The following technical report is available from http://aib.informatik.rwth-aachen.de: Strategy Machines and their Complexity Marcus Gelderie AIB 2012-04 We introduce a machine model for the execution of strategies in (regular) infinite games that refines the standard model of Mealy automata. This model of controllers is formalized in the terminological framework of Turing machines. We show how polynomially sized controllers can be found for Muller and Streett games. We are able to distinguish aspects of executing strategies ("size", "latency", "space consumption") that are not visible in Mealy automata. We show upper and lower bounds for these parameters for several classes of omega-regular games.