Distributed Algorithms for Aggregative Games on Graphs

We consider a class of Nash games, termed as aggregative games, being played over a networked system. In an aggregative game, a player’s objective is a function of the aggregate of all the players’ decisions. Every player maintains an estimate of this aggregate, and the players exchange this information with their local neighbors over a connected network. We study distributed synchronous and asynchronous algorithms for information exchange and equilibrium computation over such a network. Under standard conditions, we establish the almost-sure convergence of the obtained sequences to the equilibrium point. We also consider extensions of our schemes to aggregative games where the players’ objectives are coupled through a more general form of aggregate function. Finally, we present numerical results that demonstrate the performance of the proposed schemes.

[1]  J. Nash.  Equilibrium Points in N-Person Games. , 1950, Proceedings of the National Academy of Sciences of the United States of America.

[2]  Ιωαννησ Τσιτσικλησ,et al.  PROBLEMS IN DECENTRALIZED DECISION MAKING AND COMPUTATION , 1984 .

[3]  W. Novshek.  On the Existence of Cournot Equilibrium , 1985 .

[4]  F. Szidarovszky,et al.  The Theory of Oligopoly with Multi-Product Firms , 1990 .

[5]  Drew Fudenberg,et al.  Game theory (3. pr.) , 1991 .

[6]  On Bounded Rationality And Computational Complexity , 1994 .

[7]  L. Shapley,et al.  Potential Games , 1994 .

[8]  M. Anthony,et al.  Mathematics for Economics and Finance: Introduction to optimisation , 1996 .

[9]  H. Simon,et al.  The sciences of the artificial (3rd ed.) , 1996 .

[10]  L. Shapley,et al.  REGULAR ARTICLEPotential Games , 1996 .

[11]  D. Fudenberg,et al.  The Theory of Learning in Games , 1998 .

[12]  T. Basar,et al.  A game-theoretic framework for congestion control in general topology networks , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[13]  Dudley,et al.  Real Analysis and Probability: Measurability: Borel Isomorphism and Analytic Sets , 2002 .

[14]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[15]  Ana B. Ania,et al.  The evolutionary stability of perfectly competitive behavior , 2005 .

[16]  Tansu Alpcan,et al.  Distributed Algorithms for Nash Equilibria of Flow Control Games , 2005 .

[17]  Stephen P. Boyd,et al.  Randomized gossip algorithms , 2006, IEEE Transactions on Information Theory.

[18]  Roni Abiri,et al.  Introduction to Optimisation of the UMTS Radio Network , 2006 .

[19]  Benjamin Van Roy,et al.  Consensus Propagation , 2005, IEEE Transactions on Information Theory.

[20]  Lacra Pavel,et al.  A noncooperative game approach to OSNR optimization in optical networks , 2006, IEEE Transactions on Automatic Control.

[21]  Pradeep Dubey,et al.  Strategic complements and substitutes, and potential games , 2006, Games Econ. Behav..

[22]  Lacra Pavel,et al.  An extension of duality to a game-theoretic framework , 2007, Autom..

[23]  Slawomir Stanczak,et al.  Unifying View on Min-Max Fairness, Max-Min Fairness, and Utility Optimization in Cellular Networks , 2007, EURASIP J. Wirel. Commun. Netw..

[24]  Minyi Huang,et al.  Large-Population Cost-Coupled LQG Problems With Nonuniform Agents: Individual-Mass Behavior and Decentralized $\varepsilon$-Nash Equilibria , 2007, IEEE Transactions on Automatic Control.

[25]  Anders Rantzer,et al.  Using Game Theory for Distributed Control Engineering , 2008 .

[26]  Björn Johansson,et al.  On Distributed Optimization in Networked Systems , 2008 .

[27]  A. Juditsky,et al.  Solving variational inequalities with Stochastic Mirror-Prox algorithm , 2008, 0809.0815.

[28]  Mikael Johansson,et al.  A Randomized Incremental Subgradient Method for Distributed Optimization in Networked Systems , 2009, SIAM J. Optim..

[29]  Angelia Nedic,et al.  Asynchronous gossip algorithms for stochastic optimization , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[30]  Anders Rantzer,et al.  Dynamic dual decomposition for distributed control , 2009, 2009 American Control Conference.

[31]  Prashant G. Mehta,et al.  Nash equilibrium problems with congestion costs and shared constraints , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

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

[33]  Lacra Pavel,et al.  Games with coupled propagated constraints in optical networks with multi-link topologies , 2009, Autom..

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

[35]  M. K. Jensen.  Aggregative games and best-reply potentials , 2010 .

[36]  Francisco Facchinei,et al.  Nash equilibria: the variational approach , 2010, Convex Optimization in Signal Processing and Communications.

[37]  Tansu Alpcan,et al.  A system performance approach to OSNR optimization in optical networks , 2010, IEEE Transactions on Communications.

[38]  Benjamin Van Roy,et al.  Convergence of Min-Sum Message-Passing for Convex Optimization , 2010, IEEE Transactions on Information Theory.

[39]  A. Nedić,et al.  Asynchronous Gossip Algorithm for Stochastic Optimization: Constant Stepsize Analysis* , 2010 .

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

[41]  Angelia Nedic,et al.  Distributed Stochastic Subgradient Projection Algorithms for Convex Optimization , 2008, J. Optim. Theory Appl..

[42]  Angelia Nedic,et al.  Single timescale regularized stochastic approximation schemes for monotone Nash games under uncertainty , 2010, 49th IEEE Conference on Decision and Control (CDC).

[43]  S. Meyn,et al.  Synchronization of coupled oscillators is a game , 2010, ACC 2010.

[44]  Uday V. Shanbhag,et al.  Distributed iterative regularization algorithms for monotone Nash games , 2010, 49th IEEE Conference on Decision and Control (CDC).

[45]  Angelia Nedic,et al.  Asynchronous Broadcast-Based Convex Optimization Over a Network , 2011, IEEE Transactions on Automatic Control.

[46]  Prashant G. Mehta,et al.  Nash Equilibrium Problems With Scaled Congestion Costs and Shared Constraints , 2011, IEEE Transactions on Automatic Control.

[47]  Angelia Nedic,et al.  Distributed Asynchronous Constrained Stochastic Optimization , 2011, IEEE Journal of Selected Topics in Signal Processing.

[48]  Lacra Pavel,et al.  A Lyapunov-Krasovskii stability analysis for game-theoretic based power control in optical links , 2011, Telecommun. Syst..

[49]  D. Martimort,et al.  Aggregate Representations of Aggregate Games , 2011 .

[50]  Asuman E. Ozdaglar,et al.  Distributed Subgradient Methods for Convex Optimization Over Random Networks , 2011, IEEE Transactions on Automatic Control.

[51]  Pascal Bianchi,et al.  On the Convergence of a Multi-Agent Projected Stochastic Gradient Algorithm , 2011, ArXiv.

[52]  Uday V. Shanbhag,et al.  Distributed Computation of Equilibria in Monotone Nash Games via Iterative Regularization Techniques , 2012, SIAM J. Optim..

[53]  Angelia Nedic,et al.  A new class of distributed optimization algorithms: application to regression of distributed data , 2012, Optim. Methods Softw..

[54]  Ali H. Sayed,et al.  Diffusion Strategies Outperform Consensus Strategies for Distributed Estimation Over Adaptive Networks , 2012, IEEE Transactions on Signal Processing.

[55]  Jong-Shi Pang,et al.  Joint Sensing and Power Allocation in Nonconvex Cognitive Radio Games: Nash Equilibria and Distributed Algorithms , 2012, IEEE Transactions on Information Theory.

[56]  Angelia Nedic,et al.  Regularized Iterative Stochastic Approximation Methods for Stochastic Variational Inequality Problems , 2013, IEEE Transactions on Automatic Control.

[57]  Jason R. Marden,et al.  Designing Games for Distributed Optimization , 2013, IEEE J. Sel. Top. Signal Process..

[58]  Ali H. Sayed,et al.  On the Influence of Informed Agents on Learning and Adaptation Over Networks , 2012, IEEE Transactions on Signal Processing.

[59]  Dusan M. Stipanovic,et al.  Distributed Bregman-Distance Algorithms for Min-Max Optimization , 2013, Agent-Based Optimization.

[60]  Pascal Bianchi,et al.  Convergence of a Multi-Agent Projected Stochastic Gradient Algorithm for Non-Convex Optimization , 2011, IEEE Transactions on Automatic Control.

[61]  Angelia Nedic,et al.  A regularized smoothing stochastic approximation (RSSA) algorithm for stochastic variational inequality problems , 2013, 2013 Winter Simulations Conference (WSC).

[62]  Jason R. Marden,et al.  A Model-Free Approach to Wind Farm Control Using Game Theoretic Methods , 2013, IEEE Transactions on Control Systems Technology.

[63]  M. K. Jensen.  Aggregative games: Theory , 2018 .

[64]  Augustin M. Cournot.  Cournot, Antoine Augustin: Recherches sur les principes mathématiques de la théorie des richesses , 2019, Die 100 wichtigsten Werke der Ökonomie.

[65]  CONTROL AND GAME-THEORETIC TOOLS FOR COMMUNICATION NETWORKS SURVEY , 2022 .