Coherence of countably many bets

TL;DR

This paper explores conditions under which countably many bets can justify countable additivity of probabilities, extending de Finetti's finite betting argument to infinite scenarios and analyzing convergence conditions.

Contribution

It introduces two new convergence conditions for countably many bets and compares them with existing ones, advancing the understanding of probability justification in infinite betting frameworks.

Findings

Two new convergence conditions proposed and analyzed.

Comparison of new conditions with existing literature.

Insights into when countably many bets justify countable additivity.

Abstract

De Finetti's betting argument is used to justify finitely additive probabilities when only finitely many bets are considered. Under what circumstances can countably many bets be used to justify countable additivity? In this framework, one faces issues such as the convergence of the returns of the bet. Generalizations of de Finetti's argument depend on what type of conditions on convergence are required of the bets under consideration. Two new such conditions are compared with others presented in the literature.