An Analytical Approach to Parallel Repetition via CSP Inverse Theorems

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) ≤ 1loglog⋯logC times n, where N=N(G) and 1≤ C≤ kO(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].