Monotonicity and Rigidity in Gaussian Inverse Regression: The One-Period Kyle Model Has a Unique Equilibrium

Let $V$ and $U$ be independent standard normal random variables. For a Borel function $\phi: \mathbb{R} \to \mathbb{R}$, let $P_\phi$ be a version of the inverse regression $P_\phi(y) = E[V \mid \phi(V)+U = y]$, and let $F_\phi(x) = E[P_\phi(x+U)]$ be its Gaussian smoothing. We prove that $\phi(v) \in \mathrm{argmax}_x \{ xv - x F_\phi(x) \}$ for every real $v$ if and only if $\phi = \mathrm{id}$, the identity function. This is the pointwise best-response condition of the normalised one-period Kyle insider trading model; consequently the affine equilibrium strategy of Kyle (1985) is unique amongst all strategies. This settles the uniqueness question for the one-period Gaussian Kyle model. The additional ingredient relative to the McLennan, Monteiro and Tourky (2017) analytic framework is probabilistic: an exchangeable pair, obtained by resampling the value from the market makers' posterior, whose balance identities, combined with a Gaussian inequality for monotone functions, make the posterior mean affine.