  In this work we show how a vector-valued version of Schechtman's empirical
method can be used to reduce the number of inputs in a nonlocal game $G$ while
preserving the quotient $\beta^*(G)/\beta(G)$ of the quantum over the classical
bias. We apply our method to the Khot-Vishnoi game, with exponentially many
questions per player, to produce another game with polynomially many ($N\approx
n^8$) questions so that the quantum over the classical bias is $\Omega
(n/\log^2 n)$.
