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.

翻译：我们继续研究双边交易问题中固定价格机制的性能，并在多种设定下改进了社会福利最优机制的近似比。具体而言，在仅知晓买方分布的情况下，我们证明存在一个在固定价格机制上的概率分布，使得其近似比位于区间[0.71, 0.7381]内。此外，我们证明当买方和卖方分布均已知时，最优固定价格机制具有相同的近似比。从而，先前最优的 (1 - 1/e+0.0001) 近似比可改进为 $0.71$。进一步地，我们考察了在仅从卖方分布获取单个样本时（涵盖对称与非对称设定）的随机化固定价格机制。研究结果表明，在所有随机化固定价格机制中，将该单一样本设为报价仍为最优策略。