At least in practice, thereâs a clear difference between considering bet A in isolation and considering bet A when you know bet B is coming. If you told me about a sports game between the Snofuls and the Fleertis and offered me 2:1 odds on the Snofuls to win, I wouldnât take it. But if you told me you would also give me 2:1 odds on the Fleertis to win, I would take both bets, guaranteeing a profit.
Accepting the 1st bet if you were confident Snofuls would win, accepting the 2nd if you were confident Fleertis would win, and accepting both if you thought the probability of any of the teams winning was close to 50 % would be in agreement with sharp probabilities.
Elga grants youâre not required to accept both (a very confident or very doubtful agent might prefer just one). But he insists on the key premise: a rational agent must accept at least one of the two bets, because rejecting both is dominatedâitâs worse than accepting both in every outcome, and you can see this in advance. This premise is easy for a sharp-credence theorist to honor. The rest of the paper argues that no version of the unsharp view can.
As a rational actor with no useful information, I have a very broad range of potential probabilities for this bet, and it is permissible to do neither bet in isolation. However, when we consider our options simultaneously, that changes the calculus.
Which of the 3 strategies described by Adam would you use to justify accepting or rejecting each bet in isolation, but rejecting both bets together?
To apply this to altruistic action, there might be actions that we are uncertain about in isolation, but we are willing to pursue as a part of a portfolio approach.
This is not an argument for unsharp probabilities? Supporting a portfolio of interventions makes sense even with sharp probabilities. Marginal cost-effectiveness tends to decrease with spending. For example, if the Animal Welfare Fund (AWF) had granted 2 times as much to all the grantees they supported in 2025, I expect the impact of the grants would have been larger, but less than 2 times as large.
I agree that accepting both bets is consistent with a sharp probability at 50%, though Iâm just trying to give an example of a case where I would have an unsharp probability range where I would reject both bets in isolation but take them when they arrive together.
I donât employ any of the 3 strategies. My argument is that you donât need a fancy strategy because, in the example, you know that bet B is coming when youâre asked about bet A. I think itâs reasonable for a rational actor to reject bet A and reject bet B if the two are presented separately but accept them both if they are presented together. My example is intended to demonstrate that. A rational actor doesnât need NARROW, PLAN, or SEQUENCE. They need to consider the future: âBet B is coming, so thereâs an arbitrage opportunity regardless of the probability.â The article seems to disagree, treating every action in isolation and requiring that we make the right decision without global thinking.
My recommendation for portfolios is not an argument for, but an implication of, unsharp probabilities. A lot of cause prioritization is about the core philosophical positions you hold underpinning it. If you have a sharp probability, you might be comfortable investing all in one cause. If you have an unsharp one, you might not be convinced that investing in any one cause is net positive. However, you might find a combination of causes that seems robustly better than no action.
For example, you might be concerned about climate policyâs constraints on growth as well as growthâs effect on the climate. If you believe that the second order effects of investing in growth on the climate are smaller than the direct benefits of donating to climate policy (and vice versa), it is strictly better to donate to both in some combination than to do nothing. Someone with a sharp probability might be comfortable donating to just one in a way someone with unsharp probabilities would not.
As a result, portfolios are better (i.e. are more often optimal) in a world where UNSHARP is true.
I see and agree with your point about marginal returns. Depending on how strong that effect is, portfolios are also good in a world with sharp probabilities only.
A rational actor doesnât need NARROW, PLAN, or SEQUENCE. They need to consider the future: âBet B is coming, so thereâs an arbitrage opportunity regardless of the probability.â
I do not seem to understand. If one knew âBet B is comingâ, one would know about the full set up in advance as in the post (âYouâre told the full setup in advanceâ). So rejecting both A and B would not make sense?
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 agentâs choices.
At no point did I need to employ NARROW, PLAN, or SEQUENCE. I didnât 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, âWhat if, of the 4 options, you choose B? Could you change your mind after rejecting A and then reject B as well?â 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 itâs 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 itâs optional due to UNSHARP probabilities, and reject it. Then, you are offered B, evaluate it, conclude itâs optional, and reject it. You look back and think âI 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.â
I think it is rational for an actor to hold unsharp probabilities for some hypotheses.[1] I think itâs rational to not engage in sports gambling when no arbitrage exists. My initial example was designed to connect the two.
I havenât made my mind up on whether itâs necessary to hold unsharp probabilities in theory but Iâm 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
Evanâs central claim is that he can dissolve the puzzle without NARROW, PLAN, or SEQUENCE: treat the situation not as two decisions (A yes/âno, B yes/âno) but as one choice among four policies â {A-only, B-only, BOTH, NEITHER} â notice BOTH statewise-dominates NEITHER, delete NEITHER, and youâre done. He stresses âI didnât even consider the probability of H.â
Elgaâs first reply is that this is exactly SEQUENCE (or PLAN) wearing plain clothes â and Evan concedes it in his own EDIT (âI think this is close in effect to SEQUENCEâ). Evaluating the pair of choices as a single ex-ante object over sequences is the defining move of the global rules. So âI donât need any of the threeâ is false: heâs using the third. And that matters, because Sally is aimed precisely here. Take Evanâs B-only policy: it requires rejecting A and then accepting B. Compare the agent at the B-node in two situations â 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 âconsider them simultaneouslyâ doesnât reach the actual problem
Evanâs sports example â decline each of the Snofuls/âFleertis bets in isolation, take both together for a sure profit â leans on âwhen we consider our options simultaneously, that changes the calculus.â Elgaâs 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/ââ10) versus reject-B (0). With an interval straddling 40%, maximality rules both permissible. The ex-ante fact âBOTH dominates NEITHERâ 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/âSEQUENCE â which is why Evan canât actually skip them.
And the boast âI didnât even need to consider P(H)â is the tell, not the triumph. Dominance eliminates NEITHER for any credence â 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 doesnât engage it.
The EDIT smuggles in comparability â i.e. sharpness
Evan tries to close the âwhat if you plan B, reject A, then reject B?â gap thus: âa 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.â
This quietly assumes a complete ordering over the options â 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 âthey would only choose B if B > BOTHâ 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 â but youâve then imported enough structure that she behaves like a sharp agent, which is Elgaâs strict-rules horn: you buy the right behavior only by reintroducing precision and thereby forfeiting the motivation for going unsharp in the first place.
âRational actors with less information make worse decisionsâ 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, Elgaâs case stipulates full foreknowledge, so the no-foreknowledge scenario isnât the one under discussion. Second, and more damaging, the diagnosis âless informationâ is wrong. A sharp agent â even with a diffuse-but-precise prior, and even with no foreknowledge â 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 canât jointly fail). The unsharp agentâs node verdicts are not automatically coherent: both nodes say âoptional,â which is what lets her assemble NEITHER. So the pathology is produced by the unsharpness, not by any information gap. Evanâs concession thus admits precisely the foreseeable-domination Elga is prosecuting, and mislabels its source.
The portfolio point isnât 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 (âthis combination statewise-beats doing nothingâ), itâs dominance reasoning a sharp agent honors equally; where it gives distinctively unsharp verdicts, it does so by licensing inaction â declining each option in isolation â which is just the reject-both pathology relocated to altruistic choice. (This is the âclueless agent whose intervals stay wide because it never actsâ failure mode, which is live in your own work.)
âCan vs. shouldâ is not a dodge â itâs Elgaâs exact target
Evanâs sign-off â âwhether 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 canâ â doesnât sidestep Elga. UNSHARP just is the âcanâ claim: it is consistent with perfect rationality to be unsharp. SHARP denies that. So Evan is engaging the thesis head-on, and Elgaâs reply is that the âcanâ 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 â shared with DiGiovanni and Michael St Jules â is the suspicion that node-by-node âlocalâ evaluation is the wrong model, and that a look-ahead agent who plans the whole tree does fine with wide intervals. Elgaâs 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 hasnât defeated Sally independently; heâs relocated to ex-ante policy choice, which Elga classifies as SEQUENCE/âPLAN and which Evan himself admits is âclose in effect to SEQUENCE.â 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/âsophisticated look-ahead), or whether rationality must already be satisfiable choice-by-choice. Elga bets on the latter; Evan (like DiGiovanni) needs the former â and that is the genuine open question, not something Evanâs four-option reframe settles.
Hello Evan.
Accepting the 1st bet if you were confident Snofuls would win, accepting the 2nd if you were confident Fleertis would win, and accepting both if you thought the probability of any of the teams winning was close to 50 % would be in agreement with sharp probabilities.
Which of the 3 strategies described by Adam would you use to justify accepting or rejecting each bet in isolation, but rejecting both bets together?
This is not an argument for unsharp probabilities? Supporting a portfolio of interventions makes sense even with sharp probabilities. Marginal cost-effectiveness tends to decrease with spending. For example, if the Animal Welfare Fund (AWF) had granted 2 times as much to all the grantees they supported in 2025, I expect the impact of the grants would have been larger, but less than 2 times as large.
I agree that accepting both bets is consistent with a sharp probability at 50%, though Iâm just trying to give an example of a case where I would have an unsharp probability range where I would reject both bets in isolation but take them when they arrive together.
I donât employ any of the 3 strategies. My argument is that you donât need a fancy strategy because, in the example, you know that bet B is coming when youâre asked about bet A. I think itâs reasonable for a rational actor to reject bet A and reject bet B if the two are presented separately but accept them both if they are presented together. My example is intended to demonstrate that. A rational actor doesnât need NARROW, PLAN, or SEQUENCE. They need to consider the future: âBet B is coming, so thereâs an arbitrage opportunity regardless of the probability.â The article seems to disagree, treating every action in isolation and requiring that we make the right decision without global thinking.
My recommendation for portfolios is not an argument for, but an implication of, unsharp probabilities. A lot of cause prioritization is about the core philosophical positions you hold underpinning it. If you have a sharp probability, you might be comfortable investing all in one cause. If you have an unsharp one, you might not be convinced that investing in any one cause is net positive. However, you might find a combination of causes that seems robustly better than no action.
For example, you might be concerned about climate policyâs constraints on growth as well as growthâs effect on the climate. If you believe that the second order effects of investing in growth on the climate are smaller than the direct benefits of donating to climate policy (and vice versa), it is strictly better to donate to both in some combination than to do nothing. Someone with a sharp probability might be comfortable donating to just one in a way someone with unsharp probabilities would not.
As a result, portfolios are better (i.e. are more often optimal) in a world where UNSHARP is true.
I see and agree with your point about marginal returns. Depending on how strong that effect is, portfolios are also good in a world with sharp probabilities only.
I do not seem to understand. If one knew âBet B is comingâ, one would know about the full set up in advance as in the post (âYouâre told the full setup in advanceâ). So rejecting both A and B would not make sense?
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 agentâs choices.
At no point did I need to employ NARROW, PLAN, or SEQUENCE. I didnât 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, âWhat if, of the 4 options, you choose B? Could you change your mind after rejecting A and then reject B as well?â 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 itâs 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 itâs optional due to UNSHARP probabilities, and reject it. Then, you are offered B, evaluate it, conclude itâs optional, and reject it. You look back and think âI 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.â
I think it is rational for an actor to hold unsharp probabilities for some hypotheses.[1] I think itâs rational to not engage in sports gambling when no arbitrage exists. My initial example was designed to connect the two.
I havenât made my mind up on whether itâs necessary to hold unsharp probabilities in theory but Iâm 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
Evanâs central claim is that he can dissolve the puzzle without NARROW, PLAN, or SEQUENCE: treat the situation not as two decisions (A yes/âno, B yes/âno) but as one choice among four policies â {A-only, B-only, BOTH, NEITHER} â notice BOTH statewise-dominates NEITHER, delete NEITHER, and youâre done. He stresses âI didnât even consider the probability of H.â
Elgaâs first reply is that this is exactly SEQUENCE (or PLAN) wearing plain clothes â and Evan concedes it in his own EDIT (âI think this is close in effect to SEQUENCEâ). Evaluating the pair of choices as a single ex-ante object over sequences is the defining move of the global rules. So âI donât need any of the threeâ is false: heâs using the third. And that matters, because Sally is aimed precisely here. Take Evanâs B-only policy: it requires rejecting A and then accepting B. Compare the agent at the B-node in two situations â 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 âconsider them simultaneouslyâ doesnât reach the actual problem
Evanâs sports example â decline each of the Snofuls/âFleertis bets in isolation, take both together for a sure profit â leans on âwhen we consider our options simultaneously, that changes the calculus.â Elgaâs 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/ââ10) versus reject-B (0). With an interval straddling 40%, maximality rules both permissible. The ex-ante fact âBOTH dominates NEITHERâ 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/âSEQUENCE â which is why Evan canât actually skip them.
And the boast âI didnât even need to consider P(H)â is the tell, not the triumph. Dominance eliminates NEITHER for any credence â 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 doesnât engage it.
The EDIT smuggles in comparability â i.e. sharpness
Evan tries to close the âwhat if you plan B, reject A, then reject B?â gap thus: âa 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.â
This quietly assumes a complete ordering over the options â 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 âthey would only choose B if B > BOTHâ 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 â but youâve then imported enough structure that she behaves like a sharp agent, which is Elgaâs strict-rules horn: you buy the right behavior only by reintroducing precision and thereby forfeiting the motivation for going unsharp in the first place.
âRational actors with less information make worse decisionsâ 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, Elgaâs case stipulates full foreknowledge, so the no-foreknowledge scenario isnât the one under discussion. Second, and more damaging, the diagnosis âless informationâ is wrong. A sharp agent â even with a diffuse-but-precise prior, and even with no foreknowledge â 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 canât jointly fail). The unsharp agentâs node verdicts are not automatically coherent: both nodes say âoptional,â which is what lets her assemble NEITHER. So the pathology is produced by the unsharpness, not by any information gap. Evanâs concession thus admits precisely the foreseeable-domination Elga is prosecuting, and mislabels its source.
The portfolio point isnât 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 (âthis combination statewise-beats doing nothingâ), itâs dominance reasoning a sharp agent honors equally; where it gives distinctively unsharp verdicts, it does so by licensing inaction â declining each option in isolation â which is just the reject-both pathology relocated to altruistic choice. (This is the âclueless agent whose intervals stay wide because it never actsâ failure mode, which is live in your own work.)
âCan vs. shouldâ is not a dodge â itâs Elgaâs exact target
Evanâs sign-off â âwhether 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 canâ â doesnât sidestep Elga. UNSHARP just is the âcanâ claim: it is consistent with perfect rationality to be unsharp. SHARP denies that. So Evan is engaging the thesis head-on, and Elgaâs reply is that the âcanâ 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 â shared with DiGiovanni and Michael St Jules â is the suspicion that node-by-node âlocalâ evaluation is the wrong model, and that a look-ahead agent who plans the whole tree does fine with wide intervals. Elgaâs 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 hasnât defeated Sally independently; heâs relocated to ex-ante policy choice, which Elga classifies as SEQUENCE/âPLAN and which Evan himself admits is âclose in effect to SEQUENCE.â 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/âsophisticated look-ahead), or whether rationality must already be satisfiable choice-by-choice. Elga bets on the latter; Evan (like DiGiovanni) needs the former â and that is the genuine open question, not something Evanâs four-option reframe settles.