  In a recent work, Moshkovitz [FOCS '14] presented a transformation on
two-player games called "fortification", and gave an elementary proof of an
(exponential decay) parallel repetition theorem for fortified two-player
projection games. In this paper, we give an analytic reformulation of
Moshkovitz's fortification framework, which was originally cast in
combinatorial terms. This reformulation allows us to expand the scope of the
fortification method to new settings.
  First, we show any game (not just projection games) can be fortified, and
give a simple proof of parallel repetition for general fortified games. Then,
we prove parallel repetition and fortification theorems for games with players
sharing quantum entanglement, as well as games with more than two players. This
gives a new gap amplification method for general games in the quantum and
multiplayer settings, which has recently received much interest.
  An important component of our work is a variant of the fortification
transformation, called "ordered fortification", that preserves the entangled
value of a game. The original fortification of Moshkovitz does not in general
preserve the entangled value of a game, and this was a barrier to extending the
fortification framework to the quantum setting.
