Vasco Grilo🔸 comments on Subjective Probabilities should be Sharp

[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.