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.
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.
I see. Thanks for clarifying. Below is how Claude thinks Adam (the author of the article) would object to your comments. The objections make sense to me. Any reactions?
The unifying objection: the four-option reframe is one of the three rules
Evan’s central claim is that he can dissolve the puzzle without NARROW, PLAN, or SEQUENCE: treat the situation not as two decisions (A yes/no, B yes/no) but as one choice among four policies — {A-only, B-only, BOTH, NEITHER} — notice BOTH statewise-dominates NEITHER, delete NEITHER, and you’re done. He stresses “I didn’t even consider the probability of H.”
Elga’s first reply is that this is exactly SEQUENCE (or PLAN) wearing plain clothes — and Evan concedes it in his own EDIT (“I think this is close in effect to SEQUENCE”). Evaluating the pair of choices as a single ex-ante object over sequences is the defining move of the global rules. So “I don’t need any of the three” is false: he’s using the third. And that matters, because Sally is aimed precisely here. Take Evan’s B-only policy: it requires rejecting A and then accepting B. Compare the agent at the B-node in two situations — one where she reached it by rejecting A, one where B is offered alone. For a money-only agent these are identical in everything she cares about, yet the reframe must call rejecting-B impermissible in the first (it would complete NEITHER) and permissible in the second. That is the SEQUENCE verdict, and it fails for the SEQUENCE reason.
Why “consider them simultaneously” doesn’t reach the actual problem
Evan’s sports example — decline each of the Snofuls/Fleertis bets in isolation, take both together for a sure profit — leans on “when we consider our options simultaneously, that changes the calculus.” Elga’s rejoinder: in his setup the bets are not simultaneous. You settle A, and only then face B. So the live question is what binds you at the B-node, where A is already done and the only comparison is accept-B (+15/−10) versus reject-B (0). With an interval straddling 40%, maximality rules both permissible. The ex-ante fact “BOTH dominates NEITHER” is true but does not, by itself, reach into the B-node and make accepting B required there. Supplying that reach is the whole job of PLAN/SEQUENCE — which is why Evan can’t actually skip them.
And the boast “I didn’t even need to consider P(H)” is the tell, not the triumph. Dominance eliminates NEITHER for any credence — a sharp agent excludes it too. So the four-option elimination is entirely neutral between SHARP and UNSHARP; it was never the point in dispute. The dispute is about the sequential assembly of a dominated outcome from two individually-licensed choices, and the reframe simply doesn’t engage it.
The EDIT smuggles in comparability — i.e. sharpness
Evan tries to close the “what if you plan B, reject A, then reject B?” gap thus: “a rational actor does not change their mind without new information. They would only choose B if they believe B > BOTH > NEITHER. Any rational actor who believes B > NEITHER would end up betting B.”
This quietly assumes a complete ordering over the options — exactly what UNSHARP denies. B-only beats BOTH only when P(H) > 60%; with the interval [10%, 80%], B-only and BOTH are incomparable under maximality, as are A-only and BOTH. So “they would only choose B if B > BOTH” presupposes the agent can rank options the way a sharp credence lets her. Grant that comparability and of course she never lands on a dominated outcome — but you’ve then imported enough structure that she behaves like a sharp agent, which is Elga’s strict-rules horn: you buy the right behavior only by reintroducing precision and thereby forfeiting the motivation for going unsharp in the first place.
“Rational actors with less information make worse decisions” gives the game away
Evan concedes that without foreknowledge an UNSHARP agent can reject A as optional, reject B as optional, land on NEITHER, and shrug it off as an information deficit. Two problems. First, Elga’s case stipulates full foreknowledge, so the no-foreknowledge scenario isn’t the one under discussion. Second, and more damaging, the diagnosis “less information” is wrong. A sharp agent — even with a diffuse-but-precise prior, and even with no foreknowledge — never rejects both, because her node-by-node expected-value verdicts are automatically time-coherent (reject A only if P(H) > 60%, accept B only if P(H) > 40%, and these can’t jointly fail). The unsharp agent’s node verdicts are not automatically coherent: both nodes say “optional,” which is what lets her assemble NEITHER. So the pathology is produced by the unsharpness, not by any information gap. Evan’s concession thus admits precisely the foreseeable-domination Elga is prosecuting, and mislabels its source.
The portfolio point isn’t an argument for UNSHARP
Vasco already made the core objection and Evan half-conceded it: diversification falls straight out of sharp EV reasoning with diminishing marginal returns and cross-correlations. Elga would add the sharper version: where the portfolio reasoning gives sensible verdicts (“this combination statewise-beats doing nothing”), it’s dominance reasoning a sharp agent honors equally; where it gives distinctively unsharp verdicts, it does so by licensing inaction — declining each option in isolation — which is just the reject-both pathology relocated to altruistic choice. (This is the “clueless agent whose intervals stay wide because it never acts” failure mode, which is live in your own work.)
“Can vs. should” is not a dodge — it’s Elga’s exact target
Evan’s sign-off — “whether we should have unsharp probabilities is beside the point; my argument is about whether we can have them without sacrificing rationality, and I believe we can” — doesn’t sidestep Elga. UNSHARP just is the “can” claim: it is consistent with perfect rationality to be unsharp. SHARP denies that. So Evan is engaging the thesis head-on, and Elga’s reply is that the “can” fails for the reasons above: every route Evan takes either collapses into SEQUENCE (Sally sinks it) or into sharp-style comparability (motivation lost).
The honest crux
Where Evan has a real point — shared with DiGiovanni and Michael St Jules — is the suspicion that node-by-node “local” evaluation is the wrong model, and that a look-ahead agent who plans the whole tree does fine with wide intervals. Elga’s whole case does assume that a theory of rational credence must deliver correct verdicts at each actual choice node, not merely over ex-ante policies. Evan is, in effect, denying that assumption. But he hasn’t defeated Sally independently; he’s relocated to ex-ante policy choice, which Elga classifies as SEQUENCE/PLAN and which Evan himself admits is “close in effect to SEQUENCE.” So the disagreement bottoms out exactly where it did in the DiGiovanni thread [this one]: whether an idealized agent is entitled to bind her future choices (resolute/sophisticated look-ahead), or whether rationality must already be satisfiable choice-by-choice. Elga bets on the latter; Evan (like DiGiovanni) needs the former — and that is the genuine open question, not something Evan’s four-option reframe settles.
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.
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.
I see. Thanks for clarifying. Below is how Claude thinks Adam (the author of the article) would object to your comments. The objections make sense to me. Any reactions?
The unifying objection: the four-option reframe is one of the three rules
Evan’s central claim is that he can dissolve the puzzle without NARROW, PLAN, or SEQUENCE: treat the situation not as two decisions (A yes/no, B yes/no) but as one choice among four policies — {A-only, B-only, BOTH, NEITHER} — notice BOTH statewise-dominates NEITHER, delete NEITHER, and you’re done. He stresses “I didn’t even consider the probability of H.”
Elga’s first reply is that this is exactly SEQUENCE (or PLAN) wearing plain clothes — and Evan concedes it in his own EDIT (“I think this is close in effect to SEQUENCE”). Evaluating the pair of choices as a single ex-ante object over sequences is the defining move of the global rules. So “I don’t need any of the three” is false: he’s using the third. And that matters, because Sally is aimed precisely here. Take Evan’s B-only policy: it requires rejecting A and then accepting B. Compare the agent at the B-node in two situations — one where she reached it by rejecting A, one where B is offered alone. For a money-only agent these are identical in everything she cares about, yet the reframe must call rejecting-B impermissible in the first (it would complete NEITHER) and permissible in the second. That is the SEQUENCE verdict, and it fails for the SEQUENCE reason.
Why “consider them simultaneously” doesn’t reach the actual problem
Evan’s sports example — decline each of the Snofuls/Fleertis bets in isolation, take both together for a sure profit — leans on “when we consider our options simultaneously, that changes the calculus.” Elga’s rejoinder: in his setup the bets are not simultaneous. You settle A, and only then face B. So the live question is what binds you at the B-node, where A is already done and the only comparison is accept-B (+15/−10) versus reject-B (0). With an interval straddling 40%, maximality rules both permissible. The ex-ante fact “BOTH dominates NEITHER” is true but does not, by itself, reach into the B-node and make accepting B required there. Supplying that reach is the whole job of PLAN/SEQUENCE — which is why Evan can’t actually skip them.
And the boast “I didn’t even need to consider P(H)” is the tell, not the triumph. Dominance eliminates NEITHER for any credence — a sharp agent excludes it too. So the four-option elimination is entirely neutral between SHARP and UNSHARP; it was never the point in dispute. The dispute is about the sequential assembly of a dominated outcome from two individually-licensed choices, and the reframe simply doesn’t engage it.
The EDIT smuggles in comparability — i.e. sharpness
Evan tries to close the “what if you plan B, reject A, then reject B?” gap thus: “a rational actor does not change their mind without new information. They would only choose B if they believe B > BOTH > NEITHER. Any rational actor who believes B > NEITHER would end up betting B.”
This quietly assumes a complete ordering over the options — exactly what UNSHARP denies. B-only beats BOTH only when P(H) > 60%; with the interval [10%, 80%], B-only and BOTH are incomparable under maximality, as are A-only and BOTH. So “they would only choose B if B > BOTH” presupposes the agent can rank options the way a sharp credence lets her. Grant that comparability and of course she never lands on a dominated outcome — but you’ve then imported enough structure that she behaves like a sharp agent, which is Elga’s strict-rules horn: you buy the right behavior only by reintroducing precision and thereby forfeiting the motivation for going unsharp in the first place.
“Rational actors with less information make worse decisions” gives the game away
Evan concedes that without foreknowledge an UNSHARP agent can reject A as optional, reject B as optional, land on NEITHER, and shrug it off as an information deficit. Two problems. First, Elga’s case stipulates full foreknowledge, so the no-foreknowledge scenario isn’t the one under discussion. Second, and more damaging, the diagnosis “less information” is wrong. A sharp agent — even with a diffuse-but-precise prior, and even with no foreknowledge — never rejects both, because her node-by-node expected-value verdicts are automatically time-coherent (reject A only if P(H) > 60%, accept B only if P(H) > 40%, and these can’t jointly fail). The unsharp agent’s node verdicts are not automatically coherent: both nodes say “optional,” which is what lets her assemble NEITHER. So the pathology is produced by the unsharpness, not by any information gap. Evan’s concession thus admits precisely the foreseeable-domination Elga is prosecuting, and mislabels its source.
The portfolio point isn’t an argument for UNSHARP
Vasco already made the core objection and Evan half-conceded it: diversification falls straight out of sharp EV reasoning with diminishing marginal returns and cross-correlations. Elga would add the sharper version: where the portfolio reasoning gives sensible verdicts (“this combination statewise-beats doing nothing”), it’s dominance reasoning a sharp agent honors equally; where it gives distinctively unsharp verdicts, it does so by licensing inaction — declining each option in isolation — which is just the reject-both pathology relocated to altruistic choice. (This is the “clueless agent whose intervals stay wide because it never acts” failure mode, which is live in your own work.)
“Can vs. should” is not a dodge — it’s Elga’s exact target
Evan’s sign-off — “whether we should have unsharp probabilities is beside the point; my argument is about whether we can have them without sacrificing rationality, and I believe we can” — doesn’t sidestep Elga. UNSHARP just is the “can” claim: it is consistent with perfect rationality to be unsharp. SHARP denies that. So Evan is engaging the thesis head-on, and Elga’s reply is that the “can” fails for the reasons above: every route Evan takes either collapses into SEQUENCE (Sally sinks it) or into sharp-style comparability (motivation lost).
The honest crux
Where Evan has a real point — shared with DiGiovanni and Michael St Jules — is the suspicion that node-by-node “local” evaluation is the wrong model, and that a look-ahead agent who plans the whole tree does fine with wide intervals. Elga’s whole case does assume that a theory of rational credence must deliver correct verdicts at each actual choice node, not merely over ex-ante policies. Evan is, in effect, denying that assumption. But he hasn’t defeated Sally independently; he’s relocated to ex-ante policy choice, which Elga classifies as SEQUENCE/PLAN and which Evan himself admits is “close in effect to SEQUENCE.” So the disagreement bottoms out exactly where it did in the DiGiovanni thread [this one]: whether an idealized agent is entitled to bind her future choices (resolute/sophisticated look-ahead), or whether rationality must already be satisfiable choice-by-choice. Elga bets on the latter; Evan (like DiGiovanni) needs the former — and that is the genuine open question, not something Evan’s four-option reframe settles.