  Brassard et al. [Phys. Rev. Lett. 96, 250401 (2006)] showed that shared
nonlocal boxes with the CHSH probability greater than $\frac{3+\sqrt{6}}6$
yields trivial communication complexity. There still exists the gap with the
maximum CHSH probability $\frac{2+\sqrt{2}}4$ achievable by quantum mechanics.
It is an interesting open question to determine the exact threshold for the
trivial communication complexity. Brassard et al.'s idea is based on the
recursive bias amplification by the 3-input majority function. It was not
obvious if other choice of function exhibits stronger bias amplification. We
show that the 3-input majority function is the unique optimal, so that one
cannot improve the threshold $\frac{3+\sqrt{6}}6$ by Brassard et al.'s bias
amplification. In this work, protocols for computing the function used for the
bias amplification are restricted to be non-adaptive protocols or particular
adaptive protocol inspired by Paw{\l}owski et al.'s protocol for information
causality [Nature 461, 1101 (2009)]. We first show a new adaptive protocol
inspired by Paw{\l}owski et al.'s protocol, and then show that the new adaptive
protocol is better than any non-adaptive protocol. Finally, we show that the
3-input majority function is the unique optimal for the bias amplification if
we apply the new adaptive protocol to each step of the bias amplification.
