Countable Additivity and Subjective Probability

While there are several arguments on either side, it is far from clear as to whether or not countable additivity is an acceptable axiom of subjective probability. I focus here on de Finetti’s central argument against countable additivity and provide a new Dutch book proof of the principle, to argue that if we accept the Dutch book foundations of subjective probability, countable additivity is an unavoidable constraint. 1 Guess the number 2 The betting set-up 3 Consequences versus foundations 4 Uniform distributions 5 A Dutch book argument 6 Subjectivity ensured 7 Summary Largely due to Kolmogorov’s influence ([1933]), axiomatizations of the mathematical theory of probability now include the principle of countable additivity, or an equivalent principle, as standard. But de Finetti, one of the pioneers of the subjective interpretation of probability, argued against its acceptance (de Finetti [1970]). I shall give a brief overview of some of the central arguments for and against the adoption of countable additivity for subjective probability, and I will argue that they are inconclusive. I will then go on to look more closely at de Finetti’s view, before presenting a simple Dutch book argument in favour of the principle, which will mean that we cannot reject countable additivity without abandoning the simplest and most intuitive foundations for subjective probability: the betting set-up and the ensuing Dutch book argument. Ironically from de Finetti’s point of view, accepting countable additivity ensures that the interpretation retains its subjective flavour, for the principle prevents subjective probability being bolstered into a logical interpretation.