  We study two-player zero-sum concurrent stochastic games with finite state
and action space played for an infinite number of steps. In every step, the two
players simultaneously and independently choose an action. Given the current
state and the chosen actions, the next state is obtained according to a
stochastic transition function. An objective is a measurable function on plays
(or infinite trajectories) of the game, and the value for an objective is the
maximal expectation that the player can guarantee against the adversarial
player. We consider: (a) stateful-discounted objectives, which are similar to
the classical discounted-sum objectives, but states are associated with
different discount factors rather than a single discount factor; and (b) parity
objectives, which are a canonical representation for $\omega$-regular
objectives. For stateful-discounted objectives, given an ordering of the
discount factors, the limit value is the limit of the value of the
stateful-discounted objectives, as the discount factors approach zero according
to the given order.
  The computational problem we consider is the approximation of the value
within an arbitrary additive error. The above problem is known to be in
EXPSPACE for the limit value of stateful-discounted objectives and in PSPACE
for parity objectives. The best-known algorithms for both the above problems
are at least exponential time, with an exponential dependence on the number of
states and actions. Our main results for the value approximation problem for
the limit value of stateful-discounted objectives and parity objectives are as
follows: (a) we establish TFNP[NP] complexity; and (b) we present algorithms
that improve the dependency on the number of actions in the exponent from
linear to logarithmic. In particular, if the number of states is constant, our
algorithms run in polynomial time.
