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.

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