  Given a target set $A\subseteq \mathbb{R}^d$ and a real number $\beta\in
(0,1)$, McMullen introduced the notion of $A$ being an absolutely
$\beta$-winning set. This involves a two player game which we call the
$\beta$-McMullen game. We consider the version of this game in which the
parameter $\beta$ is removed, which we call the no-$\beta$ McMullen game. More
generally, we consider the game with respect to arbitrary norms on
$\mathbb{R}^d$, and even more generally with respect to general convex sets. We
show that for strictly convex sets in $\mathbb{R}^d$, polytopes in
$\mathbb{R}^d$, and general convex sets in $\mathbb{R}^2$, that player
$\boldsymbol{I}$ wins the no-$\beta$ McMullen game iff $A$ contains a perfect
set and player $\boldsymbol{I}\kern-0.05cm\boldsymbol{I}$ wins iff $A$ is
countable. So, the no-$\beta$ McMullen game is equivalent to the perfect set
game for $A$ in these cases. The proofs of these results use a connection
between the geometry of the game and techniques from logic. Because of the
geometry of this game, this result has strong implications for the geometry of
uncountable sets in $\mathbb{R}^d$. We also present an example of a compact,
convex set in $\mathbb{R}^3$ to which our methods do not apply, and also an
example due to D.\ Simmons of a closed, convex set in $\ell_2(\mathbb{R})$
which illustrate the obstacles in extending the results further.
