  We introduce a notion of strategies based on averaging for nonlocal games in
quantum information theory. These so-called statistical strategies come in a
commuting type and a more specific spatial type, which are respectively special
cases of the quantum commuting and quantum spatial strategies commonly
considered in the field. We prove a theorem that the sets of statistical
commuting strategies and statistical spatial strategies are respectively equal
to the sets of quantum commuting strategies and quantum spatial strategies for
any nonlocal game. Thus we are able to use the recent negative solution of
Tsirelson's problem to obtain a statistical analog showing that there exists a
nonlocal game where the set of statistical commuting strategies properly
contains the closure of the set of statistical spatial strategies. The proof of
this theorem involves development of statistical replicas for numerous
constructions in quantum information theory, in particular for the Fourier-type
duality between observation structures and dynamical structures. The main point
of the argument is to apply the established theory of approximating unitary
representations of countable discrete groups by ergodic measure preserving
actions of such groups. We note that the relevant groups are nonamenable. We
also give an explicit description of a statistical strategy to win the CHSH
game from Aspect's experiment with a probability exceeding the maximum possible
value for a classical strategy.
