  In the context of multiplayer games, the parallel repetition problem can be
phrased as follows: given a game $G$ with optimal winning probability
$1-\alpha$ and its repeated version $G^n$ (in which $n$ games are played
together, in parallel), can the players use strategies that are substantially
better than ones in which each game is played independently? This question is
relevant in physics for the study of correlations and plays an important role
in computer science in the context of complexity and cryptography. In this work
the case of multiplayer non-signalling games is considered, i.e., the only
restriction on the players is that they are not allowed to communicate during
the game. For complete-support games (games where all possible combinations of
questions have non-zero probability to be asked) with any number of players we
prove a threshold theorem stating that the probability that non-signalling
players win more than a fraction $1-\alpha+\beta$ of the $n$ games is
exponentially small in $n\beta^2$, for every $0\leq \beta \leq \alpha$. For
games with incomplete support we derive a similar statement, for a slightly
modified form of repetition. The result is proved using a new technique, based
on a recent de Finetti theorem, which allows us to avoid central technical
difficulties that arise in standard proofs of parallel repetition theorems.
