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A symbolic shortest path algorithm for computing subgame-perfect Nash equilibria

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Języki publikacji
EN
Abstrakty
EN
Consider games where players wish to minimize the cost to reach some state. A subgame-perfect Nash equilibrium can be regarded as a collection of optimal paths on such games. Similarly, the well-known state-labeling algorithm used in model checking can be viewed as computing optimal paths on a Kripke structure, where each path has a minimum number of transitions. We exploit these similarities in a common generalization of extensive games and Kripke structures that we name “graph games”. By extending the Bellman–Ford algorithm for computing shortest paths, we obtain a model-checking algorithm for graph games with respect to formulas in an appropriate logic. Hence, when given a certain formula, our model-checking algorithm computes the subgame-perfect Nash equilibrium (as opposed to simply determining whether or not a given collection of paths is a Nash equilibrium). Next, we develop a symbolic version of our model checker allowing us to handle larger graph games. We illustrate our formalism on the critical-path method as well as games with perfect information. Finally, we report on the execution time of benchmarks of an implementation of our algorithms.
Rocznik
Strony
577--596
Opis fizyczny
Bibliogr. 30 poz., rys., tab., wykr.
Twórcy
  • Institute for Research in Applied Mathematics and Systems, National Autonomous University of Mexico, A.P. 20-126, C.P. 01000, Mexico D.F., Mexico
  • Institute for Research in Applied Mathematics and Systems, National Autonomous University of Mexico, A.P. 20-126, C.P. 01000, Mexico D.F., Mexico
Bibliografia
  • [1] Bahar, R., Frohm, E., Gaona, C., Hachtel, G., Macii, E., Pardo, A. and Somenzi, F. (1997). Algebraic decision diagrams and their applications, Formal Methods in System Design 10(2–3): 171–206.
  • [2] Baier, C. and Katoen, J.-P. (2008). Principles of Model Checking, MIT Press, New York, NY.
  • [3] Berghammer, R. and Bolus, S. (2012). On the use of binary decision diagrams for solving problems on simple games, European Journal of Operational Research 222(3): 529–541.
  • [4] Bolus, S. (2011). Power indices of simple games and vector-weighted majority games by means of binary decision diagrams, European Journal of Operational Research 210(2): 258–272.
  • [5] Bonanno, G. (2001). Branching time, perfect information games, and backward induction, Games and Economic Behavior 36(1): 57–73.
  • [6] Bryant, R.E. (1986). Graph-based algorithms for Boolean function manipulation, IEEE Transactions on Computers 35(8): 677–691.
  • [7] Burch, J., Clarke, E., McMillan, K., Dill, D. and Hwang, L. (1992). Symbolic model checking: [...] states and beyond, Information and Computation 98(2): 142–170.
  • [8] Clarke, E. and Emerson, E. (1982). Design and synthesis of synchronization skeletons using branching time temporal logic, in D. Kozen (Ed.), Workshop on Logics of Programs, Lecture Notes in Computer Science, Vol. 131, Springer, Berlin/Heidelberg, pp. 52–71.
  • [9] Clarke, E.M., Emerson, E.A. and Sistla, A.P. (1986). Automatic verification of finite-state concurrent systems using temporal logic specifications, ACM Transactions on Programming Languages and Systems 8(2): 244–263.
  • [10] Clarke, E.M., Grumberg, O. and Peled, D.A. (1999). Model Checking, MIT Press, London.
  • [11] Clarke, E., McMillan, K., Zhao, X., Fujita, M. and Yang, J. (1993). Spectral transforms for large Boolean functions with applications to technology mapping, 30th Conference on Design Automation, Dallas, TX, USA, pp. 54–60.
  • [12] Cormen, T.H., Leiserson, C.E., Rivest, R.L. and Stein, C. (2009). Introduction to Algorithms, 3rd Edn., MIT Press, Cambridge, MA.
  • [13] Dasgupta, P., Chakrabarti, P.P., Deka, J.K. and Sankaranarayanan, S. (2001). Min-max computation tree logic, Artificial Intelligence 127(1): 137–162.
  • [14] Dsouza, A. and Bloom, B. (1995). Generating BDD models for process algebra terms, Proceedings of the 7th International Conference on Computer Aided Verification, Liège, Belgium, pp. 16–30.
  • [15] Enders, R., Filkorn, T. and Taubner, D. (1992). Generating BDDs for symbolic model checking in CCS, Proceedings of the 3rd International Workshop on Computer Aided Verification, CAV’91, London, UK, pp. 203–213.
  • [16] Fujita, M., McGeer, P.C. and Yang, J.C.-Y. (1997). Multi-terminal binary decision diagrams: An efficient data structure for matrix representation, Formal Methods in System Design 10(2–3): 149–169.
  • [17] Garroppo, R.G., Giordano, S. and Tavanti, L. (2010). A survey on multi-constrained optimal path computation: Exact and approximate algorithms, Computer Networks 54(17): 3081–3107.
  • [18] Harrenstein, P., van der Hoek, W., Meyer, J.-J.C. and Witteveen, C. (2003). A modal characterization of Nash equilibrium, Fundamenta Informaticae 57(2–4): 281–321.
  • [19] Hermanns, H., Meyer-Kayser, J. and Siegle, M. (1999). Multi terminal binary decision diagrams to represent and analyse continuous time Markov chains, in B. Plateau, W.J. Stewart and M. Silva (Eds.), 3rd International Workshop on the Numerical Solution of Markov Chains, Zaragoza, Spain, Prensas Universitarias de Zaragoza, Zaragoza, pp. 188–207.
  • [20] Kelley, Jr, J.E. and Walker, M.R. (1959). Critical-path planning and scheduling, Eastern Joint IRE-AIEE-ACM Computer Conference, Boston, MA, USA, pp. 160–173.
  • [21] Lozovanu, D. and Pickl, S. (2009). Optimization and Multiobjective Control of Time-Discrete Systems, Springer, Berlin/Heidelberg.
  • [22] McKelvey, R.D. and McLennan, A. (1996). Computation of equilibria in finite games, in H.M. Amman, D.A. Kendrick and J. Rust (Eds.), Handbook of Computational Economics, Vol. 1, Elsevier, North Holland, Chapter 2, pp. 87–142.
  • [23] McKelvey, R.D., McLennan, A.M. and Turocy, T.L. (2014). Gambit: Software tools for game theory, version 13.1.2, http://www.gambit-project.org.
  • [24] Meinel, C. and Theobald, T. (1998). Algorithms and Data Structures in VSLI Design: OBDD—Foundations and Applications, Springer-Verlag, Berlin.
  • [25] MSDN (2013). Memory limits for windows releases, Microsoft Developer Network, http://msdn.microsoft.com/en-us/library/windows/desktop/aa366778(v=vs.85).aspx.
  • [26] Osborne, M.J. and Rubinstein, A. (1994). A Course in Game Theory, The MIT Press, Cambridge, MA.
  • [27] Raimondi, F. and Lomuscio, A. (2007). Automatic verification of multi-agent systems by model checking via ordered binary decision diagrams, Journal of Applied Logic 5(2): 235–251.
  • [28] Russell, S. and Norvig, P. (2003). Artificial Intelligence. A Modern Approach, Prentince Hall, Englewood Cliffs, NJ.
  • [29] Sawitzki, D. (2004). Experimental studies of symbolic shortest-path algorithms, in C.C. Ribeiro and S.L. Martins (Eds.), Experimental and Efficient Algorithms, Lecture Notes in Computer Science, Vol. 3059, Springer, Berlin/Heidelberg, pp. 482–497.
  • [30] Tarapata, Z. (2007). Selected multicriteria shortest path problems: An analysis of complexity, models and adaptation of standard algorithms, International Journal of Applied Mathematics and Computer Science 17(2): 269–287, DOI: 10.2478/v10006-007-0023-2.
Typ dokumentu
Bibliografia
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