  The second author introduced with I. T\"orm\"a a two-player word-building
game [Playing with Subshifts, Fund. Inform. 132 (2014), 131--152]. The game has
a predetermined (possibly finite) choice sequence $\alpha_1$, $\alpha_2$,
$\ldots$ of integers such that on round $n$ the player $A$ chooses a subset
$S_n$ of size $\alpha_n$ of some fixed finite alphabet and the player $B$ picks
a letter from the set $S_n$. The outcome is determined by whether the word
obtained by concatenating the letters $B$ picked lies in a prescribed target
set $X$ (a win for player $A$) or not (a win for player $B$). Typically, we
consider $X$ to be a subshift. The winning shift $W(X)$ of a subshift $X$ is
defined as the set of choice sequences for which $A$ has a winning strategy
when the target set is the language of $X$. The winning shift $W(X)$ mirrors
some properties of $X$. For instance, $W(X)$ and $X$ have the same entropy.
Virtually nothing is known about the structure of the winning shifts of
subshifts common in combinatorics on words. In this paper, we study the winning
shifts of subshifts generated by marked uniform substitutions, and show that
these winning shifts, viewed as subshifts, also have a substitutive structure.
Particularly, we give an explicit description of the winning shift for the
generalized Thue-Morse substitutions. It is known that $W(X)$ and $X$ have the
same factor complexity. As an example application, we exploit this connection
to give a simple derivation of the first difference and factor complexity
functions of subshifts generated by marked substitutions. We describe these
functions in particular detail for the generalized Thue-Morse substitutions.
