双边贸易中随机报价机制的更紧边界
Tighter Bounds for the Random-Offerer Mechanism in Bilateral Trade
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中文总结 AI 辅助
研究双边贸易中随机报价机制，设\(\rho_{\rm RO}\)为该机制贸易收益与最优贸易收益之比下确界。证明了\(\frac{1}{\pi}\leq\rho_{\rm RO}<0.460242308085529\)，改进了上下界，分别采用参数化拉格朗日边界和构造明确族的方法。
中文摘要 AI 辅助
双边贸易的随机报价机制均匀选择卖家或买家，让所选代理做出利润最大化的要么接受要么放弃的报价。设\(\rho_{\rm RO}\)为该机制的贸易收益与最优贸易收益之比在独立价值分布下的下确界。我们证明了\(\frac{1}{\pi}\leq\rho_{\rm RO}<0.460242308085529\)。对于下界，将先前的保证从约\(0.317844\)提高到\(\frac{1}{\pi}\approx0.318310\)，证明使用了逐点单调分配的参数化拉格朗日边界。对于上界，构造了一个明确的族，该族满足\(\frac{\text{FB}}{\text{RO}}>2.17276852308451\)，改进了先前的明确比率\(2.0749\)。
英文摘要
The random-offerer mechanism for bilateral trade selects the seller or the buyer uniformly and lets the selected agent make a profit-maximizing take-it-or-leave-it offer. Let $ρ_{\rm RO}$ be the infimum, over independent value distributions, of the mechanism's gains from trade divided by first-best gains from trade. We prove $\frac1π\le ρ_{\rm RO}<0.460242308085529$. For the lower bound, we improve the previous guarantee from approximately $0.317844$ to $1/π\approx 0.318310$. The proof uses a parameterized Lagrangian bound for pointwise-monotone allocations. At multiplier one, this bound has coefficient $2/π$, and the Lagrangian separates into two terms controlled by the optimal seller-offering and buyer-offering profits. For the upper bound, we construct an explicit family consisting of a truncated equal-revenue buyer and a seller distribution with a tilted power-law lower tail and a constant-virtual-cost segment. The family satisfies $\operatorname{FB}/\operatorname{RO}>2.17276852308451$, improving the previous explicit ratio $2.0749$; rigorous interval arithmetic certifies the numerical inequality.
