Decision-implementation complexity of cooperative game systems


This paper focuses on the decision-implementation complexity (DIC) of cooperative games. The complexity of a control law is a fundamental issue in practice because it is closely related to control cost. First, we formulate implementation measures of strategies as system control protocols. Then, for a class of cooperative games, a decision-implementation system model is established, and an energy-based DIC index is given as the energy expectation under Nash equilibrium strategies. A definition of DIC is presented to describe the optimal values of the DIC index. DIC is calculated by the exhaust algorithm in some specific cases, whereas the one for general cases is too complex to be calculated. In order to obtain a general calculation method, the problem is described in the form of matrices; an analytical expression describing DIC is obtained by using the properties of matrix singular values. Furthermore, when only partial information of actions is shared among players, DIC can be reduced and an improved protocol can be designed as a two-phase protocol. A numerical example is given to show that the DIC obtained in this study is the same as the one obtained by the exhaust algorithm, and that the calculation complexity of the proposed algorithm is much lower.

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  1. 1

    de Deus T F, Lopes P F. A game about biology for biology students cell life as a learning tool. In: Proceedings of the 8th Iberian Conference on Information Systems and Technologies, Lisboa, 2013. 1–6

    Google Scholar 

  2. 2

    Chen L, Yu N, Su Y. The application of game theory in the Hercynian economic development. In: Proceedings of International Conference on Management Science and Engineering, Helsinki, 2014. 794–799

    Google Scholar 

  3. 3

    Ott U F. International business research and game theory: looking beyond the prisoner’s dilemma. Int Bus Rev, 2013, 22: 480–491

    Article  Google Scholar 

  4. 4

    Chou H M, Zhou L. A game theory approach to deception strategy in computer mediated communication. In: Proceedings of IEEE International Conference on Intelligence and Security Informatics, Arlington, 2012. 7–11

    Google Scholar 

  5. 5

    Abellanas M, López M D, Rodrigo J. Searching for equilibrium positions in a game of political competition with restrictions. Eur J Oper Res, 2010, 201: 892–896

    MathSciNet  Article  MATH  Google Scholar 

  6. 6

    Yao P, Jiang D, Zhu T. On optimization of auction mechanism in the military logistics based on the game theory. In: Proceedings of the 2nd International Conference on Information Science and Engineering, Hangzhou, 2010. 5940–5943

    Google Scholar 

  7. 7

    Liu T F, Lu X S, Jiang Z P. A junction-by-junction feedback-based strategy with convergence analysis for dynamic traffic assignment. Sci China Inf Sci, 2016, 59: 010203

    Google Scholar 

  8. 8

    Qi L Y, Hu X M. Design of evacuation strategies with crowd density feedback. Sci China Inf Sci, 2016, 59: 010204

    Article  Google Scholar 

  9. 9

    Sasaki Y, Kijima K. Hierarchical hypergames and Bayesian games: a generalization of the theoretical comparison of hypergames and Bayesian games considering hierarchy of perceptions. J Syst Sci Complex, 2016, 29: 187–201

    MathSciNet  Article  Google Scholar 

  10. 10

    Yuan Y. A dynamic games approach to H control design of DoS with application to longitudinal flight control. Sci China Inf Sci, 2015, 58: 092202

    MathSciNet  Google Scholar 

  11. 11

    Mesmer B L, Bloebaum C L. Incorporation of decision, game, and Bayesian game theory in an emergency evacuation exit decision model. Fire Safety J, 2014, 67: 121–134

    Article  Google Scholar 

  12. 12

    Qi X, Zhang Y, Meng J. Game theory model and equilibrium analysis of peasant’s production decision. J Northeast Agr Univ (English Edition), 2012, 19: 71–73

    Article  Google Scholar 

  13. 13

    Bando K. On the existence of a strictly strong Nash equilibrium under the student-optimal deferred acceptance algorithm. Games Econ Behav, 2014, 87: 269–287

    MathSciNet  Article  MATH  Google Scholar 

  14. 14

    Dai Y, Yang H, Wu J, et al. The joint non-inferior Nash equilibrium analysis of electricity futures and spot market. In: Proceedings of Transmission and Distribution Conference and Exposition: Asia and Pacific, Seoul, 2009. 1–4

    Google Scholar 

  15. 15

    Zhang C, Gao J, Liu Z, et al. Nash equilibrium strategy of two-person bilinear-quadratic differential game: a recursive approach. Intell Control Automat, 2006, 1: 1075–1079

    Google Scholar 

  16. 16

    Sintchenko V, Coiera E. Decision complexity affects the extent and type of decision support use. AMIA Annu Symp Proc, 2006, 2006: 724–728

    Google Scholar 

  17. 17

    Dabbagh S R, Sheikh-El-Eslami M K. Risk-based profit allocation to DERs integrated with a virtual power plant using cooperative game theory. Electr Power Syst Res, 2015, 121: 368–378

    Article  Google Scholar 

  18. 18

    Lozano S. Information sharing in DEA: a cooperative game theory approach. Eur J Oper Res, 2012, 222: 558–565

    MathSciNet  Article  MATH  Google Scholar 

  19. 19

    Tao Z, Liu X, Chen H, et al. Group decision making with fuzzy linguistic preference relations via cooperative games method. Comput Ind Eng, 2015, 83: 184–192

    Article  Google Scholar 

  20. 20

    Rashedi N, Kebriaei H. Cooperative and non-cooperative Nash solution for linear supply function equilibrium game. Appl Math Comput, 2014, 244: 794–808

    MathSciNet  MATH  Google Scholar 

  21. 21

    Yao A C C. Some complexity questions related to distributive computing. In: Proceedings of the 11th Annual ACM Symposium on Theory of Computing, Atlanta, 1979

    Google Scholar 

  22. 22

    Impagliazzo R, Williams R. Communication complexity with synchronized clocks. In: Proceedings of IEEE 25th Annual Conference on Computational Complexity, Cambridge, 2010. 259–269

    Google Scholar 

  23. 23

    Klauck H. Rectangle size bounds and threshold covers in communication complexity. In: Proceedings of the 18th IEEE Annual Conference on Computational Complexity, Aarhus, 2003. 118–134

    Google Scholar 

  24. 24

    Kushilevitz E, Nisan N. Communication complexity. Sci Comput Program, 1999, 33: 215–216

    Article  MATH  Google Scholar 

  25. 25

    Shigeta M, Amano K. Ordered biclique partitions and communication complexity problems. Discrete Appl Math, 2015, 184: 248–252

    MathSciNet  Article  MATH  Google Scholar 

  26. 26

    Baillieul J, Wong W S. The standard parts problem and the complexity of control communication. In: Proceedings of the 48th IEEE Conference on Decision and Control, Shanghai, 2009. 2723–2728

    Google Scholar 

  27. 27

    Wong W S. Control communication complexity of distributed control systems. SIAM J Control Optimiz, 2009, 48: 1722–1742

    MathSciNet  Article  MATH  Google Scholar 

  28. 28

    Wong W S, Baillieul J. Control communication complexity of nonlinear systems. Commun Inf Syst, 2009, 9: 103–140

    MathSciNet  MATH  Google Scholar 

  29. 29

    Baillieul J, Özcimder K. The control theory of motion-based communication: problems in teaching robots to dance. Am Control Conf, 2011, 50: 4319–4326

    Google Scholar 

  30. 30

    Yao Y, Gehrke J. Query processing for sensor networks. In: Proceedings of the 1st Biennial Conference in Innovative Asta Systems Research (CIDR), Asilomar, 2003. 1364

    Google Scholar 

  31. 31

    Wong W S, Baillieul J. Control communication complexity of distributed actions. IEEE Trans Automat Control, 2012, 57: 2731–2745

    MathSciNet  Article  Google Scholar 

  32. 32

    Liu Z, Wong W S. Choice-based cluster consensus in multi-agent systems. In: Proceedings of the 32nd Chinese Control Conference, Xi’an, 2013. 7285–7290

    Google Scholar 

  33. 33

    Liu Z, Wong W S, Guo G. Cooperative control of linear systems with choice actions. In: Proceedings of American Control Conference, Washington, 2013. 5374–5379

    Google Scholar 

  34. 34

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

    Article  Google Scholar 

  35. 35

    Zhang H, Jiang C, Beaulieu N C, et al. Resource allocation for cognitive small cell networks: a cooperative bargaining game theoretic approach. IEEE Trans Wirel Commun, 2015, 14: 3481–3493

    Article  Google Scholar 

  36. 36

    Zheng W, Su T, Zhang H, et al. Distributed power optimization for spectrum-sharing femtocell networks: a fictitious game approach. J Netw Comput Appl, 2014, 37: 315–322

    Article  Google Scholar 

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This work was supported by National Basic Research Program of China (973 Program) (Grant No. 2014CB845301) and National Natural Science Foundation of China (Grant No. 61227902).

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Correspondence to Ji-Feng Zhang.

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Xu, C., Zhao, Y. & Zhang, JF. Decision-implementation complexity of cooperative game systems. Sci. China Inf. Sci. 60, 112201 (2017).

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  • cooperative game
  • distributed protocol
  • optimal control
  • decision-implementation complexity
  • twophase protocol