Does hiding trade direction make markets cheaper or just shift costs?

When a market maker can't see the true direction of trades (only a noisy version due to privacy tech), the bid-ask spread shrinks to μ(1−2η)Δ. But this isn't free: a "privacy subsidy" worth μηΔ per trade flows from the liquidity pool to traders. The finding shows privacy mechanisms in markets have a precise, measurable cost hidden in spreads.