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：我们推导了当做市商观察到被二进制翻转信道（概率η）扰动的交易方向时的封闭形式买卖价差与福利分解——这是针对作用于方向信号的隐私机制的自然信息论模型。在贝叶斯承诺做市商定价规则下，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到交易者的每笔交易转移μηΔ——即"隐私补贴"，这与先前研究中建立的高斯-凯尔模型类似。该结果将隐私补贴概念从连续高斯模型扩展到离散双状态微观结构，证明了该概念在两个经典模型间的稳健性。主要应用：采用ε-差分隐私方向披露且基于噪声方向信号定价的MPC匹配引擎。