A counterexample in linear-quadratic games: Existence of nonlinear nash solutions

1974 ◽  
Vol 14 (4) ◽  
pp. 425-430 ◽  
Author(s):  
T. Basar
Keyword(s):  
Linear Quadratic ◽  
Linear Quadratic Games

scholarly journals Mean Field Linear Quadratic Games with Set Up Costs

2012 ◽  
Vol 3 (1) ◽  
pp. 89-104 ◽  
Author(s):  
Dario Bauso ◽  
Raffaele Pesenti
Keyword(s):  
Mean Field ◽  
Linear Quadratic ◽  
Set Up ◽  
Linear Quadratic Games

Minimax methods for open-loop equilibria in N-person differential games. Part I: Linear quadratic games and constrained games

1986 ◽  
Vol 103 (1-2) ◽  
pp. 15-34
Author(s):  
Goong Chen ◽  
Quan Zheng ◽  
Jian-Xin Zhou
Keyword(s):  
Differential Games ◽  
Compact Convex ◽  
Dimensional Subspace ◽  
Open Loop ◽  
Finite Dimensional Subspace ◽  
Convex Constraints ◽  
Linear Quadratic ◽  
Matrix Games ◽  
Minimax Methods ◽  
Linear Quadratic Games

SynopsisIn [13], Nikaido and Isoda generalised von Neumann's symmetrisation method for matrix games. They showed that N-person noncooperative games can be treated by a minimax method.We apply this method to N-person differential games. Lukes and Russell [11] first studied N-person nonzero sum linear quadratic games in 1971. Here we have reproduced and strengthened their results. The existence and uniqueness of equilibria are completely determined by the invertibility of the decision operator, and the nonuniqueness of equilibrium strategies is only up to a finite dimensional subspace of the space of all admissible strategies.In the constrained case, we have established an existence result for games with a much weaker convexity assumption subject to compact convex constraints. We have also derived certain results for games with noncompact constraints. Several examples of quadratic and non-quadratic games are given to illustrate the theorem.Numerical computations are also possible and are given in the sequel [3].

ON THE STABILITY OF THE BEST REPLY MAP FOR NONCOOPERATIVE DIFFERENTIAL GAMES

2012 ◽  
Vol 10 (02) ◽  
pp. 113-132 ◽  
Author(s):  
ALBERTO BRESSAN ◽  
ZIPENG WANG
Keyword(s):  
Present Note ◽  
Nonlinear Perturbations ◽  
Linear Quadratic ◽  
Infinite Time ◽  
Feedback Controls ◽  
Affine Functions ◽  
The Stability ◽  
Discounted Costs ◽  
Noncooperative Differential Games ◽  
Linear Quadratic Games

Consider a differential game for two players in infinite time horizon, with exponentially discounted costs. A pair of feedback controls [Formula: see text] is Nash equilibrium solution if [Formula: see text] is the best strategy for Player 1 in reply to [Formula: see text], and [Formula: see text] is the best strategy for Player 2, in reply to [Formula: see text]. The aim of the present note is to investigate the stability of the best reply map: [Formula: see text]. For linear-quadratic games, we derive a condition which yields asymptotic stability, within the class of feedbacks which are affine functions of the state x ∈ ℝn. An example shows that stability is lost, as soon as nonlinear perturbations are considered.

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