Subjective Probabilities should be Sharp

[Vasco Grilo🔸]

Link post

This is a linkpost for Subjective Probabilities should be Sharp by Adam Elga, which was originally published in Philosophers’ Imprint in May 2010. Here is an errata for it. Below is a summary from Claude Opus 4.8 High. Adam said “I took a quick look and at first glance I saw nothing wrong with the summary”. I also think the summary is accurate based on my read of the article. I used the following prompt. “Hi. Make an in-depth summary of the paper “Subjective Probabilities should be Sharp”, which I send attached”.

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.

The target debate

Elga takes aim at a popular view in formal epistemology about how to respond to weak or unspecific evidence. He opens with a contrast between three kinds of evidential situations. Sometimes evidence is sharp (you’ve watched a biased coin land heads 83% of thousands of tosses, so your credence should be ~83%). Sometimes it’s sparse but with a clear upshot (you have almost no evidence about whether the number of humans born in 1984 was even, but symmetry still pushes you to ~50%). And sometimes it’s sparse and unspecific—his memorable example is a stranger pulling objects from a bag (a regular tube of toothpaste, a live jellyfish, a travel-sized tube of toothpaste), where there’s no obvious basis for any particular credence that the next object is toothpaste.

In that third kind of case, many philosophers find it natural to say your degree of belief shouldn’t be any single precise number but rather should be indeterminate, vague, or interval-valued—represented by a range like [10%, 80%] (or, more powerfully, by a set of probability functions rather than one). Elga cites Joyce, Levi, Sturgeon, and Walley as defenders of this idea. Their shared claim is not that some computational or representational limitation stops you from forming a precise credence; it’s that the evidence itself makes a precise credence unwarranted, no matter how idealized the agent is.

Elga distills the view into a deliberately cautious thesis:

• UNSHARP: It is consistent with perfect rationality to have unsharp degrees of belief.

His own position is the negation:

• SHARP: Perfect rationality requires sharp degrees of belief.

An important clarification

Before arguing, Elga separates SHARP from a stronger doctrine called Uniqueness (the idea that each body of evidence permits exactly one rational credence function). SHARP does not say there’s only one permissible function per evidential situation; it allows that several functions might be permissible. It just insists that whichever one(s) you adopt must each be perfectly precise. So this is not a defense of a uniquely rational credence—only of precision.

The strategy: demand a decision rule

Elga’s whole case rests on a challenge. On the standard story (expected utility theory), a rational agent’s beliefs are a probability function, and she acts to maximize expected utility—and the paper simplifies by assuming utility is linear in dollars. Anyone who says rational agents can have unsharp credences owes a corresponding account of how unsharp probabilities constrain rational action. Elga argues that no acceptable such account exists. If there’s no good answer to “how do unsharp credences guide choice?”, then the view collapses.

The “great series of bets”

The engine of the argument is a sequential betting setup on some proposition H (say, that it rains tomorrow):

• Bet A: If H is true you lose $10; otherwise you win $15.

• Bet B: If H is true you win $15; otherwise you lose $10.

You’re told the full setup in advance, Bet A is offered first and Bet B immediately after, and crucially your opinion about H won’t change during the process (no new evidence, no reinterpretation, just the passage of time). The bets are mirror images, so accepting both guarantees a net $5 gain no matter how H turns out.

Elga grants you’re not required to accept both (a very confident or very doubtful agent might prefer just one). But he insists on the key premise: a rational agent must accept at least one of the two bets, because rejecting both is dominated—it’s worse than accepting both in every outcome, and you can see this in advance. This premise is easy for a sharp-credence theorist to honor. The rest of the paper argues that no version of the unsharp view can.

To set up the problem: suppose your credence is the wide interval P(H) = [10%, 80%]. A sharp agent evaluates Bet A by a clean threshold—accept if P(H) < 60%, reject if P(H) > 60%, optional at exactly 60%. (Bet A’s expected value is positive precisely when P(H) is below 60%.) Your interval straddles 60%, and that’s where the trouble starts.

Permissive rules are too permissive

The first and most natural family of rules: since your interval spans the 60% threshold, neither accepting nor rejecting Bet A is mandatory—the bet is optional. By the same reasoning Bet B is optional too. But then it’s permissible to reject both. Elga says this is plainly absurd: a money-loving agent who knowingly walks away from a guaranteed $5 has departed from perfect rationality. He notes that a wide swath of decision rules in the literature (Levi, Walley, Good, Seidenfeld, Gärdenfors–Sahlin, Gilboa–Schmeidler) deliver exactly this verdict, and so are unacceptable as accounts of ideal rationality. (He’s careful to flag that some of these authors—e.g. Gilboa–Schmeidler—may only be theorizing about non-ideal agents, in which case his critique doesn’t touch them; and that Levi explicitly embraces the reject-both consequence.)

Strict rules are too strict

At the opposite extreme is the midpoint rule: evaluate bets using the midpoint of your interval, so [10%, 80%] behaves like a precise 45%. This does yield the correct verdict (you’d never reject both bets, since no sharp agent does). But Elga argues it’s self-defeating for the unsharp camp. The original motivation for unsharp credences was that the evidence fails to “nail down” any exact probability. Yet the midpoint rule lets the evidence nail down a completely precise pattern of betting odds: an H-ticket worth $100 if true gets valued at exactly $45.000…. If it’s fishy for rationality to require an exact credence of 45%, it’s equally fishy to require valuing the ticket at exactly $45. So strict rules buy the right behavior only by smuggling precise constraints back in—which removes any reason to have rejected precise credences in the first place. The midpoint rule (and its strict relatives) thus robs unsharpness of its point.

What’s needed, and the three “global” attempts

So the unsharp theorist needs a rule that is strict enough to forbid rejecting both bets, yet permissive enough to leave a range of options open when, say, Bet B is offered alone—and that motivates both verdicts naturally. Elga sees only three candidate strategies, all “global” in that they assess choices in light of other (past or future) choices.

1. NARROW — Acting sharpens your interval. Reject Bet A and your P(H) narrows (e.g. to [60%, 80%]), which then disposes you to accept Bet B; in general your intervals contract so as to block predictably inferior sequences. This delivers the right strictness/​permissiveness mix. Elga’s objection: it forces a rational agent to change her opinion without any change in relevant evidence. His illustration: if your credence about rain is unsharp enough that wearing either a rain-poncho or a non-water-resistant suede jacket is permissible, NARROW says choosing the poncho makes you confident it’ll rain and choosing the jacket makes you confident it won’t—even though your choice of jacket carries no evidence about the weather (you have no rain-sensing powers, and your clothing doesn’t control the sky). Rationality never requires revising an opinion when relevant evidence is unchanged, so NARROW fails.

2. PLAN — When you act, you simultaneously form a plan binding your later choices to cohere with it (reject Bet A → plan to accept Bet B → follow through), but without changing any beliefs. Elga refutes this with the case of Sally, who cares only about money and has a highly unsharp credence about rain. Compare two scenarios: in the first she rejected Bet A and is now offered Bet B; in the second she’s offered Bet B alone. PLAN permits rejecting Bet B in the second but not the first. Yet the monetary consequences of accepting and of rejecting Bet B are identical across the two scenarios, and her beliefs are identical, and money is all she cares about—so the situations are alike in every respect she cares about. Rationality can’t impose different requirements on choices that are identical in all relevant respects. To the rejoinder “but rejecting Bet B would break her plan,” Elga replies that either plan-breaking is something Sally finds costly (contradicting the stipulation that it’s costless for her), or it isn’t—in which case “Don’t break plans!” is as groundless a constraint as “Don’t break mirrors!” He flags but sets aside the resolute-choice tradition (Gauthier, McClennen) that would defend plan-following.

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.

Conclusion

Every candidate account of how unsharp probabilities guide action falls into one of the traps: permissive rules wrongly license rejecting both bets; strict rules pin down precise betting odds and thereby destroy the motivation for unsharpness; and the global rules (NARROW, PLAN, SEQUENCE) either demand belief change without evidence change or treat choice-identical situations differently. Since there’s no good answer to how unsharp credences constrain rational action, UNSHARP is false—and so perfect rationality requires perfectly sharp probabilities.

[Jim Buhler]

Do you agree with the other’s (EDIT: authors’) non-endorsement of Uniqueness? My impression was that you’d endorse SHARP because you think your sharp credence is uniquely appropriate. If not, why endorse this one rather than another sharp one that isn’t any less appropriate?

[Vasco Grilo🔸]

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?

[Jim Buhler]

Helpful thanks! Related thoughts from Clifton, here. But you actually do not object to UNSHARP (to some degree) for limited agents like us, then, right?

[Vasco Grilo🔸]

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.

[Jim Buhler]

just getting wide distributions for the cost-effectiveness

A normal Gaussian distribution? If so, then you still think the value in the middle of the curve is uniquely appropriate (even if barely so). To me, that’s the key difference between A) imprecision and B) precision with severe credal fragility. The former assumes you can’t non-arbitrarily pin down a precise credence at all, while the latter assumes you still can.

If VOI is overwhelmingly high, both A and B might recommend research, such that the difference doesn’t matter. But it matters a lot at least in situations where actors want to fund non-research things (because they think VOI is not that high or whatever). Then, A and B deeply disagree on what should be done.

[Vasco Grilo🔸]

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.

[Jim Buhler]

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.

One thing is whoever does not reject UNSHARP might not have severely imprecise credences about everything. I might believe that

• intervention 1 has severely indeterminate but astronomically high (positive or negative) EV.

• intervention 2 seems overall good, although it has lower EV.

Then, I’d probably prioritize intervention 2. If I instead endorsed SHARP, I might favor intervention 1 (because of a sufficient 51% credence 1 is good). (I’m actually not sure about this, though. One could argue that 1 and 2 remain incomparable and that I have no reason to favor 2 over 1.)

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.

[Vasco Grilo🔸]

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.

[Anthony DiGiovanni 🔸]

Thanks Vasco. I’ve summarized my reply on LessWrong here (figured that this might be of (more?) interest to LW readers).

[Vasco Grilo🔸]

Hi Anthony. Thanks. I followed up on LessWrong.

[Michael St Jules 🔸]

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.

[Vasco Grilo🔸]

Hi Michael.

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.

Why does Parfit’s hitchhiker pose a problem? I would think my chance of survival is equal to my chance of keeping the commitment. So I would simply aim to commit as much as possible if I wanted to maximise my chances of survival. I understand the dilemma is that it would make sense for me to break the committment after I was driven to town, but my decision and thoughts in the town would be constrained from my chat with the driver in the desert. If the driver predicted I was 90 % likely to keep the commitment, and their predictions were calibrated, I would be 90 % likely to keep the commitment, and my thoughts would have to be compatible with this? If the driver predicted I was certain to keep the commitment, I would not consider breaking it in town? Otherwise, the predictions of the driver would not be accurate, which violates the set up of the thought experiment? Here is the description of the thought experiment for readers’ context.

You are stranded in the desert, running out of water, and soon to die. Someone in a motor vehicle drives up to you. The driver of the motor vehicle is a selfish ideally game-theoretical agent, and what’s more, so are you. Furthermore, the driver is Paul Ekman who has spent his whole life studying facial microexpressions and is extremely good at reading people’s honesty by looking at their faces.

The driver says, “Well, as an ideal selfish rational agent, I’ll convey you into town if it’s in my own interest to do so. I don’t want to bother dragging you to Small Claims Court if you don’t pay up. So I’ll just ask you this question: Can you honestly say that you’ll give me $1,000 from an ATM after we reach town?”

[...]

We may assume that Parfit’s driver also asks you questions like “Have you really thought through what you’ll do?” and “Are you trying to think one thing now, knowing that you’ll probably think something else in the city?” and watches your facial expression on those answers as well.

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

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?

Adam’s argument holds as long as, given 2 sharp states of the world A and B, A is better, worse, or as good as B? In Sally’s case, her money is the only thing that matters. For realistic cases, many other factors will contribute to the value of a sharp world state.

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

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

[Michael St Jules 🔸]

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.

[Vasco Grilo🔸]

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

Solution to which problem? I am not sure what is supposed to be problematic. As far as I understand, one should just commit as much as possible to maximise the chances of survival.

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).

I agree there is a probability above 0 of (counterfactual) impact being larger than X for any X. So I think impact can be arbitrarily large. 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). The function g(x) = 1/​x can take an arbitrary small value, but not a value of exactly 0 (its range is the set of real numbers besides 0).

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?

It is very counterintuitive that could matter for Sally for reasons that do not have to do with money.

[Michael St Jules 🔸]

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.

[Vasco Grilo🔸]

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.

[Vasco Grilo🔸]

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

1. “What if I do care about the differences?”

Michael’s most direct hit: Elga’s Sally argument needs the two B-situations to be identical in everything she cares about, and Michael asks why the agent can’t just care about whether she’s about to complete a dominated sequence — “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?”

Elga has a ready answer, and it’s the one he actually gives in the paper against the parallel “but rejecting B would break her plan” rejoinder. It splits into a dilemma:

Either this caring is a genuine, independent source of value for Sally — in which case the case has been changed, not answered. Elga’s Sally is stipulated to care only about money, with reconsideration costless. If you smuggle in a taste for sequence-completion or an aversion to regret, you’re no longer discussing Elga’s agent; you’re conceding that a purely money-motivated unsharp agent is stuck, and rescuing a different agent who has been given an extra terminal value precisely engineered to patch the hole. That’s ad hoc: the value exists only to deliver the verdict UNSHARP needs.

Or the caring is not an independent value but just tracks “this would be irrational” — in which case it’s viciously circular. “I disprefer rejecting B because rejecting B here is irrational” cannot be what makes it irrational; the account owes us a prior reason, and this isn’t one. Elga’s “Don’t break plans!”-is-like-”Don’t break mirrors!” point applies verbatim: either breaking the sequence is independently costly (then say so, and it’s a different case) or it isn’t (then “avoid completing dominated sequences” is a bare, unmotivated constraint dressed up as a preference).

The regret variant is especially weak. Regret is backward-looking; at the B-node the money consequences of accept-B and reject-B are fixed and identical across the two situations. If anticipated regret genuinely moves her, it’s doing so as a real (dis)utility — back to horn one, the case is changed. Vasco’s reply on the forum (“it is very counterintuitive that this could matter for Sally for reasons that don’t have to do with money”) is exactly Elga’s point, just stated flatly.

2. Michael’s “treat them fairly” /​ Parfit’s-hitchhiker parity argument

This is Michael’s best move, and it’s really DiGiovanni’s commitment point [made here] sharpened into a parity charge: there are cases everyone agrees call for binding commitments you’ll later be inclined to break — Parfit’s hitchhiker, St. Petersburg with unbounded utility — so the same “commit and rule out the bad branch” solution should be available to the unsharp agent, if you’re treating her fairly. And he uses this to answer Vasco’s “but unsharp probabilities are supposed to allow rejecting A”: “They don’t have to in every case. If it were A in isolation, both would be permissible. But that’s not the case presented to us.”

Elga would grant the parity and then deny it helps — for two reasons.

First, notice what Michael has conceded. He now says the unsharp agent is required to accept A (to zero out the chance of the dominated branch). But that is Elga’s whole thesis about this case: rationality forces a determinate verdict at the A-node. The disagreement was never “can she avoid NEITHER?” — of course she can. It’s whether the unsharp credence leaves A genuinely optional. Michael answers “no, not here,” which means the interval straddling 60% is not translating into optionality on A. So the imprecision is doing no work at the node where it was supposed to; the commitment (or the statewise argument, see below) is doing all of it. That’s confirmation of Elga’s challenge — “how do unsharp credences constrain action?” — with the answer “they don’t; something bolted on top does.”

Second, the Parfit’s-hitchhiker analogy cuts the wrong way for him. In the hitchhiker case the commitment is valuable because the two situations genuinely differ in a consequence the agent cares about: keep-the-commitment vs break-it have different payoffs (you live vs you die, or the predictor’s reading changes your prospects). That’s exactly what legitimizes binding there. In Sally’s case Elga has stipulated the two B-situations don’t differ in any consequence she cares about. So the disanalogy is precisely the feature that makes hitchhiker-style commitment rational: where binding pays, it pays because of a real downstream difference; strip that difference out (as Sally’s stipulation does) and the rationale for binding evaporates. Michael can restore the rationale only by putting a real difference back in — which is move 1′s first horn again, changing the case.

Vasco’s exchange on the hitchhiker actually pins this down: he points out that if you just “commit as much as possible,” your chance of survival tracks your commitment probability and there’s no residual puzzle. Michael’s reply — “the same solution is available to the unsharp agent if you treat them fairly” — is true but double-edged: yes, the resolute solution is available, and invoking it is the concession that local unsharp verdicts had to be overridden.

3. The statewise /​ maximality argument for accepting A

Michael’s most technical contribution (in the top comment) is a way for the unsharp agent to derive “accept A first” without any of NARROW/​PLAN/​SEQUENCE: comparing “accept A now” (call it 1) against “reject A and hope to accept B” (2), he says 1 statewise-beats 2 with some probability and they’re incomparable otherwise — so under maximality 1 is permissible and he’ll take it, killing the dominated branch.

Elga’s objection: look at what’s actually being compared. Option 2 as Michael frames it is “reject A and if I can’t guarantee I’ll accept B, risk the dominated sequence.” To get 1 to dominate 2, he has to treat 2 as carrying a live risk of ending in NEITHER — i.e. he has to already be modeling his own future B-node choice as possibly landing on reject-B. But that’s the entire question. If the agent could guarantee she’ll accept B after rejecting A (which is just the commitment), then 2 = B-only, which does not dominate 1 = A-only (they’re incomparable, as their EVs cross at 60%), and the argument for being required to accept A collapses. So the statewise argument works only on the assumption that she cannot bind her future self — in which case Elga simply agrees the sequence is a problem and asks what makes each local rejection rational — or it works by importing the commitment, in which case the imprecise credence is again idle and we’re at move 2′s concession [see here]. Either way it doesn’t vindicate UNSHARP; it either restates the problem or resolves it by non-credal means.

There’s also a subtler point. Maximality, applied node-by-node, is precisely the permissive rule Elga says is too permissive: at the B-node in isolation it licenses reject-B. Michael’s statewise argument applies maximality to the ex-ante policy comparison instead. Switching the object of maximization from acts to policies is, once more, the SEQUENCE/​PLAN move — so Elga files it there and runs Sally. Michael’s is the most resourceful version because he’s derived the ex-ante verdict from a dominance relation rather than asserting a plan-norm, but the structural commitment (evaluate policies, not nodes) is identical, and it’s that commitment Sally targets.

4. The “arbitrary precision” tu quoque

Michael’s jab — isn’t requiring sharpness “any worse than picking numbers to ensure precision for no better reason than that they occurred to you”? — is a real objection to SHARP, but Elga would note it’s an objection to the plausibility/​motivation of sharpness, not to the bet argument. And SHARP has a specific shield here: recall it explicitly does not entail Uniqueness. Elga isn’t claiming the evidence picks out one number 45.000%; he allows a range of sharp functions to be permissible responses to the toothpaste evidence. So “you’re forcing a spuriously exact number” misfires — SHARP permits you to adopt any of many precise credences; it just denies that your state can itself be spread out. The charge of false precision is aimed at Uniqueness, which Elga has already disowned. What SHARP does insist is that whatever you land on functions as a sharp probability for the purpose of guiding action — and the bet argument is what supports that, independently of how you chose the number.

The bottom line on Michael

Michael is the only one of the three [Anthony, Evans, and you] who attacks the load-bearing premise directly (“what if she cares about the difference?”) rather than trying to route around it, and he’s right that Elga’s argument stands or falls on the stipulation that the two situations are identical in all respects the agent values. But Elga’s reply is stable: every way of making the difference “matter” either (i) reintroduces a genuine downstream (dis)utility — which changes Sally’s case and concedes that the money-only unsharp agent is stuck — or (ii) makes the mattering parasitic on “it would be irrational,” which is circular. The Parfit’s-hitchhiker parity and the statewise argument both turn out to require the commitment capacity, and invoking it is precisely the admission Elga wants: that unsharp credences, left to constrain action on their own, deliver the wrong verdict and must be overridden by a resolute policy that behaves like a determinate disposition.

So against all three of your interlocutors the dialectic funnels to the same joint: is a rational ideal agent to be assessed choice-by-choice (Elga) or entitled to bind herself and be assessed over policies (DiGiovanni’s commitment, Evan’s four-option reframe, Michael’s statewise/​parity argument)? Michael states the crux most honestly — he’ll happily say the unsharp agent is required to accept A here — and that very concession is what Elga reads as victory: the imprecision has stopped doing the one thing it was introduced to do.

[Michael St Jules 🔸]

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).

[Vasco Grilo🔸]

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.)

[Michael St Jules 🔸]

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.

[Evan LaForge]

This seems to be a breakdown with the consideration of actions in complete isolation rather than with having coarse probability estimates.

At least in practice, there’s a clear difference between considering bet A in isolation and considering bet A when you know bet B is coming. If you told me about a sports game between the Snofuls and the Fleertis and offered me 2:1 odds on the Snofuls to win, I wouldn’t take it. But if you told me you would also give me 2:1 odds on the Fleertis to win, I would take both bets, guaranteeing a profit.

As a rational actor with no useful information, I have a very broad range of potential probabilities for this bet, and it is permissible to do neither bet in isolation. However, when we consider our options simultaneously, that changes the calculus.

To apply this to altruistic action, there might be actions that we are uncertain about in isolation, but we are willing to pursue as a part of a portfolio approach.

[Michael St Jules 🔸]

As a rational actor with no useful information

mood

[Vasco Grilo🔸]

Funny, and relatable.

[Vasco Grilo🔸]

Hello Evan.

At least in practice, there’s a clear difference between considering bet A in isolation and considering bet A when you know bet B is coming. If you told me about a sports game between the Snofuls and the Fleertis and offered me 2:1 odds on the Snofuls to win, I wouldn’t take it. But if you told me you would also give me 2:1 odds on the Fleertis to win, I would take both bets, guaranteeing a profit.

Accepting the 1st bet if you were confident Snofuls would win, accepting the 2nd if you were confident Fleertis would win, and accepting both if you thought the probability of any of the teams winning was close to 50 % would be in agreement with sharp probabilities.

Elga grants you’re not required to accept both (a very confident or very doubtful agent might prefer just one). But he insists on the key premise: a rational agent must accept at least one of the two bets, because rejecting both is dominated—it’s worse than accepting both in every outcome, and you can see this in advance. This premise is easy for a sharp-credence theorist to honor. The rest of the paper argues that no version of the unsharp view can.

---

As a rational actor with no useful information, I have a very broad range of potential probabilities for this bet, and it is permissible to do neither bet in isolation. However, when we consider our options simultaneously, that changes the calculus.

Which of the 3 strategies described by Adam would you use to justify accepting or rejecting each bet in isolation, but rejecting both bets together?

To apply this to altruistic action, there might be actions that we are uncertain about in isolation, but we are willing to pursue as a part of a portfolio approach.

This is not an argument for unsharp probabilities? Supporting a portfolio of interventions makes sense even with sharp probabilities. Marginal cost-effectiveness tends to decrease with spending. For example, if the Animal Welfare Fund (AWF) had granted 2 times as much to all the grantees they supported in 2025, I expect the impact of the grants would have been larger, but less than 2 times as large.

[Evan LaForge]

I agree that accepting both bets is consistent with a sharp probability at 50%, though I’m just trying to give an example of a case where I would have an unsharp probability range where I would reject both bets in isolation but take them when they arrive together.

I don’t employ any of the 3 strategies. My argument is that you don’t need a fancy strategy because, in the example, you know that bet B is coming when you’re asked about bet A. I think it’s reasonable for a rational actor to reject bet A and reject bet B if the two are presented separately but accept them both if they are presented together. My example is intended to demonstrate that. A rational actor doesn’t need NARROW, PLAN, or SEQUENCE. They need to consider the future: “Bet B is coming, so there’s an arbitrage opportunity regardless of the probability.” The article seems to disagree, treating every action in isolation and requiring that we make the right decision without global thinking.

My recommendation for portfolios is not an argument for, but an implication of, unsharp probabilities. A lot of cause prioritization is about the core philosophical positions you hold underpinning it. If you have a sharp probability, you might be comfortable investing all in one cause. If you have an unsharp one, you might not be convinced that investing in any one cause is net positive. However, you might find a combination of causes that seems robustly better than no action.

For example, you might be concerned about climate policy’s constraints on growth as well as growth’s effect on the climate. If you believe that the second order effects of investing in growth on the climate are smaller than the direct benefits of donating to climate policy (and vice versa), it is strictly better to donate to both in some combination than to do nothing. Someone with a sharp probability might be comfortable donating to just one in a way someone with unsharp probabilities would not.

As a result, portfolios are better (i.e. are more often optimal) in a world where UNSHARP is true.

I see and agree with your point about marginal returns. Depending on how strong that effect is, portfolios are also good in a world with sharp probabilities only.

[Vasco Grilo🔸]

A rational actor doesn’t need NARROW, PLAN, or SEQUENCE. They need to consider the future: “Bet B is coming, so there’s an arbitrage opportunity regardless of the probability.”

I do not seem to understand. If one knew “Bet B is coming”, one would know about the full set up in advance as in the post (“You’re told the full setup in advance”). So rejecting both A and B would not make sense?

[Evan LaForge]

I agree that rejecting both A and B would not make sense, if you are informed of both. I think the author is wrong to treat A and B as separate decisions, when the agent knows about both in advance.

Knowing that you have the option to take bet B later fundamentally changes the considerations for bet A. As a result, we are not making 2 independent decisions (A: yes or no, and B: yes or no). We are making 4 (A, B, BOTH, NEITHER).

When considering that list, we can see that BOTH is strictly greater than NEITHER in all worlds and rule out NEITHER. We are left with A, B, and BOTH to choose from, all of which might make sense depending on the agent’s choices.

At no point did I need to employ NARROW, PLAN, or SEQUENCE. I didn’t even consider the probability of H, let alone whether that probability is sharp. I just considered the available options differently.

EDIT: I think this is close in effect to SEQUENCE. As a result, there might be the objection, “What if, of the 4 options, you choose B? Could you change your mind after rejecting A and then reject B as well?” To this I would say that a rational actor does not change their mind without new information. They would only choose B if they believe B > BOTH > NEITHER. Any rational actor who believes B > NEITHER would end up betting B. They would never bet NEITHER.

What might have muddied the waters:

I separately considered how I might deal with these probabilities separately, WITHOUT knowledge that one will follow the other. This is a distinct problem from the original dilemma. However, I think it’s the only situation where a rational actor who follows UNSHARP might behave differently.

Without knowledge beforehand, if you hold UNSHARP, the following can happen:

You receive A, evaluate it, conclude it’s optional due to UNSHARP probabilities, and reject it. Then, you are offered B, evaluate it, conclude it’s optional, and reject it. You look back and think “I wish I would have known beforehand. I would have taken advantage of the arbitrage. Oh well. I guess rational actors with less information make worse decisions.”

I think it is rational for an actor to hold unsharp probabilities for some hypotheses.^[1] I think it’s rational to not engage in sports gambling when no arbitrage exists. My initial example was designed to connect the two.

1. ^

   I haven’t made my mind up on whether it’s necessary to hold unsharp probabilities in theory but I’m much more confident in practice.

   When you see a new opportunity that you know very little about that might be massively valuable, using your minimally informed baseline model to direct action seems irresponsible. Upon further investigation, everything regresses to the mean.

   In the sports gambling example I gave, you should reject unless you see arbitrage because ~all available information is priced in. In the case of impact, new opportunities look more exciting than reality due to (e.g.) selection effects and stable equilibria.

   This discussion of whether or not we should have unsharp probabilities is beside the point. My argument is about whether we can have unsharp probabilities without sacrificing rationality. I believe we can.

[Vasco Grilo🔸]

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

The unifying objection: the four-option reframe is one of the three rules

Evan’s central claim is that he can dissolve the puzzle without NARROW, PLAN, or SEQUENCE: treat the situation not as two decisions (A yes/​no, B yes/​no) but as one choice among four policies — {A-only, B-only, BOTH, NEITHER} — notice BOTH statewise-dominates NEITHER, delete NEITHER, and you’re done. He stresses “I didn’t even consider the probability of H.”

Elga’s first reply is that this is exactly SEQUENCE (or PLAN) wearing plain clothes — and Evan concedes it in his own EDIT (“I think this is close in effect to SEQUENCE”). Evaluating the pair of choices as a single ex-ante object over sequences is the defining move of the global rules. So “I don’t need any of the three” is false: he’s using the third. And that matters, because Sally is aimed precisely here. Take Evan’s B-only policy: it requires rejecting A and then accepting B. Compare the agent at the B-node in two situations — one where she reached it by rejecting A, one where B is offered alone. For a money-only agent these are identical in everything she cares about, yet the reframe must call rejecting-B impermissible in the first (it would complete NEITHER) and permissible in the second. That is the SEQUENCE verdict, and it fails for the SEQUENCE reason.

Why “consider them simultaneously” doesn’t reach the actual problem

Evan’s sports example — decline each of the Snofuls/​Fleertis bets in isolation, take both together for a sure profit — leans on “when we consider our options simultaneously, that changes the calculus.” Elga’s rejoinder: in his setup the bets are not simultaneous. You settle A, and only then face B. So the live question is what binds you at the B-node, where A is already done and the only comparison is accept-B (+15/​−10) versus reject-B (0). With an interval straddling 40%, maximality rules both permissible. The ex-ante fact “BOTH dominates NEITHER” is true but does not, by itself, reach into the B-node and make accepting B required there. Supplying that reach is the whole job of PLAN/​SEQUENCE — which is why Evan can’t actually skip them.

And the boast “I didn’t even need to consider P(H)” is the tell, not the triumph. Dominance eliminates NEITHER for any credence — a sharp agent excludes it too. So the four-option elimination is entirely neutral between SHARP and UNSHARP; it was never the point in dispute. The dispute is about the sequential assembly of a dominated outcome from two individually-licensed choices, and the reframe simply doesn’t engage it.

The EDIT smuggles in comparability — i.e. sharpness

Evan tries to close the “what if you plan B, reject A, then reject B?” gap thus: “a rational actor does not change their mind without new information. They would only choose B if they believe B > BOTH > NEITHER. Any rational actor who believes B > NEITHER would end up betting B.”

This quietly assumes a complete ordering over the options — exactly what UNSHARP denies. B-only beats BOTH only when P(H) > 60%; with the interval [10%, 80%], B-only and BOTH are incomparable under maximality, as are A-only and BOTH. So “they would only choose B if B > BOTH” presupposes the agent can rank options the way a sharp credence lets her. Grant that comparability and of course she never lands on a dominated outcome — but you’ve then imported enough structure that she behaves like a sharp agent, which is Elga’s strict-rules horn: you buy the right behavior only by reintroducing precision and thereby forfeiting the motivation for going unsharp in the first place.

“Rational actors with less information make worse decisions” gives the game away

Evan concedes that without foreknowledge an UNSHARP agent can reject A as optional, reject B as optional, land on NEITHER, and shrug it off as an information deficit. Two problems. First, Elga’s case stipulates full foreknowledge, so the no-foreknowledge scenario isn’t the one under discussion. Second, and more damaging, the diagnosis “less information” is wrong. A sharp agent — even with a diffuse-but-precise prior, and even with no foreknowledge — never rejects both, because her node-by-node expected-value verdicts are automatically time-coherent (reject A only if P(H) > 60%, accept B only if P(H) > 40%, and these can’t jointly fail). The unsharp agent’s node verdicts are not automatically coherent: both nodes say “optional,” which is what lets her assemble NEITHER. So the pathology is produced by the unsharpness, not by any information gap. Evan’s concession thus admits precisely the foreseeable-domination Elga is prosecuting, and mislabels its source.

The portfolio point isn’t an argument for UNSHARP

Vasco already made the core objection and Evan half-conceded it: diversification falls straight out of sharp EV reasoning with diminishing marginal returns and cross-correlations. Elga would add the sharper version: where the portfolio reasoning gives sensible verdicts (“this combination statewise-beats doing nothing”), it’s dominance reasoning a sharp agent honors equally; where it gives distinctively unsharp verdicts, it does so by licensing inaction — declining each option in isolation — which is just the reject-both pathology relocated to altruistic choice. (This is the “clueless agent whose intervals stay wide because it never acts” failure mode, which is live in your own work.)

“Can vs. should” is not a dodge — it’s Elga’s exact target

Evan’s sign-off — “whether we should have unsharp probabilities is beside the point; my argument is about whether we can have them without sacrificing rationality, and I believe we can” — doesn’t sidestep Elga. UNSHARP just is the “can” claim: it is consistent with perfect rationality to be unsharp. SHARP denies that. So Evan is engaging the thesis head-on, and Elga’s reply is that the “can” fails for the reasons above: every route Evan takes either collapses into SEQUENCE (Sally sinks it) or into sharp-style comparability (motivation lost).

The honest crux

Where Evan has a real point — shared with DiGiovanni and Michael St Jules — is the suspicion that node-by-node “local” evaluation is the wrong model, and that a look-ahead agent who plans the whole tree does fine with wide intervals. Elga’s whole case does assume that a theory of rational credence must deliver correct verdicts at each actual choice node, not merely over ex-ante policies. Evan is, in effect, denying that assumption. But he hasn’t defeated Sally independently; he’s relocated to ex-ante policy choice, which Elga classifies as SEQUENCE/​PLAN and which Evan himself admits is “close in effect to SEQUENCE.” So the disagreement bottoms out exactly where it did in the DiGiovanni thread [this one]: whether an idealized agent is entitled to bind her future choices (resolute/​sophisticated look-ahead), or whether rationality must already be satisfiable choice-by-choice. Elga bets on the latter; Evan (like DiGiovanni) needs the former — and that is the genuine open question, not something Evan’s four-option reframe settles.

[simon]

from the paper:

Any perfectly rational agent who is sequentially offered
bets A and B in the above circumstances (full disclosure in
advance about the whole setup, no change of belief in H
during the whole process, utilities linear in dollars) will ac-
cept at least one of the bets.

I kind of struggle to understand what “full disclosure in advance” really means and how it does not invalidate the sequential structure itself. Perhaps I’m missing something.

Any agent who knows the entire set up in advance could just interpret this as “if you take both bets you will get $5 regardless of your probability estimates of anything or lack thereof” so it would be irrational not to do so. Is this in the “Plan” or “Sequence” category?
I suspect “full disclosure” is less strict than what I’m interpreting it to be here?

[Vasco Grilo🔸]

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.

[simon]

Yeah good point I should’ve asked Claude, too!

My sense is that (perhaps similar to others in the comment section?) I don’t buy that this problem set up is a sequential problem at all.

[simon]

I burned some tokens iterating with Claude fable and ended up learning quite a bit here:

**The background: two ways of describing one problem**

A sequential decision problem can be written in *extensive form* — a tree with choice nodes (where you act) and chance nodes (where nature acts), with consequences at the leaves — or in *normal form*, where you list every complete contingency plan (“strategy”) and treat the problem as a single one-shot choice among those plans. Elga’s two-bet setup in extensive form is: choose accept/​reject A, then choose accept/​reject B. In normal form it’s one choice among four strategies: {BOTH, A-only, B-only, NEITHER}.

The anti-sequentialist move you were making in this thread is: since nothing happens between Elga’s nodes, the extensive form is a misleading picture and the normal form is the *real* problem. Call this **normal-form reduction**: rational choice in a tree must agree with rational choice among the tree’s strategies. It feels like a free, innocent principle — surely how a problem is *typeset* can’t matter.

Hammond’s theorem is the demonstration that it is not free. It is roughly the most expensive principle in decision theory.

**Hammond’s conditions**

In “Consequentialist Foundations for Expected Utility” (*Theory and Decision*, 1988), Peter Hammond starts not from preference axioms but from *behavior*: a “behavior norm” that specifies, for every finite decision tree, which moves are acceptable at each node. He then imposes three structural conditions:

*Consequentialism.* Acceptable behavior in a tree depends only on the consequences the available strategies deliver — not on the tree’s shape. Two trees whose strategies map to the same consequences must license the same consequences. This *is* normal-form reduction, stated as an axiom.

*Dynamic consistency.* The plan you’d endorse at the start is the plan your later selves actually continue; there is no predictable defection from your own strategy.

*Separability.* Behavior at a node in the middle of a tree is the same as behavior in the “snipped-off” subtree treated as a standalone problem starting there. Your past — the branch you traveled to get here — is normatively inert.

Plus an *unrestricted domain* assumption: the norm must deliver verdicts for every finite tree, not just convenient ones.

**The theorem, and why completeness falls out**

Hammond proves that any behavior norm satisfying these conditions is representable as maximization of expected utility: a complete, transitive preference ordering satisfying the independence axiom and — once chance nodes with subjective uncertainty enter — a single, precise probability function. Sharp credences, derived, not assumed.

The philosophically interesting part is where *completeness* comes from, since that’s what the imprecise probabilist denies. It comes almost embarrassingly cheaply, and seeing why is the heart of the counterattack. Behavior, unlike preference, is always decisive: put an agent in a tree offering x or y and she leaves the room having done *something*. Define “x is weakly preferred to y” as “x is choosable from some tree offering exactly {x, y}.” Since something is always choosable, this revealed relation is complete *by construction*. That’s trivial so far — a genuinely conflicted agent also picks something, perhaps arbitrarily, and we shouldn’t read a considered ranking off one forced pick.

Here’s the catch: incompleteness that can’t show up in any single choice can only ever show up as a *pattern across contexts* — choosing x over y inside one tree but y over x inside another, or refusing at a later node to complete a sequence your earlier choice began. Those context-sensitive patterns are precisely what consequentialism and separability forbid. Consequentialism says the embedding tree can’t matter; separability says your history can’t matter; dynamic consistency says your plan can’t come apart from your conduct. Jointly they seal every exit through which incomparability could behaviorally *express itself*. Whatever tie-breaking the agent does under conflict gets laminated, by the consistency conditions, into a single coherent ordering across all trees — and Hammond shows the standard money-pump/​Dutch-book tree constructions then force transitivity, independence, and precise probabilities. An agent who satisfies all of Hammond’s conditions is *behaviorally indistinguishable from a sharp expected-utility maximizer*, whatever fuzzy inner life she reports.

**Why this is a counterattack on the anti-sequential polemic specifically**

Recall your polemic’s shape: Elga’s tree has no decision-relevant events between nodes, so reduce it to the normal form, pick from {BOTH, A-only, B-only, NEITHER} by policy-level maximality, exclude only NEITHER, and declare that imprecision survives with exactly the permissiveness Elga demanded.

Hammond’s theorem says: you just endorsed consequentialism (the reduction) and dynamic consistency (your policy-follow-through). If you *also* accept separability and unrestricted domain, the theorem grinds forward and hands you back completeness — and with it sharp probabilities. Your hammer, swung with full force, rebuilds Elga’s conclusion from the opposite direction. The reduction move is therefore not a safe harbor; it’s the first premise of a proof of SHARP.

So the anti-sequentialist *must* locate a Hammond condition to reject, and the only live candidate is **separability** — she must say that what’s rationally choosable at the B-node genuinely depends on the tree it sits in (whether an A-node preceded it). And now look at what Elga’s Sally argument actually is: it is a direct intuition pump *for separability*. Sally at the B-node after rejecting A, and Sally offered B alone, are stipulated identical in beliefs, options, and everything she cares about; Elga insists rationality must treat them alike. That is separability, stated in a vignette. The dialectic thus closes into a perfect circle: the polemic escapes Sally by reduction, Hammond shows reduction-plus-separability yields sharpness, so the escape requires denying separability, and Sally is the argument for separability. The whole debate was never about probabilities at all. It is a debate about one axiom of dynamic choice, and both sides’ machinery is downstream of it.

This is exactly the lesson of Teddy Seidenfeld’s companion paper from the same year (“Decision Theory Without ‘Independence’ or Without ‘Ordering’,” *Economics and Philosophy* 1988): once you take dynamic coherence seriously, you face a forced trilemma — give up completeness/​ordering, give up independence, or give up one of the dynamic conditions (separability or consistency or reduction). You cannot keep everything, and no position gets to win by framing. McClennen’s resolute choice keeps ordering and reduction but drops separability; Levi and Seidenfeld keep separability-ish structure but drop ordering and accept lumpy behavior; Hammond keeps all the dynamic conditions and is thereby *committed* to sharp EU.

**The honest bottom line**

Hammond doesn’t refute the anti-sequentialist. His conditions are premises, and separability in particular is deniable — plenty of serious people deny it. What the theorem does is strip the polemic of its air of costlessness. “It’s obviously one decision typeset on two lines” presents normal-form reduction as bookkeeping; Hammond shows it is one third of an engine that manufactures completeness, so anyone wielding it owes an account of which sibling condition they’re rejecting and why the money-pump constructions that enforce it are toothless against them. That’s a real debt, payable in the separability literature — Seidenfeld, McClennen, Rabinowicz, Steele — rather than in polemic. The promotion I mentioned before stands, though: “which of Hammond’s axioms is false?” is a far better fight for the imprecise probabilist than “why did you burn $5?”

[Vasco Grilo🔸]

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