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

Technical Summary: The Privacy Subsidy in Glosten-Milgrom

Problem Statement
The paper investigates the impact of privacy-preserving mechanisms on market microstructure within the canonical Glosten-Milgrom (GM) 1985 sequential-trading model. In the standard GM framework, a risk-neutral market maker (MM) observes the exact direction of an incoming trade (buy or sell) to set bid and ask prices, resulting in a bid-ask spread that compensates for adverse selection from informed traders.

The authors address a gap in contemporary exchange design (e.g., MPC-based matching, sealed-bid RFQs), where privacy mechanisms intentionally obscure the trade direction. Specifically, the paper models the MM's observation of trade direction as passing through a binary symmetric channel (a "flip-noise" channel). With probability $1-\eta$, the MM observes the true direction; with probability $\eta$, the direction is flipped. The core question is how this information-theoretic leakage affects the equilibrium bid-ask spread and the welfare distribution among informed traders, noise traders, and the protocol (MM).

Methodology
The study employs a discrete-state, sequential-trading model with the following setup:

1. Asset and Agents: A single risky asset with value $v \in \{v_H, v_L\}$ and prior probability $\pi = 1/2$. Traders arrive sequentially; a fraction $\mu$ are informed (know $v$), and $1-\mu$ are noise traders (choose direction uniformly at random).
2. Privacy Mechanism: The true trade direction $d$ is perturbed by a flip probability $\eta \in [0, 1/2]$ before reaching the MM. The observed signal is $\tilde{d}$.
3. Pricing Rule: The authors adopt a committed Bayesian pricing rule. Unlike the classical competitive MM who ensures zero expected profit conditional on the actual trade direction, this MM commits to a pricing rule based on the observed noisy signal:
   • $Ask = E[v \mid \tilde{d} = \text{buy}]$
   • $Bid = E[v \mid \tilde{d} = \text{sell}]$
     The MM bears the realized expected loss on the actual flow without a zero-profit constraint on the actual trade direction.
4. Analysis: The authors derive closed-form expressions for the equilibrium spread and decompose the per-trade welfare (P&L) for informed traders ($\pi_I$), noise traders ($\pi_N$), and the market maker ($\pi_M$).

Key Results

1. Modified Bid-Ask Spread (Theorem 1):
   The equilibrium spread is derived as:
   $$\text{Spread} = \mu(1 - 2\eta)\Delta$$
   where $\Delta = v_H - v_L$.

   • When $\eta = 0$ (no privacy), the spread recovers the textbook GM value $\mu\Delta$.
   • When $\eta = 1/2$ (perfect privacy noise), the spread collapses to 0, as the observed signal becomes independent of the asset value.

2. Welfare Decomposition and the Privacy Subsidy (Theorem 2):
   The paper identifies a non-zero expected loss for the protocol/MM per trade:
   $$\pi_M = -\mu\eta\Delta$$
   This loss is defined as the privacy subsidy. It represents a per-trade transfer from the protocol's liquidity pool to the aggregate of traders (both informed and uninformed).

   • Informed Traders: Their profit increases as $\eta$ increases because the spread narrows, reducing the cost of extracting rent.
   • Noise Traders: Contrary to intuition, noise traders also benefit. Their expected loss per trade decreases (becomes less negative) as $\eta$ increases.
   • Zero-Sum: The sum of welfare components is zero: $\mu\pi_I + (1-\mu)\pi_N + \pi_M = 0$.

3. Comparative Statics:

   • The spread decreases linearly with $\eta$.
   • The privacy subsidy increases linearly with $\eta$.
   • Both trader types gain from the introduction of privacy noise, while the protocol bears the entire cost.

Significance and Claims
The paper claims to establish the robustness of the "privacy subsidy" phenomenon across the two canonical microstructure models:

• Extension of Prior Work: It extends the privacy-subsidy concept, previously established for continuous-Gaussian Kyle markets under additive noise (Nakamura [9]), to the discrete two-state Glosten-Milgrom model under binary symmetric noise.
• Conceptual Unification: The result demonstrates that the privacy subsidy is a fundamental feature of committed Bayesian pricing under information asymmetry, regardless of whether the noise is Gaussian (continuous) or binary (discrete).
• Application to MPC: The authors position the result as directly applicable to MPC-based matching engines where trade directions are disclosed via $\epsilon$-differentially private mechanisms. In this context, the privacy subsidy quantifies the welfare cost (or required fee floor) of the privacy budget.
• Relation to LVR: The privacy subsidy is placed in the same conceptual family as Loss-Versus-Rebalancing (LVR) and the Gaussian privacy subsidy, representing a closed-form, per-period adverse-selection cost borne by an automated pricing mechanism.

Limitations and Future Work
The authors note that the results rely on a symmetric prior ($\pi=1/2$) and a committed Bayesian pricing rule. They explicitly state that asymmetric priors would require future work to derive precise closed-forms, though the qualitative results are expected to persist. Additionally, the analysis is a partial-equilibrium study; a full fee-equilibrium analysis where fees distort trader participation is left for future investigation. The paper also delineates that mechanisms involving batched aggregation or delayed reveal (temporal asymmetry) fall outside this specific flip-noise framework.