Stochastic Limit-Average Games are in EXPTIME
  The value of a finite-state two-player zero-sum stochastic game with limit-average payoff can be approximated to within $\epsilon$ in time exponential in a polynomial in the size of the game times polynomial in logarithmic in $\frac{1}{\epsilon}$, for all $\epsilon>0$. 
Krishnendu Chatterjee, Rupak Majumdar and Thomas A. Henzinger