  We investigate the value of parallel repetition of one-round games with any
number of players $k\ge 2$. It has been an open question whether an analogue of
Raz's Parallel Repetition Theorem holds for games with more than two players,
i.e., whether the value of the repeated game decays exponentially with the
number of repetitions. Verbitsky has shown, via a reduction to the density
Hales-Jewett theorem, that the value of the repeated game must approach zero,
as the number of repetitions increases. However, the rate of decay obtained in
this way is extremely slow, and it is an open question whether the true rate is
exponential as is the case for all two-player games.
  Exponential decay bounds are known for several special cases of multi-player
games, e.g., free games and anchored games. In this work, we identify a certain
expansion property of the base game and show all games with this property
satisfy an exponential decay parallel repetition bound. Free games and anchored
games satisfy this expansion property, and thus our parallel repetition theorem
reproduces all earlier exponential-decay bounds for multiplayer games. More
generally, our parallel repetition bound applies to all multiplayer games that
are connected in a certain sense.
  We also describe a very simple game, called the GHZ game, that does not
satisfy this connectivity property, and for which we do not know an exponential
decay bound. We suspect that progress on bounding the value of this the
parallel repetition of the GHZ game will lead to further progress on the
general question.
