Modeling and Stability Analysis of Delayed Game Network

Authors: Zixin Liu, Jian Yu, Daoyun Xu

Abstract:

This paper aims to establish a delayed dynamical relationship between payoffs of players in a zero-sum game. By introducing Markovian chain and time delay in the network model, a delayed game network model with sector bounds and slope bounds restriction nonlinear function is first proposed. As a result, a direct dynamical relationship between payoffs of players in a zero-sum game can be illustrated through a delayed singular system. Combined with Finsler-s Lemma and Lyapunov stable theory, a sufficient condition guaranteeing the unique existence and stability of zero-sum game-s Nash equilibrium is derived. One numerical example is presented to illustrate the validity of the main result.

Keywords: Game networks, zero-sum game, delayed singular system, nonlinear perturbation, time delay.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1330989

References:

[1] J. Von Neumann, O. Morgenstern, H.W. Kuhn, A. Rubinstein, Theory of Games and Economic Behavior, Princeton university press, Princeton, NJ, 1947.
[2] J. Nash, Equilibrium points in n-person games, Proc. Nat. Acad. Sci. U.S.A. 36 (1) (1950) 48-49.
[3] L.S. Shapley, Stochastic Games, Proc. Nat. Acad. Sci. U.S.A. 39 (1953) 1095-1100.
[4] Harsanyi, C. John, An equilibrium point interpretation of stable sets, Manage. SCI. 20 (11) (1974) 1472-1495.
[5] S. Maynard, John, Evolution and the theory of games, Cambridge University Press, 1982.
[6] Skyrms, Brian, The stag hunt and the evolution of social structure, Cambridge University Press, 2004.
[7] Z. Rong, X. Li, X. Wang, Roles of mixing patterns in cooperation on a scale-free networked game, Phys. Rev. E. 76 (2) (2007) 027101.
[8] J. Hofbauer, K. Sigmund, Evolutionary game dynamics, Bull. Am. Math. Soc. 40 (4) (2003) 479-519.
[9] E. Semsar-Kazerooni, K. Khorasani, Multi-agent team cooperation:a game theory approach, Automatica 45 (10) (2009) 2205-2213.
[10] Z. Varga, A. Scarelli, R. Cressman, J. Garay, Evolutionary game model for a marketing cooperative with penalty for unfaithfulness, Nonlinear Anal. Real World Appl. 11 (2) (2010) 742-749.
[11] J. Yu, Slightly altruistic equilibria of n-person noncooperative game, J. of Systems Science and Mathematical Sciences, 31(5) (2011) 534-539.
[12] J.Wang, F. Fu, and L.Wang, Effects of heterogeneous wealth distribution on public cooperation with collective risk, Phys. Rev. E. 82(1) (2010) 16102.
[13] F.V. Jennifer, Policy: Global cooperation game, Nature Climate Chan. 1 (2011) DOI: doi:10.1038/nclimate1038.
[14] J.Q. Liu, X.D. Liu, Fuzzy extensions of bargaining sets and their existence in cooperative fuzzy games, Fuzzy Sets Syst. 188 (1) (2012) 88-101.
[15] W.J. Xiong, W.H. Daniel, J.D. Cao, Dynamic alanalysis of a game network, Nonlinear Anal. Real World Appl. 12 (2011) 2286-2293.
[16] H.L. Gao, B.G. Xu, Delay-dependent state feedback robust stabilization for uncertain singular time-delay systems, J. Syst. Eng. Electron. 19(4) (2008) 758-765.
[17] Y.Q. Xia, et al., Stability and stabilization of continuous-time singular hybrid systems, Automatica, 45(6) (2009) 1504-1509.
[18] S.L. Tung, et al., An improved particle swarm optimization for exponential stabilization of a singular linear time-varying system, Expert Syst. Appl. 38 (10) (2011) 13425-13431.
[19] R.E. Skelton, T. Lwasaki, K.M. Grigoradis, A unified algebraic approach to linear control design, Taylor and Francis, New York, 1997.
[20] J. Sun, G.P. Liu, J. Chen, Delay-dependent stability and stabilization of neutral time-delay systems, Int. J. Roubst Nonlinear Control. 19 (2009) 1364-1375.
[21] J.H. Park , O.M. Kwon , S.M. Lee, LMI optimization approach on stability for delayed neural networks of neutral-type, Appl. Math. Comput. 196 (2008) 236-244.
[22] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnam, Linear Matrix Inequalities in System and Control Theory, SIMA, Philadelphia, 1994.