Zhou, X., Li, J., Shen, H.T., Kitsuregawa, M., Zhang, Y. (eds.) Proceedings of the 8th Asia-Pacific Web Conference on Frontiers of WWW Research and Development, APWeb 2006, Harbin, China, January 16–18, 2006. Lecture Notes in Computer Science, vol. 3841, pp. 121–132. Springer, Berlin (2006) CrossRefGoogle Scholar
  25. 25.
    Garcia, A.: The complexity of angel-daemons and game isomorphism. Ph.D. Thesis, Universitat Politècnica de Catalunya (Barcelona Tech.) (2012) Google Scholar
  26. 26.
    Gärtner, F.C.: Fundamentals of fault-tolerant distributed computing in asynchronous environments. ACM Comput. Surv. 31(1), 1–26 (1999) CrossRefGoogle Scholar
  27. 27.
    Gottlob, G., Greco, G., Scarcello, F.: Pure Nash equilibria: hard and easy games. J. Artif. Intell. Res. 24, 215–230 (2005) MathSciNetGoogle Scholar
  28. 28.
    Greenlaw, R., Hoover, H.J., Ruzzo, W.L.: Limits to Parallel Computation: P-Completeness Theory. Oxford University Press, New York (1995) zbMATHGoogle Scholar
  29. 29.
    Johnson, B., Grossklags, J., Christin, N., Chuang, J.: Uncertainty in interdependent security games. In: Alpcan, T., Buttyán, L., Baras, J.S. (eds.) Proceedings of the First International Conference on Decision and Game Theory for Security, GameSect. 2010, Berlin, Germany, November 22–23, 2010. Lecture Notes in Computer Science, vol. 6442, pp. 234–244. Springer, Berlin (2010) CrossRefGoogle Scholar
  30. 30.
    Karakostas, G., Viglas, A.: Equilibria for networks with malicious users. Math. Program. 110(3), 591–613 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Kearns, M.J., Littman, M.L., Singh, S.P.: Graphical models for game theory. In: Breese, J.S., Koller, D. (eds.) UAI’01: Proceedings of the 17th Conference in Uncertainty in Artificial Intelligence, University of Washington, Seattle, Washington, USA, August 2–5, 2001, pp. 253–260. Morgan Kaufmann, San Mateo (2001) Google Scholar
  32. 32.
    Keynes, J.: The General Theory of Employment, Interest and Money. Palgrave Macmillan, Basingstoke, UK (2007) Google Scholar
  33. 33.
    Kitchin, D., Quark, A., Cook, W.R., Misra, J.: The Orc programming language. In: Lee, D., Lopes, A., Poetzsch-Heffter, A. (eds.) Proceedings of the Joint 11th IFIP WG 6.1 International Conference FMOODS 2009 and 29th IFIP WG 6.1 International Conference FORTE 2009 on Formal Techniques for Distributed Systems. Lecture Notes in Computer Science, vol. 5522, pp. 1–25. Springer, Berlin (2009) Google Scholar
  34. 34.
    Knight, F., Risk, uncertainty and profit. Houghton Mifflin, free electronic access in: (1921)
  35. 35.
    Kokash, N., D’Andrea, V.: Evaluating quality of web services: a risk-driven approach. In: Abramowicz, W. (ed.) Proceedings of the 10th International Conference on Business Information Systems, BIS 2007, Poznan, Poland, April 25–27, 2007. Lecture Notes in Computer Science, vol. 4439, pp. 180–194. Springer, Berlin (2007) Google Scholar
  36. 36.
    Koo, C.Y.: Broadcast in radio networks tolerating byzantine adversarial behavior. In: Chaudhuri, S., Kutten, S. (eds.) Proceedings of the Twenty-Third Annual ACM Symposium on Principles of Distributed Computing, PODC 2004, St. John’s, Newfoundland, Canada, July 25–28, 2004, pp. 275–282. ACM, New York (2004) CrossRefGoogle Scholar
  37. 37.
    Ladner, R.E.: The circuit value problem is log space complete for p. SIGACT News 7(1), 18–20 (1975) CrossRefMathSciNetGoogle Scholar
  38. 38.
    Lamport, L., Shostak, R.E., Pease, M.C.: The byzantine generals problem. ACM Trans. Program. Lang. Syst. 4(3), 382–401 (1982) CrossRefzbMATHGoogle Scholar
  39. 39.
    Liu, A., Li, Q., Huang, L., Ying, S., Xiao, M.: Coalitional game for community-based autonomous web services cooperation. IEEE Trans. Serv. Comput. 99 (2012). Preprint. doi: 10.1109/TSC.2012.12
  40. 40.
    Malkhi, D., Reiter, M.K.: Byzantine quorum systems. Distrib. Comput. 11(4), 203–213 (1998) CrossRefGoogle Scholar
  41. 41.
    Mavronicolas, M., Monien, B., Wagner, K.W.: Weighted boolean formula games. In: Deng, X., Graham, F.C. (eds.) Proceedings of the Third International Workshop on Internet and Network Economics, WINE 2007, San Diego, CA, USA, December 12–14, 2007. Lecture Notes in Computer Science, vol. 4858, pp. 469–481 (2007) Google Scholar
  42. 42.
    Misra, J.: A programming model for the orchestration of web services. In: Software Engineering and Formal Methods, SEFM’04, pp. 2–11. IEEE Comput. Soc., Los Alamitos (2004) Google Scholar
  43. 43.
    Misra, J., Cook, W.: Computation orchestration: a basis for wide-area computing. Softw. Syst. Model. 6(1), 83–110 (2007) CrossRefGoogle Scholar
  44. 44.
    Moscibroda, T., Schmid, S., Wattenhofer, R.: The price of malice: a game theoretic framework for malicious behaviour in distributed systems. Internet Econ. 6(2), 125–155 (2009) zbMATHMathSciNetGoogle Scholar
  45. 45.
    von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944) zbMATHGoogle Scholar
  46. 46.
    Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V.: Algorithmic Game Theory. Cambridge University Press, Cambridge (2007) CrossRefzbMATHGoogle Scholar
  47. 47.
    Nisan, N., Rougarden, T., Tardos, E., Vazirani, V.V. (eds.): Algorithmic Game Theory. Cambridge University Press, Cambridge (2007) zbMATHGoogle Scholar
  48. 48.
    Niyato, D., Vasilakos, A.V., Zhu, K.: Resource and revenue sharing with coalition formation of cloud providers: game theoretic approach. In: 11th IEEE/ACM International Symposium on Cluster, Cloud and Grid Computing, CCGrid 2011, Newport Beach, CA, USA, May 23–26, 2011, pp. 215–224. IEEE Press, New York (2011) CrossRefGoogle Scholar
  49. 49.
    Osborne, M.: An Introductions to Game Theory. Oxford University Press, London (2004) Google Scholar
  50. 50.
    Osborne, M., Rubinstein, A.: A Course in Game Theory. MIT Press, Cambridge (1994) zbMATHGoogle Scholar
  51. 51.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994) zbMATHGoogle Scholar
  52. 52.
    Papadimitriou, C.H.: The complexity of finding Nash equilibria. In: Nisan, N., Rougarden, T., Tardos, E., Vazirani, V.V. (eds.) Algorithmic Game Theory, pp. 29–52. Cambridge University Press, Cambridge (2007) CrossRefGoogle Scholar
  53. 53.
    Papadimitriou, C.H., Roughgarden, T.: Computing correlated equilibria in multi-player games. J. ACM 55(3), 14:1–14:29 (2008) MathSciNetGoogle Scholar
  54. 54.
    Pease, M.C., Shostak, R.E., Lamport, L.: Reaching agreement in the presence of faults. J. ACM 27(2), 228–234 (1980) zbMATHMathSciNetGoogle Scholar
  55. 55.
    Rosario, S., Benveniste, A., Jard, C.: Flexible probabilistic QoS management of orchestrations. Int. J. Web Serv. Res. 7(2), 21–42 (2010) CrossRefGoogle Scholar
  56. 56.
    Rosenthal, R.: A class of games possessing pure-strategy Nash equilibria. Int. J. Game Theory 2, 65–67 (1973) CrossRefzbMATHGoogle Scholar
  57. 57.
    Roth, A.: The price of malice in linear congestion games. In: Papadimitriou, C.H., Zhang, S. (eds.) Proceedings of the 4th International Workshop on Internet and Network Economics, WINE 2008, Shanghai, China, December 17–20, 2008. Lecture Notes in Computer Science, vol. 5385, pp. 118–125. Springer, Berlin (2008) Google Scholar
  58. 58.
    Roughgarden, T.: Stackelberg scheduling strategies. In: Vitter, J.S., Spirakis, P.G., Yannakakis, M. (eds.) Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, Heraklion, Crete, Greece, July 6–8, 2001, pp. 104–113. ACM, New York (2001) Google Scholar
  59. 59.
    Schoenebeck, G., Vadhan, S.P.: The computational complexity of Nash equilibria in concisely represented games. In: 7th ACM Conference on Electronic Commerce (EC 2006), pp. 270–279 (2006) Google Scholar
  60. 60.
    Srikanth, T.K., Toueg, S.: Simulating authenticated broadcasts to derive simple fault-tolerant algorithms. Distrib. Comput. 2(2), 80–94 (1987) CrossRefGoogle Scholar
  61. 61.
    Wang, X., Vitvar, T., Kerrigan, M., Toma, I.: A QoS-aware selection model for semantic web services. In: Dan, A., Lamersdorf, W. (eds.) Proceedings of the 4th International Conference on Service-Oriented Computing, ICSOC 2006, Chicago, IL, USA, December 4–7, 2006. Lecture Notes in Computer Science, vol. 4294, pp. 390–401. Springer, Berlin (2006) CrossRefGoogle Scholar
  62. 62.
    Watson, J.: Strategy: An Introduction to Game Theory. Norton, New York (2002) Google Scholar
  63. 63.
    Wehrman, I., Kitchin, D., Cook, W.R., Misra, J.: A timed semantics of orc. Theor. Comput. Sci. 402(2–3), 234–248 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  64. 64.
    Welch, J.L., Lynch, N.A.: A new fault-tolerance algorithm for clock synchronization. Inf. Comput. 77(1), 1–36 (1988) CrossRefzbMATHMathSciNetGoogle Scholar
  65. 65.
    Yang, X., Cui, W., Liu, Z., Ouyang, F.: Study on uncertainty of geospatial semantic web services composition based on broker approach and Bayesian networks. In: Proceedings of the SPIE 7143 on Geoinformatics 2008 and Joint Conference on GIS and Built Environment: Geo-Simulation and Virtual GIS Environments (2008) Google Scholar
  66. 66.
    Zheng, X., Martin, P., Powley, W., Brohman, K.: Applying bargaining game theory to web services negotiation. In: IEEE International Conference on Services Computing (SCC), pp. 218–225 (2010) Google Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.ALBCOM Research GroupUniversitat Politècnica de CatalunyaBarcelonaSpain

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