The One-Period Gaussian Kyle Model Has Exactly One Equilibrium

Paulo K. Monteiro and Rabee Tourky

ANU Working Papers in Economics and Econometrics from Australian National University, College of Business and Economics, School of Economics

Abstract: 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.

Keywords: Gaussian conditional expectation; posterior-mean pricing; exchangeable pairs; monotone functions; Kyle model; equilibrium uniqueness. (search for similar items in EconPapers)
JEL-codes: C62 D82 G14 (search for similar items in EconPapers)
Date: 2026-08
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