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.Comment: 28 pages, 5 figures, 1 tabl