  We continue the study of the performance for fixed-price mechanisms in the
bilateral trade problem, and improve approximation ratios of welfare-optimal
mechanisms in several settings. Specifically, in the case where only the buyer
distribution is known, we prove that there exists a distribution over different
fixed-price mechanisms, such that the approximation ratio lies within the
interval of [0.71, 0.7381]. Furthermore, we show that the same approximation
ratio holds for the optimal fixed-price mechanism, when both buyer and seller
distributions are known. As a result, the previously best-known (1 -
1/e+0.0001)-approximation can be improved to $0.71$. Additionally, we examine
randomized fixed-price mechanisms when we receive just one single sample from
the seller distribution, for both symmetric and asymmetric settings. Our
findings reveal that posting the single sample as the price remains optimal
among all randomized fixed-price mechanisms.
