A Dutch Book theorem for partial subjective probability

TL;DR

This paper extends the Dutch Book theorem to partial subjective probabilities, showing how partial probability can be justified through betting arguments similar to classical probability, and compares different bet concepts.

Contribution

It introduces the concept of partial bets and partial Dutch Books, proving a Ramsey-De Finetti type theorem for partial probability, and analyzes different notions of betting in this context.

Findings

Partial probability can be justified via subjective betting arguments.

A partial Dutch Book theorem analogous to classical results is established.

Different concepts of bets on events versus sentences are compared in partial probability.

Abstract

The aim of this paper is to show that partial probability can be justified from the standpoint of subjective probability in much the same way as classical probability does. The seminal works of Ramsey and De Finetti have furnished a method for assessing subjective probabilities: ask about the bets the decision-maker would be willing to place. So we introduce the concept of partial bet and partial Dutch Book and prove for partial probability a result similar to the Ramsey-De Finetti theorem. Finally, we make a comparison between two concepts of bet: we can bet our money on a sentence describing an event, or we can bet our money on the event itself, generally conceived as a set. These two ways of understanding a bet are equivalent in classical probability, but not in partial probability.