  It is shown that Matet's characterization of the Ramsey property relative to
a selective co-ideal $\mathcal{H}$, in terms of games of Kastanas, still holds
if we consider semiselectivity instead of selectivity. Moreover, we prove that
a co-ideal $\mathcal{H}$ is semiselective if and only if Matet's game-theoretic
characterization of the $\mathcal{H}$-Ramsey property holds. This lifts
Kastanas's characterization of the classical Ramsey property to its optimal
setting, from the point of view of the local Ramsey theory and gives a
game-theoretic counterpart to a theorem of Farah \cite{far}, asserting that a
co-ideal $\mathcal{H}$ is semiselective if and only if the family of
$\mathcal{H}$-Ramsey subsets of $\N^{[\infty]}$ coincides with the family of
those sets having the abstract $\mathcal{H}$-Baire property. Finally, we show
that under suitable assumptions, for every semiselective co-ideal $\mathcal H$
all sets of real numbers are $\mathcal H$-Ramsey.
