Chapter 2. Rationality and Dynamic Consistency Under Risk and Uncertainty For choice with deterministic consequences the standard rationality hypothesis is ordinality i.e. maximization of a weak preference ordering. For choice under risk (resp. uncertainty) preferences are assumed to be represented by the objectively (resp. subjectively) expected value of a von Neumann Morgenstern utility function. For choice under risk this implies a key independence axiom; under uncertainty it implies some version of Savage’s sure thing principle. This chapter investigates the extent to which ordinality independence and the sure thing principle can be derived from more fundamental axioms concerning behavior in decision trees. Following Cubitt (1996) these principles include dynamic consistency separability and reduction of sequential choice which can be derived in turn from one consequentialist hypothesis applied to continuation subtrees as well as entire decision trees. Examples of behavior violating these principles are also reviewed as are possible explanations of why such violations are often observed in experiments.