Arhangel'skii proved that if a first countable Hausdorff space is Lindelöf, then its cardinality is at most 2 א‎0. Such a clean upper bound for Lindelöf spaces in the larger class of spaces whose points are Gδ 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, are essential tools in our investigations.