Imprecise probabilities, bets and functional analytic methods in Łukasiewicz logic

Abstract

In his foundation of probability theory, Bruno de Finetti devised a betting scheme where a bookmaker offers bets on the outcome of events φ 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 Lukasiewicz 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 Lukasiewicz 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 (see Section 5). Our main result is Theorem 8.3 which 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 Lukasiewicz 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 C(X). Functional analytic methods, developed in Section 6, are used for the proof of our main result.

Hahn–Banach theorems for MV-algebras

“…½0; 1 is an MV-seminorm then define u :¼ fs : A ! ½0; 1js ug: As in Lemma 4.2 of Fedel et al (2012), the maps u 7 ! u and X 7 !…”

“…We give in Remark 2.4 a probabilistic interpretation of MVseminorms as 'imprecise (upper) probabilities'. One can see Fedel et al (2012) for an analysis of 'imprecise probabilities' in Łukasiewicz logic.…”

“…½0; 1 is an MV-seminorm. By Theorem 6.3 of Fedel et al (2012), u ¼ u if and only if u is strongly normalized, i.e. uða þ q Á 1Þ ¼ uðaÞ þ q whenever q 2 ½0; 1 such that a þ q Á 1 is defined in A.…”

Stable Non-standard Imprecise Probabilities

“…We begin by recalling a characterisation of coherence as avoiding inadmissible bets given in [6] (the terminology "bad bet" was used there.) Definition 5.…”

“…In [6] it is shown that an assessment of upper probability avoids inadmissible bets iff it can be extended to an upper prevision. The result was shown first, although in a different setting, by Walley in [12].…”