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Generalising diagonal strict concavity property for uniqueness of Nash equilibrium

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Abstract

In this paper, we extend the notion of diagonally strictly concave functions and use it to provide a sufficient condition for uniqueness of Nash equilibrium in some concave games. We then provide an alternative proof of the existence and uniqueness of Nash equilibrium for a network resource allocation game arising from the so-called Kelly mechanism by verifying the new sufficient condition. We then establish that the equilibrium resulting from the differential pricing in the Kelly mechanism is related to a normalised Nash equilibrium of a game with coupled strategy space.

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References

  1. P. J. Kolodzy, Interference temperature: A metric for dynamic spectrum utilization, International Journal of Network Management, 16(2) (2006), 103–113.

    Article  Google Scholar 

  2. F. P. Kelly, Charging and rate control for elastic traffic, European Transactions on Telecommunications, 8 (1997), 33–37.

    Article  Google Scholar 

  3. B. Hajek and G. Gopalakrishnan, Do greedy autonomous systems make for a sensible Internet? Presented at the Conference on Stochastic Networks, Stanford University, CA, 2002.

    Google Scholar 

  4. R. Johari and J. N. Tsitsiklis, Efficiency loss in a network resource allocation game, Mathematics of Operations Research, 29(3) (2004), 407–435.

    Article  MathSciNet  MATH  Google Scholar 

  5. R. Johari, S. Mannor and J. N. Tsitsiklis, Efficiency loss in a network resource allocation game: The case of elastic supply, IEEE Transactions on Automatic Control, 50(11) (2005), 1712–1724.

    Article  MathSciNet  Google Scholar 

  6. R. Johari and J. N. Tsitsiklis, Efficiency of scalar-parameterized mechanisms, Operations Research, 57(4) (2009), 823–839.

    Article  MathSciNet  MATH  Google Scholar 

  7. R. Ma, D. M. Chiu, J. Lui, V. Mishra and D. Rubenstein, Price differentiation in the Kelly mechanism, in Workshop on Pricing and Incentives in Networks, W-PIN, London, UK, June 2012.

    Google Scholar 

  8. J. Rosen, Existence and uniqueness of equilibrium points for concave n-person games, Econometrica, 33(3) (1965), 520–534.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Orda, N. Rom and N. Shimkin, Competitive routing in multi-user communication networks, IEEE/ACM Transaction on Networking, 1(5) (1993), 614–627.

    Article  Google Scholar 

  10. N. Shimkin, A survey of uniqueness results for selfish routing, in Proceedings of NETCOOP 2007, LNCS 4465, (2007), 33–42.

    Google Scholar 

  11. L. S. Shapley, A solution containing an arbitrary closed component, Contribution to the theory of games, 4 (1959), 87–93.

    MathSciNet  MATH  Google Scholar 

  12. D. Monderer and L. S. Shapley, Potential games, Games and Economic Behavior, 14 (1996), 124–143.

    Article  MathSciNet  MATH  Google Scholar 

  13. D. M. Topkis, Equilibrium points in nonzero-sum-person submodular games, SIAM Journal on Control Optimization, 17 (1979), 773–778.

    Article  MathSciNet  MATH  Google Scholar 

  14. R. D. Yates, A framework for uplink power control in cellular radio systems, IEEE Journal on Selected Areas in Communications, 13 (1995), 1341–1347.

    Article  Google Scholar 

  15. C. U. Saraydar, N. B. Mandayam and D. J. Goodman, Efficient power control via pricing in wireless data networks, IEEE Transactions on Communications, 50(2) (2002), 291–303.

    Article  Google Scholar 

  16. A. A. Kulkarni and U. V. Shanbhag, On the variational equilibrium as a refinement of the generalized Nash equilibrium, Automatica, 48 (2012), 45–55.

    Article  MathSciNet  MATH  Google Scholar 

  17. E. Altman and E. Solan, Constrained games: The impact of the attitude to adversary’s constraints, IEEE Transactions on Automatic Control, 54(10) (2009), 2435–2440.

    Article  MathSciNet  Google Scholar 

  18. W. Leininer and C. Yang, Dynamic rent seeking games, Games and Economic Behavior, 7 (1994), 406–427.

    Article  MathSciNet  MATH  Google Scholar 

  19. O. Richman and N. Shimkin, Topological uniqueness of the Nash equilibrium for selfish routing, Mathematics of Operations Research INFORMS, 32(1) (2007), 215–232.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. INRIA Sophia Antipolis, 2004 Route des Lucioles, B.P. 93 06902, Sophia Antipolis Cedex, France

    Eitan Altman

  2. Industrial Engineering and Operations Research, Mechanical Engineering Building, Indian Institute of Technology Bombay, Powai, Mumbai, 400 076, India

    Manjesh Kumar Hanawal

  3. Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore, 560 012, India

    Rajesh Sundaresan

  4. LINCS, 23 Avenue d’Italie, 75013, Paris, France

    Eitan Altman

Authors
  1. Eitan Altman
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  2. Manjesh Kumar Hanawal
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  3. Rajesh Sundaresan
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Corresponding author

Correspondence to Eitan Altman.

Additional information

Eitan Altman and Rajesh Sundaresan were supported by IFCAM (Indo-French Centre for Applied Mathematics). Manjesh Kumar Hanawal was supported by the European Commission within the framework of the CONGAS project FP7-ICT-2011-8-317672.

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Altman, E., Hanawal, M.K. & Sundaresan, R. Generalising diagonal strict concavity property for uniqueness of Nash equilibrium. Indian J Pure Appl Math 47, 213–228 (2016). https://doi.org/10.1007/s13226-016-0185-4

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  • DOI: https://doi.org/10.1007/s13226-016-0185-4

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