The Privacy Subsidy in Glosten-Milgrom: Bid-Ask Spread and Welfare under Flip-Noise Direction Observation
Pith reviewed 2026-05-20 01:43 UTC · model grok-4.3
The pith
In the Glosten-Milgrom model with privacy noise on trade directions, the equilibrium spread becomes μ(1-2η)Δ and creates a privacy subsidy of μηΔ per trade from the liquidity pool.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Under a committed Bayesian market-maker pricing rule applied to the trade direction perturbed by a binary flip channel of probability η, the equilibrium spread is μ(1-2η)Δ. The welfare decomposition identifies a per-trade transfer μηΔ from the protocol's liquidity pool to traders, termed the privacy subsidy. This extends the privacy-subsidy concept from the continuous Gaussian-Kyle model to the discrete Glosten-Milgrom setting.
What carries the argument
The binary flip channel of probability η that models the privacy mechanism on the direction signal and enables the committed Bayesian pricing rule.
Load-bearing premise
The market maker commits to Bayesian pricing based on the noisy direction signal from the independent binary flip channel while traders act strategically under that fixed rule.
What would settle it
Simulate the sequential trading game with fixed values of μ, η, and Δ, then check whether the realized spread converges to μ(1-2η)Δ and the average per-trade transfer to traders equals μηΔ.
read the original 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper extends the Glosten-Milgrom 1985 sequential trading model by modeling the market maker's observation of trade direction as output from a binary symmetric flip channel with error probability η. Under the assumption of a committed Bayesian market maker who prices solely on the noisy direction signal, the authors derive a closed-form equilibrium bid-ask spread of μ(1−2η)Δ, where μ is the fraction of informed traders and Δ = v_H − v_L. They further decompose welfare to identify a per-trade transfer μηΔ from the liquidity pool to traders, termed the 'privacy subsidy,' and note that this mirrors an earlier result in the Gaussian Kyle model.
Significance. If the derivation is correct, the result establishes that the privacy-subsidy phenomenon is robust across both the continuous Gaussian Kyle setting and the canonical discrete two-state Glosten-Milgrom setting. The closed-form expression supplies a transparent welfare accounting for privacy mechanisms (such as ε-differentially private direction disclosure in MPC matching engines) without introducing additional free parameters, which is a concrete strength for applied market-design analysis.
major comments (1)
- [§3] §3 (equilibrium derivation): the claim that the posterior probability an observed buy is informed equals ½ + ½μ(1−2η) follows immediately once the market maker is restricted to the noisy direction signal, but the manuscript must explicitly verify that this posterior remains consistent with traders' strategic best responses under the same pricing rule; otherwise the fixed-point argument for equilibrium is incomplete.
minor comments (2)
- [§2] The notation for the flip probability η is introduced in the abstract but should be restated with the channel definition at the beginning of §2 to avoid any ambiguity for readers unfamiliar with information-theoretic privacy models.
- [Figure 1] Figure 1 (welfare decomposition diagram) would benefit from an explicit arrow or label indicating the direction of the μηΔ transfer from the liquidity pool.
Simulated Author's Rebuttal
We thank the referee for the positive evaluation, the recognition of the result's robustness across models, and the constructive comment on the equilibrium fixed-point argument. We address the single major comment below and will incorporate the suggested clarification.
read point-by-point responses
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Referee: [§3] §3 (equilibrium derivation): the claim that the posterior probability an observed buy is informed equals ½ + ½μ(1−2η) follows immediately once the market maker is restricted to the noisy direction signal, but the manuscript must explicitly verify that this posterior remains consistent with traders' strategic best responses under the same pricing rule; otherwise the fixed-point argument for equilibrium is incomplete.
Authors: We agree that an explicit verification strengthens the presentation of the fixed-point equilibrium. In the revised manuscript we will add a short paragraph in §3, immediately after the posterior derivation, that confirms consistency: under the candidate pricing rule with spread μ(1−2η)Δ, each informed trader's optimal action (buy if private value = v_H, sell if = v_L) generates an aggregate informed-trade probability whose image under the binary flip channel recovers exactly the posterior ½ + ½μ(1−2η). This closes the loop and shows that the pricing rule is indeed a best-response equilibrium for the market maker given the traders' strategies. No changes to the model, assumptions, or closed-form results are required. revision: yes
Circularity Check
No significant circularity; derivation is self-contained within standard Glosten-Milgrom Bayesian update
full rationale
The paper applies the classic Glosten-Milgrom sequential trading model under the explicit assumption of a committed Bayesian market maker who prices using only the direction signal after it has passed through an independent binary symmetric flip channel of probability η. The equilibrium spread μ(1-2η)Δ is obtained by direct Bayesian updating: the posterior that an observed buy is informed becomes ½ + ½μ(1-2η), which is exactly the standard Glosten-Milgrom formula with the effective informed fraction reduced by the channel. The per-trade privacy subsidy μηΔ is then the mechanical difference between the full-information spread and this noisy spread, arising as the expected loss to the market maker from residual uncertainty. No equation reduces to a fitted parameter renamed as a prediction, no self-referential definition appears, and the reference to the Gaussian-Kyle analog is contextual rather than load-bearing for the present closed-form result. The derivation therefore stands on its own modeling assumptions and standard Bayesian mechanics without circular reduction.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption Market maker commits to Bayesian pricing on the noisy direction signal.
- domain assumption Noise is an independent binary flip channel with fixed probability η.
discussion (0)
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