The Uniqueness of Exponential Second-Order Expected Utility

Exponential Second-Order Expected Utility (SOEU) underlies the entropic approach to model uncertainty. This paper explores in what sense that functional form is essential. In the misspecification-robust Smooth Ambiguity criterion, let a single parameter govern both the model-level robustness and the ambiguity-averse aggregation across models Cerreia-Vioglio, Hansen, Maccheroni, and Marinacci (2026). The two-layer criterion then equals Exponential SOEU for every compact set of models and every second-order prior, with the Bayesian predictive measure as the baseline. The main results are converses. On the aggregator side, matched curvature is \emph{necessary}: at a fixed curvature no other continuous, strictly increasing aggregator delivers the reduction, and with mismatched curvature there are a model set and a prior for which no single-layer entropic value, at any curvature and any baseline, reproduces the criterion. On the cost side, within the power-divergence family, which contains chi-squared and reverse Kullback--Leibler (KL), only KL has a dual of the log-sum-exp form, and the other members first depart from it at the third cumulant. Finally, the value dual to Exponential SOEU is a robust-control value: it is motivated by Rational Inattention, but no Bayes-plausible information-acquisition problem about a fixed act generates it.