Applied Logics Seminar Series (XXVII) -- A Simpler More Realistic Form of Savage's System
- LecturerProf. Haim Gaifman (Philosophy Department, Columbia University, USA)
Host: Churn-Jung Liau - Time2016-03-25 (Fri.) 15:30 ~ 17:30
- LocationAuditorium 106 at IIS new Building
Abstract
The talk will present two mathematical results that lead to considerable simplifications of Savage’s well known system, in which subjective probabilities and utilities are derived from the agent preference relation over acts. Using axioms P1-P6, Savage first derives a qualitative probability and then shows that it is fully represented by a unique, quantitative finitely-additive probability distribution. In order to do this he has to assume that the underlying Boolean algebra of events is a σ-algebra (i.e., closed under countable unions). Savage is not happy with this assumption; he justifies it on expediency grounds: there seems to be no way of avoiding it in his proof. The first mathematical result, on which I shall focus most, is a technique by which the desired quantitative probability is derivable without presupposing the σ-algebra assumption. Savage would have been happy with this simplification, which strengthens his system.
The other simplification consists in eliminating the constant-acts assumption, which states that for every consequence there is a constant act that has this consequence in every state. This assumption is used by Savage in order to assign utilities to the consequences, so that the preference over acts is determined by their expected utilities. There are well-known difficulties with the constant act assumptions since it often has counter-intuitive implications. The proposed simplification consists in restricting the set of acts, but leaving it sufficiently rich for handling all scenarios which an idealized human agent can conceivably face.
These results are in two joint works with Yang Liu. I shall try to give a self-contained presentation, including a rough outline of Savage’s system.
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