Algorithms for computing Nash equilibria in deterministic LQ games

In this paper we review a number of algorithms to compute Nash equilibria in deterministic linear quadratic differential games. We will review the open-loop and feedback information case. In both cases we address both the finite and the infinite-planning horizon.

[1]  D. Tabak Numerical solutions of differential game problems , 1975 .

[2]  A. Laub Invariant Subspace Methods for the Numerical Solution of Riccati Equations , 1991 .

[3]  Jacob Engwerda,et al.  LQ Dynamic Optimization and Differential Games , 2005 .

[4]  J. Verschelde,et al.  Homotopies exploiting Newton polytopes for solving sparse polynomial systems , 1994 .

[5]  V. Mehrmann The Autonomous Linear Quadratic Control Problem: Theory and Numerical Solution , 1991 .

[6]  R. Neck,et al.  Solving dynamic macroeconomic policy games using the algorithm OPTGAME 1.0 , 2001 .

[7]  Tomasz P. Michalak,et al.  Dynamic Modeling of Monetary and Fiscal Cooperation Among Nations , 2006 .

[8]  Leiba Rodman,et al.  Algebraic Riccati equations , 1995 .

[9]  H. Abou-Kandil,et al.  Analytic solution for a class of linear quadratic open-loop Nash games , 1986 .

[10]  Huibert Kwakernaak,et al.  Linear Optimal Control Systems , 1972 .

[11]  J. B. CruzJr.,et al.  On the existence of Nash strategies and solutions to coupled riccati equations in linear-quadratic games , 1979 .

[12]  David A. McAllester,et al.  Solving Polynomial Systems Using a Branch and Prune Approach , 1997 .

[13]  P. Dooren A Generalized Eigenvalue Approach for Solving Riccati Equations , 1980 .

[14]  J. M. Schumacher,et al.  An equivalence result in linear-quadratic theory , 2003, Autom..

[15]  T.-Y. Li,et al.  Lyapunov Iterations for Solving Coupled Algebraic Riccati Equations of Nash Differential Games and Algebraic Riccati Equations of Zero-Sum Games , 1995 .

[16]  Jennifer F. Reinganum,et al.  Oligopoly Extraction of a Common Property Natural Resource: The Importance of the Period of Commitment in Dynamic Games , 1985 .

[17]  W. Reid,et al.  Riccati Differential Equations , 1975, IEEE Transactions on Systems, Man, and Cybernetics.

[18]  G. Jank,et al.  Iterative solution of algebraic matrix Riccati equations in open loop Nash games , 2003 .

[19]  E. Dockner,et al.  Differential Games in Economics and Management Science , 2001 .

[20]  G. Olsder,et al.  On the linear-quadratic, closed-loop, no-memory Nash game , 1984 .

[21]  Y. Ho,et al.  Nonzero-sum differential games , 1969 .

[22]  Georges Zaccour,et al.  Differential Games in Marketing , 2003 .

[23]  Ngo Van Long,et al.  Differential Games in Economics and Management Science: List of tables , 2000 .

[24]  Y. Ho On the Linear-Quadratic , Closed-Loop , No-Memory Nash Game , 2004 .

[25]  Jacob C. Engwerda A Numerical Algorithm to Find Soft-Constrained Nash Equilibria in Scalar Lq-Games , 2004 .

[26]  J. Cruz,et al.  On the solution of the open-loop Nash Riccati equations in linear quadratic differential games† , 1973 .

[27]  Gerhard Jank,et al.  Iterative solution of algebraic matrix Riccati equations in open loop Nash games , 2005 .

[28]  Y. Ho,et al.  Further properties of nonzero-sum differential games , 1969 .

[29]  C. Loan,et al.  A Schur decomposition for Hamiltonian matrices , 1981 .

[30]  A. Laub A schur method for solving algebraic Riccati equations , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[31]  T. Başar,et al.  Dynamic Noncooperative Game Theory , 1982 .

[32]  T.-P. Azevedo-Perdicoúlis,et al.  Linear Quadratic Nash Games on Positive Linear Systems , 2005, Eur. J. Control.

[33]  Z. Rekasius,et al.  On the solution of the optimal linear control problems under conflict of interest , 1971 .

[34]  L. Rodman,et al.  Matrix Riccati Equations in Control and Systems Theory [Book Report] , 2005, IEEE Control Systems.

[35]  Hiroaki Mukaidani,et al.  A numerical analysis of the Nash strategy for weakly coupled large-scale systems , 2006, IEEE Transactions on Automatic Control.

[36]  B. Anderson,et al.  Linear Optimal Control , 1971 .

[37]  H. Abou-Kandil,et al.  Matrix Riccati Equations in Control and Systems Theory , 2003, IEEE Transactions on Automatic Control.