Subjective Probabilities should be Sharp

By Vasco Grilo🔸 @ 2026-07-10T16:47 (+25)

This is a linkpost to https://www.princeton.edu/~adame/papers/sharp/elga-subjective-probabilities-should-be-sharp.pdf

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:

His own position is the negation:

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):

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.

Michael St Jules 🔸 @ 2026-07-11T09:06 (+7)

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🔸 @ 2026-07-11T11:24 (+2)

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 🔸 @ 2026-07-11T19:27 (+6)

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🔸 @ 2026-07-12T11:14 (+2)

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 🔸 @ 2026-07-12T16:03 (+4)

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🔸 @ 2026-07-12T19:08 (+2)

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. 

Michael St Jules 🔸 @ 2026-07-22T18:25 (+4)

The expected value of the St. Petersburg lottery is 1 + 1 + ... = +inf. It involves finite terms, but infinitely many terms. I meant to relate f(x) = x in my comment to the expected value of the St. Petersburg lottery. If this involved an arbitrarily large number of terms, its expected value would be arbitrarily large, but not infinite. 

I don't understand what you're objecting to. Are you saying all probability distributions should only take finitely many possible values? If so, do you have an upper bound on what the total value should be over all of the future and possible existence, that you're 100% confident in, under your course of action if where you do nothing? (It doesn't have to be the least upper bound / tight.)

Vasco Grilo🔸 @ 2026-07-23T08:55 (+2)

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

I do not have an upper bound for counterfactual impact, but this does not mean it can be infinite. A normal distribution does not have a maximum value. It can take an arbitrarily large value. However, it cannot take an infinite value. Its range is the set of real numbers.

Michael St Jules 🔸 @ 2026-07-23T16:57 (+4)

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

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

 

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

Vasco Grilo🔸 @ 2026-07-24T12:53 (+2)

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

I also believe 2 actions can have infinite expected cost-effectiveness considering all future effects, and still be comparable. Imagine actions A and B have expected cost-effectiveness of CE_A(t) and CE_B(t) considering effects in the next t years (after a given date), and that CE_A and CE_B tend to positive infinity as t tends to infinity. If the ratio R(t) = CE_A(t)/CE_B(t) tends to:

  • 0, B is infinitely more cost-effective than A.
  • N, A is N times as cost-effective as B.
  • Positive infinity, A is infinitely more cost-effective than B.

I am sceptical of the 1st and last cases in practice. However, in any of the above cases, there would be a clear answer about which action has the highest expected cost-effectiveness considering all future effects.

There would be a dilemma if the expected cost-effectiveness growing faster kept oscillating forever between positive and negative forever. For example, if CE_A(t) = t, and CE_B(t) = t^2*sin(t), R(t) = 1/(t*sin(t)), which tends to 0 as t goes to infinity, but keeps oscillating between an infinitesimally positive and negative value. However, in the real world, one could reasonably assume actions grow similarly fast sufficiently far into the future when there would be exactly the same evidence about the effects of A and B? In this case, R always tends to a finite single number as t goes to infinity.

Michael St Jules 🔸 @ 2026-07-27T04:05 (+6)

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

 

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

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

  1. ^
    1. I think falsifiability for a Bayesian could just mean we can imagine evidence that would warrant assigning very very low credences to the hypothesis. Actual infinities are often falsifiable in principle in this way, in specific cases. I think the probability you should assign to you being infinite in spatial size is extremely small. Similarly for the Earth being infinite in size.
    2. In the case of the spatial extent of the universe being infinite, it seems hard to falsify now only because all the evidence we've gathered so far is in fact consistent with it (or no less consistent than with a bounded/finite universe), and it remains one of the standard models used by experts. My own view is that it’s more likely than not infinite in spatial extent, because it’s the simplest model consistent with the evidence.
    3. I think our credences that the universe is infinite in spatial extent should increase with our credences that the universe is globally flat (0 global curvature), which has been measurable. If we had good evidence of nonzero global curvature, that would be decent (but not overwhelming) evidence that the universe is finite/bounded, because it would rule out the most plausible infinite models of the universe.
    4. I agree that we can sometimes compare actions with infinite expected cost-effectiveness. I'm most partial to expansionist views of some kind, as the most complete (given precise probabilities).
Vasco Grilo🔸 @ 2026-07-27T09:41 (+2)

I think falsifiability for a Bayesian could just mean we can imagine evidence that would warrant assigning very very low credences to the hypothesis.

Makes sense. I cannot imagine any evidence that would update me. However, if I did, there would be falsifiability.

Actual infinities are often falsifiable in principle in this way, in specific cases. I think the probability you should assign to you being infinite in spatial size is extremely small. Similarly for the Earth being infinite in size.

The way I see it, the probability of Earth having a radius larger than X tends to 0 as X goes to infinity. So I would say the probability of Earth having an infinite radius is exactly 0.

I think our credences that the universe is infinite in spatial extent should increase with our credences that the universe is globally flat (0 global curvature), which has been measurable.

Do we really have any evidence that the universe is globally flat? From Wikipedia's page on the shape of the universe:

Final results of the Planck mission, released in 2018, show the cosmological curvature parameter, 1 − Ω = ΩK = −Kc2/a2H2, to be 0.0007±0.0019, consistent with a flat universe.[47] (i.e. positive curvature: K = +1, ΩK < 0, Ω > 1, negative curvature: K = −1, ΩK > 0, Ω < 1, zero curvature: K = 0, ΩK = 0, Ω = 1).

We have evidence that the universe is close to flat. However, there are infinitely many values arbitrary close to exactly 0. So applying some sort of principle of indifference results in a probability of exactly 0 of curvature being exactly 0 (or any other sharp value)?

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

Michael St Jules 🔸 @ 2026-07-27T17:50 (+6)

What do you think about the St Petersburg problem now?

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

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

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

 

EDIT:

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

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

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

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

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

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

Vasco Grilo🔸 @ 2026-07-28T15:15 (+2)

What do you think about the St Petersburg problem now?

Thanks for pushing me to think about this more. I had only looked into your post a few days after you published it around 3 years ago, but just had a look again. I agree the money pump you described there does not require prospect which could have an infinite value. It only requires prospects with infinite expected value as you have been saying.

I think there is exactly 0 empirical evidence for distributions with infinite expected value for the same reasons I believe there is exactly 0 empirical evidence for infinities. As far as I can tell, exactly 100 % of the empirical evidence that could ever be gathered in principle could be exactly 100 % explained by distributions with finite expected value. Do you agree? I agree distributions should not have a maximum because one cannot be exactly 100 % confident there are not higher values. However, a lack of maximum does not imply infinite expected value.

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

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

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

I would apply the principle of indifference to models which explain the same empirical evidence. If a curvature of 0 had a probability above 0, and the values of the curvature just above 0 followed a continuous distribution, the curvature of 0 would be infinitely more likely than a positive curvature arbitrarily close to 0. This is very counterintuitive to me because the curvatures would have an arbitrarily close explanatory power. I would rather concede all universe models are wrong with probability 1 while acknowledging simpler ones are more useful for further scientific progress all else equal.

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

Yes. I think there will always be infinitely many values arbitrarily close to 0 which explain exactly the same empirical evidence as a value of 0.

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

Yes. Edit after Michael's comment just below. I would assign a probability of exactly 0 to any physical law because there are arbitrarily many physical laws arbitrarily close to any physical law. So I would also assign a probability of exactly 0 to any set of physical laws, including the set of laws involving any given number of fundamental forces.

Exactly 0 probability to there not being ghosts [ghosts existing with probability of exactly 1], because their effects could just be very small or rare?

Yes, but the effects of the ghosts would have to be sufficiently small or rare to be unfalsifiable. I assume the existence of ghosts is falsifiable under some typical definitions. Likewise for some defitions of God. Edit after Michael's comment just below. I would not assign a probability of exactly 1 to something falsifiable.

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

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

Michael St Jules 🔸 @ 2026-07-28T16:00 (+6)

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

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

 

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

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

 

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

Vasco Grilo🔸 @ 2026-07-28T17:56 (+2)

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

Why do you think I should expect all distributions to have infinite or undefined expected value instead of rejecting such distributions?

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

I corrected my answers in my past comment. You can see what I crossed out, and wrote after "Edit after Michael's comment just below".

Some could be vengeful and want harm to fall upon those that have caused them harm in their lives (whether or not they enact it themselves).

For the "ghosts [I mentioned in my last comment] which have exactly 0 measurable effects on the world", the benefit and harm they could cause would be sufficiently small to be practically negligible.

Vasco Grilo🔸 @ 2026-08-09T08:57 (+2)

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

There is also Hitchens's razor.

What can be asserted without evidence can also be dismissed without evidence.

Evan LaForge @ 2026-07-10T19:29 (+7)

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 🔸 @ 2026-07-11T09:08 (+7)

As a rational actor with no useful information

mood

VeryJerry @ 2026-07-16T00:34 (+4)

You guys get to be rational actors??

Vasco Grilo🔸 @ 2026-07-12T11:15 (+2)

Funny, and relatable.

Vasco Grilo🔸 @ 2026-07-11T10:07 (+2)

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 @ 2026-07-11T12:43 (+6)

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🔸 @ 2026-07-11T13:36 (+2)

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 @ 2026-07-12T01:58 (+6)

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.

Jim Buhler @ 2026-07-13T09:05 (+6)

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🔸 @ 2026-07-13T13:39 (+2)

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 @ 2026-07-13T14:11 (+6)

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🔸 @ 2026-07-13T19:07 (+2)

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 @ 2026-07-14T09:39 (+4)

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🔸 @ 2026-07-14T11:21 (+3)

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 @ 2026-07-14T17:06 (+4)

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, e.g., some (other) form of bracketing.

Vasco Grilo🔸 @ 2026-07-14T18:54 (+2)

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 🔸 @ 2026-07-11T11:11 (+6)

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

Vasco Grilo🔸 @ 2026-07-11T13:58 (+2)

Hi Anthony. Thanks. I followed up on LessWrong.

simon @ 2026-07-12T14:53 (+4)

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🔸 @ 2026-07-12T15:38 (+2)

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 @ 2026-07-12T16:19 (+3)

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 @ 2026-07-12T17:36 (+10)

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🔸 @ 2026-07-12T19:03 (+2)

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

Vasco Grilo🔸 @ 2026-08-09T08:38 (+2)

I just linkposted a summary of the setup, premises, and conclusions of the theorems presented in Consequentialist Foundations for Expected Utility by Peter J. Hammond.