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Effects of Players’ Random Participation to the Stability in LQ Games

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Advances in Dynamic and Mean Field Games (ISDG 2016)

Part of the book series: Annals of the International Society of Dynamic Games ((AISDG,volume 15))

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Abstract

We consider a linear quadratic (LQ) game with randomly arriving players, staying in the game for a random period of time. The Nash equilibrium of the game is characterized by a set of coupled Riccati-type equations for Markovian jump linear systems (MJLS), and the existence of a Nash equilibrium is proved using Brouwer’s fixed point theorem. We then consider the game, in the limit as the number of players becomes large, assuming a partially Kantian behavior. We then focus on the effects of the random entrance, random exit, and partial Kantian cooperation to the stability of the overall system. Some numerical results are also presented. It turns out that in the noncooperative case, the overall system tends to become more unstable as the number of players increases and tends to stabilize as the expected time horizon increases. In the partially cooperative case, an explicit relation of the expected time horizon of each player with the minimum amount of cooperation, sufficient to stabilize the closed loop system, is derived.

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Notes

  1. 1.

    Regarding the relationship of [5] with the current work, [5] has clearly a more general framework. Some of the results of Section 3 of the current work and particularly Proposition 1 can be obtained using a modified version of Theorem 2.4 of [5]. Furthermore, this modification is described in [5]. However, the manipulations leading to Proposition 1 are necessary for the rest of the paper and thus they are included. Let us note that the first version of the current work, including Proposition 1, appeared in the ISDG Symposium paper in Urbino, Italy, July 2016, and we become aware of [5] after the presentation.

References

  1. R. J. Aumann. Markets with a continuum of traders. Econometrica, 32:39–50, 1964.

    Google Scholar 

  2. Yves Balasko and Karl Shell. The overlapping-generations model, I: The case of pure exchange without money. Journal of Economic Theory, 23(3):281–306, 1980.

    Google Scholar 

  3. T Basar. On the uniqueness of the nash solution in linear-quadratic differential games. International Journal of Game Theory, 5(2–3):65–90, 1976.

    Google Scholar 

  4. Pierre Bernhard and Marc Deschamps. On dynamic games with randomly arriving players. Dynamic Games and Applications, pages 1–26, 2015.

    Google Scholar 

  5. Pierre Bernhard and Marc Deschamps. Dynamic equilibrium in games with randomly arriving players. [Research Report] RR-8961 Université de la Côte d’Azur-INRIA, 2016, pp.38. < hal-01379644 > , 2016.

    Google Scholar 

  6. Pierre Bernhard and Frédéric Hamelin. Sharing a resource with randomly arriving foragers. Mathematical biosciences, 273:91–101, 2016.

    Google Scholar 

  7. Behram Homi Bharucha. On the stability of randomly varying systems. Technical report, DTIC Document, 1961.

    Google Scholar 

  8. El-Kébir Boukas, Alain Haurie, and Philippe Michel. An optimal control problem with a random stopping time. Journal of optimization theory and applications, 64(3):471–480, 1990.

    Google Scholar 

  9. Oswaldo Luiz Valle Costa, Marcelo Dutra Fragoso, and Ricardo Paulino Marques. Discrete-time Markov jump linear systems. Springer Science & Business Media, 2006.

    Google Scholar 

  10. Jacob Engwerda. Properties of feedback nash equilibria in scalar lq differential games. Automatica, 69:364–374, 2016.

    Google Scholar 

  11. A. Ghosh and N. Van Long. Kant’s rule of behavior and kant-nash equilibria in games of contributions to public goods. 2015.

    Google Scholar 

  12. Ekaterina Viktorovna Gromova and José Daniel López-Barrientos. A differential game model for the extraction of non renewable resources with random initial times. Contributions to Game Theory and Management, 8(0):58–63, 2015.

    Google Scholar 

  13. Garrett Hardin. The tragedy of the commons. Science, 162(3859):1243–1248, 1968.

    Google Scholar 

  14. A Haurie. Piecewise deterministic differential games. In Differential Games and Applications, pages 114–127. Springer, 1989.

    Google Scholar 

  15. Steffen Jørgensen and David WK Yeung. Inter-and intragenerational renewable resource extraction. Annals of Operations Research, 88:275–289, 1999.

    Google Scholar 

  16. Marc Jungers, George P Papavassilopoulos, and Hisham Abou-Kandil. Feedback nash strategy for games with player-dependent time horizons. IFAC Proceedings Volumes, 45(25):45–50, 2012.

    Google Scholar 

  17. Immanuel Kant. Groundwork for the metaphysics of morals. 1785.

    Google Scholar 

  18. Alan F Karr. Weak convergence of a sequence of markov chains. Probability Theory and Related Fields, 33(1):41–48, 1975.

    Google Scholar 

  19. Ioannis Kordonis. A model for partial kantian cooperation. arXiv preprint arXiv:1609.01921, 2016.

    Google Scholar 

  20. Ioannis Kordonis and George P. Papavassilopoulos. Lq nash games with players participating in the game for random time intervals. In The 15 th International Symposium on Dynamic Games and Applications, Bysice, Czech Republic, July 18–22, 2012.

    Google Scholar 

  21. Ioannis Kordonis and George P Papavassilopoulos. On stability and lq control of mjls with a markov chain with general state space. IEEE Transactions on Automatic Control, 59(2): 535–540, 2014.

    Google Scholar 

  22. Ioannis Kordonis and George P Papavassilopoulos. Lq nash games with random entrance: An infinite horizon major player and minor players of finite horizons. IEEE Transactions on Automatic Control, 60(6):1486–1500, 2015.

    Google Scholar 

  23. Jesús Marín-Solano and Ekaterina V Shevkoplyas. Non-constant discounting and differential games with random time horizon. Automatica, 47(12):2626–2638, 2011.

    Google Scholar 

  24. Michel Mariton. Jump linear systems in automatic control. M. Dekker New York, 1990.

    Google Scholar 

  25. J. W. Milnor and L. S. Shapley. Values of large games ii: Oceanic games. Mathematics of Operations Research, 3(4):290–307, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  26. Roger B Myerson. Population uncertainty and poisson games. International Journal of Game Theory, 27(3):375–392, 1998.

    Google Scholar 

  27. Roger B Myerson. Large poisson games. Journal of Economic Theory, 94(1):7–45, 2000.

    Google Scholar 

  28. G. Owen. Game Theory. W.B. Saunders Company, 1969.

    Google Scholar 

  29. George P. Papavassilopoulos. University-students game. Dynamic Games and Applications, 3(3):387–418, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  30. George P Papavassilopoulos, Hisham Abou-Kandil, and Marc Jungers. A nash game with long-term and short-term players. In 52nd IEEE Conference on Decision and Control, pages 1726–1731. IEEE, 2013.

    Google Scholar 

  31. Leon A Petrosjan and EKATERINA V Shevkoplyas. Cooperative solutions for games with random duration. Game theory and applications, 9:125–139, 2003.

    Google Scholar 

  32. John E Roemer. Kantian equilibrium. The Scandinavian Journal of Economics, 112(1):1–24, 2010.

    Google Scholar 

  33. John E Roemer. How we (do and could) cooperate. 2015.

    Google Scholar 

  34. John E Roemer. Kantian optimization: A microfoundation for cooperation. Journal of Public Economics, 127:45–57, 2015.

    Google Scholar 

  35. Paul A Samuelson. An exact consumption-loan model of interest with or without the social contrivance of money. The journal of political economy, pages 467–482, 1958.

    Google Scholar 

  36. David Schmeidler. Equilibrium points of nonatomic games. Journal of statistical Physics, 7(4):295–300, 1973.

    Article  MathSciNet  MATH  Google Scholar 

  37. Ngo Van Long. Kant-nash equilibrium in a dynamic game of climate change mitigations. 2015.

    Google Scholar 

  38. Ngo Van Long. Kant-nash equilibria in a quantity-setting oligopoly. In Equilibrium Theory for Cournot Oligopolies and Related Games, pages 179–201. Springer, 2016.

    Google Scholar 

  39. John Glen Wardrop. Road paper. some theoretical aspects of road traffic research. Proceedings of the institution of civil engineers, 1(3):325–362, 1952.

    Google Scholar 

  40. Agnieszka Wiszniewska-Matyszkiel. Static and dynamic equilibria in games with continuum of players. Positivity, 6(4):433–453, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  41. Agnieszka Wiszniewska-Matyszkiel. Static and dynamic equilibria in stochastic games with continuum of players. Control and Cybernetics, 32(1):103–126, 2003.

    MathSciNet  MATH  Google Scholar 

  42. Agnieszka Wiszniewska-Matyszkiel. Open and closed loop nash equilibria in games with a continuum of players. Journal of Optimization Theory and Applications, 160(1):280–301, 2014.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. CentraleSupélec, Avenue de la Boulaie, 35576, Cesson-Sévigné, France

    Ioannis Kordonis

  2. School of Electrical and Computer Engineering, National Technical University of Athens, Iroon Polytechniou 9, Zografou, 15780, Athens, Greece

    George P. Papavassilopoulos

Authors
  1. Ioannis Kordonis
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  2. George P. Papavassilopoulos
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Correspondence to Ioannis Kordonis .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, Antigonish, Nova Scotia, Canada

    Joseph Apaloo

  2. Department of Mathematics, University of Padova, Padova, Italy

    Bruno Viscolani

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Kordonis, I., Papavassilopoulos, G.P. (2017). Effects of Players’ Random Participation to the Stability in LQ Games. In: Apaloo, J., Viscolani, B. (eds) Advances in Dynamic and Mean Field Games. ISDG 2016. Annals of the International Society of Dynamic Games, vol 15. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-70619-1_12

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