Additively Coupled Sum Constrained Games
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 EquilibriumPreview
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References
- 1.Fudenberg, D., Tirole, J.: Game Theory. MIT Press, Cambridge (1991)zbMATHGoogle Scholar
- 2.Rosen, J.: Existence and uniqueness of equilibrium points for concave n-person games. Econometrica 33, 520–534 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
- 3.Topkis, D.M.: Supermodularity and Complementarity. Princeton Univ. Press, Princeton (1998)Google Scholar
- 4.Yao, D.: S-modular games with queueing applications. Queueing Syst. 21, 449–475 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
- 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.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.Palomar, D.P., Chiang, M.: A tutorial on decomposition methods for network utility maximization. IEEE JSAC 24(8), 1439–1451 (2006)Google Scholar
- 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.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.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.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.Huang, J., Berry, R., Honig, M.: Distributed interference compensation for wireless networks. IEEE JSAC 24(5), 1074–1084 (2006)Google Scholar
- 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.Lasaulce, S., Debbah, M., Altman, E.: Methodologies for analyzing equilibria in wireless games. IEEE Signal Process. Magazine 26, 41–52 (2009)CrossRefGoogle Scholar
- 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.Mo, J., Walrand, J.: Fair end-to-end window-based congestion control. IEEE Trans. on Networking 8(5), 556–567 (2000)CrossRefGoogle Scholar
- 17.Bertsekas, D.P., Tsitsiklis, J.N.: Parallel and Distributed Computation. Prentice Hall, Englewood Cliffs (1997)zbMATHGoogle Scholar
- 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.Su, Y., van der Schaar, M.: Structural Solutions For Additively Coupled Sum Constrained Games, UCLA Tech. Rep. (2010), http://arxiv.org/abs/1005.0880