  The Hausdorff $\delta$-dimension game was introduced by Das, Fishman, Simmons
and {Urba{\'n}ski} and shown to characterize sets in $\mathbb{R}^d$ having
Hausdorff dimension $\leq \delta$. We introduce a variation of this game which
also characterizes Hausdorff dimension and for which we are able to prove an
unfolding result similar to the basic unfolding property for the Banach-Mazur
game for category. We use this to derive a number of consequences for Hausdorff
dimension. We show that under $\mathsf{AD}$ any wellordered union of sets each
of which has Hausdorff dimension $\leq \delta$ has dimension $\leq \delta$. We
establish a continuous uniformization result for Hausdorff dimension. The
unfolded game also provides a new proof that every $\boldsymbol{\Sigma}^1_1$
set of Hausdorff dimension $\geq \delta$ contains a compact subset of dimension
$\geq \delta'$ for any $\delta'<\delta$, and this result generalizes to
arbitrary sets under $\mathsf{AD}$.
