  We introduce a new, game-theoretic approach to anti-classification results
for orbit equivalence relations. Within this framework, we give a short
conceptual proof of Hjorth's turbulence theorem. We also introduce a new
dynamical criterion providing an obstruction to classification by orbits of CLI
groups. We apply this criterion to the relation of equality of countable sets
of reals, and the relations of unitary conjugacy of unitary and selfadjoint
operators on the separable infinite-dimensional Hilbert space.
