Improved Parallel Repetition for GHZ-Supported Games via Spreadness

2026-02-10

AI Summary

This work addresses three-player nonlocal games whose support matches that of the GHZ game, for which only polynomial decay bounds were previously known for the value under parallel repetition. The paper establishes, for the first time, exponential decay of the success probability after $n$ parallel repetitions, providing an upper bound of $\exp(-n^c)$ for some constant $c > 0$, along with concentration bounds quantifying deviations from the baseline success probability of $3/4 + \varepsilon$. The core technique leverages the algebraic regularity theory developed by Kelley and Meka, integrated with probabilistic and combinatorial analysis to achieve a refined characterization of the parallel repetition structure. This result significantly improves upon existing polynomial bounds and delivers the first exponential decay guarantee for parallel repetition in multi-player nonlocal games.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Multiagent Systems: Mechanism Design

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• parallel repetition
• GHZ game
• exponential decay
• concentration bound
• nonlocal games

Innovation

• algebraic spreadness
• parallel repetition
• GHZ game
• exponential decay
• concentration bound