In this chapter, we present linear complementarity problems, and use them to provide sufficient conditions that guarantee the existence of an undiscounted $\ep$-equilibrium in quitting games.

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.

In this chapter, we proveKakutani's Fixed Point Theorem, which is an extension of Brouwer's Fixed Point Theoremto correspondences (set-valued mappings). We then define the concept of $\lambda$-discounted equilibrium, and using Kakutani's Fixed Point Theorem we prove that every multiplayer stochastic game admits a stationary $\lambda$-discounted equilibrium, for every discount factor $\lambda \in (0,1]$.