  To each one-dimensional subshift $X$, we may associate a winning shift $W(X)$
which arises from a combinatorial game played on the language of $X$.
Previously it has been studied what properties of $X$ does $W(X)$ inherit. For
example, $X$ and $W(X)$ have the same factor complexity and if $X$ is a sofic
subshift, then $W(X)$ is also sofic. In this paper, we develop a notion of
automaticity for $W(X)$, that is, we propose what it means that a vector
representation of $W(X)$ is accepted by a finite automaton.
  Let $S$ be an abstract numeration system such that addition with respect to
$S$ is a rational relation. Let $X$ be a subshift generated by an $S$-automatic
word. We prove that as long as there is a bound on the number of nonzero
symbols in configurations of $W(X)$ (which follows from $X$ having sublinear
factor complexity), then $W(X)$ is accepted by a finite automaton, which can be
effectively constructed from the description of $X$. We provide an explicit
automaton when $X$ is generated by certain automatic words such as the
Thue-Morse word.
