Skip to main content
SpringerLink
Log in

Orthogonal Decomposition of Incomplete-Profile Finite Game Space

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

This work studies the orthogonal decomposition of the incomplete-profile normal finite game (IPNFG) space using the method of semi-tensor product (STP) of matrices. Firstly, by calculating the rank of the incomplete-profile potential matrix, the bases of incomplete-profile potential game subspace (\((\cal{G}^{\Omega}_{P})\)) and incomplete-profile non-strategic game subspace (\((\cal{N}^{\Omega})\)) are obtained. Then the bases of incomplete-profile pure potential game subspace (\((\cal{P}^{\Omega})\)) and incomplete-profile pure harmonic game subspace (\((\cal{H}^{\Omega})\)) are also revealed. These bases offer an expression for the orthogonal decomposition. Finally, an example is provided to verify the theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aydogmus O, Discovering the effect of nonlocal payoff calculation on the stabilty of ESS: Spatial patterns of Hawk-Dove game in metapopulations, Journal of Theortical Biology, 2018, 442: 87–97.

    Article  MathSciNet  MATH  Google Scholar 

  2. Samadi A H, Montakhab A, and Marzban H, Quantum Barro-Gordon game in monetary economics, Physica A-Statistical Mechanics and Its Applications, 2018, 489: 94–101.

    Article  MathSciNet  MATH  Google Scholar 

  3. Taghizadeh A, Kebriaei H, and Niyato D, Mean field game for equilibrium analysis of mining computational power in blockchains, IEEE Internet of Things Journal, 2020, 7(8): 7625–7635.

    Article  Google Scholar 

  4. Zhang X, Hao Y Q, and Cheng D Z, Incomplete-profile potential games, Journal of the Franklin Institute-Engineering and Applied Mathematics, 2018, 355(2): 862–877.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Xiao Y F and Dougherty E R, The impact of function perturbations in Boolean networks, Bioinformatics, 2007, 23(10): 1265–1273.

    Article  Google Scholar 

  7. Li H T and Wang Y Z, Minimum-time state feedback stabilization of constrained Boolean control networks, Asian Journal of Control, 2016, 8(5): 1688–1697.

    Article  MathSciNet  MATH  Google Scholar 

  8. Rosenthal R W, A class of games possessing pure-strategy Nash equilibria, International Journal of Game Theory, 1973, 2(1): 65–67.

    Article  MathSciNet  MATH  Google Scholar 

  9. Monderer D and Shapley L S, Potential games, Games and Economic Behavior, 1996, 14(1): 124–143.

    Article  MathSciNet  MATH  Google Scholar 

  10. Marden J R, Arslan G, and Shamma J S, Cooperative control and potential games, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetic, 2009, 39(6): 1393–1407.

    Article  Google Scholar 

  11. Yaziciǒglu A Y, Egerstedt M, and Shamma J S, A game theoretic approach to distributed coverage of graphs by heterogeneous mobile agents, Estimation and Control of Networked Systems, 2013, 4(1): 309–315.

    Google Scholar 

  12. Liang Y L, Liu F, and Wang C, Distributed demand-side energy management scheme in residential smart grids: An ordinal state-based potential game approach, Applied Energy, 2017, 206: 991–1008.

    Article  Google Scholar 

  13. Heikkinen T, A potential game approach to distributed power control and scheduling, Computer Networks, 2006, 50(13): 2295–2311.

    Article  MATH  Google Scholar 

  14. Marden J R, State based potential games, Automatica, 2012, 48(12): 3075–3088.

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang X, Xiao N, Wongpiromsarn T, et al., Distributed consensus in noncooperative congestion games: An application to road pricing, Proceedings of the 10th IEEE International Conference on Control and Automation, 2013, 1668–1673.

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Cheng D Z, Liu T, Zhang K Z, et al., On decomposed subspaces of finite games, IEEE Transactions on Automatic Control, 2016, 61(11): 3651–3656.

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang Y H, Liu T, and Cheng D Z, From weighted potential game to weighted harmonic game, IET Control Theory & Applications, 2017, 11(13): 2161–2169.

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  20. 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 

  21. Meng M, Lam J, and Feng J E, Stability and stabilization of Boolean networks with stochastic delays, IEEE Transactions on Automatic Control, 2019, 64(2): 790–796.

    MathSciNet  MATH  Google Scholar 

  22. Li H T, Xie L H, and Wang Y Z, On robust control invariance of Boolean control networks, Automatica, 2016, 68: 392–396.

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhu S Y, Lu J Q, Liu Y, et al., Output tracking of probabilistic Boolean networks by output feedback control, Information Sciences, 2019, 483: 96–105.

    Article  MathSciNet  MATH  Google Scholar 

  24. Qi H S, Wang Y H, Liu T, et al., Vector space structure of finite evoulutionary games and its application to strategy profile convergence, Journal of Systems Science & Complexity, 2016, 29(3): 602–628.

    Article  MathSciNet  MATH  Google Scholar 

  25. Li C X, Xing Y, and He F H, A strategic learning algorithm for state-based games, Automatica, 2020, 133: 108615.

    Article  MathSciNet  MATH  Google Scholar 

  26. Mei S W, Liu F, and Xue A C, Semi-Tensor Product Method in Power System Transient Analysis, Tsinghua University Press, Beijing, 2010 (in Chinese).

    Google Scholar 

  27. Wang G R, Wei Y M, and Qiao S Z, Generalized Inverses: Theory and Computations, Science Press, Beijing, 2018.

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Science, Hebei University of Technology, Tianjin, 300401, China

    Xiaoyan Dai, Jinhuan Wang & Yong Xu

Authors
  1. Xiaoyan Dai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jinhuan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yong Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jinhuan Wang.

Additional information

This work was supported by the Natural Science Foundation of Hebei Province under Grant Nos. F2021202032, A2019202205, and the Cultivation of Postgraduate Students Innovation Ability of Hebei Province under Grant No. CXZZSS2021045.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dai, X., Wang, J. & Xu, Y. Orthogonal Decomposition of Incomplete-Profile Finite Game Space. J Syst Sci Complex 35, 2208–2222 (2022). https://doi.org/10.1007/s11424-022-1019-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-022-1019-6

Keywords

Search

Navigation