 Central to the literature on decision theory are a number of representation theorems showing that every agent with qualitative preferences satisfying a set of rationality postulates can be represented as an expected utility maximizer (von Neumann and Morgenstern, 1944 and Savage, 1972). These axioms are controversial, and are subject to intuitive counterexamples. Allais (1953) and Ellsberg (1961) give examples in which seemingly rational agents violate the rationality postulates and therefore cannot, even in principle, be represented as expected utility maximizers. For more on the challenge of Ellsberg and Allais, see the entry on  descriptive decision theory  as well as Buchak (2013).