Can free evidence be bad? Value of information for the imprecise probabilist

This paper considers a puzzling conflict between two positions that are each compelling: (A) it is irrational for an agent to pay to avoid ‘free’ evidence before making a decision, and (B) rational agents may have imprecise beliefs and/or desires. Indeed, we show that Good’s theorem Good (1967) concerning the invariable choice-worthiness of free evidence does not generalise to the imprecise realm, given the plausible existing decision theories for handling imprecision. A key ingredient in the analysis, and a potential source of controversy, is the general approach taken for resolving sequential decision problems – we make explicit what the key alternatives are and defend our own approach. Furthermore, we endorse a resolution of the aforementioned puzzle – we privilege decision theories that merely permit avoiding free evidence over decision theories for which avoiding free evidence is uniquely admissible. Finally, we situate this particular result about free evidence within the broader ‘dynamic-coherence’ literature.