We study the problem of social welfare maximization in bilateral trade, where two agents, a buyer and a seller, trade an indivisible item. We consider arguably the simplest form of mechanisms -- the fixed-price mechanisms, where the designer offers trade at a fixed price to the seller and buyer. Besides the simple form, fixed-price mechanisms are also the only DSIC and budget balanced mechanisms in bilateral trade. We obtain improved approximation ratios of fixed-price mechanisms in different settings. In the full prior information setting where the designer has access to the value distributions of both the seller and buyer, we show that the optimal fixed-price mechanism can achieve at least $0.72$ of the optimal welfare, and no fixed-price mechanism can achieve more than $0.7381$ of the optimal welfare. Prior to our result the state of the art approximation ratio was $1 - 1/e + 0.0001 \approx 0.632$. Interestingly, we further show that the optimal approximation ratio achievable with full prior information is identical to the optimal approximation ratio obtainable with only one-sided prior information. We further consider two limited information settings. In the first one, the designer is only given the mean of the buyer's (or the seller's) value. We show that with such minimal information, one can already design a fixed-price mechanism that achieves $2/3$ of the optimal social welfare, which surpasses the previous state of the art ratio even when the designer has access to the full prior information. Furthermore, $2/3$ is the optimal attainable ratio in this setting. In the second one, we assume that the designer has sample access to the value distributions. We propose a new family mechanisms called order statistic mechanisms and provide a complete characterization of their approximation ratios for any fixed number of samples.

翻译：我们研究了双边贸易中社会福利最大化问题，其中两个参与者（买方和卖方）交易一件不可分割的商品。我们考虑了机制中最简单的形式——固定价格机制，即设计者以固定价格向卖方和买方提供交易。除了形式简单外，固定价格机制也是双边贸易中唯一满足占优策略激励相容（DSIC）和预算平衡的机制。我们在不同设定下改进了固定价格机制的近似比。在完全先验信息设定（设计者可获取买卖双方价值分布）下，我们证明了最优固定价格机制能达到最优社会福利的至少0.72，且任何固定价格机制无法超过最优社会福利的0.7381。此前最优近似比结果为1 − 1/e + 0.0001 ≈ 0.632。有趣的是，我们进一步发现，在完全先验信息下可获得的最优近似比与仅利用单边先验信息下的最优近似比完全相同。我们还考虑了两种有限信息设定。第一种设定中，设计者仅知道买方（或卖方）价值的均值。我们证明，凭借这种最小信息，已可设计出达到最优社会福利2/3的固定价格机制，该结果甚至超越了此前设计者掌握完全先验信息时的最优近似比。此外，2/3是该设定下可达到的最优比率。第二种设定中，我们假设设计者可通过样本访问价值分布。我们提出了一类新的机制族——顺序统计量机制，并完整刻画了其在任意固定样本数量下的近似比。