We consider the bilateral trade problem, in which two agents trade a single indivisible item. It is known that the only dominant-strategy truthful mechanism is the fixed-price mechanism: given commonly known distributions of the buyer's value $B$ and the seller's value $S$, a price $p$ is offered to both agents and trade occurs if $S \leq p \leq B$. The objective is to maximize either expected welfare $\mathbb{E}[S + (B-S) \mathbf{1}_{S \leq p \leq B}]$ or expected gains from trade $\mathbb{E}[(B-S) \mathbf{1}_{S \leq p \leq B}]$. We improve the approximation ratios for several welfare maximization variants of this problem. When the agents' distributions are identical, we show that the optimal approximation ratio for welfare is $\frac{2+\sqrt{2}}{4}$. With just one prior sample from the common distribution, we show that a $3/4$-approximation to welfare is achievable. When agents' distributions are not required to be identical, we show that a previously best-known $(1-1/e)$-approximation can be strictly improved, but $1-1/e$ is optimal if only the seller's distribution is known.
翻译:我们考虑双边贸易问题,即两个代理商交易一个单一不可分割的物品。众所周知,唯一具有支配地位的忠实战略机制是固定价格机制:考虑到买主价值美元和卖主价值美元众所周知的分配情况,我们向代理商和交易提供价格美元,如果美元S\leq p\leq B$,则交易即发生。目标是最大限度地增加预期福利$\mathbb{E}[S +(B-S)\mathbf{[B-S)\(B)1 ⁇ S\leqp p\leqB}]美元或贸易预期收益$\mathbb{E}[(B-S)\mathbf{1 ⁇ S\leq p\leqB}美元。我们改进了这一问题的若干福利最大化变数的近似比率。当代理商的分布情况相同时,我们显示,福利的最佳近似比率只有$\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\