Abstract  We investigate the expressive power of Parikh's Game Logic interpreted in Kripke structures, and show that the syntactical alternation hierarchy of this logic is strict. This is done by encoding the winning condition for parity games of rank n. It follows that Game Logic is not captured by any finite level of the modal -calculus alternation hierarchy. Moreover, we can conclude that model checking for the -calculus is efficiently solvable iff this is possible for Game Logic