Jeffrey does talk about this in his book! Denote the probability of an event as P(E), and the expected value as V(E). Now suppose we cut an event A into parts B and C. We must have that V(A) = V(B)P(B|A) + V(C)P(C|A). Using this, we can cut the world up into small events which we’re comfortable assigning values to, and then put things back together into expected values for larger events. Basically exactly what you’d do normally, but no events are distinguished as “outcomes” so you can start wherever you want.
The JB axioms don’t assume anything like countable additivity, so an event like “St. Petersburg Lottery” needs a valuation which is consistent (in the above-mentioned sense) with the value of all other events, but there isn’t (necessarily) a computation which tells you the value of the infinite sum, the way there is for finite sums. We can add axioms which constrain values in situations like that, to avoid absurd things; but since it isn’t clear how to evaluate infinite sums in general, it makes sense to keep those axioms weak.
I think of this as a true result: the value of infinite lotteries is subjective (even after we know how to value all their finite parts). It has a lot of coherence constraints, but not enough to fully pin down a value. This means we don’t have to worry about all the messy nonsense of trying to evaluate divergent sums.
However, I think if we assume that any sub-event of the st petersburg lottery in which we still have a chance of winning something has a positive value (which seems very reasonable), then we can prove that the total value of the lottery is not any real number, by splitting off more and more of the finite sub-lotteries and arguing that the total value must exceed each.
Where exactly are the guardrails between ordinary lotteries and impossible lotteries, and how might I come to realize (from within the Jeffrey-Bolker framework) that I’m not allowed to believe that a St. Petersburg lottery is real?
The continuity axiom is what stops you from having non-archimedean values (just like with VNM), so that’s where the buck stops. If our preferences respect continuity, then we have to choose between believing St-petersburg-like lotteries are possible, vs believing enough niceness axioms about the values of infinite lotteries to prove that St. Petersburg has a value greater than all the reals.
For me, it seems like a pretty obvious choice to reject continuity (as a rationality condition that must apply to all rational minds—it could very well apply to humans in a practical sense). Continuity seems poorly-motivated in the first place.
Jeffrey does talk about this in his book! Denote the probability of an event as P(E), and the expected value as V(E). Now suppose we cut an event A into parts B and C. We must have that V(A) = V(B)P(B|A) + V(C)P(C|A). Using this, we can cut the world up into small events which we’re comfortable assigning values to, and then put things back together into expected values for larger events. Basically exactly what you’d do normally, but no events are distinguished as “outcomes” so you can start wherever you want.
The JB axioms don’t assume anything like countable additivity, so an event like “St. Petersburg Lottery” needs a valuation which is consistent (in the above-mentioned sense) with the value of all other events, but there isn’t (necessarily) a computation which tells you the value of the infinite sum, the way there is for finite sums. We can add axioms which constrain values in situations like that, to avoid absurd things; but since it isn’t clear how to evaluate infinite sums in general, it makes sense to keep those axioms weak.
I think of this as a true result: the value of infinite lotteries is subjective (even after we know how to value all their finite parts). It has a lot of coherence constraints, but not enough to fully pin down a value. This means we don’t have to worry about all the messy nonsense of trying to evaluate divergent sums.
However, I think if we assume that any sub-event of the st petersburg lottery in which we still have a chance of winning something has a positive value (which seems very reasonable), then we can prove that the total value of the lottery is not any real number, by splitting off more and more of the finite sub-lotteries and arguing that the total value must exceed each.
The continuity axiom is what stops you from having non-archimedean values (just like with VNM), so that’s where the buck stops. If our preferences respect continuity, then we have to choose between believing St-petersburg-like lotteries are possible, vs believing enough niceness axioms about the values of infinite lotteries to prove that St. Petersburg has a value greater than all the reals.
For me, it seems like a pretty obvious choice to reject continuity (as a rationality condition that must apply to all rational minds—it could very well apply to humans in a practical sense). Continuity seems poorly-motivated in the first place.