Subjective Probabilities should be Sharp

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.

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.

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.

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.

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