Hi Abraham and Mal. Nice post.

"Welfare per year" = "population (animal-years per year)"*"welfare per animal-year" = "deaths per year"*"welfare per death". It is unclear to me whether "welfare per animal-year" varies more or less than "welfare per death". So it is also unclear to me whether population is a better or worse proxy for welfare than deaths per year as long as both proxies cover the same life stages.

Using more appropriate units (such as total annual deaths or days of experience) reveals that highly r-selected animals might dominate moral calculus to a greater degree than a naive estimate might suggest.

On the other hand, more r-selected animals will tend to be more abundant, and have a lower moral weight? I agree with the above because I think differences in moral weight may be very small, but I also believe they may be very large.

Standing population size is usually not a reliable proxy for comparing the likely scale of harm between different highly numerous species, due to significant differences in population throughput and life-history.

"adult population size"?

For example, for identical stable populations of ants and aphids, we might expect there to be over 200x as many aphid deaths, and over 7x as many aphid days of experience.

"stable adult populations"?

One could argue that 1 and 2 remain incomparable and that I have no reason to favor 2 over 1.

If the absolute value of the expected cost-effectiveness of 1 was astronomically larger than that of intervention 2, I think comparing the interventions would be similar to comparing intervention 1 with one with cost-effectiveness of 0 (burning money). It is very unclear whether the expected cost-effectiveness of 1 is positive or negative. So it would be close to arbitrary which intervention has the highest expected cost-effectiveness.

Another thing, assuming there is no 2-like intervention, is that the criterion to pick could be something other than "act straightforwardly as if you were endorsing SHARP". It could instead be some (other) form of bracketing.

Bracketing departs from impartiality, and I find this very unappealing.

I think cost-effectiveness accounting for effects on all organisms spans many orders of magnitude (OOMs) due to large uncertainty about how to compare welfare across species. So I expect something like a loguniform or lognormal distributions would be more appropriate. Ideally, one would model the inputs as distributions instead of assuming a distribution for the cost-effectiveness.

In the context of assessing interventions with very uncertain cost-effectiveness (in my view, practically any context), in which sense would it matter a lot whether one uses sharp or unsharp probabilities? With sharp probabilities, it would be close to arbitrary which interventions should be supported. With unsharp probabilities, it would be indeterminate which interventions should be supported, but one would still end up supporting something based on some criteria. From my perspective, it is unclear which one would lead to greater impact. Given the large uncertainty, it is not even clear to me whether any of the approaches would outperform picking interventions randomly.

So I believe the priority would be decreasing uncertainty. I expect this can be most cost-effectively achieved via research (on comparing welfare across species). However, supporting the interventions under comparison also indirectly decreases uncertainty to some extent. Funders who do not want to fund research directly decreasing the uncertainty might be open to funding research aiming to figure out how to decrease uncertainty via supporting existing interventions. They could then update to some extent towards funding interventions which look better in terms of decreasing uncertainty. I guess ones contributing to moral circle expansion help attracts resources to target more neglected animals, including to study how their welfare compares with that of other less neglected animals.

Right. I think using unsharp probabilities, and expected values is fine to highlight it is unclear which of the interventions being compared has the highest expected cost-effectiveness. However, I do not see what is the advantage of this relative to just getting wide distributions for the cost-effectiveness, and showing these overlap a lot, which would be a sign that decreasing their uncertaity may have a higher expected cost-effectiveness than picking the intervention with the highest expected cost-effectiveness. One can analyse value of information (VOI) using perfectly sharp credences.

Hi Jim. You meant "the author's non-endorsement of Uniqueness"? You said "the other's".

Adam (the author) says "It is compatible with sharp that for certain batches of evidence, there is more than one probability function it is rationally permissible to have on the basis of that evidence". However, Adam concedes in footnote 11 it may be difficult to accept sharpness, and deny uniqueness.

There may well be difficulties with accepting sharp while denying Uniqueness. But I will not press any such difficulties here. Thanks to Susanna Rinard and John Collins for pressing me on this point.

I endorse sharpness and uniqueness. As far as I can tell, the issues of unsharp probabilities would apply in the same way to non-unique probabilities. Why would this not be the case? 

At the same time, I believe there are many reasonable probabilities. Humans have a limited memory, and therefore cannot represent infinitely precise / sharp probabilities. One would need infinite resources to represent an infinitely precise probability. If I say a given event has a chance of 10 %, I mean the sharp unique probability of a rational being with the evidence I have access to is close to 10 % (how close would depend on the context). I do not mean it is exactly 10 %. So I would convey practically the same information (just in an unnecessarily precise way) if I said that same event has a chance of 10.001 %. Does this make sense?

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.

I followed up here.

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.

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. 

Very interesting. Thanks. Relatedly, you may be interested in this comment.

Here is a video I found useful that explains how to use backward induction. Below is Claude's reply to your comment after some iteration between us.

Thanks Michael — the backward-induction framing is the strongest version of the reply, and I want to grant what it gets right before saying where I think it's still exposed.

It does defuse three things at once. It needs no commitment (you predict the future Bet B choice and fold it back, rather than binding yourself), it needs no complete ordering (it runs on statewise dominance, so the Bet B node can stay genuinely unsharp), and it isn't ad hoc (backward induction is the standard discipline for sequential choice). So this isn't PLAN in disguise. Fair enough.

But I think the argument turns on a step that quietly does more than "just backward induction." Here is the full tree, with payoffs written as (if H / if not‑H). Bet A pays −10/+15 and Bet B pays +15/−10, so the four leaves are BOTH +5/+5, A-only −10/+15, B-only +15/−10, and NEITHER 0/0:

Notice both Bet B nodes are under-determined: at each, neither action statewise-dominates the other (BOTH vs A-only cross; B-only vs NEITHER cross). That is exactly the optionality unsharpness is meant to preserve, so dominance-pruning removes nothing at a Bet B node. To get a verdict on Bet A, backward induction has to fold each Bet B node back into a single continuation value — and the value of the reject-A branch depends entirely on which of its two (equally maximal) leaves you assume you'll pick.

Crucially, the accept-A node is also under-determined — it can land on BOTH or on A-only. So to compare the two root actions I have to fix a policy over both identical Bet B nodes. There are only three consistent options:

The only statewise-dominance relation anywhere in the tree is BOTH ≻ NEITHER. In particular A-only vs NEITHER crosses — A-only is worse than NEITHER in the H-state (−10 < 0) — so accepting A does not statewise-dominate rejecting A. Under either consistent policy (always-accept or always-reject), both root actions stay admissible and there's no dominance reason to prefer accepting A. And note that under "always accept B," NEITHER is never reached on either branch, so there's nothing for accepting-A to protect against in the first place.

The recommendation to accept A appears only under the third policy — the one that accepts B after accept-A but rejects B after reject-A. That is what produces the BOTH-vs-NEITHER pairing that makes accepting A look dominant. But that policy isn't backward induction resolving each node on its merits; it's a rule that makes your Bet B choice depend on whether Bet A preceded it, handing down different verdicts at two Bet B nodes that (for a money-only agent) are identical in every respect she cares about. That is precisely the SEQUENCE/PLAN pattern Elga's Sally case is built to reject.

Put differently: the recommendation to accept A materialises only when you assume you'll reject B specifically on the reject-A branch — i.e. you distrust your future self on one branch but not the other. That asymmetric self-distrust is either the sophisticated-chooser reading (treat your own future permitted choice as a hazard to steer around) or the differential treatment of identical nodes. Both are exactly the concessions at issue: if you're rationally required to prevent your future self from exercising reject-B, then reject-B was never really optional — which is just SHARP's verdict reached the long way.

So a sharper version of my earlier question: your derivation of "accept A" resolves the accept-A continuation to BOTH and the reject-A continuation to NEITHER. What consistent policy over the two identical Bet B nodes yields that pair? If "always accept B," reject-A gives B-only and the dominance is gone. If "always reject B," accept-A gives A-only and the dominance is gone. The only policy that yields it treats the two Bet B nodes differently — which is the thing an imprecise theorist owes an account of, and which Sally says you can't have.

(One aside on "you'd use backward induction even with sharp probabilities, or be worse off": agreed, but with sharp credences backward induction never has to override a node's verdict — it agrees with local EV-maximisation, and the cases where skipping it hurts are cases of myopia, not override. This is the unique setting where the rule must reverse a choice the agent's own decision rule calls permissible. That asymmetry is the tell.)

Hi Anthony. Readers of this post may be interested in this summary and discussion in the comments of Adam Elga's article Subjective Probabilities should be Sharp. I very much agree subjective probabilities should be sharp. So I am not concerned about the unawareness argument for "no impartial altruistic justification for preferring any action over another", which relies on unsharp probabilities.

Hi Simon. Below is what Claude has to say about that.

Hi Simon. I think you've actually put your finger on the load-bearing feature rather than missed something — but the tension you're sensing resolves once you separate two things that "full disclosure" runs together in your reading: what the agent knows, and when she chooses.

Full disclosure only fixes the first. At the A-node she knows the whole tree: that B will follow, the payoffs, and that her credence in H won't move. What it does not do is collapse the two choices into one simultaneous package-choice. She still acts twice, at two separate moments — accept/reject A, and then, after that's settled, accept/reject B. Foreknowledge isn't simultaneity. So the sequential structure survives full disclosure intact; the agent is fully informed and still makes two timed decisions.

That distinction is exactly why your "just interpret it as: take both, get $5, so it's irrational not to" doesn't invalidate the setup — and here's the part that I think will unstick you. That reasoning isn't a competitor to Elga's argument; it's Elga's own premise. The paper's central claim is precisely that a rational agent "will accept at least one of the bets" because rejecting both is dominated and she can see this in advance. He is not disagreeing that reject-both is irrational. He's asserting it. Your intuition and his premise are the same sentence.

So the question the paper is asking is one notch more subtle than the one you're answering. It's not "is it irrational to reject both?" (everyone says yes). It's: "what account of how unsharp credences guide action actually delivers that verdict, given that the unsharp agent's rule makes each bet, taken on its own, merely optional?" With an interval straddling 60%, rejecting A is permitted at the A-node; with the interval straddling 40%, rejecting B is permitted at the B-node. A rule that just evaluates each bet locally therefore licenses reject-both — the two "optional"s compose into the dominated outcome, foreknowledge notwithstanding. The challenge is to find a rule that blocks that without wrecking the optionality elsewhere.

Now to your direct question — is your move PLAN or SEQUENCE? Once you try to turn "take both, it's $5" from an observation into a decision rule the agent runs, you land in the global-rules family, and closest to SEQUENCE (equivalently, the "treat it as one choice among {A, B, BOTH, NEITHER} and eliminate the dominated NEITHER" reframe that Evan proposed in the other thread [here]). That rule does give the right verdict here — Elga grants that. His objection is that as a rule it fails the Sally test: it has to call "reject B" impermissible when A-was-just-rejected but permissible when B is offered alone, even though a money-only agent faces identical money-consequences at the B-node in both cases. So your reasoning isn't a fourth escape hatch; it's a restatement of what a global rule has to deliver, and Elga's argument is aimed precisely at whether any such rule survives.

On your suspicion that full disclosure "must be weaker than I'm interpreting" — I'd say the opposite. It's exactly as strong as you're reading it, and that strength is the point. Full disclosure, plus no-belief-change, plus linear-in-dollars, is Elga removing every excuse the unsharp theorist could offer for reject-both: she wasn't surprised, she didn't learn anything mid-sequence that shifted her credence, and she has no non-monetary reasons. With all the confounders stripped out, any residual permission to reject both traces to the unsharp decision rule alone — which is the whole rhetorical purpose of piling on those stipulations.

One last thing that I think will make the sequential structure feel less suspicious: Elga needs it to be sequential, and this is why he can't just let it be a simultaneous package. If the offer were genuinely one simultaneous choice ("accept both / reject both / mix"), the puzzle evaporates — even a maximally permissive rule picks BOTH over NEITHER by straight dominance, and the unsharp theorist wins for free. The problem only has teeth in the sequential version, because that's the only structure where a local rule can manufacture reject-both out of two individually-permitted rejections. So the sequential framing and full disclosure aren't in tension; they're doing complementary jobs — disclosure kills the "she didn't see it coming" defense, sequentiality is what generates the difficulty in the first place.