Multiplayer parallel repetition without dependency-breaking and anchoring variables: monotonic, concave amplification

TL;DR

This paper provides quantitative estimates on how the optimal value of multiplayer games decreases under parallel repetition, using monotonic concave functions instead of dependency-breaking variables, extending previous two-player results.

Contribution

It introduces a new approach employing monotonic concave functions for multiplayer parallel repetition, generalizing prior two-player methods and addressing open questions in the field.

Findings

Quantitative decay rates for multiplayer game values under parallel repetition.

Extension of two-player monotonic concave function techniques to multiplayer settings.

Analysis of amplification functions with complex combinatorial structures.

Abstract

We obtain quantitative estimates on the decay of the multiplayer optimal value under parallel repetition. In comparison to a previous work of the author in 2025 (arXiv: 2508.09380) which sought to generalize dependency-breaking and anchoring variables from two-player Quantum games, being able to establish quantitative estimates on the decay of the optimal value of a multiplayer game under parallel repetition is of interest to establish under different assumptions. Specifically, independently of the dependency-breaking and anchoring variables that have previously been employed to remove correlations from entangled information shared between Alice and Bob (hence removing dependencies), monotonic concave functions can be used in place of such variables to obtain rates of decay on the optimal value. The game-theoretic setting with two players was first analyzed with monotonic concave functions by Lanzenberger and Maurer. For $q_i , x_i > 0$ $\forall 1 \leq i \leq N$ where $N > 0 $ is the total number of players we adddress an open question raised in their work regarding potential generalizations of two-player monotonic concave functions, through amplification functions of the form $\Psi_{\textit{Mult}} \equiv \Psi = N - \underset{1 \leq i \leq N}{\prod} \mathrm{exp} \big[ - q_i x_i \big]$, which in the multiplayer game-theoretic setting have more intricate combinatorial structures.