  We prove that for every 3-player game with binary questions and answers and
value $<1$, the value of the $n$-fold parallel repetition of the game decays
polynomially fast to 0. That is, for every such game, there exists a constant
$c>0$, such that the value of the $n$-fold parallel repetition of the game is
at most $n^{-c}$. Along the way to proving this theorem, we prove two
additional parallel repetition theorems for multiplayer games, that may be of
independent interest:
  Playerwise Connected Games (with any number of players and any Alphabet
size): We identify a large class of multiplayer games and prove that for every
game with value $<1$ in that class, the value of the $n$-fold parallel
repetition of the game decays polynomially fast to 0. More precisely, our
result applies for playerwise connected games, with any number of players and
any alphabet size. The class of playerwise connected games is strictly larger
than the class of connected games that was defined in [DHVY17] and for which
exponentially fast decay bounds are known [DHVY17]. For playerwise connected
games that are not connected, only inverse Ackermann decay bounds were
previously known [Ver96].
  Exponential Bounds for the Anti-Correlation Game: In the 3-player
anti-correlation game, two out of three players are given $1$ as input, and the
remaining player is given $0$. The two players who were given $1$ must produce
different outputs in $\{0,1\}$. We prove that the value of the $n$-fold
parallel repetition of that game decays exponentially fast to 0. Only inverse
Ackermann decay bounds were previously known [Ver96]. This game was studied and
motivated in several previous works. In particular, Holmgren and Yang gave it
as an example for a 3-player game whose non-signaling value (is smaller than 1
and yet) does not decrease at all under parallel repetition [HY19].
