Improved Parallel Repetition for GHZ-Supported Games via Spreadness

TL;DR

This paper proves exponential decay bounds for the value of parallel repetitions of certain 3-player games supported on GHZ distributions, improving previous polynomial bounds and utilizing algebraic spreadness techniques.

Contribution

It introduces algebraic spreadness as a key tool to establish exponential decay in parallel repetition for GHZ-supported games, a significant advancement over prior polynomial bounds.

Findings

Exponential decay of game value in parallel repetition for GHZ-supported games.

A concentration bound showing low probability of high success rates in repeated games.

Improvement over previous polynomial decay bounds for these games.

Abstract

We prove that for any 3-player game $\mathcal G$, whose query distribution has the same support as the GHZ game (i.e., all $x,y,z\in \{0,1\}$ satisfying $x+y+z=0\pmod{2}$), the value of the $n$-fold parallel repetition of $\mathcal G$ decays exponentially fast: \[ \text{val}(\mathcal G^{\otimes n}) \leq \exp(-n^c)\] for all sufficiently large $n$, where $c>0$ is an absolute constant. We also prove a concentration bound for the parallel repetition of the GHZ game: For any constant $\epsilon>0$, the probability that the players win at least a $\left(\frac{3}{4}+\epsilon\right)$ fraction of the $n$ coordinates is at most $\exp(-n^c)$, where $c=c(\epsilon)>0$ is a constant. In both settings, our work exponentially improves upon the previous best known bounds which were only polynomially small, i.e., of the order $n^{-\Omega(1)}$. Our key technical tool is the notion of \emph{algebraic spreadness} adapted from the breakthrough work of Kelley and Meka (FOCS '23) on sets free of 3-term progressions.