An Analytical Approach to Parallel Repetition via CSP Inverse Theorems

2025-11-05

AI Summary

This work studies value decay in $k$-player non-cooperative games under $n$-fold parallel repetition, focusing on query distributions that admit no nontrivial Abelian embedding. We develop a unified analytical framework integrating the CSP inverse theorem, information theory, and group representation theory. Our main contribution is the first $loglog n$-scale upper bound—i.e., value $leq 1/loglog n$—on the repeated-game value for all three-player games whose query distributions are pairwise connected. This significantly improves upon the previously known inverse-Ackermann-type decay. The bound takes the form $O(1/log^{(C)} n)$, where $C leq k^{O(k)}$, subsuming and unifying multiple existing parallel repetition theorems. Crucially, our approach characterizes the structural properties of multi-player games via combinatorial and probabilistic methods—rather than relying on algebraic or geometric assumptions—thereby establishing a new paradigm for analyzing parallel repetition in high-dimensional, multi-player settings.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

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 $ngeq N$, the value of the $n$-fold parallel repetition of $mathcal{G}$ is $$ ext{val}(mathcal{G}^{otimes n}) leq frac{1}{underbrace{loglogcdotslog}_{C ext{ times}} n}, $$ where $N=N(mathcal{G})$ and $1leq Cleq k^{O(k)}$ 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].

Problem

• Analyzing parallel repetition of multiplayer games with value decay bounds
• Establishing parallel repetition theorems for specific game classes and query distributions
• Providing unified proofs for known parallel repetition results with weaker bounds

Innovation

• Using CSP inverse theorems for parallel repetition
• Analyzing games with no Abelian embeddings
• Establishing decay bounds for multi-player games