Articles

THE OPTIMAL CONTROL FOR PROMOTING THE COOPERATION IN EVOLUTION GAME GENERATED BY PRISONER'S DILEMMA

  • Xian-jia WANG ,
  • Rui DONG ,
  • Lin CHEN
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  • Institute of Systems Engineering, Wuhan University, Wuhan 430072, China

Received date: 2017-01-20

  Online published: 2018-02-25

Supported by

This work is supported by NSFC (71231007, 71501149).

Abstract

Natural selection opposes the evolution of cooperation unless specific mechanisms are at work in Prisoner's Dilemma. By taking advantage of the modern control theory, the controller design is discussed and the optimal control is designed for promoting cooperation based on the recent advances in mechanisms for the evolution of cooperation. Two control strategies are proposed:compensation control strategy for the cooperator when playing against a defector and reward control strategy for cooperator when playing against a cooperator. The feasibility and effectiveness of these control strategies for promoting cooperation in different stages are analyzed. The reward for cooperation can't prevent defection from being evolutionary stable strategy (ESS). On the other hand, compensation for the cooperator can't prevent defection from emerging and sustaining. By considering the effect and the cost, an optimal control scheme with constraint on the admissible control set is put forward. By analyzing the special nonlinear system of replicator dynamics, the exact analytic solution of the optimal control scheme is obtained based on the maximum principle. Finally, the effectiveness of the proposed method is illustrated by examples.

Cite this article

Xian-jia WANG , Rui DONG , Lin CHEN . THE OPTIMAL CONTROL FOR PROMOTING THE COOPERATION IN EVOLUTION GAME GENERATED BY PRISONER'S DILEMMA[J]. Acta mathematica scientia, Series B, 2018 , 38(1) : 73 -92 . DOI: 10.1016/S0252-9602(17)30118-2

References

[1] Nowak M A. Evolving cooperation. J Theor Biol, 2012, 299:1-8
[2] Nowak M A. Five rules for the evolution of cooperation. Science, 2006, 314(5805):1560-1563
[3] Axelrod R, Hamiton W D. The evolution of cooperation. Science, 1981, 211(4489):1390-1396
[4] Li J. The effect of memory size on the evolutionary stability of strategies in iterated Prisoner's Dilemma. IEEE Tran Evol Comput, 2014, (186):819-826
[5] Nowak M A, Sigmund K. Evolution of indirect reciprocity. Nature, 2005, 437(7063):1291-1298
[6] Wedekind C, Milinski M. Cooperation through image scoring in humans. Science, 2000, 288(5467):850-852
[7] Nowak M A, Robert M M. Evolutionary games and spatial chaos. Nature, 1992, 359(6398):826-829
[8] Taylor P D, Day T, Wild G. Evolution of cooperation in a finite homogeneous graph. Nature, 2007, 447(7143):469-472
[9] Taylor C, Nowak M A. Transforming the dilemma. Evolution, 2007, 61(10):2281-2292
[10] Goodnight C J. Multilevel selection:the evolution of cooperation in non-kin groups. Popul Ecol, 2005, 471:3-12
[11] Lamba S. Social learning in cooperative dilemmas. Proc Roy Soc B:Biol Sci, 2014, 281(1787):4739-4753
[12] Benjamin A, Nowak M A. Games among relatives revisited. J Theor Biol, 2015, 378:103-116
[13] Vincent T L, Vincent, T L S. Evolution and control system design. IEEE Control Syst Mag, 2000, 20(5):20-35
[14] Cheng D Z, He F H, Qi H S. Modeling, analysis and control of networked evolutionary games. IEEE Trans Autom Control, 2015, 60(9):2402-2415
[15] Wang L, Wu T, Zhang Y L. Feedback mechanism in coevolutionary games. Control Theory & Applications, 2014, 31(7):823-836
[16] Liang H L, Cao M, Wang X F. Analysis and shifting, of stochastically stable equilibria for evolutionary snowdrift games. Systems & Control Letters, 2015, 85:16-22
[17] Sigmund K, De Silva H, Hauert C, Traulsen A. Social learning promotes institutions for governing the commons. Nature, 2010, 466:861-863
[18] Tatsuya S, Tatsuo U. Replicator dynamics in public goods games with reward funds. J Theor Biol, 2011, 287:109-114
[19] Liu Q L, Li X C. Effective stability control research of evolutionary game in China's coal mine safety supervision. J of Beijing Institute of Technology (Social Sciences Edition), 2015, 17(4):49-56
[20] Pan F, Xi B, Wang L. Analysis on environmental regulation strategy of local government based on evolutionary game theory. Syst Engi-Theo Prac, 2015, 35(6):1393-1404
[21] Tilman A R, Watson J R, Levin S. Maintaining cooperation in social-ecological systems. Theor Ecol, 2016:1-11
[22] Chiang A C. Fundamental Methods of Mathematical Economics. McGraw-Hill/Irwin, 2005
[23] Nuño G, Moll B. Controlling a Distribution of Heterogeneous Agents. Social Science Electronic Publishing, 2015
[24] Chaluba F A C C, and Souza M O. Discrete versus continuous models in evolutionary dynamics:from simple to simpler-and even simpler-models. Math Comput Model, 2006, 47(7):743-754
[25] Hofbauer J, Sigmund K. Evolutionary Games and Population Dynamics. Cambridge University Press, 1998
[26] Luo S. A unifying framework reveals key properties of multilevel selection. J Theor Biol, 2014, 341(2):41-52
[27] Wang X J, Lan J, Dong Q J, Lei G L. Evolutionary dynamics on one-dimensional cycle with shifting mechanism and tiny mutation rate. Acta Math Sci (English Series), 2015, 35(1):95-104
[28] Maynard-Smith J, Price G R. The logic of animal conflict. Nature, 1973, 246(5427):15-18
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