Additively Coupled Sum Constrained Games

  • Yi Su
  • Mihaela van der Schaar
Conference paper
Part of the Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering book series (LNICST, volume 75)

Abstract

We propose and analyze a broad family of games played by resource-constrained players, which are characterized by the following central features: 1) each user has a multi-dimensional action space, subject to a single sum resource constraint; 2) each user’s utility in a particular dimension depends on an additive coupling between the user’s action in the same dimension and the actions of the other users; and 3) each user’s total utility is the sum of the utilities obtained in each dimension. Familiar examples of such multi-user environments in communication systems include power control over frequency-selective Gaussian interference channels and flow control in Jackson networks. In settings where users cannot exchange messages in real-time, we study how users can adjust their actions based on their local observations. We derive sufficient conditions under which a unique Nash equilibrium exists and the best-response algorithm converges globally and linearly to the Nash equilibrium. In settings where users can exchange messages in real-time, we focus on user choices that optimize the overall utility. We provide the convergence conditions of two distributed action update mechanisms, gradient play and Jacobi update.

Keywords

Nash Equilibrium Convergence Condition Pure Nash Equilibrium Strategic Substitute Unique Nash Equilibrium 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Fudenberg, D., Tirole, J.: Game Theory. MIT Press, Cambridge (1991)MATHGoogle Scholar
  2. 2.
    Rosen, J.: Existence and uniqueness of equilibrium points for concave n-person games. Econometrica 33, 520–534 (1965)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Topkis, D.M.: Supermodularity and Complementarity. Princeton Univ. Press, Princeton (1998)Google Scholar
  4. 4.
    Yao, D.: S-modular games with queueing applications. Queueing Syst. 21, 449–475 (1995)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Altman, E., Altman, Z.: S-modular games and power control in wireless networks. IEEE Trans. Automatic Control 48(5), 839–842 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chiang, M., Low, S.H., Calderbank, A.R., Doyle, J.C.: Layering as optimization decomposition. Proceedings of the IEEE 95, 255–312 (2007)CrossRefGoogle Scholar
  7. 7.
    Palomar, D.P., Chiang, M.: A tutorial on decomposition methods for network utility maximization. IEEE JSAC 24(8), 1439–1451 (2006)Google Scholar
  8. 8.
    Yu, W., Ginis, G., Cioffi, J.: Distributed multiuser power control for digital subscriber lines. IEEE JSAC 20(5), 1105–1115 (2002)Google Scholar
  9. 9.
    Chung, S.T., Seung, J.L., Kim, J., Cioffi, J.: A game-theoretic approach to power allocation in frequency-selective Gaussian interference channels. In: Proc. IEEE Int. Symp. on Inform. Theory, p. 316 (June 2003)Google Scholar
  10. 10.
    Cendrillon, R., Huang, J., Chiang, M., Moonen, M.: Autonomous spectrum balancing for digital subscriber lines. IEEE Trans. on Signal Process. 55(8), 4241–4257 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Scutari, G., Palomar, D.P., Barbarossa, S.: Optimal linear precoding strategies for wideband noncooperative systems based on game theory - Part II: Algorithms. IEEE Trans. Signal Process. 56(3), 1250–1267 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Huang, J., Berry, R., Honig, M.: Distributed interference compensation for wireless networks. IEEE JSAC 24(5), 1074–1084 (2006)Google Scholar
  13. 13.
    Shi, C., Berry, R., Honig, M.: Distributed interference pricing for OFDM wireless networks with non-seperable utilities. In: Proc. of Conference on Information Sciences and Systems (CISS), March 19-21, pp. 755–760 (2008)Google Scholar
  14. 14.
    Lasaulce, S., Debbah, M., Altman, E.: Methodologies for analyzing equilibria in wireless games. IEEE Signal Process. Magazine 26, 41–52 (2009)CrossRefGoogle Scholar
  15. 15.
    Bulow, J., Geanakoplos, J., Klemperer, P.: Multimarket oligopoly: strategic substitutes and strategic complements. Journal of Political Economy 93, 488–511 (1985)CrossRefGoogle Scholar
  16. 16.
    Mo, J., Walrand, J.: Fair end-to-end window-based congestion control. IEEE Trans. on Networking 8(5), 556–567 (2000)CrossRefGoogle Scholar
  17. 17.
    Bertsekas, D.P., Tsitsiklis, J.N.: Parallel and Distributed Computation. Prentice Hall, Englewood Cliffs (1997)MATHGoogle Scholar
  18. 18.
    La, R., Anantharam, V.: Utility based rate control in the internet for elastic traffic. IEEE/ACM Trans. Networking 10(2), 271–286 (2002)CrossRefGoogle Scholar
  19. 19.
    Su, Y., van der Schaar, M.: Structural Solutions For Additively Coupled Sum Constrained Games, UCLA Tech. Rep. (2010), http://arxiv.org/abs/1005.0880

Copyright information

© ICST Institute for Computer Science, Social Informatics and Telecommunications Engineering 2012

Authors and Affiliations

  • Yi Su
    • 1
  • Mihaela van der Schaar
    • 1
  1. 1.Department of Electrical EngineeringUCLAUSA

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