Modeling and measuring the effects of vagueness in decision models

Because subjective probability assessments do not represent measurements of empirical quantities, they often exhibit vagueness, or ambiguity. We adopt a standard approach to modeling vagueness as a second-order probability (SOP) on a first-order probability. We define the expected value of including vagueness (EVIV) as the change in expected value due to explicit probabilistic representation of vagueness. We show that, under general conditions, the EVIV is always zero, regardless of the SOP or the loss or utility function. This shows that representing higher-order uncertainty as SOP never changes the output of a decision model. We analyze the value of reducing vagueness, and we provide a framework for measuring the expected value of reducing vagueness (EVRV) on a probability assessment using equivalent sample size and relative information multiple (RIM). We use influence diagrams to represent the flow of potential but unobserved evidence from a vagueness-reducing activity. We demonstrate our results for a simple problem implemented in Demos, a decision modeling software environment