Book contents
- Frontmatter
- Dedication
- Contents
- Introduction
- 1 Markov Decision Problems
- 2 A Tauberian Theorem and Uniform ∈-Optimality in Hidden Markov Decision Problems
- 3 Strategic-Form Games: A Review
- 4 Stochastic Games: The Model
- 5 Two-Player Zero-Sum Discounted Games
- 6 Semi-Algebraic Sets and the Limit of the Discounted Value
- 7 B-Graphs and the Continuity of the Limit limλ→0 ʋλ(s;q,r)
- 8 Kakutani’s Fixed-Point Theorem and Multiplayer Discounted Stochastic Games
- 9 Uniform Equilibrium
- 10 The Vanishing Discount Factor Approach and Uniform Equilibrium in Absorbing Games
- 11 Ramsey’s Theorem and Two-Player Deterministic Stopping Games
- 12 Infinite Orbits and Quitting Games
- 13 Linear Complementarity Problems and Quitting Games
- References
- Index
7 - B-Graphs and the Continuity of the Limit limλ→0 ʋλ(s;q,r)
Published online by Cambridge University Press: 05 May 2022
Book contents
- Frontmatter
- Dedication
- Contents
- Introduction
- 1 Markov Decision Problems
- 2 A Tauberian Theorem and Uniform ∈-Optimality in Hidden Markov Decision Problems
- 3 Strategic-Form Games: A Review
- 4 Stochastic Games: The Model
- 5 Two-Player Zero-Sum Discounted Games
- 6 Semi-Algebraic Sets and the Limit of the Discounted Value
- 7 B-Graphs and the Continuity of the Limit limλ→0 ʋλ(s;q,r)
- 8 Kakutani’s Fixed-Point Theorem and Multiplayer Discounted Stochastic Games
- 9 Uniform Equilibrium
- 10 The Vanishing Discount Factor Approach and Uniform Equilibrium in Absorbing Games
- 11 Ramsey’s Theorem and Two-Player Deterministic Stopping Games
- 12 Infinite Orbits and Quitting Games
- 13 Linear Complementarity Problems and Quitting Games
- References
- Index
Summary
In Section~\ref{continuity} we proved that the discounted value is continuous in the parameters of the game, see Theorem~\ref{theorem7}.
One weakness of this result is that it does not bound the Lipschitz constant of the value function $(\lambda,q,r) \mapsto v_\lambda(s;q,r)$.
In this chapter, we will strengthen Theorem~\ref{theorem7}, and, using the concept of $B$-graphs, develop a bound on the Lipschitz constant of the value function.
Our technique will allow us to study the continuityof the limit $\lim_{\lambda \to 0} v_\lambda(s;q,r)$ as a function of $q$ and $r$.
- Type
- Chapter
- Information
- A Course in Stochastic Game Theory , pp. 94 - 110Publisher: Cambridge University PressPrint publication year: 2022