Shortly after the introduction of Expected Utility Theory (EUT), economists and psychologists began publishing results that showed choices made by experimental subjects which apparently violate one or more of the EUT axioms. I discuss economists' responses to this evidence. These vary from developing new theoretical models, models that nest EUT as a special case, such as Rank Dependent Utility (RDU) and Regret Theory, as well as models that do not nest EUT, such as Cumulative Prospect Theory, to critiques of experimental methods and scope, to the promotion of stochastic models of choice. I discuss popular stochastic choice models in depth and evaluate their normative coherence. I find that the "Random Preferences" stochastic model fails to make normatively coherent statements, in contrast to the "Random Error" and "Tremble" models, which do so. I demonstrate a method to calculate the unconditional likelihood of choice errors for populations of EUT-compliant and RDU-compliant agents, and show how certain characteristics of the population relate to the likelihood of these choice errors and their costliness in terms of forgone welfare. I find that elements of the stochastic model that are not related to preference relations tend to have a greater influence on unconditional welfare estimates than the preference parameters themselves. Finally, I conduct a power analysis of the ability of a lottery battery instrument to correctly classify experimental subjects as employing either EUT or RDU, and the effect of this classification on the accuracy of the estimates of welfare surplus for the subjects. For large ranges of parameter values for these models, I find that the probability of type I and type II errors in the classification process are not trivial, and can be very costly in terms of welfare surplus. Additionally, I show that for a hypothetical population comprising subjects employing EUT or RDU, we can arrive at more accurate welfare surplus estimates on average by assuming that every subject employs the RDU functional, rather than by first trying to differentiate RDU subjects from EUT subjects.