The One-Period Gaussian Kyle Model Has Exactly One Equilibrium

Let V and U be independent standard normal random variables. For every Borel-measurable map ϕ: R → R, let Pϕ be a version of the inverse regression Pϕ(y) = E[V | ϕ(V ) + U = y], and let Fϕ(x) := E[Pϕ(x + U)] be its Gaussian smoothing. We prove that ϕ(v) ∈ arg max x∈R xv - xFϕ(x), for every v ∈ R, if and only if ϕ = idR, the identity function. This rigidity theorem implies uniqueness of equilibrium in the one-period Gaussian Kyle (1985) model. An informed trader observes an asset value V ~ N (µ, τ2) and submits demand ϕ(V ). Independent noise demand U ~ N (0, σ2) is submitted at the same time. Competitive market makers observe aggregate order flow Y := ϕ(V ) + U and set the execution price according to P(Y ) = E[V | Y ] almost surely. For each observed value v, the insider chooses an order x to maximise E[(v - P(x + U))x], taking the pricing rule P as given; equilibrium requires ϕ(v) to be a maximiser for every v ∈ R. The Kyle (1985) affine equilibrium is the unique equilibrium among all Borel-measurable strategies, and the equilibrium pricing rule is unique up to almost-sure equality. Unlike earlier uniqueness results, ours leaves the original Gaussian Kyle model unchanged and imposes no restriction on admissible strategies beyond Borel measurability. We therefore resolve a long-standing open question. The proof is probabilistic and convex-analytic and uses no complex analysis.