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SCIENCE CHINA Information Sciences, Volume 62 , Issue 2 : 022201(2019) https://doi.org/10.1007/s11432-018-9414-5

Matrix expression of Shapley values and its application to distributed resource allocation

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  • ReceivedJan 27, 2018
  • AcceptedMar 5, 2018
  • PublishedDec 25, 2018

Abstract


Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant No. 61773371).


References

[1] Mei S W, Wang Y Y, Liu F. Game approaches for hybrid power system planning. IEEE Trans Sustain Energ, 2012, 3: 506-517 CrossRef ADS Google Scholar

[2] Zhu M H, Martinez S. Distributed coverage games for energy-aware mobile sensor networks. SIAM J Control Optim, 2013, 51: 1-27 CrossRef Google Scholar

[3] Gopalakrishnan R, Marden J R, Wierman A. An architectural view of game theoretic control. SIGMETRICS Perform Eval Rev, 2011, 38: 31-36 CrossRef Google Scholar

[4] Marden J R, Wierman A. Distributed welfare games. Oper Res, 2013, 61: 155-168 CrossRef Google Scholar

[5] Shapley L S. A value for $n$-person games. Contrib Theory Games, 1953, 2: 307--317. Google Scholar

[6] Perez-Castrillo D, Wettstein D. Bidding for the surplus: a non-cooperative approach to the Shapley value. J Econ Theory, 2001, 100: 274-294 CrossRef Google Scholar

[7] Shapley L S. Additive and non-additioe set functions. Dissertation for Ph.D. Degree. Princeton: Princeton University, 1953. Google Scholar

[8] Kalai E, Samet D. On weighted Shapley values. Int J Game Theory, 1987, 16: 205-222 CrossRef Google Scholar

[9] Chun Y. On the symmetric and weighted shapley values. Int J Game Theory, 1991, 20: 183-190 CrossRef Google Scholar

[10] Nowak A S, Radzik T. On axiomatizations of the weighted Shapley values. Games Econ Behav, 1995, 8: 389-405 CrossRef Google Scholar

[11] Cheng D Z, Qi H S, Li Z Q. Analysis and Control of Boolean Networks: A Semi-tensor Product Approach. Berlin: Springer, 2011. Google Scholar

[12] Li F F, Sun J T. Stability and stabilization of Boolean networks with impulsive effects. Syst Control Lett, 2012, 61: 1-5 CrossRef Google Scholar

[13] Li H T, Zhao G D, Meng M. A survey on applications of semi-tensor product method in engineering. Sci China Inf Sci, 2018, 61: 010202 CrossRef Google Scholar

[14] Meng M, Liu L, Feng G. Stability and $l_1$ gain analysis of Boolean networks with Markovian jump parameters. IEEE Trans Autom Control, 2017, 62: 4222-4228 CrossRef Google Scholar

[15] Lu J Q, Zhong J, Huang C. On pinning controllability of Boolean control networks. IEEE Trans Autom Control, 2016, 61: 1658-1663 CrossRef Google Scholar

[16] Zhao G D, Fu S H. Matrix approach to trajectory control of higher-order k-valued logical control networks. IET Control Theory Appl, 2017, 11: 2110-2115 CrossRef Google Scholar

[17] Liu Z B, Wang Y Z, Li H T. Two kinds of optimal controls for probabilistic mix-valued logical dynamic networks. Sci China Inf Sci, 2014, 57: 052201 CrossRef Google Scholar

[18] Zheng Y T, Li H T, Ding X Y. Stabilization and set stabilization of delayed Boolean control networks based on trajectory stabilization. J Franklin Inst, 2017, 354: 7812-7827 CrossRef Google Scholar

[19] Wang Y Z, Zhang C H, Liu Z B. A matrix approach to graph maximum stable set and coloring problems with application to multi-agent systems. Automatica, 2012, 48: 1227-1236 CrossRef Google Scholar

[20] Zhong J, Lu J Q, Huang C. Finding graph minimum stable set and core via semi-tensor product approach. Neurocomputing, 2016, 174: 588-596 CrossRef Google Scholar

[21] Zhao J T, Chen Z Q, Liu Z X. Modeling and analysis of colored petri net based on the semi-tensor product of matrices. Sci China Inf Sci, 2018, 61: 010205 CrossRef Google Scholar

[22] Cheng D Z, Xu T T. Application of STP to cooperative games. In: Proceedings of 10th IEEE International Conference on Control and Automation (ICCA), Hangzhou, 2013. 1680--1685. Google Scholar

[23] Cheng D Z. On finite potential games. Automatica, 2014, 50: 1793-1801 CrossRef Google Scholar

[24] Wang Y H, Liu T, Cheng D Z. From weighted potential game to weighted harmonic game. IET Control Theor Appl, 2017, 11: 2161-2169 CrossRef Google Scholar

[25] Cheng D Z, He F H, Qi H S. Modeling, analysis and control of networked evolutionary games. IEEE Trans Autom Control, 2015, 60: 2402-2415 CrossRef Google Scholar

[26] 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 CrossRef Google Scholar

[27] Guo P L, Zhang H X, Alsaadi F E. Semi-tensor product method to a class of event-triggered control for finite evolutionary networked games. IET Control Theory Appl, 2017, 11: 2140-2145 CrossRef Google Scholar

[28] Branzei R, Dimitrov D, Tijs S. Models in Cooperative Game Theory. Berlin: Springer, 2008. Google Scholar

[29] Sedgewick R. Permutation generation methods. ACM Comput Surv, 1977, 9: 137-164 CrossRef Google Scholar

[30] Rosen K H, Michaels J G, et al. Handbook of Discrete and Combinatorial Mathematics. Boca Raton: CRC Press, 1999. Google Scholar

  • Table 1   Payoff function using symmetric Shapley value
    $r_2$$r_3$
    $r_1$$1,~1$$1,~0.5$
    $r_2$$1.5,~0.5$$2,~0.5$
  • Table 2   Global objective $W(a)$
    $r_2$$r_3$
    $r_1$$2$$1.5$
    $r_2$$2$$2.5$
  • Table 3   Payoff function using the weighted Shapley value
    $r_2$$r_3$
    $r_1$$1,~1$$1,~0.5$
    $r_2$$1.8,~0.2$$2,~0.5$