On Basis and Pure Nash Equilibrium of Finite Pure Harmonic Games

LIU Aixin, LI Haitao, LI Ping, YANG Xinrong

1. School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China
• Received:2020-02-22 Revised:2021-04-21 Online:2022-08-25 Published:2022-08-02
• Contact: LI Haitao,Email:haitaoli09@gmail.com
• Supported by:
The paper was supported by the National Natural Science Foundation of China under Grant No. 62073202, and the Young Experts of Taishan Scholar Project under Grant No. tsqn201909076.

LIU Aixin, LI Haitao, LI Ping, YANG Xinrong. On Basis and Pure Nash Equilibrium of Finite Pure Harmonic Games[J]. Journal of Systems Science and Complexity, 2022, 35(4): 1415-1428.

This paper investigates the basis and pure Nash equilibrium of finite pure harmonic games (FPHGs) based on the vector space structure. First, a new criterion is proposed for the construction of pure harmonic subspace, based on which, a more concise basis is constructed for the pure harmonic subspace. Second, based on the new basis of FPHGs and auxiliary harmonic vector, a more easily verifiable criterion is presented for the existence of pure Nash equilibrium in basis FPHGs. Third, by constructing a pure Nash equilibrium cubic matrix, the verification of pure Nash equilibrium in three-player FPHGs is given.
 [1] Friedman J W, Game Theory with Application to Economics, New York: Oxford University Press, New York, 1986.[2] Nash J, Non-cooperative games, Annals of Mathematics: Second Series, 1951, 54(2): 286-295.[3] Farrell J, Communication, coordination and Nash equilibrium, Economics Letters, 1988, 27(3): 209-214.[4] Rosenthal R W, A class of games possessing pure-strategy Nash equilibria, International Journal of Game Theory, 1973, 2(1): 65-67.[5] Monderer D and Shapley L S, Potential games, Games and Economic Behavior, 1996, 14(1): 124-143.[6] Cheng D Z, On finite potential games, Automatica, 2014, 50: 1793-1801.[7] Liu X Y and Zhu J D, On potential equations of finite games, Automatica, 2016, 68: 245-253.[8] Candogan O, Menache I, Ozdaglar A, et al., Flows and decompositions of games: Harmonic and potential games, Mathematics of Operations Research, 2011, 36(3): 474-503.[9] Hao Y Q and Cheng D Z, On skew-symmetric games, Journal of the Franklin Institute, 2018, 355(6): 3196-3220.[10] Li C X, He F H, Liu T, et al., Symmetry-based decomposition of finite games, Science China Information Sciences, 2019, 62(1): 012207.[11] Liu T, Qi H S, and Cheng D Z, Dual expressions of decomposed subspaces of finite games, Proceedings of the 34th Chinese Control Conference, Hangzhou, 2015, 9146-9151.[12] Li C X, Liu T, He F H, et al., On finite harmonic games, The 55th IEEE Conference on Decision and Control, Las Vegas, NV, 2016, 7024-7029.[13] 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.[14] Cheng D Z and Liu T, Linear representation of symmetric games, IET Control Theory & Applications, 2017, 11(18): 3278-3287.[15] 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 Control Theory & Applications, 2018, 12(16): 2269-2275.[16] Qi H S, Wang Y H, Liu T, et al., Vector space structure of finite evolutionary games and its application to strategy profile convergence, Journal of Systems Science & Complexity, 2016, 29(3): 602-628.[17] Li C X, He F H, Liu T, et al., Verification and dynamics of group-based potential games, IEEE Transactions on Control of Network Systems, 2019, 6(1): 215-224.[18] Mao Y, Wang L Q, Liu Y, et al., Stabilization of evolutionary networked games with length-r information, Applied Mathematics and Computation, 2018, 337: 442-451.[19] Jiang K C and Wang J H, Stabilization of a class of congestion games via intermittent control, Science China Information Sciences, 2022, 65: 149203.[20] Zhang X and Cheng D Z, Profile-dynamic based fictitious play, Science China Information Sciences, 2021, 64: 169202.[21] Guo P L and Wang Y Z, The computation of Nash equilibrium in fashion games via semi-tensor product method, Journal of Systems Science & Complexity, 2016, 29(4): 881-896.[22] Li H T, Zhao G D, Guo P L, et al., Analysis and Control of Finite-Value Systems, CRC Press, Florida, 2018.[23] Zou Y L and Zhu J D, Graph theory methods for decomposition w.r.t. outputs of Boolean control networks, Journal of Systems Science & Complexity, 2015, 30(3): 519-534.[24] Li C X, Xing Y, He F H, et al., A strategic learning algorithm for state-based games, Automatica, 2020, 113: 108615.[25] Liang J L, Chen H W, and Liu Y, On algorithms for state feedback stabilization of Boolean control networks, Automatica, 2017, 84: 10-16.[26] Wang H Y, Zhong J H, and Lin D D, Linearization of multi-valued nonlinear feedback shift registers, Journal of Systems Science & Complexity, 2016, 30(2): 494-509.[27] Guo Y Q, Zhou R P, Wu Y H, et al., Stability and set stability in distribution of probabilistic Boolean networks, IEEE Transactions on Automatic Control, 2019, 64(2): 736-742.[28] Meng M, Lam J, Feng J, et al., l1-gain analysis and model reduction problem for Boolean control networks, Information Sciences, 2016, 348: 68-83.[29] Jiang D P and Zhang K Z, Observability of Boolean control networks with time-variant delays in states, Journal of Systems Science & Complexity, 2018, 31(2): 436-445.[30] Wang S L and Li H T, Aggregation method to reachability and optimal control of large-size Boolean control networks, Science China Information Sciences, 2022, DOI: 10.1007/s11432-021- 3388-y.[31] Lu J Q, Li H T, Liu Y, et al., Survey on semi-tensor product method with its applications in logical networks and other finite-valued systems, IET Control Theory & Applications, 2017, 11(13): 2040-2047.[32] Li H T, Zhao G D, Meng M, et al., A survey on applications of semi-tensor product method in engineering, Science China Information Sciences, 2018, 61: 010202.[33] Fornasini E and Valcher M E, Recent developments in Boolean networks control, Journal of Control & Decision, 2016, 3(1): 1-18.[34] 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.[35] Bates D M and Watts D G, Relative curvature measures of nonlinearity, Journal of the Royal Statistical Society, 1980, 42(1): 1-25.
 [1] FAN Naqi, ZHANG Lijun, ZHANG Shenggui, LIU Jiuqiang. Matching Algorithms of Minimum Input Selection for Structural Controllability Based on Semi-Tensor Product of Matrices [J]. Journal of Systems Science and Complexity, 2022, 35(5): 1808-1823. [2] GUO Peilian,WANG Yuzhen. The Computation of Nash Equilibrium in Fashion Games via Semi-Tensor Product Method [J]. Journal of Systems Science and Complexity, 2016, 29(4): 881-896. [3] LI Haitao, WANG Yuzhen, LIU Zhenbin. ON THE OBSERVABILITY OF FREE BOOLEAN NETWORKS VIA THE SEMI-TENSOR PRODUCT METHOD [J]. Journal of Systems Science and Complexity, 2014, 27(4): 666-678.
Viewed
Full text

Abstract