Should You Train for Decisions? A Decision-Adequacy Test and Stein Shrinkage for Predict-then-Optimize

A cost model for an optimization problem can be fitted by least squares and then optimized (two-stage) or trained on the decisions it induces (decision-focused). Which is better depends on an unknown misspecification, so the two are mixed by cross-validation or by hand, with no guarantee. We ask instead whether the model is decision-adequate: would decision-focused training change its decisions? We test this with a Hausman-type statistic, the gap between the two fits in the geometry of the decision loss rather than the Wald geometry, which overweights directions that change no decision. Smoothing the SPO+ loss gives both fits a closed-form covariance, and the same gap is the unknown coefficient of the risk along the path between them, so one statistic says whether to move and how far: a closed-form James-Stein weight that, asymptotically in the risk of that geometry, is never much worse than the better fit and, under a regime condition, beats the decision-focused fit. In preregistered experiments the test has size 0.051 under correct specification and rejects in most splits on two electricity markets where decision-focused training cuts regret by up to a factor of six. In practice this gives a p-value for decision adequacy and, for methods that interpolate between the fits, a closed-form step that requires no refits and no prior knowledge of which fit is better. Every failed criterion is reported: the shrinkage gain is small, the rule trails cross-validation where the smoothing scale is coarse, and the test over-rejects under heavy tails at small samples and reads skewed noise as misspecification.