  Strategy Logic (SL) is a logical formalism for strategic reasoning in
multi-agent systems. Its main feature is that it has variables for strategies
that are associated to specific agents with a binding operator. We introduce
Graded Strategy Logic (GradedSL), an extension of SL by graded quantifiers over
tuples of strategy variables, i.e., "there exist at least g different tuples
(x_1,...,x_n) of strategies" where g is a cardinal from the set N union
{aleph_0, aleph_1, 2^aleph_0}. We prove that the model-checking problem of
GradedSL is decidable. We then turn to the complexity of fragments of GradedSL.
When the g's are restricted to finite cardinals, written GradedNSL, the
complexity of model-checking is no harder than for SL, i.e., it is
non-elementary in the quantifier rank. We illustrate our formalism by showing
how to count the number of different strategy profiles that are Nash equilibria
(NE), or subgame-perfect equilibria (SPE). By analyzing the structure of the
specific formulas involved, we conclude that the important problems of checking
for the existence of a unique NE or SPE can both be solved in 2ExpTime, which
is not harder than merely checking for the existence of such equilibria.
