Probability weighting functions

Expected utility (EU) theory is unable to accommodate the observed nonlinear weighting of probabilities. We outline three stylized facts on nonlinear weighting that a theory of risk must ideally address. These are that people overweight small probabilities and underweight large ones (S1), do not choose stochastically dominated options when such dominance is obvious (S2), and ignore very small probabilities and code extremely large probabilities as one (S3). We then show that the concept of a probability weighting function (PWF) is crucial in addressing S1–S3. A PWF is not, however, in itself, a theory of risk. PWFs need to be embedded within some theory of risk in order to have significant predictive content. The two main alternative theories that are relevant in this regard are rank-dependent utility (RDU) and cumulative prospect (CP) theory. RDU and CP explain S1, S2 but not S3. We outline the recent proposal of al-Nowaihi and Dhami for composite prospect (CPP) theory that uses the composite Prelec probability weighting function (CPF). CPF is axiomatically founded, is flexible, and, is parsimonious. CPP can explain all three stylized facts S1, S2, and S3.