  It is conjectured that the game domination number is at most $3n/5$ for every
$n$-vertex graph which does not contain isolated vertices. It was proved in the
recent years that the conjecture holds for several graph classes, including the
class of forests and that of graphs with minimum degree at least two. Here we
prove that the slightly bigger upper bound $5n/8$ is valid for every
isolate-free graph.
