Characterization and computation of local Nash equilibria in continuous games

We present derivative-based necessary and sufficient conditions ensuring player strategies constitute local Nash equilibria in non-cooperative continuous games. Our results can be interpreted as generalizations of analogous second-order conditions for local optimality from nonlinear programming and optimal control theory. Drawing on this analogy, we propose an iterative steepest descent algorithm for numerical approximation of local Nash equilibria and provide a sufficient condition ensuring local convergence of the algorithm. We demonstrate our analytical and computational techniques by computing local Nash equilibria in games played on a finite-dimensional differentiable manifold or an infinite-dimensional Hilbert space.

[1]  J. Neumann,et al.  Theory of Games and Economic Behavior. , 1945 .

[2]  J. Neumann,et al.  Theory of games and economic behavior, 2nd rev. ed. , 1947 .

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

[4]  I. Ekeland.  TOPOLOGIE DIFFERENTIELLE ET THEORIE DES JEUX , 1974 .

[5]  T. Eisele.  Nonexistence and nonuniqueness of open-loop equilibria in linear-quadratic differential games , 1982 .

[6]  Tamer Basar,et al.  Distributed algorithms for the computation of noncooperative equilibria , 1987, Autom..

[7]  T. Basar,et al.  Relaxation techniques and asynchronous algorithms for on-line computation of noncooperative equilibria , 1987, 26th IEEE Conference on Decision and Control.

[8]  R. Rubinstein,et al.  On relaxation algorithms in computation of noncooperative equilibria , 1994, IEEE Trans. Autom. Control..

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

[10]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

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

[12]  Elijah Polak,et al.  Optimization: Algorithms and Consistent Approximations , 1997 .

[13]  Sjur Didrik Flåm,et al.  Restricted attention, myopic play, and thelearning of equilibrium , 1998, Ann. Oper. Res..

[14]  Daniel E. Koditschek,et al.  Phase Regulation of Decentralized Cyclic Robotic Systems , 2002, Int. J. Robotics Res..

[15]  B. Dundas,et al.  DIFFERENTIAL TOPOLOGY , 2002 .

[16]  John M. Lee.  Introduction to Smooth Manifolds , 2002 .

[17]  Magnus Egerstedt,et al.  Decentralized Coordination with Local Interactions: Some New Directions , 2003 .

[18]  J. Krawczyk,et al.  Numerical solutions to Nash-Cournot equilibria in coupled constraint electricity markets , 2004, IEEE Transactions on Power Systems.

[19]  S. Smale.  Global analysis and economics , 1975, Synthese.

[20]  W. Marsden.  I and J , 2012 .

[21]  S. Shankar Sastry,et al.  Pricing in linear-quadratic dynamic games , 2012, 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

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

[23]  S. Shankar Sastry,et al.  Genericity and structural stability of non-degenerate differential Nash equilibria , 2014, 2014 American Control Conference.