  In this paper we connect selection principles on a topological space to
corresponding selection principles on one of its hyperspaces. We unify
techniques and generalize theorems from the known results about selection
principles for common hyperspace constructions. This includes results of
Lj.D.R. Ko\v{c}inac, Z. Li, and others. We use selection games to generalize
selection principles and we work with strategies of various strengths for these
games. The selection games we work with are primarily abstract versions of the
selection principles of Rothberger, Menger, and Hurewicz type, as well as games
of countable fan tightness and selective separability. The hyperspace
constructions that we work with are the Vietoris and Fell topologies, both
upper and full, generated by ideals of closed sets. Using a new technique we
are able to extend straightforward connections between topological constructs
to connections between selection games related to those constructs. This
extension process works regardless of the length of the game, the kind of
selection being performed, or the strength of the strategy being considered.
