I grappled with Savage a few years ago, to re-establish the possibility of unbounded utility functions and a proper treatment of paradoxical games like St. Petersburg within a minimally modified version of his framework. It will take me a while to reload that material into my brain.
Meanwhile, your construction in the case of countable S depends on having only finite additivity, to construct a countable set of null sets whose union is not null. Finite vs. countable additivity was a live issue when Savage was writing, but my impression (as a non-professional in this area) is that finite additivity eventually fell out of favour. Too much becomes awkward or goes wrong, and non-measurable sets and the Banach-Tarski paradox are a small price to pay.
Is there still a counterexample to Strong Dominance in the setting of countable additivity, and taking S to be a measurable space with events limited to measurable subsets?
ETA: On countable additivity, Savage mentions that his construction might be carried out in that context also and cites Villegas has having done so (pp 43-44 of the 1972 edition; Villegas “On qualitative probability sigma-algebras,” Annals of Mathematical Statistics, 35, 1787-1796.).
Hi! Thank you so much for this thoughtful comment. It’s great connecting someone who has spent time deep in the weeds of Savage. It took me a long time to understand St. Petersburg, Banach-Tarski, and Villegas-Debreu: definitely not easy!
Loosely speaking, the tension here comes down to two ingredients implied by Savage’s axioms:
Universal domain (all possible mappings from states to outcomes are valid acts).
Expected Utility (EU)
I agree Countable Additivity (CA) is the workhorse of most fields, including empirical ML. The exceptions are decision theorists and philosophers, who remain enthusiastic about the assumption of the Universal Domain. I personally think preserving the Universal Domain is specifically important for the philosophy of AGI decision-making: an AGI, by definition, should be general and capable of evaluating all acts.
Finite Additivity (FA) is a consequence of the Universal Domain alongside other assumptions. Savage himself noted that his probability measures have to be FA. In our work, we further prove that Savage’s probability measures must also be locally strongly finitely additive under the axiom of constructibility.
As you pointed out, Villegas (and Debreu) provide the axiomatic foundation of Subjective Expected Utility with Countable Additivity by abandoning the universal domain, and our impossbility does not apply.
To summarize the theoretical landscape of what is and isn’t possible:
Non-universal domain + EU + CA + Strong Dominance: Totally possible (and widely used).
Universal Domain + EU + CA: Not mathematically possible.
Universal Domain + EU + FA + Strong Dominance: Not possible (our result).
Universal Domain + non-EU + FA + Strong Dominance: Explored by Machina and Schmeidler (1992). To my knowledge, it is unknown if Strong Dominance is compatible here.
My own taste is towards countable additivity and measurable events. You might be interested in the reformulation of Savage on that basis that I mentioned, motivated by my presumption that deducing bounded utility from Savage’s axioms indicated a problem with those axioms rather than demonstrating that utility must be bounded. I put it on ResearchGate, as the only journal that sent it out for review, while not having any technical objections, decided that it wasn’t substantial enough to publish.
But the paper is all in that countable additivity framework, so may not be close to your own concerns.
I grappled with Savage a few years ago, to re-establish the possibility of unbounded utility functions and a proper treatment of paradoxical games like St. Petersburg within a minimally modified version of his framework. It will take me a while to reload that material into my brain.
Meanwhile, your construction in the case of countable S depends on having only finite additivity, to construct a countable set of null sets whose union is not null. Finite vs. countable additivity was a live issue when Savage was writing, but my impression (as a non-professional in this area) is that finite additivity eventually fell out of favour. Too much becomes awkward or goes wrong, and non-measurable sets and the Banach-Tarski paradox are a small price to pay.
Is there still a counterexample to Strong Dominance in the setting of countable additivity, and taking S to be a measurable space with events limited to measurable subsets?
ETA: On countable additivity, Savage mentions that his construction might be carried out in that context also and cites Villegas has having done so (pp 43-44 of the 1972 edition; Villegas “On qualitative probability sigma-algebras,” Annals of Mathematical Statistics, 35, 1787-1796.).
Hi! Thank you so much for this thoughtful comment. It’s great connecting someone who has spent time deep in the weeds of Savage. It took me a long time to understand St. Petersburg, Banach-Tarski, and Villegas-Debreu: definitely not easy!
Loosely speaking, the tension here comes down to two ingredients implied by Savage’s axioms:
Universal domain (all possible mappings from states to outcomes are valid acts).
Expected Utility (EU)
I agree Countable Additivity (CA) is the workhorse of most fields, including empirical ML. The exceptions are decision theorists and philosophers, who remain enthusiastic about the assumption of the Universal Domain. I personally think preserving the Universal Domain is specifically important for the philosophy of AGI decision-making: an AGI, by definition, should be general and capable of evaluating all acts.
Finite Additivity (FA) is a consequence of the Universal Domain alongside other assumptions. Savage himself noted that his probability measures have to be FA. In our work, we further prove that Savage’s probability measures must also be locally strongly finitely additive under the axiom of constructibility.
As you pointed out, Villegas (and Debreu) provide the axiomatic foundation of Subjective Expected Utility with Countable Additivity by abandoning the universal domain, and our impossbility does not apply.
To summarize the theoretical landscape of what is and isn’t possible:
Non-universal domain + EU + CA + Strong Dominance: Totally possible (and widely used).
Universal Domain + EU + CA: Not mathematically possible.
Universal Domain + EU + FA + Strong Dominance: Not possible (our result).
Universal Domain + non-EU + FA + Strong Dominance: Explored by Machina and Schmeidler (1992). To my knowledge, it is unknown if Strong Dominance is compatible here.
Universal Domain + non-EU + CA + Strong Dominance: I believe this is also an open problem. For related thought, see Gul and Pesendorfer (2014) Expected Uncertain Utility.
Thanks again for engaging so deeply!
My own taste is towards countable additivity and measurable events. You might be interested in the reformulation of Savage on that basis that I mentioned, motivated by my presumption that deducing bounded utility from Savage’s axioms indicated a problem with those axioms rather than demonstrating that utility must be bounded. I put it on ResearchGate, as the only journal that sent it out for review, while not having any technical objections, decided that it wasn’t substantial enough to publish.
But the paper is all in that countable additivity framework, so may not be close to your own concerns.