An Analytical Approach to Parallel Repetition via CSP Inverse Theorems

Abstract: Let $\mathcal{G}$ 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 $n\geq N$, the value of the $n$-fold parallel repetition of $\mathcal{G}$ is $$ \text{val}(\mathcal{G}^{\otimes n}) \leq \frac{1}{\underbrace{\log\log\cdots\log}_{C\text{ times}} n}, $$ where $N=N(\mathcal{G})$ and $1\leq C\leq k^{O(k)}$ 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 $(\mathbb{Z}, +)$ [BKM23c, BBKLM25]. (4) Parallel repetition for all 3-player games with binary inputs [HR20, GHMRZ21, GHMRZ22, GMRZ22].