3. SEQUENCE â Sequences of actions can be assessed for rationality independently of their parts: each of âreject Aâ and âreject Bâ can be individually permissible while the sequence âreject-A-then-reject-Bâ 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 sequenceâso it, too, imposes different requirements across two situations Sally can see are identical in everything she cares about. Hence SEQUENCE fails.
I donât think this is a strong argument. There are other cases where you should make commitments that you would later be inclined to break, like Parfitâs 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, âimposes different requirements across two situations Sally can see are identical in everything she cares aboutâ. 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 thatâs what it takes to produce arbitrarily precise probabilities if you fix what information is available to you in realistic settings.
Also, hereâs 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:
If I accept bet A at the start, then the probability that I pick the dominated sequence (rejecting both) is 0.
If I reject bet A at the start and if I canât guarantee that I will accept bet B next, then thereâs 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.
Why does Parfitâs 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 readersâ 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 whatâs 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 peopleâs honesty by looking at their faces.
The driver says, âWell, as an ideal selfish rational agent, Iâll convey you into town if itâs in my own interest to do so. I donât want to bother dragging you to Small Claims Court if you donât pay up. So Iâll just ask you this question: Can you honestly say that youâll give me $1,000 from an ATM after we reach town?â
[...]
We may assume that Parfitâs driver also asks you questions like âHave you really thought through what youâll do?â and âAre you trying to think one thing now, knowing that youâll probably think something else in the city?â and watches your facial expression on those answers as well.
Furthermore, âimposes different requirements across two situations Sally can see are identical in everything she cares aboutâ. What if I do care about the differences?
Adamâs 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 Sallyâs 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, hereâs 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?
It doesnât require an infinite world, only that you canât 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, doesnât treat the evidence on each side fairly, and is poorly argued, imo. But I donât want to rehash this.)
In Sallyâs case, her money is the only thing that matters.
Why canât the fact that sheâd 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 donât 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 thatâs not the case presented to us.
Because the same kind of solution is available to someone with unsharp probabilities in Elgaâs scenario, if youâre 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 doesnât require an infinite world, only that you canât 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/âx can take an arbitrary small value, but not a value of exactly 0 (its range is the set of real numbers besides 0).
Why canât the fact that sheâd 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.
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 youâll change your mind, unless you can force your future self to honor a commitment (or plan) youâd no longer want to keep when it actually comes time to honor it.
EDIT: my bad, the problem is that if you donât use commitments, you could be worse off. Using backward induction in the Sally argument actually works fine, doesnât leave you (or Sally) worse off and doesnât 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 doesnât require any state to have infinite value. Its value is (canonically) 2^n with probability 1/â2^n for each n at least 1. Always finite actual value, but infinite expected value.
EDIT: my bad, the problem is that if you donât use commitments, you could be worse off. Using backward induction in the Sally argument actually works fine, doesnât leave you (or Sally) worse off and doesnât require any commitment.
St Petersburg doesnât require any state to have infinite value. Its value is (canonically) 2^n with probability 1/â2^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.
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 donât understand what youâre 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 youâre 100% confident in, under your course of action if where you do nothing? (It doesnât have to be the least upper bound /â tight.)
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.
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? Shouldnât you use some principle of indifference or symmetry here?
If thereâs 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?
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)/âCE_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/â(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.
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, donât require assigning positive probability to any possible infinities out there in the world.
Iâll leave the rest in a footnote, because itâs not that relevant to the point Iâve been making so donât plan to go further with it, but I already wrote it and it may be of interest to you.[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.
In the case of the spatial extent of the universe being infinite, it seems hard to falsify now only because all the evidence weâve gathered so far is in fact consistent with it (or no less consistent than with a bounded/âfinite universe), and it remains one of the standard models used by experts. My own view is that itâs more likely than not infinite in spatial extent, because itâs the simplest model consistent with the evidence.
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/âbounded, because it would rule out the most plausible infinite models of the universe.
I agree that we can sometimes compare actions with infinite expected cost-effectiveness. Iâm most partial to expansionist views of some kind, as the most complete (given precise probabilities).
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 Wikipediaâs page on the shape of the universe:
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.
What do you think about the St Petersburg problem now?
I think your arguments re infinities conflict with Occamâs 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 youâd guess.
A flat universe effectively has one fewer parameter and is simpler. So it shouldnât get 0 credence. Among flat universes, the bounded/âfinite 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 shouldnât 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 shouldnât this get you to infinitely many fundamental forces? For any finite set of fundamental forces, you could posit another one and just say itâs 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 Occamâs razor.
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 thereâs 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 Occamâs 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 shouldnât this get you to infinitely many fundamental forces? For any finite set of fundamental forces, you could posit another one and just say itâs very weak.
Yes. Edit after Michaelâs 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 Michaelâs 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 Occamâs 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.
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.
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 âEdit after Michaelâs comment just belowâ.
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 âghosts [I mentioned in my last comment] which have exactly 0 measurable effects on the worldâ, the benefit and harm they could cause would be sufficiently small to be practically negligible.
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 Occamâs razor.
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. âWhat if I do care about the differences?â
Michaelâs most direct hit: Elgaâs Sally argument needs the two B-situations to be identical in everything she cares about, and Michael asks why the agent canât just care about whether sheâs about to complete a dominated sequence â âwhy canât the fact that sheâd pick a dominated sequence or regret it if she rejects both bets matter to her after rejecting bet A?â
Elga has a ready answer, and itâs the one he actually gives in the paper against the parallel âbut rejecting B would break her planâ rejoinder. It splits into a dilemma:
Either this caring is a genuine, independent source of value for Sally â in which case the case has been changed, not answered. Elgaâs 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, youâre no longer discussing Elgaâs agent; youâre 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. Thatâs ad hoc: the value exists only to deliver the verdict UNSHARP needs.
Or the caring is not an independent value but just tracks âthis would be irrationalâ â in which case itâs viciously circular. âI disprefer rejecting B because rejecting B here is irrationalâ cannot be what makes it irrational; the account owes us a prior reason, and this isnât one. Elgaâs âDonât break plans!â-is-like-âDonât break mirrors!â point applies verbatim: either breaking the sequence is independently costly (then say so, and itâs a different case) or it isnât (then âavoid completing dominated sequencesâ 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, itâs doing so as a real (dis)utility â back to horn one, the case is changed. Vascoâs reply on the forum (âit is very counterintuitive that this could matter for Sally for reasons that donât have to do with moneyâ) is exactly Elgaâs point, just stated flatly.
2. Michaelâs âtreat them fairlyâ /â Parfitâs-hitchhiker parity argument
This is Michaelâs best move, and itâs really DiGiovanniâs commitment point [made here] sharpened into a parity charge: there are cases everyone agrees call for binding commitments youâll later be inclined to break â Parfitâs hitchhiker, St. Petersburg with unbounded utility â so the same âcommit and rule out the bad branchâ solution should be available to the unsharp agent, if youâre treating her fairly. And he uses this to answer Vascoâs âbut unsharp probabilities are supposed to allow rejecting Aâ: âThey donât have to in every case. If it were A in isolation, both would be permissible. But thatâs not the case presented to us.â
Elga would grant the parity and then deny it helps â 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 Elgaâs whole thesis about this case: rationality forces a determinate verdict at the A-node. The disagreement was never âcan she avoid NEITHER?â â of course she can. Itâs whether the unsharp credence leaves A genuinely optional. Michael answers âno, not here,â 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. Thatâs confirmation of Elgaâs challenge â âhow do unsharp credences constrain action?â â with the answer âthey donât; something bolted on top does.â
Second, the Parfitâs-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 predictorâs reading changes your prospects). Thatâs exactly what legitimizes binding there. In Sallyâs case Elga has stipulated the two B-situations donât 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 Sallyâs stipulation does) and the rationale for binding evaporates. Michael can restore the rationale only by putting a real difference back in â which is move 1âČs first horn again, changing the case.
Vascoâs exchange on the hitchhiker actually pins this down: he points out that if you just âcommit as much as possible,â your chance of survival tracks your commitment probability and thereâs no residual puzzle. Michaelâs reply â âthe same solution is available to the unsharp agent if you treat them fairlyâ â 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 /â maximality argument for accepting A
Michaelâs most technical contribution (in the top comment) is a way for the unsharp agent to derive âaccept A firstâ without any of NARROW/âPLAN/âSEQUENCE: comparing âaccept A nowâ (call it 1) against âreject A and hope to accept Bâ (2), he says 1 statewise-beats 2 with some probability and theyâre incomparable otherwise â so under maximality 1 is permissible and heâll take it, killing the dominated branch.
Elgaâs objection: look at whatâs actually being compared. Option 2 as Michael frames it is âreject A and if I canât guarantee Iâll accept B, risk the dominated sequence.â To get 1 to dominate 2, he has to treat 2 as carrying a live risk of ending in NEITHER â i.e. he has to already be modeling his own future B-node choice as possibly landing on reject-B. But thatâs the entire question. If the agent could guarantee sheâll accept B after rejecting A (which is just the commitment), then 2 = B-only, which does not dominate 1 = A-only (theyâre 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 â in which case Elga simply agrees the sequence is a problem and asks what makes each local rejection rational â or it works by importing the commitment, in which case the imprecise credence is again idle and weâre at move 2âČs concession [see here]. Either way it doesnât vindicate UNSHARP; it either restates the problem or resolves it by non-credal means.
Thereâs 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. Michaelâs 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/âPLAN move â so Elga files it there and runs Sally. Michaelâs is the most resourceful version because heâs 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 itâs that commitment Sally targets.
4. The âarbitrary precisionâ tu quoque
Michaelâs jab â isnât requiring sharpness âany worse than picking numbers to ensure precision for no better reason than that they occurred to youâ? â is a real objection to SHARP, but Elga would note itâs an objection to the plausibility/âmotivation of sharpness, not to the bet argument. And SHARP has a specific shield here: recall it explicitly does not entail Uniqueness. Elga isnât 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 âyouâre forcing a spuriously exact numberâ misfires â 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 â 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 (âwhat if she cares about the difference?â) rather than trying to route around it, and heâs right that Elgaâs argument stands or falls on the stipulation that the two situations are identical in all respects the agent values. But Elgaâs reply is stable: every way of making the difference âmatterâ either (i) reintroduces a genuine downstream (dis)utility â which changes Sallyâs case and concedes that the money-only unsharp agent is stuck â or (ii) makes the mattering parasitic on âit would be irrational,â which is circular. The Parfitâs-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 (DiGiovanniâs commitment, Evanâs four-option reframe, Michaelâs statewise/âparity argument)? Michael states the crux most honestly â heâll happily say the unsharp agent is required to accept A here â and that very concession is what Elga reads as victory: the imprecision has stopped doing the one thing it was introduced to do.
Taking bet A doesnât require any commitment. My argument just uses backward induction (+ignoring statewise incomparability), which you should generally use in sequential choice situations, or else youâll 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).
Here is a video I found useful that explains how to use backward induction. Below is Claudeâs reply to your comment after some iteration between us.
Thanks Michael â 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 itâs 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 isnât ad hoc (backward induction is the standard discipline for sequential choice). So this isnât PLAN in disguise. Fair enough.
But I think the argument turns on a step that quietly does more than âjust backward induction.â Here is the full tree, with payoffs written as (if H /â if notâH). Bet A pays â10/â+15 and Bet B pays +15/ââ10, so the four leaves are BOTH +5/â+5, A-only â10/â+15, B-only +15/ââ10, and NEITHER 0/â0:
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 â and the value of the reject-A branch depends entirely on which of its two (equally maximal) leaves you assume youâll pick.
Crucially, the accept-A node is also under-determined â 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 â» NEITHER. In particular A-only vs NEITHER crosses â A-only is worse than NEITHER in the H-state (â10 < 0) â so accepting A does not statewise-dominate rejecting A. Under either consistent policy (always-accept or always-reject), both root actions stay admissible and thereâs no dominance reason to prefer accepting A. And note that under âalways accept B,â NEITHER is never reached on either branch, so thereâs nothing for accepting-A to protect against in the first place.
The recommendation to accept A appears only under the third policy â 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 isnât backward induction resolving each node on its merits; itâs 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/âPLAN pattern Elgaâs Sally case is built to reject.
Put differently: the recommendation to accept A materialises only when you assume youâll reject B specifically on the reject-A branch â 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 youâre rationally required to prevent your future self from exercising reject-B, then reject-B was never really optional â which is just SHARPâs verdict reached the long way.
So a sharper version of my earlier question: your derivation of âaccept Aâ 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 âalways accept B,â reject-A gives B-only and the dominance is gone. If âalways reject B,â accept-A gives A-only and the dominance is gone. The only policy that yields it treats the two Bet B nodes differently â which is the thing an imprecise theorist owes an account of, and which Sally says you canât have.
(One aside on âyouâd use backward induction even with sharp probabilities, or be worse offâ: agreed, but with sharp credences backward induction never has to override a nodeâs verdict â 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 agentâs own decision rule calls permissible. That asymmetry is the tell.)
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 donât think this is a strong argument. There are other cases where you should make commitments that you would later be inclined to break, like Parfitâs 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, âimposes different requirements across two situations Sally can see are identical in everything she cares aboutâ. 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 thatâs what it takes to produce arbitrarily precise probabilities if you fix what information is available to you in realistic settings.
Also, hereâs 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:
If I accept bet A at the start, then the probability that I pick the dominated sequence (rejecting both) is 0.
If I reject bet A at the start and if I canât guarantee that I will accept bet B next, then thereâs 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.
Hi Michael.
Why does Parfitâs 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 readersâ context.
The St. Petersburg paradox involves an infinite expected payoff, and I reject infinite worlds.
Adamâs 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 Sallyâs 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.
I understand one should accept bet A based on that strategy. However, unsharp probabilities are supposed to allow for accepting or rejecting A?
Because the same kind of solution is available to someone with unsharp probabilities in Elgaâs scenario, if youâre treating them fairly.
It doesnât require an infinite world, only that you canât 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, doesnât treat the evidence on each side fairly, and is poorly argued, imo. But I donât want to rehash this.)
Why canât the fact that sheâd pick a dominated sequence or regret it if she rejects both bets matter to her after rejecting bet A?
They donât 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 thatâs not the case presented to us.
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.
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/âx can take an arbitrary small value, but not a value of exactly 0 (its range is the set of real numbers besides 0).
It is very counterintuitive that could matter for Sally for reasons that do not have to do with money.
That if you use backward induction on acting rationally at each step, you will be worse off. You will predict later that youâll change your mind, unless you can force your future self to honor a commitment (or plan) youâd no longer want to keep when it actually comes time to honor it.EDIT: my bad, the problem is that if you donât use commitments, you could be worse off. Using backward induction in the Sally argument actually works fine, doesnât leave you (or Sally) worse off and doesnât require any commitment.
St Petersburg doesnât require any state to have infinite value. Its value is (canonically) 2^n with probability 1/â2^n for each n at least 1. Always finite actual value, but infinite expected value.
I followed up here.
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 donât understand what youâre 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 youâre 100% confident in, under your course of action if where you do nothing? (It doesnât have to be the least upper bound /â tight.)
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.
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? Shouldnât you use some principle of indifference or symmetry here?
If thereâs 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?
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)/âCE_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/â(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.
St Petersburg-like lotteries, defined in terms of your Bayesian credences, donât require assigning positive probability to any possible infinities out there in the world.
Iâll leave the rest in a footnote, because itâs not that relevant to the point Iâve been making so donât plan to go further with it, but I already wrote it and it may be of interest to you.[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.
In the case of the spatial extent of the universe being infinite, it seems hard to falsify now only because all the evidence weâve gathered so far is in fact consistent with it (or no less consistent than with a bounded/âfinite universe), and it remains one of the standard models used by experts. My own view is that itâs more likely than not infinite in spatial extent, because itâs the simplest model consistent with the evidence.
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/âbounded, because it would rule out the most plausible infinite models of the universe.
I agree that we can sometimes compare actions with infinite expected cost-effectiveness. Iâm most partial to expansionist views of some kind, as the most complete (given precise probabilities).
Makes sense. I cannot imagine any evidence that would update me. However, if I did, there would be falsifiability.
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.
Do we really have any evidence that the universe is globally flat? From Wikipediaâs page on the shape of the universe:
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.
What do you think about the St Petersburg problem now?
I think your arguments re infinities conflict with Occamâs 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 youâd guess.
A flat universe effectively has one fewer parameter and is simpler. So it shouldnât get 0 credence. Among flat universes, the bounded/âfinite 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 shouldnât 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:
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 shouldnât this get you to infinitely many fundamental forces? For any finite set of fundamental forces, you could posit another one and just say itâs 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 Occamâs razor.
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.
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 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.
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.
Yes.Edit after Michaelâs 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.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 Michaelâs comment just below. I would not assign a probability of exactly 1 to something falsifiable.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.
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.
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.
Why do you think I should expect all distributions to have infinite or undefined expected value instead of rejecting such distributions?
I corrected my answers in my past comment. You can see what I crossed out, and wrote after âEdit after Michaelâs comment just belowâ.
For the âghosts [I mentioned in my last comment] which have exactly 0 measurable effects on the worldâ, the benefit and harm they could cause would be sufficiently small to be practically negligible.
There is also Hitchensâs razor.
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. âWhat if I do care about the differences?â
Michaelâs most direct hit: Elgaâs Sally argument needs the two B-situations to be identical in everything she cares about, and Michael asks why the agent canât just care about whether sheâs about to complete a dominated sequence â âwhy canât the fact that sheâd pick a dominated sequence or regret it if she rejects both bets matter to her after rejecting bet A?â
Elga has a ready answer, and itâs the one he actually gives in the paper against the parallel âbut rejecting B would break her planâ rejoinder. It splits into a dilemma:
Either this caring is a genuine, independent source of value for Sally â in which case the case has been changed, not answered. Elgaâs 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, youâre no longer discussing Elgaâs agent; youâre 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. Thatâs ad hoc: the value exists only to deliver the verdict UNSHARP needs.
Or the caring is not an independent value but just tracks âthis would be irrationalâ â in which case itâs viciously circular. âI disprefer rejecting B because rejecting B here is irrationalâ cannot be what makes it irrational; the account owes us a prior reason, and this isnât one. Elgaâs âDonât break plans!â-is-like-âDonât break mirrors!â point applies verbatim: either breaking the sequence is independently costly (then say so, and itâs a different case) or it isnât (then âavoid completing dominated sequencesâ 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, itâs doing so as a real (dis)utility â back to horn one, the case is changed. Vascoâs reply on the forum (âit is very counterintuitive that this could matter for Sally for reasons that donât have to do with moneyâ) is exactly Elgaâs point, just stated flatly.
2. Michaelâs âtreat them fairlyâ /â Parfitâs-hitchhiker parity argument
This is Michaelâs best move, and itâs really DiGiovanniâs commitment point [made here] sharpened into a parity charge: there are cases everyone agrees call for binding commitments youâll later be inclined to break â Parfitâs hitchhiker, St. Petersburg with unbounded utility â so the same âcommit and rule out the bad branchâ solution should be available to the unsharp agent, if youâre treating her fairly. And he uses this to answer Vascoâs âbut unsharp probabilities are supposed to allow rejecting Aâ: âThey donât have to in every case. If it were A in isolation, both would be permissible. But thatâs not the case presented to us.â
Elga would grant the parity and then deny it helps â 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 Elgaâs whole thesis about this case: rationality forces a determinate verdict at the A-node. The disagreement was never âcan she avoid NEITHER?â â of course she can. Itâs whether the unsharp credence leaves A genuinely optional. Michael answers âno, not here,â 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. Thatâs confirmation of Elgaâs challenge â âhow do unsharp credences constrain action?â â with the answer âthey donât; something bolted on top does.â
Second, the Parfitâs-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 predictorâs reading changes your prospects). Thatâs exactly what legitimizes binding there. In Sallyâs case Elga has stipulated the two B-situations donât 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 Sallyâs stipulation does) and the rationale for binding evaporates. Michael can restore the rationale only by putting a real difference back in â which is move 1âČs first horn again, changing the case.
Vascoâs exchange on the hitchhiker actually pins this down: he points out that if you just âcommit as much as possible,â your chance of survival tracks your commitment probability and thereâs no residual puzzle. Michaelâs reply â âthe same solution is available to the unsharp agent if you treat them fairlyâ â 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 /â maximality argument for accepting A
Michaelâs most technical contribution (in the top comment) is a way for the unsharp agent to derive âaccept A firstâ without any of NARROW/âPLAN/âSEQUENCE: comparing âaccept A nowâ (call it 1) against âreject A and hope to accept Bâ (2), he says 1 statewise-beats 2 with some probability and theyâre incomparable otherwise â so under maximality 1 is permissible and heâll take it, killing the dominated branch.
Elgaâs objection: look at whatâs actually being compared. Option 2 as Michael frames it is âreject A and if I canât guarantee Iâll accept B, risk the dominated sequence.â To get 1 to dominate 2, he has to treat 2 as carrying a live risk of ending in NEITHER â i.e. he has to already be modeling his own future B-node choice as possibly landing on reject-B. But thatâs the entire question. If the agent could guarantee sheâll accept B after rejecting A (which is just the commitment), then 2 = B-only, which does not dominate 1 = A-only (theyâre 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 â in which case Elga simply agrees the sequence is a problem and asks what makes each local rejection rational â or it works by importing the commitment, in which case the imprecise credence is again idle and weâre at move 2âČs concession [see here]. Either way it doesnât vindicate UNSHARP; it either restates the problem or resolves it by non-credal means.
Thereâs 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. Michaelâs 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/âPLAN move â so Elga files it there and runs Sally. Michaelâs is the most resourceful version because heâs 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 itâs that commitment Sally targets.
4. The âarbitrary precisionâ tu quoque
Michaelâs jab â isnât requiring sharpness âany worse than picking numbers to ensure precision for no better reason than that they occurred to youâ? â is a real objection to SHARP, but Elga would note itâs an objection to the plausibility/âmotivation of sharpness, not to the bet argument. And SHARP has a specific shield here: recall it explicitly does not entail Uniqueness. Elga isnât 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 âyouâre forcing a spuriously exact numberâ misfires â 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 â 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 (âwhat if she cares about the difference?â) rather than trying to route around it, and heâs right that Elgaâs argument stands or falls on the stipulation that the two situations are identical in all respects the agent values. But Elgaâs reply is stable: every way of making the difference âmatterâ either (i) reintroduces a genuine downstream (dis)utility â which changes Sallyâs case and concedes that the money-only unsharp agent is stuck â or (ii) makes the mattering parasitic on âit would be irrational,â which is circular. The Parfitâs-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 (DiGiovanniâs commitment, Evanâs four-option reframe, Michaelâs statewise/âparity argument)? Michael states the crux most honestly â heâll happily say the unsharp agent is required to accept A here â and that very concession is what Elga reads as victory: the imprecision has stopped doing the one thing it was introduced to do.
Taking bet A doesnât require any commitment. My argument just uses backward induction (+ignoring statewise incomparability), which you should generally use in sequential choice situations, or else youâll 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.
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).
Here is a video I found useful that explains how to use backward induction. Below is Claudeâs reply to your comment after some iteration between us.
Thanks Michael â 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 itâs 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 isnât ad hoc (backward induction is the standard discipline for sequential choice). So this isnât PLAN in disguise. Fair enough.
But I think the argument turns on a step that quietly does more than âjust backward induction.â Here is the full tree, with payoffs written as (if H /â if notâH). Bet A pays â10/â+15 and Bet B pays +15/ââ10, so the four leaves are BOTH +5/â+5, A-only â10/â+15, B-only +15/ââ10, and NEITHER 0/â0:
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 â and the value of the reject-A branch depends entirely on which of its two (equally maximal) leaves you assume youâll pick.
Crucially, the accept-A node is also under-determined â 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 â» NEITHER. In particular A-only vs NEITHER crosses â A-only is worse than NEITHER in the H-state (â10 < 0) â so accepting A does not statewise-dominate rejecting A. Under either consistent policy (always-accept or always-reject), both root actions stay admissible and thereâs no dominance reason to prefer accepting A. And note that under âalways accept B,â NEITHER is never reached on either branch, so thereâs nothing for accepting-A to protect against in the first place.
The recommendation to accept A appears only under the third policy â 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 isnât backward induction resolving each node on its merits; itâs 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/âPLAN pattern Elgaâs Sally case is built to reject.
Put differently: the recommendation to accept A materialises only when you assume youâll reject B specifically on the reject-A branch â 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 youâre rationally required to prevent your future self from exercising reject-B, then reject-B was never really optional â which is just SHARPâs verdict reached the long way.
So a sharper version of my earlier question: your derivation of âaccept Aâ 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 âalways accept B,â reject-A gives B-only and the dominance is gone. If âalways reject B,â accept-A gives A-only and the dominance is gone. The only policy that yields it treats the two Bet B nodes differently â which is the thing an imprecise theorist owes an account of, and which Sally says you canât have.
(One aside on âyouâd use backward induction even with sharp probabilities, or be worse offâ: agreed, but with sharp credences backward induction never has to override a nodeâs verdict â 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 agentâs own decision rule calls permissible. That asymmetry is the tell.)
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.