Abstract

In his seminal work “the Foundations of Statistics,” Savage put forward a theory of subjective probabilities. The theory is based on a well-developed axiomatic theory of rational decision making. In constructing his system, additional problematic assumptions are however required. First, there is a Boolean algebra of events on which subjective probabilities are defined. Savage’s proof requires this algebra to be a σ-algebra. However, on Savage’s view, one should not require the probability to be σ-additive. He, therefore, finds the insistence on a σ-algebra peculiar and unsatisfactory. But he sees no way of avoiding it. Second, the assignment of utilities requires the constant act assumption: for every given consequence there exists a constant act which has that consequence in every state. This assumption is known to be highly counterintuitive. The paper on which this talk is based includes two mathematical results. The first, and the more difficult one, shows that the σ-algebra assumption can be dropped. The second states that, as long as utilities are assigned to finite gambles only, the constant act assumption can be replaced by the plausible, much weaker assumption that there are at least two non-equivalent constant acts.

In this talk, I will first review Ramsey and Savage’s classical decision-theoretic models. I will then discuss the notion of “idealized agents” in standard normative decision theory. I will argue that our simplified system, which is adequate for all the actual purposes for which the system is designed, involves a more realistic notion of idealized agents. This will be followed by a brief outline of our new technique of tripartition trees which leads to the construction of quantitative probabilities in Savage-style systems without the σ-algebra assumption.