Abstract Quiggin ( J. Econ. Behav. Organization 3 (1982), 323-345) introduced anticipated ("rank-dependent") utility theory into decision making under risk. Questions have been raised about mathematical aspects of Quiggin′s analysis. This paper settles these questions and shows that a minor modification of Quiggin′s axioms leads to a useful and correct result, with features not found in other recent axiomatizations. Journal of Economic Literature Classification Number: D81.