10 - The Vanishing Discount Factor Approach and Uniform Equilibrium in Absorbing Games

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.