Advertisement

Symmetry-based decomposition of finite games

  • Changxi Li
  • Fenghua HeEmail author
  • Ting Liu
  • Daizhan Cheng
Research Paper
  • 19 Downloads

Abstract

The symmetry-based decompositions of finite games are investigated. First, the vector space of finite games is decomposed into a symmetric subspace and an orthogonal complement of the symmetric subspace. The bases of the symmetric subspace and those of its orthogonal complement are presented. Second, the potential-based orthogonal decompositions of two-player symmetric/antisymmetric games are presented. The bases and dimensions of all dual decomposed subspaces are revealed. Finally, some properties of these decomposed subspaces are obtained.

Keywords

potential game symmetric game decomposition Nash equilibrium semi-tensor product of matrices 

Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 61473099, 61273013, 61333001).

References

  1. 1.
    Nowak M A, May R M. Evolutionary games and spatial chaos. Nature, 1992, 359: 826–829CrossRefGoogle Scholar
  2. 2.
    Tembine H, Altman E, El-Azouzi R, et al. Evolutionary games in wireless networks. IEEE Trans Syst Man Cybern B, 2010, 40: 634–646CrossRefGoogle Scholar
  3. 3.
    Santos F C, Santos M D, Pacheco J M. Social diversity promotes the emergence of cooperation in public goods games. Nature, 2008, 454: 213–216CrossRefGoogle Scholar
  4. 4.
    Heikkinen T. A potential game approach to distributed power control and scheduling. Comput Netw, 2006, 50: 2295–2311CrossRefzbMATHGoogle Scholar
  5. 5.
    Marden J R, Arslan G, Shamma J S. Cooperative control and potential games. IEEE Trans Syst Man Cybern B, 2009, 39: 1393–1407CrossRefGoogle Scholar
  6. 6.
    Niyato D, Hossain E. A noncooperative game-theoretic framework for radio resource management in 4G heterogeneous wireless access networks. IEEE Trans Mobile Comput, 2008, 7: 332–345CrossRefGoogle Scholar
  7. 7.
    Arslan G, Marden J R, Shamma J S. Autonomous vehicle-target assignment: a game-theoretical formulation. J Dyn Syst Meas Control, 2007, 129: 584–596CrossRefGoogle Scholar
  8. 8.
    Candogan O, Menache I, Ozdaglar A, et al. Flows and decompositions of games: harmonic and potential games. Math Oper Res, 2011, 36: 474–503MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kalai A T, Kalai E. Cooperation and competition in strategic games with private information. In: Proceedings of the 11th ACM Conference on Electronic Commerce, Cambridge, 2010. 345–346CrossRefGoogle Scholar
  10. 10.
    Hwang S H, Rey-Bellet L. Decompositions of two player games: potential, zero-sum, and stable games. 2011. ArXiv:1106.3552Google Scholar
  11. 11.
    Wang Y H, Liu T, Cheng D Z. From weighted potential game to weighted harmonic game. IET Control Theory Appl, 2017, 11: 2161–2169MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cheng D Z, Liu T, Zhang K Z, et al. On decomposed subspaces of finite games. IEEE Trans Autom Control, 2016, 61: 3651–3656MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cheng D Z. On finite potential games. Automatica, 2014, 50: 1793–1801MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cheng D Z, Liu T. Linear representation of symmetric games. IET Control Theory Appl, 2017, 11: 3278–3287MathSciNetGoogle Scholar
  15. 15.
    Cheng D Z, He F H, Qi H S, et al. Modeling, analysis and control of networked evolutionary games. IEEE Trans Autom Control, 2015, 60: 2402–2415MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Nash J. Non-cooperative games. Ann Math, 1951, 54: 286–295MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cao Z G, Yang X G. Symmetric games revisited. 2016. https://doi.org/ssrn.com/abstract=2637225 Google Scholar
  18. 18.
    Cheng D Z, Qi H S, Zhao Y. An Introduction to Semi-Tensor Product of Matrices and Its Applications. Singapore: World Scientific, 2012CrossRefzbMATHGoogle Scholar
  19. 19.
    Wang Y H, Liu T, Cheng D Z. Some notes on semi-tensor product of matrices and swap matrix. J Syst Sci Math Sci, 2016, 36: 1367–1375MathSciNetzbMATHGoogle Scholar
  20. 20.
    Webb J N. Game Theory. Berlin: Springer, 2007zbMATHGoogle Scholar
  21. 21.
    Li C X, Liu T, He F H, et al. On finite harmonic games. In: Proceedings of the 55th Conference on Decision and Control, Las Vegas, 2016. 7024–7029Google Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Changxi Li
    • 1
  • Fenghua He
    • 1
    Email author
  • Ting Liu
    • 2
  • Daizhan Cheng
    • 2
  1. 1.School of AstronauticsHarbin Institute of TechnologyHarbinChina
  2. 2.Key Laboratory of Systems and Control, Academy of Mathematics and Systems SciencesChinese Academy of SciencesBeijingChina

Personalised recommendations