Weak Dutch Books with Imprecise Previsions

Uncertainty assessments for imprecise previsions based on coherence and related concepts require that the suprema of certain random numbers (interpreted as gains) are non-negative. The extreme situation that a supremum is zero represents what is called a Weak Dutch Book (WDB) in a betting interpretation language. While most of the previous dedicated literature focused on WDBs for de Finetti's coherence with precise probabilities, in this paper we analyse the properties of WDBs with imprecise previsions, notably for conditional (Williams') coherent lower previsions. We show that WDB assessments ensure a certain `local precision' property and imply, in the agent's evaluation, some kind of `protection' against real losses. Further, these properties vary with the consistency notion we adopt, tending to vanish with weaker ones. A generalisation of the classical strict coherence and other alternative approaches to WDBs are also discussed.