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2011 | 21 | 2 | 349-361

Tytuł artykułu

Convergence method, properties and computational complexity for Lyapunov games

Treść / Zawartość

Warianty tytułu

Języki publikacji

EN

Abstrakty

EN
We introduce the concept of a Lyapunov game as a subclass of strictly dominated games and potential games. The advantage of this approach is that every ergodic system (repeated game) can be represented by a Lyapunov-like function. A direct acyclic graph is associated with a game. The graph structure represents the dependencies existing between the strategy profiles. By definition, a Lyapunov-like function monotonically decreases and converges to a single Lyapunov equilibrium point identified by the sink of the game graph. It is important to note that in previous works this convergence has not been guaranteed even if the Nash equilibrium point exists. The best reply dynamics result in a natural implementation of the behavior of a Lyapunov-like function. Therefore, a Lyapunov game has also the benefit that it is common knowledge of the players that only best replies are chosen. By the natural evolution of a Lyapunov-like function, no matter what, a strategy played once is not played again. As a construction example, we show that, for repeated games with bounded nonnegative cost functions within the class of differentiable vector functions whose derivatives satisfy the Lipschitz condition, a complex vector-function can be built, where each component is a function of the corresponding cost value and satisfies the condition of the Lyapunov-like function. The resulting vector Lyapunov-like function is a monotonic function which can only decrease over time. Then, a repeated game can be represented by a one-shot game. The functionality of the suggested method is successfully demonstrated by a simulated experiment.

Rocznik

Tom

21

Numer

2

Strony

349-361

Opis fizyczny

Daty

wydano
2011
otrzymano
2010-01-26
poprawiono
2010-08-17
poprawiono
2010-10-15

Twórcy

  • Center for Computing Research, National Polytechnic Institute, Av. Juan de Dios Batiz s/n, Edificio CIC, Col. Nueva Industrial Valleys, 07738 Mexico City, Mexico
  • Department of Automatic Control, Center for Research and Advanced Studies, Av. IPN 2508, Col. San Pedro Zacatenco, 07360 Mexico City, Mexico

Bibliografia

  • Axelrod, R. (1984). The Evolution of Cooperation, Basic Books, New York, NY.
  • Bernheim, B.D. (1984). Rationalizable strategic behavior, Econometrica 52(4): 1007-1028.
  • Chen, X. and Deng, X. (2006). Setting the complexity of 2-player Nash equilibrium, Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2006, Berkeley, CA, USA, pp. 261-270.
  • Chen, X., Deng, X. and Tengand, S.-H. (2006). Computing Nash equilibria: Approximation and smoothed complexity, Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2006, Berkeley, CA, USA, pp. 603-612.
  • Clempner, J. (2006). Modeling shortest path games with Petri nets: A Lyapunov based theory, International Journal of Applied Mathematics and Computer Science 16(3): 387-397.
  • Daskalakis, C., Goldberg, P. and Papadimitriou, C. (2006a). The complexity of computing a Nash equilibrium, Proceedings of the 38th ACM Symposium on Theory of Computing, STOC 2006, Seattle, WA, USA, pp. 71-78.
  • Daskalakis, C., Mehta, A. and Papadimitriou, C. (2006b). A note on approximate Nash equilibria, Proceedings of the 2nd Workshop on Internet and Network Economics, WINE 06, Patras, Greece, pp. 297-306.
  • Fabrikant, A. and Papadimitriou, C. (2008). The complexity of game dynamics: BGP oscillations, sink equilibria, and beyond, Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2008, San Francisco, CA, USA, pp. 844-853.
  • Fabrikant, A., Papadimitriou, C. and Talwar, K. (2004). The complexity of pure Nash equilibria, Proceedings of the 36th ACM Symposium on Theory of Computing, STOC 2006, Chicago, IL, USA, pp. 604-612.
  • Goemans, M., Mirrokni, V. and Vetta, A. (2005). Sink equilibria and convergence, Proceedings of the 46th IEEE Symposium on Foundations of Computer Science, FOCS 2008, Pittsburgh, PA, USA, pp. 142-154.
  • Griffin, T.G. and Shepherd F.B. and Wilfong, G.W. (2002). The stable paths problem and interdomain routing, IEEE/ACM Transactions on Networking 10(2): 232-243.
  • Kakutani, S. (1941). A generalization of Brouwer's fixed point theorem, Duke Journal of Mathematics 8(3): 457-459.
  • Kontogiannis, S., Panagopoulou, P. and Spirakis, P. (2006). Polynomial algorithms for approximating nash equilibria of bimatrix games, Proceedings of the 2nd Workshop on Internet and Network Economics, WINE 06, Patras, Greece, pp. 286-296.
  • Lakshmikantham, V., Matrosov, V. and Sivasundaram, S. (1991). Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems, Kluwer Academic Publication, Dordrecht.
  • Lipton, R.J., Markakis, E. and Mehta, A. (2003). Playing large games using simple strategies, Proceedings of the 4th ACM Conference on Electronic Commerce, EC 2003, San Diego, CA, USA, pp. 36-41.
  • Mirrokni, V. and Vetta, A. (2004). Convergence issues in competitive games, Proceedings of the 7th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2004, Cambridge, MA, USA, pp. 183-194.
  • Moulin, H. (1984). Dominance solvability and Cournot stability, Mathematical Social Sciences 7(1): 83-102.
  • Myerson, R. B. (1978). Refinements of the Nash equilibrium concept, International Journal of Game Theory 7(2): 73-80.
  • Nash, J. (1996). Essays on Game Theory, Elgar, Cheltenham.
  • Nash, J. (1951). Non-cooperative games, Annals of Mathematics 54(2): 287-295.
  • Nash, J. (2002). The Essential John Nash, H.W. Kuhn and S. Nasar, Princeton, NJ.
  • Pearce, D. (1984). Rationalizable strategic behavior and the problem of perfection, Econometrica 52(4): 1029-1050.
  • Poznyak, A.S. (2008). Advance Mathematical Tools for Automatic Control Engineers, Vol. 1: Deterministic Techniques, Elsevier, Amsterdam.
  • Poznyak, A.S., Najim, K. and Gomez-Ramirez, E. (2000). SelfLearning Control of Finite Markov Chains, Marcel Dekker, New York, NY.
  • Selten, R. (1975). Reexamination of the perfectness concept for equilibrium points in extensive games, International Journal of Game Theory 4(1): 25-55.
  • 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.
  • Topkis, D. (1979). Equilibrium points in nonzero-sum n-persons submodular games, SIAM Journal of Control and Optimization 17(6): 773-787.
  • Toth, B. and Kreinovich, V. (2009). Verified methods for computing Pareto sets: General algorithmic analysis, International Journal of Applied Mathematics and Computer Science 19(3): 369-380, DOI: 10.2478/v10006/009-0031-5.

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Bibliografia

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