Abstract.
In his foundation of probability theory, Bruno de Finetti devised
a betting scheme where a bookmaker offers bets on the outcome of
events 
occurring in the future. He introduced a criterion
for coherent bookmaking, and showed that coherent betting odds are
given by some probability distribution. While de Finetti dealt
with yes-no events and boolean propositional logic, Mundici
generalized the theory to the continuous spectrum events
formalized within Łukasiewicz logic.
Both de Finetti and Mundici assume that the bookmaker/bettor roles can be interchanged. In this paper we deal with a more realistic situation, dropping the reversibility assumption. Working in the framework of Łukasiewicz logic, we introduce a coherence criterion for non-reversible bookmaking. Our main tool is given by `imprecise probabilities', which are formulated in terms either of compact convex sets of probabilities or equivalently in terms of suitable sublinear functionals. Our main result states that our coherence criterion arises from imprecise probabilities just as de Finetti's criterion arises from probabilities.
Throughout, we will work with MV-algebras. They play the same role
for Łukasiewicz logic as Boolean algebras play for classical
logic. Unital abelian lattice-ordered groups will provide an
intermediate structure: while being categorically equivalent to
MV-algebras, they are more akin to the Banach space 
.
Functional analytic methods
are used for the proof of our
main result.
© 2013 by Walter de Gruyter Berlin Boston
