Information algebras of coherent sets of gambles

In a recent paper Miranda & Zaffalon (2020) some results about compatibility or consistency of coherent sets of gambles or lower previsisons have been derived and it was remarked that these results were in fact results of the theory of information or valuation algebras (Kohlas, 2003). This point of view, however, was not worked out by Miranda & Zaffalon (2020). In this paper this issue is taken up and it is shown that coherent sets of gambles, strictly desirable sets of gambles, coherent lower and upper previsions indeed form idempotent information algebras. Like in group theory, certain results concerning particular groups follow from general group theory, so many known results about desirable gambles, lower and linear previsions are indeed properties of an information algebra and follow from the corresponding general theory. Some of these results are discussed in this paper, but there are doubtless many other properties which can be derived from the theory of information algebra.