  In the domination game, introduced by Bre\v{s}ar, Klav\v{z}ar and Rall in
2010, Dominator and Staller alternately select a vertex of a graph $G$. A move
is legal if the selected vertex $v$ dominates at least one new vertex -- that
is, if we have a $u\in N[v]$ for which no vertex from $N[u]$ was chosen up to
this point of the game. The game ends when no more legal moves can be made, and
its length equals the number of vertices selected. The goal of Dominator is to
minimize whilst that of Staller is to maximize the length of the game. The game
domination number $\gamma_g(G)$ of $G$ is the length of the domination game in
which Dominator starts and both players play optimally. In this paper we
establish an upper bound on $\gamma_g(G)$ in terms of the minimum degree
$\delta$ and the order $n$ of $G$. Our main result states that for every
$\delta \ge 4$,
  $$\gamma_g(G)\le
\frac{30\delta^4-56\delta^3-258\delta^2+708\delta-432}{90\delta^4-390\delta^3+348\delta^2+348\delta-432}\;
n.$$ Particularly, $\gamma_g(G) < 0.5139\; n$ holds for every graph of minimum
degree 4, and $\gamma_g(G)< 0.4803\; n$ if the minimum degree is greater than
4. Additionally, we prove that $\gamma_g(G) < 0.5574\; n$ if $\delta=3$.
