Recent mathematical developments in decision analysis

Decision analysts use mathematics in two conceptually distinct, though often practically indistinguishable, ways. Firstly, they use mathematics to define and explore the implications of the behavioural assumptions upon which their models are based. Secondly, decision analysts use numerical calculations, and there is a need to perform those calculations accurately and efficiently. Recent developments in decision analysis have needed mathematics in both respects. New methodologies and models based upon weaker assumptions than those implicit in subjective expected utility theory have been proposed, and debate continues on the value of these. The calculations required in the application of most methodologies are mathematically simple, but the investigation of the sensitivity of these calculations to variations in the input data is often far from simple. Only with recent advances in computing power has it become possible to consider undertaking full sensitivity analyses, which may require the solution of very many mathematical programs, some of which may be nonconvex.