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].
翻译:我们证明,对于每个有二进制问答和答案的三玩游戏,并且价值为 < 1美元, 游戏的双倍重复值为$19, 游戏的双倍重复值为1 。 也就是说, 对于每一个游戏, 都存在一个固定的 $> 0美元, 这样游戏的双倍重复值最多为 $%- c}。 在证明这个游戏的标语时, 我们证明, 多玩游戏的双倍重复性理论值为2个, 可能具有独立的兴趣: 玩者连接游戏的更小的游戏值( 任何玩家的数量和任何字母大小 ): 我们确定了一个很大的多玩游戏类别, 多玩者游戏的多倍重复值为 。 对于每个游戏的值为 < 1, $1 美元 美元, 游戏的双倍重复性重复值为 0。 更确切地说, 我们的结果适用于玩家连接过的游戏, 任何玩家和任何字母大小。