The strength of preference and the splitting of the von Neumann-Morgenstern utility function

As a conclusion to a long-lasting discussion in german literature about expected utility theory, a separation of the von Neumann-Morgenstern utility function into two parts is proposed: One part measures the preferences under certainty, the other part the pure risk preferences. This separation is proven to be a necessary and sufficient condition for an expanded expected utility to measure the strength of preference under uncertainty.