  We study the two-agent single-item bilateral trade. Ideally, the trade should
happen whenever the buyer's value for the item exceeds the seller's cost.
However, the classical result of Myerson and Satterthwaite showed that no
mechanism can achieve this without violating one of the Bayesian incentive
compatibility, individual rationality and weakly balanced budget conditions.
This motivates the study of approximating the
trade-whenever-socially-beneficial mechanism, in terms of the expected
gains-from-trade. Recently, Deng, Mao, Sivan, and Wang showed that the
random-offerer mechanism achieves at least a 1/8.23 approximation. We improve
this lower bound to 1/3.15 in this paper. We also determine the exact
worst-case approximation ratio of the seller-pricing mechanism assuming the
distribution of the buyer's value satisfies the monotone hazard rate property.
