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

TL;DR

This paper derives a unique linear Kyle equilibrium for privacy-preserving crypto exchanges observing noisy order flow, revealing how privacy noise impacts market dynamics and defining a 'privacy subsidy' for welfare analysis.

Contribution

It introduces a closed-form equilibrium model for Gaussian privacy noise in order flow observation, extending Kyle's model to privacy-preserving crypto exchange mechanisms.

Findings

Price-impact coefficient and trader strategy rescale with privacy noise

The product of impact and strategy remains invariant under privacy noise

A closed-form welfare transfer, the 'privacy subsidy', is identified

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.