The one-period Kyle model has one equilibrium

Let $V$ and $U$ be independent standard normal random variables. For each Borel-measurable function $\phi\colon\mathbb{R}\to\mathbb{R}$, let $P_\phi\colon\mathbb{R}\to\mathbb{R}$ be a Borel version of the inverse regression $y\mapsto\mathbb{E}[V\mid\phi(V)+U=y]$. We prove that $\phi(v)\in\operatorname*{arg\,max}_{x\in\mathbb{R}} \mathbb{E}[(v-P_\phi(x+U))x]$ for every $v\in\mathbb{R}$ if and only if $\phi=\operatorname{id}_{\mathbb{R}}$. The rigidity result implies that the one-period Gaussian Kyle (1985) insider-trading model has a unique Borel-measurable equilibrium strategy, namely Kyle's affine strategy. Building on preliminary results of McLennan, Monteiro, and Tourky (2017), the proof establishes that, at equilibrium, the total expected loss of noise traders attains a sharp universal upper bound and that any strategy attaining this bound must be affine almost everywhere.