The Privacy Subsidy in Glosten-Milgrom: Bid-Ask Spread and Welfare under Flip-Noise Direction Observation

Abstract

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 $\eta$ -- 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 $\mu(1-2\eta)\Delta$, where $\mu$ is the informed-trader fraction and $\Delta = v_H - v_L$ the value range. The welfare decomposition identifies a per-trade transfer $\mu\eta\Delta$ 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.