Arhangel’skii proved that if a first countable Hausdorff space is Lindelöf, then its cardinality is at most $2^{\aleph ₀}$. Such a clean upper bound for Lindelöf spaces in the larger class of spaces whose points are $G_{\delta }$ has been more elusive. In this paper we continue the agenda started by the second author, [Topology Appl. 63 (1995)], of considering the cardinality problem for spaces satisfying stronger versions of the Lindelöf property. Infinite games and selection principles, especially the Rothberger property,...