Charges and Bets: A General Characterisation of Common Priors

The seminal no betting theorem on the equivalence of common priors and absence of agreeable bets obtains only over compact state spaces. We show here that this equivalence can be generalised to any infinite space if we expand the set of priors to include probability charges as priors. Going beyond the strict prior/no common prior dichotomy, we further uncover a fine-grained decomposition of the space of type spaces into continuum many subclasses in each of which an epistemic condition approximating common priors is equivalent to a behavioural condition limiting acceptable bets. Several additional concepts relating to approximations of common priors and type spaces admitting common priors are studied, elucidating more aspects of the structure of the class of type spaces.