  Several easy to understand and computationally tractable imprecise
probability models, like the Pari-Mutuel model, are derived from a given
probability measure P_0. In this paper we investigate a family of such models,
called Nearly-Linear (NL). They generalise a number of well-known models, while
preserving a simple mathematical structure. In fact, they are linear affine
transformations of P_0 as long as the transformation returns a value in [0,1] .
We study the properties of NL measures that are (at least) capacities, and show
that they can be partitioned into three major subfamilies. We investigate their
consistency, which ranges from 2-coherence, the minimal condition satisfied by
all, to coherence, and the kind of beliefs they can represent. There is a
variety of different situations that NL models can incorporate, from
generalisations of the Pari-Mutuel model, the {\epsilon}-contamination model
and other models to conflicting attitudes of an agent towards low/high
P_0-probability events (both prudential and imprudent at the same time), or to
symmetry judgments. The consistency properties vary with the beliefs
represented, but not strictly: some conflicting and partly irrational moods may
be compatible with coherence. In a final part, we compare NL models with their
closest, but only partly overlapping, models, neo-additive capacities and
probability intervals.
