One-Player Differential Games

2017 ◽  
Author(s):  
João P. Hespanha
Keyword(s):  
Dynamical System ◽  
Differential Games ◽  
Continuous Time ◽  
Information Structure ◽  
Dynamic Games ◽  
State Feedback ◽  
Open Loop ◽  
Linear Quadratic ◽  
Time Dynamic ◽  
Termination Time

This chapter focuses on one-player continuous time dynamic games, that is, the optimal control of a continuous time dynamical system. It begins by considering a one-player continuous time differential game in which the (only) player wants to minimize either using an open-loop policy or a state-feedback policy. It then discusses continuous time cost-to-go, with the following conclusion: regardless of the information structure considered (open loop, state feedback, or other), it is not possible to obtain a cost lower than cost-to-go. It also explores continuous time dynamic programming, linear quadratic dynamic games, and differential games with variable termination time before concluding with a practice exercise and the corresponding solution.

State-Feedback Zero-Sum Dynamic Games

2017 ◽  
Author(s):  
João P. Hespanha
Keyword(s):  
Discrete Time ◽  
Information Structure ◽  
Dynamic Games ◽  
State Feedback ◽  
Linear Quadratic ◽  
Feedback Information ◽  
Time Dynamic ◽  
Finite State ◽  
Solution Methods ◽  
Zero Sum

This chapter focuses on the computation of the saddle-point equilibrium of a zero-sum discrete time dynamic game in a state-feedback policy. It begins by considering solution methods for two-player zero sum dynamic games in discrete time, assuming a finite horizon stage-additive cost that Player 1 wants to minimize and Player 2 wants to maximize, and taking into account a state feedback information structure. The discussion then turns to discrete time dynamic programming, the use of MATLAB to solve zero-sum games with finite state spaces and finite action spaces, and discrete time linear quadratic dynamic games. The chapter concludes with a practice exercise that requires computing the cost-to-go for each state of the tic-tac-toe game, and the corresponding solution.

One-Player Dynamic Games

2017 ◽  
Author(s):  
João P. Hespanha
Keyword(s):  
Dynamic Programming ◽  
Discrete Time ◽  
Information Structure ◽  
Dynamic Games ◽  
Open Loop ◽  
Linear Quadratic ◽  
Time Dynamic ◽  
Discrete Time Dynamical System ◽  
Solution Methods ◽  
The Cost

This chapter focuses on one-player discrete time dynamic games, that is, the optimal control of a discrete time dynamical system. It first considers solution methods for one-player dynamic games, which are simple optimizations, before discussing discrete time cost-to-go. It shows that, regardless of the information structure (open loop, state feedback or other), it is not possible to obtain a cost lower than the cost-to-go. A computationally efficient recursive technique that can be used to compute the cost-to-go is dynamic programming. After providing an overview of discrete time dynamic programming, the chapter explores the complexity of computing the cost-to-go at all stages, the use of MATLAB to solve finite one-player games, and linear quadratic dynamic games. It concludes with a practice exercise and the corresponding solution, along with an additional exercise.

State-Feedback Zero-Sum Differential Games

2017 ◽  
Author(s):  
João P. Hespanha
Keyword(s):  
Dynamic Programming ◽  
Saddle Point ◽  
Continuous Time ◽  
Information Structure ◽  
Dynamic Games ◽  
State Feedback ◽  
Feedback Information ◽  
Time Dynamic ◽  
Zero Sum ◽  
Saddle Point Equilibrium

This chapter focuses on the computation of the saddle-point equilibrium of a zero-sum continuous time dynamic game in a state-feedback policy. It begins by considering the solution for two-player zero sum dynamic games in continuous time, assuming a finite horizon integral cost that Player 1 wants to minimize and Player 2 wants to maximize, and taking into account a state feedback information structure. Continuous time dynamic programming can also be used to construct saddle-point equilibria in state-feedback policies. The discussion then turns to continuous time linear quadratic dynamic games and the use of dynamic programming to construct a saddle-point equilibrium in a state-feedback policy for a two-player zero sum differential game with variable termination time. The chapter also describes pursuit-evasion games before concluding with a practice exercise and the corresponding solution.

Dynamic Games

2017 ◽  
Author(s):  
João P. Hespanha
Keyword(s):  
Differential Games ◽  
Continuous Time ◽  
Dynamic Games ◽  
Information Structures ◽  
Extensive Form ◽  
New Class ◽  
Multi Stage ◽  
Stage Game ◽  
Termination Time ◽  
Action Spaces

This chapter discusses a new class of games known as dynamic games. It begins by considering a two-player multi-stage game in extensive form in which the overall tree structure can be mathematically described in a manner that actually allows for games that are more general than those typically described in extensive form. These include games described by graphs that are not trees, games with infinitely many stages, and games with action spaces that are not finite sets. The equation for solving dynamic games is often called the dynamics of the game. The chapter also examines the information structures of dynamic games and concludes with an overview of continuous time differential games and differential games with variable termination time.

Continuous-time dynamic games for the Cournot adjustment process for competing oligopolists

2013 ◽  
Vol 219 (12) ◽  
pp. 6400-6409 ◽  
Author(s):  
Brooke C. Snyder ◽  
Robert A. Van Gorder ◽  
K. Vajravelu
Keyword(s):  
Continuous Time ◽  
Dynamic Games ◽  
Adjustment Process ◽  
Time Dynamic

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].

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