  In set theory without the axiom of regularity, we consider a game in which
two players choose in turn an element of a given set, an element of this
element, etc.; a player wins if its adversary cannot make any next move. Sets
that are winning, i.e. have a winning strategy for a player, form a natural
hierarchy with levels indexed by ordinals. We show that the class of
hereditarily winning sets is an inner model containing all well-founded sets,
and that all four possible relationships between the universe, the class of
hereditarily winning sets, and the class of well-founded sets are consistent.
We describe classes of ordinals for which it is consistent that winning sets
without minimal elements are exactly in the levels indexed by ordinals of this
class. For consistency results, we propose a new method for getting
non-well-founded models. Finally, we establish a probability result by showing
that on hereditarily finite well-founded sets the first player wins almost
always.
