The One-Period Kyle (1985) Model Has a Unique Equilibrium: A Monotone Gaussian Bayes inverse-rigidity theorem

Rabee Tourky

arXiv 26 Jul 2026 · Mathematics — Probability

arXiv:2607.23585 · PDF · Extracted main text

Abstract

Let $V$ and $U$ be independent standard normal random variables. For any Borel map $φ\colon\mathbb{R}\to\mathbb{R}$, set $Y_φ=φ(V)+U$, and define $P_φ(y)=\mathbb{E}[V\mid Y_φ=y]$ and $F_φ(x)=\mathbb{E}[P_φ(x+U)]$. We prove that, if for every $v\in\mathbb{R}$, the quantity $φ(v)$ maximises $x(v-F_φ(x))$ over $x\in\mathbb{R}$, then $φ$ is the identity function. This is the normalised one-period Kyle (1985) model of insider trading. It follows that Kyle's closed-form affine strategy is the unique equilibrium of the model for arbitrary Gaussian location and scale, and that its canonical competitive pricing rule is necessarily linear. This settles a long-standing question in financial economics. Boulatov, Kyle and Livdan (2005, 2013) introduced complex-analytic techniques to the problem. McLennan, Monteiro and Tourky (2017) proved a linear growth bound for $P_φ$, real-entire regularity of $F_φ$, and uniqueness when an equilibrium strategy agrees locally with a uniquely continuable analytic function. The present proof requires no regularity assumption on $φ$ beyond Borel measurability. Its argument is real-variable and probabilistic: the maximisation forces sharp upper and lower bounds for Gaussian-randomised monotone functions to coincide, and the corresponding equality cases admit only the identity function.

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