Very interesting!

How do you recommend choosing between somewhere-uniquely-optimal actions if there are multiple? I'm imagining the worst case where we still have several left that we're clueless between.^[1]

And are all convex combinations of (lotteries over) somewhere-uniquely-optimal actions also somewhere-uniquely-optimal? (Let’s suppose our set of actions is closed under convex combinations, so somewhere-unique-optimality is defined with respect to that set.)

EDIT: The answer is no. 60-For-Sure, 100-If-True and 100-If-False are all somewhere-uniquely-optimal (under convex combinations), but 60-For-Sure dominates the convex combination of picking one of 100-If-True or 100-If-False at random, with 50% probability each.

1. ^{^}

   I suppose the actual worst case is where inaction is ~always somewhere uniquely-optimal, and especially when building a portfolio to give across organizations, but that will depend on the particulars involved.

Just to clarify the example and concept: if we added the option c-For-Sure, which gives c with 100% probability, to the set of options in your example, it would be somewhere optimal iff c is at least 50, right?

Take Pr(X)=50%. Then 50-For-Sure, 100-If-True and 100-If-False all have EV 50.

And further, any lottery/gamble between any three of them (50-For-Sure, 100-If-True and 100-If-False) has EV 50, if the draw of the option is statistically independent from X's probability distribution.

Bigger seed grants!

One could argue there is a probability above exactly 0 of any given quantity being a probabilistic mixture involving a distribution with infinite or undefined expected value. In this case, all distributions would have an infinite or undefined expected value? For me this is a bitter bullet to bite than fully rejecting distributions with infinite or undefined expected value.

I think you should just expect this, and the answer is not to deny the possibility of St Petersberg lotteries in objective quantities, but to figure out a good way to deal with them (e.g. ignore small enough probabilities, use a bounded utility function, use commitments, use bracketing of some form), or accept that they raise difficult normative problems.

Do you see any undesirable implications of believing in ghosts which have exactly 0 measurable effects on the world? I think this is effectively the same as not believing in such ghosts.

Are you saying you believe in the existence with 100% credence in anything (such as ghosts) that is not ruled out by current evidence, as long as its effects couid be arbitrarily small and are so far indistinguishable from its nonexistence? Or must it also have no important normative implications (under classical utilitarianism?)?

Ghosts could be conscious, experience pleasure and suffering and care about what you do. Some could be vengeful and want harm to fall upon those that have caused them harm in their lives (whether or not they enact it themselves). Others could want to see the happiness of loved ones. Others could want their descendants to live up to their expectations (e.g. in education, work, family, religious adherence), and not care much about their happiness. A large share could be horrified by modern secularism. It could be that every human that dies becomes a ghost indefinitely.

What do you think about the St Petersburg problem now?

I think your arguments re infinities conflict with Occam's razor, and the principle of indifference should be applied across models within the same complexity (or submodels), otherwise you will assign 0 or too little credence to simpler models that are special cases, e.g. parameter value=0 or effectively eliminating some type of feature. There are infinitely many ways the universe could be more complex, and arbitrarily more complex, than you'd guess.

A flat universe effectively has one fewer parameter and is simpler. So it shouldn't get 0 credence. Among flat universes, the bounded/finite ones also have extra parameters for the boundaries or shape of the universe (compared to something that looks like R^3), so the infinite one shouldn't get 0 credence.

(There probably are many ways for the universe to be infinite spatially in its shape, too, but those are more complex than R^3.)

EDIT:

As an aside, there have been many cases where quantities in physics were assumed to be 0, but then turned out to be just small values, like the mass of neutrinos. It often makes sense to round a quantity to 0 for simplicity, but a sufficiently small value would explain exactly the same empirical evidence.

Note that I never suggested to assign 0 probability to anything. I think this leads to further examples to illustrate my point:

Would you assign exactly 0 probability to photons having exactly 0 mass?

Exactly 0 probability to there being no additional fundamental force, because it could just be vanishingly weak? And shouldn't this get you to infinitely many fundamental forces? For any finite set of fundamental forces, you could posit another one and just say it's very weak.

Exactly 0 probability to there not being ghosts, because their effects could just be very small or rare?

I think consistently applying your arguments suggest you should assign 0 probability in these cases, and so your models blow up in complexity and you become too credulous, contrary to Occam's razor.

I like the principle of indifference. However, I think infinities are unfalsifiable in principle. In this case, does it make sense for me to attribute probabilities to them? If I did, they would be just metaphysical priors that can never be updated by any evidence.

St Petersburg-like lotteries, defined in terms of your Bayesian credences, don't require assigning positive probability to any possible infinities out there in the world.

I'll leave the rest in a footnote, because it's not that relevant to the point I've been making so don't plan to go further with it, but I already wrote it and it may be of interest to you.^[1]

1. ^{^}
   1. I think falsifiability for a Bayesian could just mean we can imagine evidence that would warrant assigning very very low credences to the hypothesis. Actual infinities are often falsifiable in principle in this way, in specific cases. I think the probability you should assign to you being infinite in spatial size is extremely small. Similarly for the Earth being infinite in size.
   2. In the case of the spatial extent of the universe being infinite, it seems hard to falsify now only because all the evidence we've gathered so far is in fact consistent with it (or no less consistent than with a bounded/finite universe), and it remains one of the standard models used by experts. My own view is that it’s more likely than not infinite in spatial extent, because it’s the simplest model consistent with the evidence.
   3. I think our credences that the universe is infinite in spatial extent should increase with our credences that the universe is globally flat (0 global curvature), which has been measurable. If we had good evidence of nonzero global curvature, that would be decent (but not overwhelming) evidence that the universe is finite/bounded, because it would rule out the most plausible infinite models of the universe.
   4. I agree that we can sometimes compare actions with infinite expected cost-effectiveness. I'm most partial to expansionist views of some kind, as the most complete (given precise probabilities).

I object to relying on infinities (not arbitrarily large finities) to guide decisions because they do not explain more empirical evidence than arbitrarily large finities.

But do they deserve to be privileged at the exclusion of the infinite EV distributions if and when the latter are at least as consistent with the evidence? Why? Shouldn't you use some principle of indifference or symmetry here?

If there's no finite upper bound on what the value the specific probability distribution can take, how can you be 100% confident it is not a probabilistic mixture with a distribution with infinite or undefined EV (but finite for every actual value), like 0.0001% probability to it being drawn from something like a St. Petersburg lottery?

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 total value should be over all of the future and possible existence, that you're 100% confident in, under your course of action if where 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).