Performance Analysis of Gradient-Based Nash Seeking Algorithms Under Quantization

This paper investigates the impact of quantized inter-agent communications on the asymptotic and transient behavior of gradient-based Nash-seeking algorithms in non-cooperative games. Using the information-theoretic notion of entropy power, we establish a universal lower bound on the asymptotic rate of exponential mean-square convergence to the Nash equilibrium (NE). This bound depends on the inter-agent data rate and the local behavior of the agents' utility functions, and is independent of the quantizer structure. Next, we study transient performance and derive an upper bound on the average time required to settle inside a specified ball around the NE, under uniform quantization. Furthermore, we establish an upper bound on the probability that agents' actions lie outside this ball, and show that this bound decays double-exponentially with time. Finally, we propose an adaptive quantization scheme that allows the gradient algorithm to converge to the NE despite quantized inter-agent communications.

[1]  R. Srikant,et al.  Quantized Consensus , 2006, 2006 IEEE International Symposium on Information Theory.

[2]  Massimo Franceschetti,et al.  Data Rate Theorem for Stabilization Over Time-Varying Feedback Channels , 2009, IEEE Transactions on Automatic Control.

[3]  Robert D. Nowak,et al.  Quantized incremental algorithms for distributed optimization , 2005, IEEE Journal on Selected Areas in Communications.

[4]  K. Johansson,et al.  Distributed positioning of autonomous mobile sensors with application to coverage control , 2011, Proceedings of the 2011 American Control Conference.

[5]  Noah D. Stein.  Characterization and computation of equilibria in infinite games , 2007 .

[6]  Milos S. Stankovic,et al.  Distributed Seeking of Nash Equilibria With Applications to Mobile Sensor Networks , 2012, IEEE Transactions on Automatic Control.

[7]  Miroslav Krstic,et al.  Stochastic Nash Equilibrium Seeking for Games with General Nonlinear Payoffs , 2011, SIAM J. Control. Optim..

[8]  J. Goodman.  Note on Existence and Uniqueness of Equilibrium Points for Concave N-Person Games , 1965 .

[9]  Asuman E. Ozdaglar,et al.  Constrained Consensus and Optimization in Multi-Agent Networks , 2008, IEEE Transactions on Automatic Control.

[10]  Jason R. Marden,et al.  Game Theory and Distributed Control , 2015 .

[11]  D. Delchamps.  Stabilizing a linear system with quantized state feedback , 1990 .

[12]  D. Bertsekas,et al.  Incremental subgradient methods for nondifferentiable optimization , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[13]  Victor Solo,et al.  Stabilization and Disturbance Attenuation Over a Gaussian Communication Channel , 2010, IEEE Transactions on Automatic Control.

[14]  Ruggero Carli,et al.  Communication constraints in the average consensus problem , 2008, Autom..

[15]  Asuman E. Ozdaglar,et al.  Distributed Subgradient Methods for Multi-Agent Optimization , 2009, IEEE Transactions on Automatic Control.

[16]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[17]  Tansu Alpcan,et al.  Noncooperative Games for Control of Networked Systems , 2006 .

[18]  John N. Tsitsiklis,et al.  On distributed averaging algorithms and quantization effects , 2007, 2008 47th IEEE Conference on Decision and Control.

[19]  Robin J. Evans,et al.  Feedback Control Under Data Rate Constraints: An Overview , 2007, Proceedings of the IEEE.

[20]  Miroslav Krstic,et al.  Nash Equilibrium Seeking in Noncooperative Games , 2012, IEEE Transactions on Automatic Control.

[21]  Lihua Xie,et al.  Network Topology and Communication Data Rate for Consensusability of Discrete-Time Multi-Agent Systems , 2011, IEEE Trans. Autom. Control..

[22]  Tansu Alpcan,et al.  Game-Theoretic Frameworks for Demand Response in Electricity Markets , 2015, IEEE Transactions on Smart Grid.

[23]  Zhiyong Geng,et al.  Design and analysis of quantizer for multi-agent systems with a limited rate of communication data , 2013, Commun. Nonlinear Sci. Numer. Simul..

[24]  D. Chattopadhyay,et al.  Multicommodity spatial Cournot model for generator bidding analysis , 2004, IEEE Transactions on Power Systems.

[25]  Thomas M. Cover,et al.  Elements of Information Theory: Cover/Elements of Information Theory, Second Edition , 2005 .

[26]  Robin J. Evans,et al.  Stabilizability of Stochastic Linear Systems with Finite Feedback Data Rates , 2004, SIAM J. Control. Optim..

[27]  John N. Tsitsiklis,et al.  On Distributed Averaging Algorithms and Quantization Effects , 2008, IEEE Trans. Autom. Control..

[28]  Vincent K. N. Lau,et al.  Convergence-Optimal Quantizer Design of Distributed Contraction-Based Iterative Algorithms With Quantized Message Passing , 2010, IEEE Transactions on Signal Processing.

[29]  John N. Tsitsiklis,et al.  Distributed subgradient methods and quantization effects , 2008, 2008 47th IEEE Conference on Decision and Control.

[30]  Tansu Alpcan,et al.  A Game-Theoretic Analysis of Wind Generation Variability on Electricity Markets , 2014, IEEE Transactions on Power Systems.

[31]  Cem U. Saraydar,et al.  Efficient power control via pricing in wireless data networks , 2002, IEEE Trans. Commun..

[32]  Asuman E. Ozdaglar,et al.  Characterization and computation of correlated equilibria in infinite games , 2007, 2007 46th IEEE Conference on Decision and Control.

[33]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.