Polynomial Bounds On Parallel Repetition For All 3-Player Games With Binary Inputs
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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.
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