Infinite games are studied in a format where two players, called Player 1 and Player 2, generate a play by building up an ω-word as they choose letters in turn. A game is specified by the ω-language which contains the plays won by Player 2. We analyze ω-languages generated from certain classes K of regular languages of finite words (called * -languages), using natural transformations of * -languages into ω-languages. Winning strategies for infinite games can be represented again in terms of * -languages. Continuing work of Selivanov (2007) and Rabinovich et al. (2007), we analyze how these "strategy * -languages" are related to the original language class K. In contrast to that work, we exhibit classes K where strategy representations strictly exceed K.

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