Decision theory and the "almost implies near" phenomenon

Traditional decision theory establishes equivalence between axioms and the functional forms of utility representations. In this paper, we develop a new framework where axioms need only be approximately satisfied, introducing a principled approach to measuring the degree of axiom violations. Our central finding is that behavior that "almost" satisfies canonical axioms induces utility representations that are "near" the standard utility forms associated with those axioms. Specifically, when agents approximately satisfy axioms like independence or stationarity, their utility functions can be approximated by the corresponding canonical utilities, with quantifiable error bounds directly linked to the magnitude of axiom violations. We apply this phenomenon across three domains: decisions under risk, uncertainty, and intertemporal choice. Our results provide theoretical foundations for approximate optimization in economics and offer a middle ground between rejecting theories and requiring exact axioms, and accommodating small deviations from the theory in empirical work.