  Infinite games (in the form of Gale-Stewart games) are studied where a play
is a sequence of natural numbers chosen by two players in alternation, the
winning condition being a subset of the Baire space $\omega^\omega$. We
consider such games defined by a natural kind of parity automata over the
alphabet $\mathbb{N}$, called $\mathbb{N}$-MSO-automata, where transitions are
specified by monadic second-order formulas over the successor structure of the
natural numbers. We show that the classical B\"uchi-Landweber Theorem (for
finite-state games in the Cantor space $2^\omega$) holds again for the present
games: A game defined by a deterministic parity $\mathbb{N}$-MSO-automaton is
determined, the winner can be computed, and an $\mathbb{N}$-MSO-transducer
realizing a winning strategy for the winner can be constructed.
