I think the more important takeaway is that the (countable) sure thing principle and transitivity together rule out preferences allowing St. Petersburg-like lotteries, and so “unbounded” preferences.
It discusses more ways preferences allowing St. Petersburg-like lotteries seem irrational, like choosing dominated strategies, dynamic inconsistency and paying to avoid information. Furthermore, they argue that the arguments supporting the finite Sure Thing Principle are generally also arguments in favour of the Countable Sure Thing Principle, because they don’t depend on the number of possibilities in a lottery being finite. So, if you reject the Countable Sure Thing Principle, you should probably reject the finite one, too, and if you accept St. Petersburg-like lotteries, you need to in principle accept behaviour that seems irrational.
They also have a general vNM-like representation theorem, dropping the Archimedean/continuity axiom, and replacing the Independence axiom with Countable Independence, and with transitivity and completeness, they get utility functions with values in lexicographically ordered ordinal sequences of bounded real utilities. (They say the sequences can have any ordinal to order them, but that seems wrong to me, since I’d think infinite length lexicographically ordered sequences get you St. Petersburg-like lotteries and violate Limitedness, but maybe I’m misunderstanding. EDIT: I think they meant you can have a an infinite sequence of dominated components, not an infinite sequence of dominating components, so you check the most important component first, and then the second, and continue for possibly infinitely many. Well-orderedness ensures there’s always a next one to check.)
Thanks for the reference, seems better than my post and I hadn’t seen it. (I think it’s the version of the argument I allude to in this comment.)
Note that if you have unboundedly positive and unboundedly negative outcomes, then you must also violate the weaker version of the countable sure thing principle with weak inequality instead of strict inequality. Violating the weak version of the sure thing principle seems much worse to me. And I think most proponents of unbounded preferences would advocate for them to point in both directions, so they run into this stronger problem.
That said, countability and strength of the sure thing principle are orthogonal: countable weak sure thing + finite strong sure thing --> countable strong sure thing. Negative utilities are only relevant as a response to someone who is OK saying “a 1% chance of an infinitely good outcome is just as good as a sure thing,” since that view is inconsistent if the 1% chance of an infinitely good outcome could be balanced out by a 2% chance of an infinitely bad outcome.
I think the more important takeaway is that the (countable) sure thing principle and transitivity together rule out preferences allowing St. Petersburg-like lotteries, and so “unbounded” preferences.
I recommend https://onlinelibrary.wiley.com/doi/abs/10.1111/phpr.12704
It discusses more ways preferences allowing St. Petersburg-like lotteries seem irrational, like choosing dominated strategies, dynamic inconsistency and paying to avoid information. Furthermore, they argue that the arguments supporting the finite Sure Thing Principle are generally also arguments in favour of the Countable Sure Thing Principle, because they don’t depend on the number of possibilities in a lottery being finite. So, if you reject the Countable Sure Thing Principle, you should probably reject the finite one, too, and if you accept St. Petersburg-like lotteries, you need to in principle accept behaviour that seems irrational.
They also have a general vNM-like representation theorem, dropping the Archimedean/continuity axiom, and replacing the Independence axiom with Countable Independence, and with transitivity and completeness, they get utility functions with values in lexicographically ordered ordinal sequences of bounded real utilities. (They say the sequences can have any ordinal to order them, but that seems wrong to me, since I’d think infinite length lexicographically ordered sequences get you St. Petersburg-like lotteries and violate Limitedness, but maybe I’m misunderstanding. EDIT: I think they meant you can have a an infinite sequence of dominated components, not an infinite sequence of dominating components, so you check the most important component first, and then the second, and continue for possibly infinitely many. Well-orderedness ensures there’s always a next one to check.)
Thanks for the reference, seems better than my post and I hadn’t seen it. (I think it’s the version of the argument I allude to in this comment.)
Note that if you have unboundedly positive and unboundedly negative outcomes, then you must also violate the weaker version of the countable sure thing principle with weak inequality instead of strict inequality. Violating the weak version of the sure thing principle seems much worse to me. And I think most proponents of unbounded preferences would advocate for them to point in both directions, so they run into this stronger problem.
That said, countability and strength of the sure thing principle are orthogonal: countable weak sure thing + finite strong sure thing --> countable strong sure thing. Negative utilities are only relevant as a response to someone who is OK saying “a 1% chance of an infinitely good outcome is just as good as a sure thing,” since that view is inconsistent if the 1% chance of an infinitely good outcome could be balanced out by a 2% chance of an infinitely bad outcome.