An Analytical Approach to Parallel Repetition via CSP Inverse Theorems

TL;DR

This paper establishes new bounds on the decay of game value under parallel repetition for certain multi-player games, using an analytical approach linked to CSP inverse theorems, unifying and extending previous results.

Contribution

It introduces a novel analytical method to prove parallel repetition bounds for multi-player games, including new bounds for 3-player games with specific query distributions and unifies prior proofs.

Findings

Bound on game value decay: inverse iterated logarithm rate.

Parallel repetition theorem for 3-player games with pairwise-connected distributions.

Unified proof framework for multiple classes of parallel repetition results.

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].