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
10 - The Vanishing Discount Factor Approach and Uniform Equilibrium in Absorbing Games
Published online by Cambridge University Press: 05 May 2022
- 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 this chapter, we present a technique to study uniform equilibria in stochastic games, called the \emph{vanishing discount factorapproach}.
This approach was developed to prove the existence of a uniform $\ep$-equilibrium in two-player nonzero-sum absorbing games using a function $\lambda \mapsto_\lambda$, which assigns a stationary $\lambda$-discounted equilibrium $x_\lambda$ to every\lambda \in (0,1]$, and analyzing the asymptotic properties of this function as $\lambda$ goes to 0.
We will use this approach to show that every absorbing game in which the probability of absorption is positive whatever the players play has a stationary uniform 0-equilibrium,and that every two-player absorbing game has a uniform $\ep$-equilibrium, which need not be stationary, for every $\ep > 0$.
To prove the second result, we will show how statistical tests are used in the construction of uniform $\ep$-equilibria.
- Type
- Chapter
- Information
- A Course in Stochastic Game Theory , pp. 173 - 194Publisher: Cambridge University PressPrint publication year: 2022