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.

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 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.

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.

Mixed Policies

2017 ◽  
Author(s):  
João P. Hespanha
Keyword(s):  
Probability Distribution ◽  
Saddle Point ◽  
Security Policy ◽  
General Type ◽  
Zero Sum Games ◽  
Security Levels ◽  
Mixed Action ◽  
Zero Sum ◽  
Saddle Point Equilibrium ◽  
Action Spaces

This chapter explores the concept of mixed policies and how the notions for pure policies can be adapted to this more general type of policies. A pure policy consists of choices of particular actions (perhaps based on some observation), whereas a mixed policy involves choosing a probability distribution to select actions (perhaps as a function of observations). The idea behind mixed policies is that the players select their actions randomly according to a previously selected probability distribution. The chapter first considers the rock-paper-scissors game as an example of mixed policy before discussing mixed action spaces, mixed security policy and saddle-point equilibrium, mixed saddle-point equilibrium vs. average security levels, and general zero-sum games. It concludes with practice exercises with corresponding solutions and an additional exercise.

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

scholarly journals A Dynamic Multi-Objective Duopoly Game with Capital Accumulation and Pollution

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1983
Author(s):  
Bertrand Crettez ◽  
Naila Hayek ◽  
Peter M. Kort
Keyword(s):  
Information Structure ◽  
Capital Accumulation ◽  
Production Capacity ◽  
Open Loop ◽  
Feedback Nash Equilibrium ◽  
Feedback Information ◽  
Time Dynamic ◽  
Clean Environment ◽  
Time Differential ◽  
Dynamic Duopoly

This paper studies a discrete-time dynamic duopoly game with homogenous goods. Both firms have to decide on investment where investment increases production capacity so that they are able to put a larger quantity on the market. The downside, however, is that a larger quantity raises pollution. The firms have multiple objectives in the sense that each one maximizes the discounted profit stream and appreciates a clean environment as well. We obtain some surprising results. First, where it is known from the continuous-time differential game literature that firms invest more under a feedback information structure compared to an open-loop one, we detect scenarios where the opposite holds. Second, in a feedback Nash equilibrium, capital stock is more sensitive to environmental appreciation than in the open-loop case.

Sign in / Sign up

Export Citation Format

Share Document