Let G be a k-player game with value <1, whose query distribution is such that no marginal on k-1 players admits a non-trivial Abelian embedding. We show that for every n N, the value of the n-fold parallel repetition of G is val (G^ n) 1₂ ₓ₈₌₄ₒ n, where N=N (G) and 1 C k^O (k) are constants. As a consequence, we obtain a parallel repetition theorem for all 3-player games whose query distribution is pairwise-connected. Prior to our work, only inverse Ackermann decay bounds were known for such games Ver96. As additional special cases, we obtain a unified proof for all known parallel repetition theorems, albeit with weaker bounds: (1) A new analytic proof of parallel repetition for all 2-player games Raz98, Hol09, DS14. (2) A new proof of parallel repetition for all k-player playerwise connected games DHVY17, GHMRZ22. (3) Parallel repetition for all 3-player games (in particular 3-XOR games) whose query distribution has no non-trivial Abelian embedding into (Z, +) BKM23c, BBKLM25. (4) Parallel repetition for all 3-player games with binary inputs HR20, GHMRZ21, GHMRZ22, GMRZ22.
Bhangale et al. (Wed,) studied this question.