  We generalise the $\alpha$-Ramsey cardinals introduced in Holy and Schlicht
(2018) for cardinals $\alpha$ to arbitrary ordinals $\alpha$, and answer
several questions posed in that paper. In particular, we show that
$\alpha$-Ramseys are downwards absolute to the core model $K$ for all $\alpha$
of uncountable cofinality, that strategic $\omega$-Ramsey cardinals are
equiconsistent with remarkable cardinals and that strategic $\alpha$-Ramsey
cardinals are equiconsistent with measurable cardinals for all $\alpha>\omega$.
We also show that the $n$-Ramseys satisfy indescribability properties and use
them to provide a game-theoretic characterisation of completely ineffable
cardinals, as well as establishing further connections between the
$\alpha$-Ramsey cardinals and the Ramsey-like cardinals introduced in Gitman
(2011), Feng (1990) and Sharpe and Welch (2011).
