For a given $\kappa > \omega$, define the game $d(\kappa)$ that runs for $\omega$ stages as follows: At stage $n$, player I chooses a sequence of elements of $\omega$, $g_n$ of length $\kappa$, and player II picks a natural number $f(n)$. Define $h_{\alpha} (n) = g_n (\alpha)$.
Player I wins the game if exists $\alpha \in \kappa$ such that $h_\alpha$ dominates $f$.
Let $\mathfrak{sd} = \min \left\lbrace \kappa : \text{player I has a winning strategy in } d(\kappa) \right\rbrace$.
$\mathfrak{sd}$ is uncountable by a diagonal argument and $\leq \mathfrak{d}$ since player I can play a dominating set.
Is $\mathfrak{sd} = \omega_1$?