Yeah, that seems like a pretty good way to go. I think creating too much imprecision might be a concern, and that there's also a concern about motivation. As I understand it, rectangularising in this case means adding probability functions to your representor on which, e.g., Pr(X | Heads)<0.01. But that seems incompatible with characterizing your representor as the set of probability functions you could settle on after ideal reflection on your current evidence, because (we can stipulate that X is such that) ideal reflection won't lead you to believe that X and Heads are so tightly anti-correlated. And given that, it seems maybe hard to justify including probability functions on which Pr(X | Heads)<0.01 in your representor (and thereby letting those probability functions affect what's permissible/impermissible for you).
I think it's reasonable to say that nowhere-optimal actions can be permissible if we don't have the dominating action in mind, but let me try push back a bit. Imagine that you're in a decision situation, thinking about what's permissible. You know that you have two options, A and B, and that neither of A and B dominates the other. However, you also remember thinking about this same decision situation in the past, where you recognized that you also have a third option C. You remember that C dominates B and that it doesn't dominate A. Unfortunately, you just can't remember what option C is. In this sort of case, I have the intuition that it'd be impermissible to choose B. If that's right, then actions can be rendered impermissible by virtue of being dominated by options that we don't have in mind.
On 3:
One thing to note here is that, although I use mixed actions to rule out nowhere-optimal actions, I don't advocate choosing mixed actions. The decision rule says you should choose somewhere-uniquely-optimal actions, and (as you say) it's impossible for a mixed action to be somewhere-uniquely-optimal. I think that helps a bit with the decision-theoretic-fishiness / dynamic inconsistency / paying to avoid information problems. See also my reply to Jesse's third point.
On 4:
Yes, it's true that more actions tend to become somewhere-uniquely-optimal when there are more probability functions in our representor. I still think that the decision rule makes lots of actions impermissible though. In particular, I think the 'flanking variants' test will rule out lots of actions. See also my reply to Jim Buhler on finding somewhere-uniquely-optimal actions.
On 5:
That's true. The preciser maximizes EV with respect to the probability function. The impreciser just has to maximize EV with respect to some probability function. But, as you say, those can be very different, so maybe it was an overstatement to say that we're almost back where we began.
And yes I think often we know more than just intervals of EVs. For example, we know whether the EV of some action increases or decreases with the probability of some proposition X.
one should never be required to strictly c-prefer a mixture of A and B to C whenever neither A nor B is c-preferred to C
There are candidate counterexamples to this claim. For example, imagine A is giving a benefit to Amy, and B and C each designate the same action of giving a benefit to Bobby. Then if you're impartial, you won't strictly c-prefer either of A or B to C, but you might strictly c-prefer the 50:50 mixture AB to C on the basis that it's fairer to randomize who gets the benefit.
Also, imprecise consequentialism (plus Dissent, Unanimity, and Justification) has an even more counterintuitive implication than 'you can be required to strictly c-prefer a mixture of AB to C even though neither A nor B is c-preferred to C.' It implies:
You can be required to strictly c-prefer a mixture of A, B, and C to D, even though (i) neither A nor B is c-preferred to D, (ii) C is c-dispreferred to D, and (iii) the mixture has an arbitrarily high probability of resulting in C.
Here's an example to illustrate:
I made the mixture have a 60% chance of C just to avoid the diagram being all bunched up. But the steeper you make the diagonals A and B, the higher you can push the probability of C and yet still have the mixture dominate D.
Thanks! Yeah, good question. The view I've sketched will say that A is c-impermissible, but we can still say that A is all-things-considered permissible after taking moral uncertainty into account. As an analogy, declining to push someone in front of a trolley to save 5 people is c-impermissible, but can be all-things-considered permissible after taking moral uncertainty into account.
On your second point, I think the thing I wrote in reply to Jim Buhler applies:
Basically I think the 'flanking variants' test will rule out a lot of actions but not all of them. In general, if our set of available actions is finite, then for each probability function there must be some action that has highest EV (and the same holds for infinite action sets modulo some small complications).
In practice, I think the easiest ones to identify will be extreme actions, like donating all your money to charity. Then we have some reason to think that the action is higher EV than its flanking variants, e.g. it's higher EV than giving less money, and it's impossible to give any more.
Of course, if we start individuating actions finely enough, then the flanking variants test might start ruling out even extreme actions. For example, it might be implausible that ideal reflection on your current evidence could lead you to judge that donating all your money at time X is higher EV than both (i) donating all your money one millisecond earlier and (ii) donating all your money one millisecond later. That would suggest donating all your money at time X is nowhere-optimal, in which case it's dominated by some mixed action, in which case it's c-impermissible.
But note that acting c-impermissibly in this way seems like an inevitable product of our cognitive limitations. So acting c-impermissibly in this way seems at least more forgiveable than choosing actions that are nowhere-optimal (and hence dominated) even on a coarse-grained individuation of our action space.
On your third point, I agree it seems kinda implausible to think you're required to strictly c-prefer the AB mixture to C even though neither A nor B is strictly c-preferred to C, but as you say denying it will have costs. We can read off one cost from my argument: you'll have to deny Dissent, Unanimity, or Justification.
Thanks! I guess given a choice between somewhere-uniquely-optimal actions, you should first check if any of them are ruled out by non-consequentialist considerations (like deontological constraints). If you still have multiple options at that stage, then the view I've sketched here will say that they're each permissible and that you can choose between them how you like. In practice, I'd try to choose the action that's somewhere-uniquely-optimal on my forced-to-say-a-number probability function, for moral uncertainty sorts of reasons.
Thanks! I wrote a reply to this but I now see it failed to send and I can't seem to recover the text.
Basically I think the 'flanking variants' test will rule out a lot of actions but not all of them. In general, if our set of available actions is finite, then for each probability function there must be some action that has highest EV (and the same holds for infinite action sets modulo some small complications).
In practice, I think the easiest ones to identify will be extreme actions, like donating all your money to charity. Then we have some reason to think that the action is higher EV than its flanking variants, e.g. it's higher EV than giving less money, and it's impossible to give any more.
Of course, if we start individuating actions finely enough, then the flanking variants test might start ruling out even extreme actions. For example, it might be implausible that ideal reflection on your current evidence could lead you to judge that donating all your money at time X is higher EV than both (i) donating all your money one millisecond earlier and (ii) donating all your money one millisecond later. That would suggest donating all your money at time X is nowhere-optimal, in which case it's dominated by some mixed action, in which case it's c-impermissible.
But note that acting c-impermissibly in this way seems like an inevitable product of our cognitive limitations, so it seems at least more forgiveable than choosing actions that are nowhere-optimal (and hence dominated) even on a coarse-grained individuation of our action space.
If you think you're justified in c-preferring the SWP donation, then Anthony's claim is disproved. But Anthony has said he doesn't consider this a counterexample, so it seems unlikely that offering more candidate counterexamples would move the debate forward.
That's especially so since I think the scope of Anthony's claim is intended to rule out other candidate counterexamples, e.g.:
You're trapped in a box that you know for sure will implode in 10 seconds. There's a puppy in there with you. You're justified in c-preferring not kicking the puppy to kicking the puppy.
That seems true to me, but I think this pair of actions falls outside the scope of Anthony's claim. He's talking about actions with effects that aren't so tightly limited in space and time.
So the debate calls for something more than just a bare counterexample. As Anthony says in another comment, I try to give that 'something more' in my post. Donating $5 to MAWF is justifiably c-dispreferred to some mixed action, as is any other action that fails the 'flanking variants' test. That likely rules out almost all actions.
Under maximality, we can't even say that [1, 10 000] is better than [-10 000, 1.0001].
That's not quite right. Maximality says an action A is impermissible when some alternative B has higher EV on every probability function in your representor. And that can be true even when A's and B's EV ranges overlap.
Example:
If A had the same range but sloped the other way, then it would be permissible by maximality:
So to figure out what's permissible under maximality, we can't just look at ranges. We need to look at the representor.
Yeah, that seems like a pretty good way to go. I think creating too much imprecision might be a concern, and that there's also a concern about motivation. As I understand it, rectangularising in this case means adding probability functions to your representor on which, e.g., Pr(X | Heads)<0.01. But that seems incompatible with characterizing your representor as the set of probability functions you could settle on after ideal reflection on your current evidence, because (we can stipulate that X is such that) ideal reflection won't lead you to believe that X and Heads are so tightly anti-correlated. And given that, it seems maybe hard to justify including probability functions on which Pr(X | Heads)<0.01 in your representor (and thereby letting those probability functions affect what's permissible/impermissible for you).
Thanks! Great points.
On 1:
I think it's reasonable to say that nowhere-optimal actions can be permissible if we don't have the dominating action in mind, but let me try push back a bit. Imagine that you're in a decision situation, thinking about what's permissible. You know that you have two options, A and B, and that neither of A and B dominates the other. However, you also remember thinking about this same decision situation in the past, where you recognized that you also have a third option C. You remember that C dominates B and that it doesn't dominate A. Unfortunately, you just can't remember what option C is. In this sort of case, I have the intuition that it'd be impermissible to choose B. If that's right, then actions can be rendered impermissible by virtue of being dominated by options that we don't have in mind.
On 3:
One thing to note here is that, although I use mixed actions to rule out nowhere-optimal actions, I don't advocate choosing mixed actions. The decision rule says you should choose somewhere-uniquely-optimal actions, and (as you say) it's impossible for a mixed action to be somewhere-uniquely-optimal. I think that helps a bit with the decision-theoretic-fishiness / dynamic inconsistency / paying to avoid information problems. See also my reply to Jesse's third point.
On 4:
Yes, it's true that more actions tend to become somewhere-uniquely-optimal when there are more probability functions in our representor. I still think that the decision rule makes lots of actions impermissible though. In particular, I think the 'flanking variants' test will rule out lots of actions. See also my reply to Jim Buhler on finding somewhere-uniquely-optimal actions.
On 5:
That's true. The preciser maximizes EV with respect to the probability function. The impreciser just has to maximize EV with respect to some probability function. But, as you say, those can be very different, so maybe it was an overstatement to say that we're almost back where we began.
Oops, pictures should be fixed now.
And yes I think often we know more than just intervals of EVs. For example, we know whether the EV of some action increases or decreases with the probability of some proposition X.
Extra stuff:
There are candidate counterexamples to this claim. For example, imagine A is giving a benefit to Amy, and B and C each designate the same action of giving a benefit to Bobby. Then if you're impartial, you won't strictly c-prefer either of A or B to C, but you might strictly c-prefer the 50:50 mixture AB to C on the basis that it's fairer to randomize who gets the benefit.
Also, imprecise consequentialism (plus Dissent, Unanimity, and Justification) has an even more counterintuitive implication than 'you can be required to strictly c-prefer a mixture of AB to C even though neither A nor B is c-preferred to C.' It implies:
Here's an example to illustrate:
I made the mixture have a 60% chance of C just to avoid the diagram being all bunched up. But the steeper you make the diagonals A and B, the higher you can push the probability of C and yet still have the mixture dominate D.
Thanks! Yeah, good question. The view I've sketched will say that A is c-impermissible, but we can still say that A is all-things-considered permissible after taking moral uncertainty into account. As an analogy, declining to push someone in front of a trolley to save 5 people is c-impermissible, but can be all-things-considered permissible after taking moral uncertainty into account.
On your second point, I think the thing I wrote in reply to Jim Buhler applies:
On your third point, I agree it seems kinda implausible to think you're required to strictly c-prefer the AB mixture to C even though neither A nor B is strictly c-preferred to C, but as you say denying it will have costs. We can read off one cost from my argument: you'll have to deny Dissent, Unanimity, or Justification.
Thanks! I guess given a choice between somewhere-uniquely-optimal actions, you should first check if any of them are ruled out by non-consequentialist considerations (like deontological constraints). If you still have multiple options at that stage, then the view I've sketched here will say that they're each permissible and that you can choose between them how you like. In practice, I'd try to choose the action that's somewhere-uniquely-optimal on my forced-to-say-a-number probability function, for moral uncertainty sorts of reasons.
Thanks! I wrote a reply to this but I now see it failed to send and I can't seem to recover the text.
Basically I think the 'flanking variants' test will rule out a lot of actions but not all of them. In general, if our set of available actions is finite, then for each probability function there must be some action that has highest EV (and the same holds for infinite action sets modulo some small complications).
In practice, I think the easiest ones to identify will be extreme actions, like donating all your money to charity. Then we have some reason to think that the action is higher EV than its flanking variants, e.g. it's higher EV than giving less money, and it's impossible to give any more.
Of course, if we start individuating actions finely enough, then the flanking variants test might start ruling out even extreme actions. For example, it might be implausible that ideal reflection on your current evidence could lead you to judge that donating all your money at time X is higher EV than both (i) donating all your money one millisecond earlier and (ii) donating all your money one millisecond later. That would suggest donating all your money at time X is nowhere-optimal, in which case it's dominated by some mixed action, in which case it's c-impermissible.
But note that acting c-impermissibly in this way seems like an inevitable product of our cognitive limitations, so it seems at least more forgiveable than choosing actions that are nowhere-optimal (and hence dominated) even on a coarse-grained individuation of our action space.
Yep, that's right!
I don't think that's so surprising. There are obvious candidate counterexamples, e.g.:
If you think you're justified in c-preferring the SWP donation, then Anthony's claim is disproved. But Anthony has said he doesn't consider this a counterexample, so it seems unlikely that offering more candidate counterexamples would move the debate forward.
That's especially so since I think the scope of Anthony's claim is intended to rule out other candidate counterexamples, e.g.:
That seems true to me, but I think this pair of actions falls outside the scope of Anthony's claim. He's talking about actions with effects that aren't so tightly limited in space and time.
So the debate calls for something more than just a bare counterexample. As Anthony says in another comment, I try to give that 'something more' in my post. Donating $5 to MAWF is justifiably c-dispreferred to some mixed action, as is any other action that fails the 'flanking variants' test. That likely rules out almost all actions.
That's not quite right. Maximality says an action A is impermissible when some alternative B has higher EV on every probability function in your representor. And that can be true even when A's and B's EV ranges overlap.
Example:
If A had the same range but sloped the other way, then it would be permissible by maximality:
So to figure out what's permissible under maximality, we can't just look at ranges. We need to look at the representor.