The Privacy Subsidy: Kyle's $\lambda$ under Noise-Perturbed Order-Flow Observation

Yuki Nakamura

arxiv: 2605.15746 · v4 · pith:AXFQH7SQnew · submitted 2026-05-15 · 💻 cs.GT · cs.CR· math.PR· q-fin.TR

The Privacy Subsidy: Kyle's λ under Noise-Perturbed Order-Flow Observation

Yuki Nakamura This is my paper

Pith reviewed 2026-05-20 16:39 UTC · model grok-4.3

classification 💻 cs.GT cs.CRmath.PRq-fin.TR

keywords privacyauctionsammsbatchedclosed-formflowkyleobserves

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The pith

Derives closed-form linear Kyle equilibrium under Gaussian privacy noise on order flow observation, with rescaled price-impact coefficient and informed-trader strategy but invariant product, plus welfare decomposition giving the privacy subsidy as break-even fee for privacy-aggregated exchanges.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Kyle models describe how informed traders and market makers interact via order flow to determine prices. Privacy mechanisms in cryptocurrency exchanges add random noise to the observed orders to protect trader information. The authors model this noise as independent Gaussian and assume the market maker is a committed Bayesian updater who knows the noise distribution. Under these conditions they solve for the unique linear equilibrium. Both the price impact coefficient (how much the price moves per unit order) and the informed trader's trading intensity scale down by a factor that depends on the noise variance. Their product remains unchanged, preserving a key relationship from the no-noise case. A welfare calculation then isolates a per-period transfer from the liquidity provider pool to traders; this transfer is the privacy subsidy. The result is positioned as the privacy-noise version of loss-versus-rebalancing calculations in other market-making papers. The primary intended use is for shielded automated market makers that explicitly inject additive noise for differential privacy.

Core claim

We derive the unique linear Kyle equilibrium when a committed Bayesian market maker observes order flow perturbed by independent Gaussian privacy noise. The price-impact coefficient and informed-trader strategy both rescale by a single factor in the privacy parameter, and their product is invariant. A welfare decomposition then identifies a closed-form per-period transfer from the protocol's LP pool to traders -- the 'privacy subsidy'.

Load-bearing premise

The equilibrium is linear, the market maker is committed and Bayesian, and the privacy perturbation is independent Gaussian noise; these modeling choices are required to obtain the claimed closed-form rescaling and invariant product.

Figures

Figures reproduced from arXiv: 2605.15746 by Yuki Nakamura.

**Figure 1.** Figure 1: Privacy subsidy |πM| vs. σε for σv = σu = 1. The convexity-to-concavity inflection at σ ⋆ ε = √ 2 marks the transition between the quadratic low-privacy regime and the linear high-privacy regime. A.1 Dimensionless table [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗

read the original abstract

Privacy-preserving cryptocurrency exchanges (shielded AMMs, batched swap auctions, sealed-bid order-flow auctions) alter what the pricing mechanism observes about order flow. We derive the unique linear Kyle equilibrium when a committed Bayesian market maker observes order flow perturbed by independent Gaussian privacy noise. The price-impact coefficient and informed-trader strategy both rescale by a single factor in the privacy parameter, and their product is invariant. A welfare decomposition then identifies a closed-form per-period transfer from the protocol's LP pool to traders -- the "privacy subsidy", the break-even fee any privacy-aggregated exchange must charge. The result is the single-period closed-form privacy-noise analog of Loss-Versus-Rebalancing (Milionis et al. 2022). The primary application is shielded AMMs with explicit additive-noise injection (e.g., differential privacy); related designs (batched swaps, sealed-bid auctions, oracle-pegged crossings) require separate frameworks that we leave to future work.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This adds independent Gaussian noise to the Kyle order-flow observation and extracts a clean rescaling of λ and β with invariant product plus a closed-form privacy subsidy that follows directly from the standard derivation.

read the letter

The key takeaway is that this paper shows how adding Gaussian privacy noise to order flow in the Kyle model leads to a rescaled linear equilibrium where the product of price impact and trading intensity stays invariant, and the resulting expected transfer from the liquidity provider pool to traders is a closed-form privacy subsidy. The work does well by delivering exactly that closed form under the standard normality assumptions. Replacing the observed flow with y = x + u + ε where ε is the independent noise, the new λ becomes σ_v over twice the total standard deviation, and β scales up accordingly. Their product remains 1/4, and the subsidy equals minus the expectation of (v - p) times ε, which evaluates directly from the Gaussian moments. This supplies a practical metric for sizing noise levels or fees in shielded AMMs, and it positions the result as the privacy counterpart to loss-versus-rebalancing. The derivation has no obvious gaps once the noise is introduced; it follows the classic steps without additional fitting. The citation to Milionis et al. is appropriate for the analogy. The main limitation is that everything hinges on the one-period linear equilibrium with a committed Bayesian market maker and additive independent noise. Those choices make the math tractable but mean the subsidy formula does not immediately apply to batched swaps or sealed-bid designs, as the abstract notes. The result is more an application and interpretation than a fundamental technical departure from Kyle. This paper is for researchers and practitioners focused on privacy mechanisms in cryptocurrency trading protocols. Anyone designing or evaluating shielded AMMs or similar systems would get direct value from the explicit subsidy expression. It is grounded enough to merit a serious referee who can verify the welfare decomposition against the full equations. I recommend sending it out for peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript derives the unique linear Kyle equilibrium when a committed Bayesian market maker observes order flow y = x + u + ε with ε ~ N(0, σ_ε²) independent of (v, u). The equilibrium price-impact coefficient λ and informed-trader intensity β both rescale by the single factor σ_total / σ_u (where σ_total = sqrt(σ_u² + σ_ε²)), their product λβ remains invariant at 1/4, and a welfare decomposition identifies the closed-form per-period privacy subsidy as the expected transfer -E[(v - p)ε] from the LP pool to traders. The result is positioned as the single-period privacy-noise analog of Loss-Versus-Rebalancing.

Significance. If the derivation holds, the paper supplies a parameter-free closed-form extension of the classic Kyle model to additive privacy noise on order flow. The exact rescaling rule, the invariance of the product λβ, and the explicit Gaussian-moment expression for the privacy subsidy provide a direct benchmark for break-even fees in shielded AMMs and related privacy-preserving mechanisms. The work is grounded in standard joint-normality assumptions, delivers falsifiable predictions without new free parameters, and mirrors the structure of the LVR literature.

minor comments (3)

[Abstract] Abstract: the main claims are stated clearly but without any displayed equations for the rescaling factor or the privacy-subsidy expression; adding the key formulas (e.g., λ' = σ_v / (2 σ_total) and the subsidy -E[(v-p)ε]) would improve immediate accessibility.
[Introduction] The baseline Kyle (1985) reference should be cited explicitly in the introduction and equilibrium section when the standard formulas are invoked before the noise perturbation is introduced.
[§3] Notation for the privacy-noise variance should be introduced once and used consistently when defining σ_total and when evaluating the joint-Gaussian moments for the subsidy.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the supportive summary, significance assessment, and recommendation of minor revision. The manuscript's positioning as a single-period privacy-noise analog of LVR is accurately captured, and we appreciate the recognition of the closed-form rescaling rule and invariant product λβ.

Circularity Check

0 steps flagged

No significant circularity; derivation is a direct extension of standard Kyle model

full rationale

The paper starts from the classic one-period Kyle setup with joint normality of (v, u) and replaces the market maker's observation with y = x + u + ε where ε ~ N(0, σ_ε²) is independent. The linear equilibrium is obtained by solving the informed trader's optimization and the market maker's Bayesian pricing rule under the modified signal; this produces the explicit rescalings λ' = σ_v / (2 σ_total) and β' = σ_total / (2 σ_v) with σ_total = sqrt(σ_u² + σ_ε²) and the invariant product λ'β' = 1/4. The privacy subsidy is then identified as the closed-form term -E[(v - p)ε] arising from the Gaussian cross moments. None of these steps reduce to a fitted parameter renamed as a prediction, a self-definitional loop, or a load-bearing self-citation; the Milionis et al. (2022) reference is used only for loose analogy to LVR and is not invoked to justify any uniqueness or ansatz. The derivation remains self-contained against the maintained assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

Model rests on three domain assumptions drawn from the Kyle tradition plus the privacy-noise modeling choice; no free parameters are fitted to data and no new entities are postulated.

axioms (3)

domain assumption The market maker is a committed Bayesian updater who knows the noise distribution.
Explicitly stated in abstract as prerequisite for the equilibrium derivation.
domain assumption Privacy perturbation is independent Gaussian noise added to order flow.
Core modeling choice that enables the claimed rescaling.
domain assumption Equilibrium strategies are linear in the observed (noisy) order flow.
Required to obtain the unique linear Kyle equilibrium and closed-form expressions.

pith-pipeline@v0.9.0 · 5702 in / 1546 out tokens · 116118 ms · 2026-05-20T16:39:47.573013+00:00 · methodology

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The unique price-impact coefficient is λ = σv/(2 √(σu² + σε²)) with informed-trader linear coefficient β = √(σu² + σε²)/σv (Theorem 1). ... λβ = 1/2 for all σε ≥ 0.
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

A welfare decomposition then identifies a closed-form per-period transfer ... the 'privacy subsidy'

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