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Dynamics and stability for a class of evolutionary games with time delays in strategies

一类具有时滞策略的演化博弈的动力学和稳定性

Abstract

This paper investigates the modeling and stability of a class of finite evolutionary games with time delays in strategies. First, the evolutionary dynamics of a sequence of strategy profiles, named as the profile trajectory, is proposed to describe the strategy updating process of the evolutionary games with time delays. Using the semi-tensor product of matrices, the profile trajectory dynamics with two kinds of time delays are converted into their algebraic forms respectively. Then a sufficient condition is obtained to assure the stability of the delayed evolutionary potential games at a pure Nash equilibrium.

摘要

创新点

针对策略存在时间延迟的有限演化博弈, 本文研究了其动态演化过程和稳定性。 首先, 利用矩阵的半张量积这一数学工具, 将策略存在常时滞和变时滞两种不同情况下的逻辑动态演化过程分别转化为相应的代数表达形式。 其次, 利用势博弈的优点, 给出了时滞演化的有限势博弈收敛到纳什均衡点的充分条件。

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References

  1. 1

    Maynard S J, Price G R. The logic of animal conflict. Nature, 1973, 246: 15–18

  2. 2

    Madeo D, Mocenni C. Game interactions and dynamics on networked populations. IEEE Trans Automat Contr, 2015, 60: 1801–1810

  3. 3

    Parke W R, Waters G A. An evolutionary game theory explanation of ARCH effects. J Econ Dynam Control, 2007, 31: 2234–2262

  4. 4

    Ohtsuki H, Hauert C, Lieberman E, et al. A simple rule for the evolution of cooperation on graphs and social networks. Nature, 2006, 441: 502–505

  5. 5

    Mojica-Nava E, Macana C A, Quijano N. Dynamic population games for optimal dispatch on hierarchical microgrid control. IEEE Trans Syst Man Cybern: Syst, 2014, 44: 306–317

  6. 6

    Alboszta J, Miekisz J. Stability and evolutionary stable strategies in discrete replicator dynamics with delay. J Theor Biol, 2004, 231: 175–179

  7. 7

    Taylor P D, Jonker L. Evolutionary stable strategies and game dynamics. Math Biosci, 1978, 16: 76–83

  8. 8

    Szabó G, Fath G. Evolutionary games on graphs. Phys Rep, 2007, 446: 97–216

  9. 9

    Suzuki S, Akiyama E. Evolutionary stability of first-order-information indirect reciprocity in sizable groups. Theor Popul Biol, 2008, 73: 426–436

  10. 10

    Arino J, Wang L, Wolkowicz G. An alternate formulation for a delayed logistic equation. J Theor Biol, 2006, 241: 109–119

  11. 11

    Miekisz J, Wesolowski S. Stochasticity and time delays in evolutionary games. Dynam Games Appl, 2011, 1: 440–448

  12. 12

    Tao Y, Wang Z. Effect of time delay and evolutionarily stable strategy. J Theor Biol, 1997, 187: 111–116

  13. 13

    Tembine H, Altman E, El-Azouzi R. Delayed evolutionary game dynamics applied to medium access control. In: Proceedings of IEEE International Conference on Mobile Adhoc and Sensor Systems, Pisa, 2007. 1–6

  14. 14

    Brown G W. Iterative solution of games by fictitious play. Activ Anal Prod Allocation, 1951, 13: 374–376

  15. 15

    Candogan O, Menache I, Ozdaglar A, et al. Flows and decompositions of games: harmonic and potential games. Math Oper Res, 2011, 36: 474–503

  16. 16

    Morris S, Ui T. Generalized potentials and robust sets of equilibria. J Econ Theory, 2005, 124: 45–78

  17. 17

    Ui T. Robust equilibria of potential games. Econometrica, 2001, 69: 1373–1380

  18. 18

    Sandholm W H. Large population potential games. J Econ Theory, 2009, 144: 1710–1725

  19. 19

    Cheng D, Qi H, Li Z. Analysis and Control of Boolean Networks: a Semi-tensor Product Approach. London: Springer, 2011

  20. 20

    Li Z Q, Song J L. Controllability of Boolean control networks avoiding states set. Sci China Inf Sci, 2014, 57: 032205

  21. 21

    Feng J, Yao J, Cui P. Singular Boolean networks: semi-tensor product approach. Sci China Inf Sci, 2013, 56: 112203

  22. 22

    Li H, Wang Y, Xie L. Output tracking control of Boolean control networks via state feedback: constant reference signal case. Automatica, 2015, 59: 54–59

  23. 23

    Zhao Y, Cheng D Z. On controllability and stabilizability of probabilistic Boolean control networks. Sci China Inf Sci, 2014, 57: 012202

  24. 24

    Liu Z B, Wang Y Z, Li H T. Two kinds of optimal controls for probabilistic mix-valued logical dynamic networks. Sci China Inf Sci, 2014, 57: 052201

  25. 25

    Wang Y, Zhang C, Liu Z. A matrix approach to graph maximum stable set and coloring problems with application to multi-agent systems. Automatica, 2012, 48: 1227–1236

  26. 26

    Cheng D, He F, Qi H, et al. Modeling, analysis and control of networked evolutionary games. IEEE Trans Automat Contr, 2015, 60: 2402–2415

  27. 27

    Guo P, Wang Y, Li H. Algebraic formulation and strategy optimization for a class of evolutionary networked games via semi-tensor method. Automatica, 2013, 49: 3384–3389

  28. 28

    Zhang L J, Zhang K Z. Controllability of time-variant Boolean control networks and its application to Boolean control networks with finite memories. Sci China Inf Sci, 2013, 56: 108201

  29. 29

    Rasmusen E. Games and Information: an Introduction to Game Theory. Oxford: Basil Blackwell, 2007

  30. 30

    Monderer D, Shapley L S. Potential games. Games Econ Behav, 1996, 14: 124–143

  31. 31

    Cheng D. On finite potential games. Automatica, 2014, 50: 1793–1801

  32. 32

    Young H P. The evolution of conventions. Econometrica, 1993, 61: 57–84

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Correspondence to Yuanhua Wang.

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Wang, Y., Cheng, D. Dynamics and stability for a class of evolutionary games with time delays in strategies. Sci. China Inf. Sci. 59, 92209 (2016). https://doi.org/10.1007/s11432-016-5532-x

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Keywords

  • evolutionary game
  • time delays
  • potential game
  • semi-tensor product of matrices
  • stability

关键词

  • 演化博弈
  • 时滞
  • 势博弈
  • 矩阵的半张量积
  • 稳定性