We derive a closed-form bid-ask spread and welfare decomposition for the Glosten-Milgrom 1985 sequential-trading model when the market maker observes the trade direction perturbed by a binary flip channel of probability $η$ -- a natural information-theoretic model of privacy mechanisms acting on the direction signal. Under a committed Bayesian market-maker pricing rule, the equilibrium spread is $μ(1-2η)Δ$, where $μ$ is the informed-trader fraction and $Δ= v_H - v_L$ the value range. The welfare decomposition identifies a per-trade transfer $μηΔ$ from the protocol's liquidity pool to traders -- the "privacy subsidy", mirroring the Gaussian-Kyle analog established in prior work. The result extends the privacy-subsidy concept from continuous Gaussian to discrete two-state microstructure, demonstrating robustness across both classical models. Primary application: MPC-based matching engines with $\varepsilon$-differentially-private direction disclosure, where the engine prices on a noisy direction signal.

翻译：我们推导了Glosten-Milgrom（1985）序贯交易模型在如下设定下的封闭形式买卖价差与福利分解：做市商观测到的交易方向受到概率为$η$的二进制翻转信道的扰动——这是一种自然的隐私机制作用于方向信号的信息论模型。在承诺贝叶斯做市商定价规则下，均衡价差为$μ(1-2η)Δ$，其中$μ$为知情交易者比例，$Δ = v_H - v_L$为价值区间。福利分解识别出从协议流动性池向交易者的每笔交易转移$μηΔ$——即"隐私补贴"，与先前工作中建立的Gaussian-Kyle类比相呼应。该结果将隐私补贴概念从连续高斯微观结构扩展至离散两状态微观结构，证明了该概念在两个经典模型中的稳健性。主要应用：基于MPC的匹配引擎采用$\varepsilon$-差分隐私的方向披露，此时引擎基于含噪方向信号定价。