The expected value of the St. Petersburg lottery is 1 + 1 + ... = +inf. It involves finite terms, but infinitely many terms. I meant to relate f(x) = x in my comment to the expected value of the St. Petersburg lottery. If this involved an arbitrarily large number of terms, its expected value would be arbitrarily large, but not infinite. 

I don't understand what you're objecting to. Are you saying all probability distributions should only take finitely many possible values? If so, do you have an upper bound on what the values should be over all of the future and possible existence, that you're 100% confident in, if you do nothing? (It doesn't have to be the least upper bound / tight.)

Below is how Claude thinks Adam (the author of the article) would object to your comments. The objections make sense to me. Any reactions?

Claude is dumb (at least without further critique and verification, and usually with), and your prompt basically put it on the task of defending the position, not actually assessing the arguments fairly. So it turned up bad arguments.

I doubt the author would respond this badly.

Taking bet A doesn’t require any commitment. My argument just uses backward induction (+ignoring statewise incomparability), which you should generally use in sequential choice situations, or else you'll be worse off in many situations, even with sharp probabilities.

It allows unsharpness. Having unsharp probabilities does not require sequential decisions to be made independently.

that very concession is what Elga reads as victory: the imprecision has stopped doing the one thing it was introduced to do.

The argument against unsharp probabilities is defeated. We just have to treat them in certain ways. The summary of the paper here missed one way we could treat them, and claimed too much against another (if we accept commitments or resolute choice in other cases).

Solution to which problem? I am not sure what is supposed to be problematic.

That if you use backward induction on acting rationally at each step, you will be worse off. You will predict later that you'll change your mind, unless you can force your future self to honor a commitment (or plan) you'd no longer want to keep when it actually comes time to honor it.

EDIT: my bad, the problem is that if you don't use commitments, you could be worse off. Using backward induction in the Sally argument actually works fine, doesn't leave you (or Sally) worse off and doesn't require any commitment.

However, I do not think it can be infinite. The function f(x) = x can take an arbitrarily large value, but not an infinite value (its range is the set of real numbers).

St Petersburg doesn't require any state to have infinite value. Its value is (canonically) 2^n with probability 1/2^n for each n at least 1. Always finite actual value, but infinite expected value.

Why does Parfit's hitchhiker pose a problem?

Because the same kind of solution is available to someone with unsharp probabilities in Elga's scenario, if you're treating them fairly.

The St. Petersburg paradox involves an infinite expected payoff, and I reject infinite worlds.

It doesn't require an infinite world, only that you can't be 100% confident in any finite upper bound on your impact that you specify, and that there are infinitely many ways that the world could be (due largely to not full certainty about physics).

(But also 0% to infinite worlds seems epistemically immodest, doesn't treat the evidence on each side fairly, and is poorly argued, imo. But I don’t want to rehash this.)

In Sally's case, her money is the only thing that matters.

Why can't the fact that she'd pick a dominated sequence or regret it if she rejects both bets matter to her after rejecting bet A?

I understand one should accept bet A based on that strategy. However, unsharp probabilities are supposed to allow for accepting or rejecting A?

They don't have to in every case. If it was A in isolation, and no other decisions, then yes, both rejecting and accepting should be permissible. But that's not the case presented to us.

As a rational actor with no useful information

mood

3. SEQUENCE — Sequences of actions can be assessed for rationality independently of their parts: each of "reject A" and "reject B" can be individually permissible while the sequence "reject-A-then-reject-B" is impermissible. Elga turns the same Sally argument on it. SEQUENCE makes rejecting Bet B fine when no Bet A preceded it but irrational when it would complete the bad sequence—so it, too, imposes different requirements across two situations Sally can see are identical in everything she cares about. Hence SEQUENCE fails.

I don't think this is a strong argument. There are other cases where you should make commitments that you would later be inclined to break, like Parfit's hitchhiker, and St. Petersburg lotteries with unbounded utility functions. The latter is an argument that unbounded utility functions are irrational, based on similar logic.

Furthermore, "imposes different requirements across two situations Sally can see are identical in everything she cares about". What if I do care about the differences? Or, is this any worse than picking numbers to ensure precision for no better reason than that they occured to you? Because that's what it takes to produce arbitrarily precise probabilities if you fix what information is available to you in realistic settings.

Also, here's another way someone with unsharp probabilities might handle this situation. In summary, I should accept bet A at the start to rule out the possibility of picking a dominated sequence:

1. If I accept bet A at the start, then the probability that I pick the dominated sequence (rejecting both) is 0.
2. If I reject bet A at the start and if I can't guarantee that I will accept bet B next, then there's some chance that I pick the dominated sequence.

If I compare 1 and 2 statewise, then 1 > 2 with some probability, and 1 and 2 are incomparable otherwise. In other words, either 1 beats 2, or I have no decisive reasons favouring either and I can ignore those cases. So I decide on the cases where 1 beats 2 and accept bet A at the start.

What would you recommend to handle cluelessness and unawareness about the far future and acausal influence, where contact and feedback may not be very practical or sufficiently informative? Maybe with the far future, we should just be extremely patient, possibly over 1000 years or more, and wait for that feedback anyway, but during that time, lots of confounders and clueless-making events could come up.

Or we simulate things in a lot of detail to try to get that feedback artificially, although that may mean simulating and realizing a lot of suffering.

(1) not "impartial" in the sense that some moral patients/consequences are bracketed out for not very well-motivated reasons, or (2) not action-guiding.

If you think it's fair to map the "same" moral patient across worlds to themselves, assuming at least some transworld identity, then I'd guess we could work out counterpart relations extending transworld identity to cover contingent moral patients, maybe along the lines of Thomas, 2019, pp.30-31 where, roughly, we map each contingent moral patient under one choice to the statistical average of the contingent moral patients — across all of spacetime and the whole multiverse if we're in one — under the other choice.^[1]

At first pass, if we're clueless about the total welfare effects for any action or comparing any two actions, I'd guess we'd be clueless about the average contingent moral patient, and, at least on bottom-up bracketing, just bracket them out and ignore them. This would mean the theory acts like the strict person-affecting view, where only currently existing (or at least "necessary") moral patients count, and we ignore (at least) the rest. That might sound bad, but we ignore them because we're consistently clueless about whether their counterparts are better or worse off in expectation, which seems like a good principled reason to do that.

This could be practically action-guiding. You'd probably focus on near-term human welfare, because currently existing non-human animals that you could "affect" cost-effectively don't typically live long enough for your actions today to help them (e.g. most live <2 years before slaughter, often <1 year), and you're clueless about the welfare of the counterparts of the future/contingent animals you could "affect" cost-effectively.

But maybe speculative scenarios, including how TAI might play out, also make you clueless about how to benefit currently existing humans if, before bracketing, you're aggregating and taking expectations over their entire futures or their current preferences pointing at things at least a few years away, and can't bracket out the speculative scenarios. I imagine you could get around this by using transworld identity at each point of time^[2] and do bracketing before aggregating over their futures, so you can bracket out the periods of time you're clueless about.

1. ^{^}

   Maybe with some extra accounting for the number of moral patients to do. And this could get trickier with infinities.

2. ^{^}

   Could need to account for relativity theory in some way.

Would you say bracketing isn't fully impartial? It doesn’t seem less impartial than Scanlon's Greater Burden Principle (or Tom Regan's harm principle), which roughly says to choose between two actions, you should prioritize the individual(s) with the strongest claim or who has the most to lose counterfactually between options (not necessarily the worst off, to contrast with Rawls' Difference Principle/maximin/leximin). 

I guess methods of bracketing where admissible bracketings must group certain individuals, e.g. spatiotemporally or by causal paths affecting them, are more partial. But if you go with the unconstrained version, it seems basically impartial?

It is sensitive to differences between worlds beyond the number of individuals at each welfare level, because it has to identify each individual across worlds to take individual differences.