An Analytical Approach to Parallel Repetition via CSP Inverse Theorems
Amey Bhangale, Mark Braverman, Subhash Khot, Yang P. Liu, Dor Minzer, Kunal Mittal
TL;DR
The paper proves a general parallel repetition theorem for k-player games under structured question-distributions, showing that if the distribution is pairwise-connected and has no-marginal-Abelian-embeddings, then val( G^{⊗n} ) decays faster than any fixed power of a polylog, specifically val( G^{⊗n} ) ≤ 1 / log log ... log n with a constant number of logarithms. The authors develop an analytic framework based on CSP inverse theorems (notably BKLM24b), product pseudorandomness, and generalized random restrictions to convert complex k-wise correlations into effectively independent behavior across coordinates. The main contributions include a unified route to derive known parallel repetition results as special cases, and new bounds for classes such as 3-player pairwise-connected games and binary-input 3-player games, under the stated distributional assumptions. The approach has potential implications for hardness amplification, PCP constructions, and high-dimensional combinatorics, linking parallel repetition to inverse theorems and random-restriction techniques. Overall, the work provides a versatile, largely analytic toolkit for understanding how structured question distributions control amplification in multiplayer games.
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].