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A Comprehensive Survey on STP Approach to Finite Games

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Abstract

Nowadays the semi-tensor product (STP) approach to finite games has become a promising new direction. This paper provides a comprehensive survey on this prosperous field. After a brief introduction for STP and finite (networked) games, a description for the principle and fundamental technique of STP approach to finite games is presented. Then several problems and recent results about theory and applications of finite games via STP are presented. A brief comment about the potential use of STP to artificial intelligence is also proposed.

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References

  1. Wiener N, Cybernetics or Control of and Communication in the Animal and the Machine, Hermann & Camb. Press, Paris, 1948.

    Google Scholar 

  2. Neumann J V and Morgenstern O, Theory of Games and Economic Behavior, Princeton University Press, Princeton, New Jersey, 1944.

    MATH  Google Scholar 

  3. Nash J, Non-cooperative game, The Annals of Mathematics, 1951, 54(2): 286–295.

    Article  MathSciNet  MATH  Google Scholar 

  4. Gale D and Shapley L S, Colle admissions and the stability of marriage, American Math. Monthly, 1962, 69: 9–15.

    Article  MathSciNet  MATH  Google Scholar 

  5. Guo L, Some comments on the development of control theory, Journal of Systems Science and Mathematical Sciences, 2011, 31(9): 1014–1018 (in Chinese).

    MathSciNet  Google Scholar 

  6. Başar T and Olsder G J, Dynamic Noncooperative Game Theory, 2nd Edition, Philadeiphia, SIAM, 1999.

    MATH  Google Scholar 

  7. Zhang R R, Wang F, and Guo L, On game-based control systems and beyond, Nat. Sci. Rev., 2020, 7: 1116–1117.

    Article  Google Scholar 

  8. Marden J R, Arslan G, and Shamma J S, Cooperative control and potential games, IEEE Trans. Sys., Man, Cybernetcs, Part B, 2009, 39(6): 1393–1407.

    Article  Google Scholar 

  9. Gibbons R, A Primer in Game Theory, Prentice Hall, London, 1992.

    MATH  Google Scholar 

  10. Bilbao J M, Cooperative Games on Combinatorial Structures, Kluwer Acad. Pub., Boston, 2000.

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  12. Cheng D Z, Qi H S, and Zhao Y, An Introduction to Semi-tensor Product of Matrices and Its Applications, World Scientific, Singapore, 2012.

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Cheng D Z, He F H, Qi H S, et al., Modeling, analysis and control of networked evolutionary games, IEEE Trans. Aut. Contr., 2015, 60(9): 2402–2415.

    Article  MathSciNet  MATH  Google Scholar 

  15. Zhao G D, Li H T, Sun W W, et al., Modelling and strategy consensus for a class of networked evolutionary games, Int. J. Sys. Sci., 2018, 49(12): 2548–2557.

    Article  MathSciNet  Google Scholar 

  16. Monderer D and Shapley L S, Potential games, Games and Economic Behavior, 1996, 14: 124–143.

    Article  MathSciNet  MATH  Google Scholar 

  17. Gopalakrishnan R, Marden J R, and Wierman A, An architectural view of game theoretic control, Performance Evaluation Review, 2011, 38(3): 31–36.

    Article  Google Scholar 

  18. Hino Y, An improved algorithm for detecting potential games, Int. J. Game Theory, 2011, 40: 199–205.

    Article  MathSciNet  MATH  Google Scholar 

  19. Hofbauer J and Sorger G, A differential game approach to evolutionary equilibrium selection, Int. Game Theory Rev., 2002, 4: 17–31.

    Article  MathSciNet  MATH  Google Scholar 

  20. Cheng D Z, On finite potential games, Automatica, 2014, 50(7): 1793–1801.

    Article  MathSciNet  MATH  Google Scholar 

  21. Cheng D Z, Xu T T, and Qi H S, Evolutionarily stable strategy of networked evolutionary games, IEEE Trans. Neu. Net. Learn. Sys., 2014, 25(7): 1335–1345.

    Article  Google Scholar 

  22. Liu X Y and Zhu J D, On potential equitions of finite games, Automatica, 2016, 68: 245–253.

    Article  Google Scholar 

  23. Wang Y H and Cheng D Z, On coset weighted potential game, J. Franklin Inst., 2020, 357(9): 5523–5540.

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang Y H, Liu T, and Cheng D Z, From weighted potential game to weighted harmonic game, IET Cont. theor. Appl., 2017, 11(13): 2161–2169.

    Article  MathSciNet  Google Scholar 

  25. Cheng D Z and Ji Z P, Weighted and near weighted potential games with application to game theoretic control, 2021, http://arxiv.org/abs/2106.12335 (submitted for pub.).

  26. Rosenthal R W, A class of games possessing pure-strategy Nash equilibria, Int. J. Game Theory, 1973, 2: 65–67.

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhang L, Gong K, and Xu M Z, Congestion control in charging stations allocation with Q-learning, Sustainability, 2019, 11: 3900.

    Article  Google Scholar 

  28. Zhou B, Song Q K, Zhao Z J, et al., A reinforcement learning scheme for the equilibrium of the in-vehicle route choice problem based on congestion game, Appl. Math. Comp., 2020, 371: 124895.

    Article  MathSciNet  MATH  Google Scholar 

  29. Le S T, Wu Y H, and Sun X M, Congestion games with player-specific utility functions and its application to NFV networks, IEEE Trans. Aut. Sci. Eng., 2019, 16: 1870–1881.

    Article  Google Scholar 

  30. Zhang J, Lu J Q, Cao J D, et al., Traffic congestion procing via network congestion game approach, Dis. Cont. Dynam. Sys. Series S, 2021, 14(4): 1553–1567.

    Google Scholar 

  31. Zhang J, Lou J G, Qiu J L, et al., Dynamics and cnovergence of huper-networked evolutionary games with time delay is strategies, Information Sciences, 2021, 563: 166–182.

    Article  MathSciNet  Google Scholar 

  32. Gao X Y, Wang J H, and Zhang K Z, Dynamics and control of evolutionary congestion games, Sci. China Inform. Sci., 2020, 63(6): 169203.

    Article  MathSciNet  Google Scholar 

  33. Candogan O, Menache I, Ozdaglar A, et al., Flows and decompositions of games: Harmonic and potential games, Mathematcs of Operations Research, 2011, 36(3): 474–503.

    Article  MathSciNet  MATH  Google Scholar 

  34. Cheng D Z, Liu T, and Zhang K Z, On decomposed subspaces of finite games, IEEE Trans. Aut. Contr., 2016, 61(11): 3651–3656.

    Article  MathSciNet  MATH  Google Scholar 

  35. Cheng D Z and Liu T, Linear representation of symmetric games, IET Contr. Theory Appl., 2017, 11(18): 3278–3287.

    Article  MathSciNet  Google Scholar 

  36. Hao Y Q and Cheng D Z, On skew-symmetric games, Journal of the Franklin Institute, 2018, 355: 3196–3220.

    Article  MathSciNet  MATH  Google Scholar 

  37. Cheng D Z and Liu T, From Boolean game to potential game, Automatica, 2018, 96: 51–60.

    Article  MathSciNet  MATH  Google Scholar 

  38. Wang Y H and Cheng D Z, Dynamics and stability for a class of evolutionary games with time delays in strategies, Sci China Inf. Sci., 2016, 59(9): 092209.

    Article  Google Scholar 

  39. Wang Y H and Cheng D Z, Stability and stabilization of a class of finite evolutionary games, J. Franklin Institute, 2017, 354(3): 1603–1617.

    Article  MathSciNet  MATH  Google Scholar 

  40. Fu S H, Zhao G D, Li H T, et al., Model and control for a class of networked evolutionary games with finite memories and time-varying networks, Circ. Sys. Sign. Proc., 2018, 37(7): 3093–3114.

    Article  MathSciNet  MATH  Google Scholar 

  41. Wang J J, Leone D R, Fu S H, et al., Event-triggered control design for networked evolutionary games with time invariant delay in strategies, Int. J. Sys. Sci., 2021, 52(3): 493–504.

    Article  MathSciNet  Google Scholar 

  42. Ding X Y, Li H T, Yang Q Q, et al., Stochastic stability and stabilization of n-person random evolutionary Boolean games, Applied Math. Comp., 2017, 306: 1–12.

    Article  MathSciNet  MATH  Google Scholar 

  43. Li H T, Ding X Y, Alsaedi A, et al., Stochastic set stabilization of n-person random evolutionary Boolean games and its applications, IET Contr. Theor. Appl., 2017, 11(13): 2152–2160.

    Article  Google Scholar 

  44. Zhang X, Hao Y Q, and Cheng D Z, Incomplete-profile potential games, J. Franklin Inst., 2018, 355: 862–877.

    Article  MathSciNet  MATH  Google Scholar 

  45. Pan Y N, Fu S H, and Zhao J L, Weighted potential incomplete-profile games, IEEE Access, 2020, 8: 67408–67415.

    Article  Google Scholar 

  46. Deng L, Fu S H, and Zhu P Y, State feedback control design to avoid players going bankrupt, Asian J. Contr., 2019, 21(6): 2551–2558.

    Article  MathSciNet  MATH  Google Scholar 

  47. Fu S H, Wang Y Z, and Zhao G D, A matrix approach to the analysis and control of networked evolutionary games with bankruptcy mechanism, Asian J. Contr., 2017, 19(2): 717–727.

    Article  MathSciNet  MATH  Google Scholar 

  48. Fu S H, Li H T, and Zhao G D, Modeling and strategy optimization for a kind of networked evolutionary games with memories under the bankruptcy mechanism, Int. J. Contr., 2018, 91(5): 1104–1117.

    Article  MATH  Google Scholar 

  49. Li H T, Ding X Y, Yang Q Q, et al., Algebraic formulation and Nash equilibrium of competitive diffusion games, Dyn. Game Appl., 2018, 8: 423–433.

    Article  MathSciNet  MATH  Google Scholar 

  50. Li Y L, Li H T, Xu X J, et al., Semi-tensor product approach to minimal-agent consensus control of networked evolutionary games, IET Contr. Theor. Appl., 2018, 12(16): 2269–2275.

    Article  MathSciNet  Google Scholar 

  51. Zhao G D, Wang Y Z, and Li H, A matrix approach to the modeling and analysis of networked evolutionary games with time delays, IEEE/CAA J. Aut. Sinica, 2018, 5(4): 818–826.

    Article  MathSciNet  Google Scholar 

  52. Ding X Y, Li H T, and Alsaadi F E, Regulation of game result for n-person random evolutionary Boolean games, Asian J. Contr., 2020, 22: 2353–2362.

    Article  MathSciNet  Google Scholar 

  53. Zheng Y T, Li C X, and Feng J, Modeling and dynamics of networked evolutionary game with switched time delay, IEEE Trans. Cont. Netw. Sys., 2021, DOI: https://doi.org/10.1109/TCNS.2021.3084548.

  54. Zhang X and Cheng D Z, Profile-dynamic based fictitious play, Science China Inform. Sci., 2021, 64(6): 169202:1–169202:3.

    Article  MathSciNet  Google Scholar 

  55. Zhao Y, Li Z Q, and Cheng D Z, Optiaml control of logical control networks, IEEE Trans. Aut. Contr., 2011, 56(8): 1766–1776.

    Article  MATH  Google Scholar 

  56. Wang L Q, Liu Y, Wu Z G, et al., Strategy optimization for static games based on STP method, Appl. Math. Comp., 2018, 316: 390–399.

    Article  MathSciNet  MATH  Google Scholar 

  57. Fu S H, Pan Y N, Feng J, et al., Strategy optimisation for coupled evolutionary public good games with threshold, Int. J. Contr., 2020, DOI: https://doi.org/10.1080/00207179.2020.1803411.

  58. Cheng D Z and Liu Z Q, Optimization via game theoretic control, Natio. Sci. Rev., 2010, 7(7): 1120–1122.

    Article  Google Scholar 

  59. Liu T T, Wang Y H, and Cheng D Z, Dynamics and stability of potential hyper-networked evolutionary games, Int. J. Aut. Comput., 2017, 14(2): 229–238.

    Article  Google Scholar 

  60. Hao Y Q and Cheng D Z, Finite element approach to continuous potential games, Science China Inform. Sci., 2021, 64: 149202:1–149202:3.

    Article  MathSciNet  Google Scholar 

  61. Cheng D Z and Li C X, Matrix expression of Bayesian game, 2021, http://arxiv.org/abs/2106.12161.

  62. Cheng D Z and Xu T T, Application of STP to coooperative games, Proc. 10th IEEE ICCA, Hangzhou, IEEE, 2013, 1680–1685.

  63. Wang Y H, Cheng D Z, and Liu X Y, Matrix expression of Shapley values and its application to distributed resource allocation, Sci. China Inform. Sci., 2019, 62: 022201:1–022201:11.

    MathSciNet  Google Scholar 

  64. Li H T, Wang S L, Liu A X, et al., Simplification of Shapley value for cooperative games via minimum carrier, Contr. Theor. Tech., 2021, 19: 157–169.

    Article  MathSciNet  MATH  Google Scholar 

  65. Xia X, Li H, Ding X, et al., Matrix approach to calculation of Banzhaf value with applications, Contr. Theor. Appl., 2020, 37(2): 446–452.

    MATH  Google Scholar 

  66. Wu Y H, Toyoda M, and Shen T L, Linear dynamic games with polytope strategy sets, IET Contr. Theor. Appl., 2017, 11(13): 2146–2151.

    Article  MathSciNet  Google Scholar 

  67. Le S T, Wu Y H, and Toyoda M, A congestion game framework for service chain composition in NFV with function benefit, Information Sciences, 2020, 514: 512–522.

    Article  MathSciNet  MATH  Google Scholar 

  68. Le S T, Wu Y H, Guo Y Q, et al., Game theoretic approach for a service function chain routing in NFV with coupled constraints, IEEE Trans. Corc. Sys. II, 2021, DOI: https://doi.org/10.1109/TC-SII.2021.3070025.

  69. Zhang X, Wang Y H, and Cheng D Z, Incomplete logical control system and its application to some intellectual problems, Asian J. Contr., 2018, 20(2): 697–706.

    Article  MathSciNet  MATH  Google Scholar 

  70. Zhang L Q and Feng J, Mix-valued logic-based formation control, Int. J. Contr., 2013, 86(6): 1191–1199.

    Article  MathSciNet  MATH  Google Scholar 

  71. Hao Y, Pan S, Qiao Y, et al., Cooperative control vial congestion game, IEEE Trans. Aut. Contr., 2018, 63(12): 4361–4366.

    Article  MATH  Google Scholar 

  72. Li C X, Xing Y, He F, et al., A strategic learning algorithm for state-based games, Automatica, 2020, 113: 108615.

    Article  MathSciNet  MATH  Google Scholar 

  73. Liu T, Wang J H, and Cheng D Z, Game theoretic control of multi-agent systems, SIAM J. Contr. Opt., 2019, 37(3): 1691–1709.

    Article  MATH  Google Scholar 

  74. Wang J H, Gao X Y, and Xu Y, Intermittent control for demand-side management of a class of networked smart grids, IET Contr. Theory Appl., 2019, 13(8): 1166–1172.

    Article  MathSciNet  MATH  Google Scholar 

  75. Wu Y H, Cheng D Z, and Ghosh B, Recent advances in optimization and game theoretic control for networked systems, Asian J. Contr., 2019, 21: 2493–2512.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors thank Prof. Jun-e Feng, Prof. Jianquan Lu, Prof. Jiandong Zhu, Prof. Min Meng and Dr. Xiao Zhang for their valuable comments on the manuscript.

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Authors and Affiliations

  1. Key Laboratory of Systems and Control, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing, 100190, China

    Daizhan Cheng

  2. Key Laboratory of Intelligent Control and Optimization for Industrial Equipment of Ministry of Education, Dalian University of Technology, Dalian, 116081, China

    Yuhu Wu

  3. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, China

    Guodong Zhao

  4. School of Mathematical Sciences, Liaocheng University, Liaocheng, 252000, China

    Shihua Fu

Authors
  1. Daizhan Cheng
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  2. Yuhu Wu
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  3. Guodong Zhao
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  4. Shihua Fu
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Corresponding authors

Correspondence to Daizhan Cheng, Yuhu Wu, Guodong Zhao or Shihua Fu.

Additional information

This work is supported partly by the National Natural Science Foundation of China (NSFC) under Grant Nos. 62073315, 61074114, and 61273013.

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Cheng, D., Wu, Y., Zhao, G. et al. A Comprehensive Survey on STP Approach to Finite Games. J Syst Sci Complex 34, 1666–1680 (2021). https://doi.org/10.1007/s11424-021-1232-8

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