  We suggest a new algorithm for two-person zero-sum undiscounted stochastic
games focusing on stationary strategies. Given a positive real $\epsilon$, let
us call a stochastic game $\epsilon$-ergodic, if its values from any two
initial positions differ by at most $\epsilon$. The proposed new algorithm
outputs for every $\epsilon>0$ in finite time either a pair of stationary
strategies for the two players guaranteeing that the values from any initial
positions are within an $\epsilon$-range, or identifies two initial positions
$u$ and $v$ and corresponding stationary strategies for the players proving
that the game values starting from $u$ and $v$ are at least $\epsilon/24$
apart. In particular, the above result shows that if a stochastic game is
$\epsilon$-ergodic, then there are stationary strategies for the players
proving $24\epsilon$-ergodicity. This result strengthens and provides a
constructive version of an existential result by Vrieze (1980) claiming that if
a stochastic game is $0$-ergodic, then there are $\epsilon$-optimal stationary
strategies for every $\epsilon > 0$. The suggested algorithm is based on a
potential transformation technique that changes the range of local values at
all positions without changing the normal form of the game.
