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Semi-tensor product approach to networked evolutionary games

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Abstract

In this paper a comprehensive introduction for modeling and control of networked evolutionary games (NEGs) via semi-tensor product (STP) approach is presented. First, we review the mathematical model of an NEG, which consists of three ingredients: network graph, fundamental network game, and strategy updating rule. Three kinds of network graphs are considered, which are i) undirected graph for symmetric games; ii) directed graph for asymmetric games, and iii) d-directed graph for symmetric games with partial neighborhood information. Three kinds of fundamental evolutionary games (FEGs) are discussed, which are i) two strategies and symmetric (S-2); ii) two strategies and asymmetric (A-2); and iii) three strategies and symmetric (S-3). Three strategy updating rules (SUR) are introduced, which are i) Unconditional Imitation (UI); ii) Fermi Rule(FR); iii) Myopic Best Response Adjustment Rule (MBRA). First, we review the fundamental evolutionary equation (FEE) and use it to construct network profile dynamics (NPD)of NEGs.

To show how the dynamics of an NEG can be modeled as a discrete time dynamics within an algebraic state space, the fundamental evolutionary equation (FEE) of each player is discussed. Using FEEs, the network strategy profile dynamics (NSPD) is built by providing efficient algorithms. Finally, we consider three more complicated NEGs: i) NEG with different length historical information, ii) NEG with multi-species, and iii) NEG with time-varying payoffs. In all the cases, formulas are provided to construct the corresponding NSPDs. Using these NSPDs, certain properties are explored. Examples are presented to demonstrate the model constructing method, analysis and control design technique, and to reveal certain dynamic behaviors of NEGs.

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Authors and Affiliations

  1. Institute of Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Daizhan Cheng, Hongsheng Qi & Tingting Xu

  2. Institute of Astronautics, Harbin Institute of Technology, Harbin Heilongjiang, 150080, China

    Fehuang He

  3. State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, 100044, China

    Hairong Dong

Authors
  1. Daizhan Cheng
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  2. Hongsheng Qi
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  3. Fehuang He
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  4. Tingting Xu
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  5. Hairong Dong
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Corresponding author

Correspondence to Daizhan Cheng.

Additional information

This work was partially supported by National Natural Science Foundation of China (Nos. 61273013, 61333001, 61104065, 61322307).

Daizhan CHENG graduated from Department of Mechanics, Tsinghua University in 1970, received the M.S. degree from Graduate School of Chinese Academy of Sciences in 1981, and the Ph.D. degree from Washington University, St. Louis in 1985. Since 1990, he has been a professor with Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences. His research interests include nonlinear control systems, switched systems, Hamiltonian systems, logical dynamic systems, and numerical realization for control design. He is the author/coauthor of 12 books, over 240 journal papers and over 130 conference papers. He was Associate Editor of International Journal of Mathematical Systems, Estimation and Control (1990–1993), Automatica (1999–2002), Asian Journal of Control (2001–2004), Subject Editor of International Journal of Robust and Nonlinear Control (2005–2008), and Associate Editor of Journal of Systems Science and Complexity, Journal of Control Theory and Applications, Journal Systems Science and Mathematics, Editorin- Chief of Journal of Control Theory and Applications. He currently is the Deputy Editor-in-Chief of Control and Decision. He was the Chairman of IEEE CSS Beijing Chapter (2006–2008), Chairman of Technical Committee on Control Theory of Chinese Association of Automation (2003–2009). He was the first recipient of a second National Natural Science Award in China in 2008 and the recipient of Automatica 2008–2010 Theory/Methodology Best Paper Prize in 2011. He is IEEE Fellow (2005–) and IFAC Fellow (2008–).

Hongsheng QI received the B.S. degree in Mathematics and Applied Mathematics from Anhui University in 2003 and received the Ph.D. degree in Systems Theory from Academy of Mathematics and Systems Science, Chinese Academy of Sciences in 2008. From July 2008 to May 2010, he was a postdoctoral fellow in the Key Laboratory of Systems Control, Chinese Academy of Sciences. In June 2010, he joined the Academy of Mathematics and Systems Science, Chinese Academy of Sciences as an assistant professor. He was the recipient of Automatica 2008–2010 Theory/Methodology Best Paper Prize in 2011. His research interests include nonlinear control, complex systems, and game theory.

Fenghua HE received her Ph.D. degree from Harbin Institute of Technology in 2005, where she is currently an associate professor in the Control and Simulation Center. Her current research interests include guidance and control of flight vehicles, cooperative control and game theory.

Tingting XU received her B.S. degree in Information and Computing Sciences from China University of Mining & Technology, Beijing in 2011. In September 2011, she became a M.S. candidate at the Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences. Her research interests include game theory, genetic regulatory network and multi-agent systems.

Hairong DONG received her B.S. and M.S. degrees from Zhengzhou University, Zhengzhou, China, in 1996 and 1999, respectively, and the Ph.D. degree from Peking University, Beijing, China, in 2002. She is currently a professor with the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University. Her research interests include optimal timetable design of railway systems, pedestrian dynamics and emergency evacuation, and multi-objective control of complex systems.

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Cheng, D., Qi, H., He, F. et al. Semi-tensor product approach to networked evolutionary games. Control Theory Technol. 12, 198–214 (2014). https://doi.org/10.1007/s11768-014-0038-9

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