Polynomial Bounds on Parallel Repetition for All 3-Player Games with Binary Inputs
We prove that for every 3-player (3-prover) game G with value less than one, whose query distribution has the support S = {(1,0,0), (0,1,0), (0,0,1)} of Hamming weight one vectors, the value of the n-fold parallel repetition G^{⊗n} decays polynomially fast to zero; that is, there is a constant c = c(G) > 0 such that the value of the game G^{⊗n} is at most n^{-c}.
Following the recent work of Girish, Holmgren, Mittal, Raz and Zhan (STOC 2022), our result is the missing piece that implies a similar bound for a much more general class of multiplayer games: For every 3-player game G over binary questions and arbitrary answer lengths, with value less than 1, there is a constant c = c(G) > 0 such that the value of the game G^{⊗n} is at most n^{-c}.
Our proof technique is new and requires many new ideas. For example, we make use of the Level-k inequalities from Boolean Fourier Analysis, which, to the best of our knowledge, have not been explored in this context prior to our work.
Parallel repetition
Multi-prover games
Fourier analysis
Theory of computation~Computational complexity and cryptography
6:1-6:17
RANDOM
We thank Justin Holmgren for important conversations and collaboration in early stages of this work.
Uma
Girish
Uma Girish
Princeton University, NJ, USA
Research supported by the Simons Collaboration on Algorithms and Geometry, by a Simons Investigator Award, by the National Science Foundation grants No. CCF-1714779, CCF-2007462 and by the IBM Phd Fellowship.
Kunal
Mittal
Kunal Mittal
Princeton University, NJ, USA
Research supported by the Simons Collaboration on Algorithms and Geometry, by a Simons Investigator Award and by the National Science Foundation grants No. CCF-1714779, CCF-2007462.
Ran
Raz
Ran Raz
Princeton University, NJ, USA
Research supported by the Simons Collaboration on Algorithms and Geometry, by a Simons Investigator Award and by the National Science Foundation grants No. CCF-1714779, CCF-2007462.
Wei
Zhan
Wei Zhan
Princeton University, NJ, USA
Research supported by the Simons Collaboration on Algorithms and Geometry, by a Simons Investigator Award and by the National Science Foundation grants No. CCF-1714779, CCF-2007462.
10.4230/LIPIcs.APPROX/RANDOM.2022.6
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