Several consistency notions for lower previsions (coherence, convexity, others) require that the suprema of certain gambles, having the meaning of gains, are non-negative. The limit situation that a gain supremum is zero is termed Weak Dutch Book (WDB). In the literature, the special case of WDBs with precise probabilities has mostly been analysed, and strict coherence has been proposed as a radical alternative. In this paper the focus is on WDBs and generalised strict coherence, termed strict consistency, with imprecise previsions. We discuss properties of lower previsions incurring WDBs and conditions for strict consistency, showing in both cases how they are differentiated by the degree of consistency of the given uncertainty assessment.
