Meager-nowhere dense games (v): Coding strategies again

Players ONE and TWO play the following game of length ω: In the n-th inning ONE first chooses a meager subset of the real line; TWO responds with a nowhere dense set. TWO wins only if the union of TWO'S nowhere dense sets is exactly equal to the union of ONE'S first category sets. We prove that TWO has a winning strategy, even if TWO remembers only the most recent two moves each inning (Corollary 8). We show that in a closely related game, the assertion that TWO has a winning strategy depending on only the most recent two moves each inning is equivalent to a weak version of the Singular Cardinals Hypothesis (Theorem 1).