Decision theory and the "almost implies near" phenomenon

We examine behavioral axioms in decision theory that are satisfied approximately rather than exactly. We demonstrate that in key domains -- decisions under risk, uncertainty, and intertemporal choice -- behavior that \emph{almost} satisfies an axiom implies the existence of a utility function that is \emph{near} one that adheres to the standard theoretical representation (e.g., expected utility, or exponentially discounted utility). We explicitly quantify the distance between the utility that captures actual behavior and the ideal theoretical utility as a function of the measured deviation from the axiom. This result formally connects two distinct quantitative exercises: measuring empirical deviations from theory and utilizing approximate optimization. Effectively, we show that small deviations from behavioral axioms rationalize the use of standard models as valid approximations.