  We present a generic strategy iteration algorithm (GSIA) to find an optimal
strategy of a simple stochastic game (SSG). We prove the correctness of GSIA,
and derive a general complexity bound, which implies and improves on the
results of several articles. First, we remove the assumption that the SSG is
stopping, which is usually obtained by a polynomial blowup of the game. Second,
we prove a tight bound on the denominator of the values associated to a
strategy, and use it to prove that all strategy iteration algorithms are in
fact fixed parameter tractable in the number of random vertices. All known
strategy iteration algorithms can be seen as instances of GSIA, which allows to
analyze the complexity of converge from below by Condon and to propose a class
of algorithms generalising Gimbert and Horn's algorithm. These algorithms
require less than $r!$ iterations in general and less iterations than the
current best deterministic algorithm for binary SSGs given by Ibsen-Jensen and
Miltersen.
