  Several variants of nonlocal games have been considered in the study of
quantum entanglement and nonlocality. This paper concerns two of these
variants, called quantum-classical games and extended nonlocal games. We give a
construction of an extended nonlocal game from any quantum-classical game that
allows one to translate certain facts concerning quantum- classical games to
extended nonlocal games. In particular, based on work of Regev and Vidick, we
conclude that there exist extended nonlocal games for which no
finite-dimensional entangled strategy can be optimal. While this conclusion is
a direct consequence of recent work of Slofstra, who proved a stronger,
analogous result for ordinary (non-extended) nonlocal games, the proof based on
our construction is considerably simpler, and the construction itself might
potentially have other applications in the study of entanglement and
nonlocality.
