**MSC:**- 90D15 Stochastic games, See also {93E05}

We give an alternative proof that

every two-person

non-zero-sum absorbing positive recursive

stochastic game with finitely many states

has approximate equilibria, a result proven

by Nicolas Vieille.

Our proof uses

a state specific discount factor which is similar to the

conventional discount factor only when there is only one non-absorbing

state. Additionally we show

that if the players engage in time homogeneous Markovian

behavior relative to some finite state space of size $n$ then

for the existence of an $\ep$-equilibrium it

suffices that one-stage deviation brings no more than

an $\ep^3/(nM)$ gain to a player,

where $M$ is a bound on the maximal difference between

any two payoffs.