  For a proper, geodesic, Gromov hyperbolic metric space X, a discrete subgroup
of isometries \Gamma whose limit set is uniformly perfect, and a disjoint
collection of horoballs {H_j}, we show that the set of limit points badly
approximable by {H_j} is absolutely winning in the limit set. As an
application, we deduce that for a geometrically finite Kleinian group acting on
H^{n+1}, the limit points badly approximable by parabolics is absolutely
winning, generalizing previous results of Dani and McMullen. As a consequence
of winning, we show that the set of badly approximable limit points has
dimension equal to the critical exponent of the group. Since this set can
alternatively be described as the limit points representing bounded geodesics
in the quotient H^{n+1}/\Gamma, we recapture a result originally due to Bishop
and Jones.
