Thank you. I think that's a fair summary. To be precise, rather than claiming that the maximality rule is generally too sensitive to small changes, my critique is that it's generally too insensitive to small changes, as virtually all updates that would be registered by a graded approach make no difference at all. As a consequence of this insensitivity, maximality implies isolated points of extreme sensitivity where some tiny change can result in a categorical jump. So yes, I think it's too sensitive to small changes in the sense that it packs all its sensitivity into those sharp points where dominance is achieved.
I agree that indeterminacy can still obtain under a graded approach. However, it’s worth noting that the band in which indeterminacy obtains under a graded approach will generally be much narrower than under maximality. So moving from maximality to a graded approach is a big change.
When I try to weigh up the reasons in favor of any A vs. B w.r.t. their total consequences, fully accounting for unawareness, I really don't see why I should consider the "upper endpoint" larger or smaller in absolute value than the "lower endpoint".
I gather we see this differently, but to very briefly explain why it seems warranted to me: Because of the evidence and arguments favoring the most plausible strategies, and the lack of similarly strong evidence and arguments against them. In particular, I don’t think the arguments against the capacity-building strategies I discuss are as strong as those in their favor. Moreover, much of the value of building capacity is the value of being able to act on considerations we aren’t yet aware of. So, in my view, unawareness bears asymmetrically on these strategies rather than neutrally (I realize this latter point is stated very briefly and needs further development).
I would like to see more of an argument that this isn't our situation, perhaps building on what you say in your appendix about low-footprint capacity-building.
Without trying to build an elaborate case here, there is a simple extension that I think further strengthens the case: instead of focusing only on the best-supported strategies, we can specifically compare the seemingly best-supported strategy (say, some form of low-footprint CB) with the strategy that seems worst (I suspect it’s better not to go into details about what that might be).
When comparing two such contrasting strategies, the expected difference plausibly gets beyond the (relatively) narrow band of indeterminacy under some plausible graded approach, considering both the arguments favoring the seemingly better strategy and the arguments opposing the seemingly worse strategy. The reason this helps is that the width of the indeterminacy band doesn’t obviously scale with how contrasting the pair is, whereas the expected difference does. (Again, the expected difference between the two strategies need not be remotely as stark as in the figure below.)
You can have insensitivity to mild sweetening under vague degrees of betterness; it doesn't assume maximality.
I agree that there can be insensitivity to mild sweetening under rules other than maximality. (As noted in more general terms, the blanket objection that rejects “any evidence or proposal we might wager on … assumes the kind of categorical decision rule that collapses a broad set of evidential states into complete and undifferentiated indeterminacy.”)
But interestingly, under a graded approach, even if indeterminacy may in some sense be preserved after a mild sweetening (within the relatively narrow indeterminacy band), we still wouldn’t get complete insensitivity to mild sweetening. In that case, we would have a hazy cloud of vague expected values, and a vague degree of ex ante betterness, but that vague degree would still slightly increase from the mild sweetening, even if the change isn’t large enough to yield a determinately higher degree of betterness for one option. The hazy cloud and degree do move.
A justified wager is not guaranteed under such an approach, as our evidence could in principle be closely balanced, but the bar for finding one is relatively low (since the approach doesn’t collapse a broad set of distinct evidential states into indeterminacy, cf. the figure here). It seems to me we can make such justified wagers given both how mild sweetenings do incrementally move ex ante degrees of betterness and the, in my view, sufficiently weighty arguments and considerations that support the most plausible strategies over the worst ones.
Regarding unappealing properties, I think it’s worth thinking about this in comparative terms. Properties that seem unappealing in some absolute sense may still be comparatively appealing when compared to properties or implications of alternative views. For example, compared to the (in my view extremely unappealing) implications of maximality discussed in the torture cases and footnote 17, the ostensibly unappealing properties of graded betterness may be highly appealing by comparison.
It is an open question whether all graded views of betterness have unappealing properties (which is not an easy case to make given that it’s a very broad category), and even if they do, the bar for having properties that are less unappealing than maximality’s does not appear that high.
In terms of CD in particular (which is just a simple example of a graded index), it seems debatable whether the possibility of a sign-flip from adding the same background consequences is desirable or not. This is related to the approach mentioned in footnote 37 about dropping the dominance clause and relying on CD alone. That is, if we decide based on the range of plausible expected values, and we do not privilege the profile of pairwise differences across probability functions, a sign-flip from adding the same background can be rationally defensible (when the background shifts the range of plausible expected values differently for the two actions).
For a concrete example, suppose A and B have expected values (1, −1, −1, −1) and (0, 0, 0, −2) at the four extreme points of P, so their ranges are [−1, 1] and [−2, 0], and A is better at both ends. Now add a background prospect Z with expected values (−1, −1, 1, 1) at those points, which is bad where A does best and good where B does worst. The resulting ranges are [−2, 0] for A + Z and [−1, 1] for B + Z, so B’s range now lies above A’s at both ends.[1]
The pairwise differences are unchanged, while the background shifts the ranges by changing which probability functions set the endpoints, a shift that independence says should be irrelevant. So the flip may be less a defect of graded views than a reflection of a potential substantive disagreement about what the comparison should be sensitive to. But again, there’s a broad range of graded views, and this simple form of midpoint ordering may not be among the most plausible such views.
Note that neither act dominates the other here, before or after Z is added, so maximality returns “both permissible” in each case. The sign-flip therefore occurs, and can only occur, entirely within the region where maximality delivers no ranking.
Thank you. I think that's a fair summary. To be precise, rather than claiming that the maximality rule is generally too sensitive to small changes, my critique is that it's generally too insensitive to small changes, as virtually all updates that would be registered by a graded approach make no difference at all. As a consequence of this insensitivity, maximality implies isolated points of extreme sensitivity where some tiny change can result in a categorical jump. So yes, I think it's too sensitive to small changes in the sense that it packs all its sensitivity into those sharp points where dominance is achieved.
Thank you for these useful comments.
I agree that indeterminacy can still obtain under a graded approach. However, it’s worth noting that the band in which indeterminacy obtains under a graded approach will generally be much narrower than under maximality. So moving from maximality to a graded approach is a big change.
I gather we see this differently, but to very briefly explain why it seems warranted to me: Because of the evidence and arguments favoring the most plausible strategies, and the lack of similarly strong evidence and arguments against them. In particular, I don’t think the arguments against the capacity-building strategies I discuss are as strong as those in their favor. Moreover, much of the value of building capacity is the value of being able to act on considerations we aren’t yet aware of. So, in my view, unawareness bears asymmetrically on these strategies rather than neutrally (I realize this latter point is stated very briefly and needs further development).
Without trying to build an elaborate case here, there is a simple extension that I think further strengthens the case: instead of focusing only on the best-supported strategies, we can specifically compare the seemingly best-supported strategy (say, some form of low-footprint CB) with the strategy that seems worst (I suspect it’s better not to go into details about what that might be).
When comparing two such contrasting strategies, the expected difference plausibly gets beyond the (relatively) narrow band of indeterminacy under some plausible graded approach, considering both the arguments favoring the seemingly better strategy and the arguments opposing the seemingly worse strategy. The reason this helps is that the width of the indeterminacy band doesn’t obviously scale with how contrasting the pair is, whereas the expected difference does. (Again, the expected difference between the two strategies need not be remotely as stark as in the figure below.)
I agree that there can be insensitivity to mild sweetening under rules other than maximality. (As noted in more general terms, the blanket objection that rejects “any evidence or proposal we might wager on … assumes the kind of categorical decision rule that collapses a broad set of evidential states into complete and undifferentiated indeterminacy.”)
But interestingly, under a graded approach, even if indeterminacy may in some sense be preserved after a mild sweetening (within the relatively narrow indeterminacy band), we still wouldn’t get complete insensitivity to mild sweetening. In that case, we would have a hazy cloud of vague expected values, and a vague degree of ex ante betterness, but that vague degree would still slightly increase from the mild sweetening, even if the change isn’t large enough to yield a determinately higher degree of betterness for one option. The hazy cloud and degree do move.
A justified wager is not guaranteed under such an approach, as our evidence could in principle be closely balanced, but the bar for finding one is relatively low (since the approach doesn’t collapse a broad set of distinct evidential states into indeterminacy, cf. the figure here). It seems to me we can make such justified wagers given both how mild sweetenings do incrementally move ex ante degrees of betterness and the, in my view, sufficiently weighty arguments and considerations that support the most plausible strategies over the worst ones.
Thank you for your comment.
Regarding unappealing properties, I think it’s worth thinking about this in comparative terms. Properties that seem unappealing in some absolute sense may still be comparatively appealing when compared to properties or implications of alternative views. For example, compared to the (in my view extremely unappealing) implications of maximality discussed in the torture cases and footnote 17, the ostensibly unappealing properties of graded betterness may be highly appealing by comparison.
It is an open question whether all graded views of betterness have unappealing properties (which is not an easy case to make given that it’s a very broad category), and even if they do, the bar for having properties that are less unappealing than maximality’s does not appear that high.
In terms of CD in particular (which is just a simple example of a graded index), it seems debatable whether the possibility of a sign-flip from adding the same background consequences is desirable or not. This is related to the approach mentioned in footnote 37 about dropping the dominance clause and relying on CD alone. That is, if we decide based on the range of plausible expected values, and we do not privilege the profile of pairwise differences across probability functions, a sign-flip from adding the same background can be rationally defensible (when the background shifts the range of plausible expected values differently for the two actions).
For a concrete example, suppose A and B have expected values (1, −1, −1, −1) and (0, 0, 0, −2) at the four extreme points of P, so their ranges are [−1, 1] and [−2, 0], and A is better at both ends. Now add a background prospect Z with expected values (−1, −1, 1, 1) at those points, which is bad where A does best and good where B does worst. The resulting ranges are [−2, 0] for A + Z and [−1, 1] for B + Z, so B’s range now lies above A’s at both ends.[1]
The pairwise differences are unchanged, while the background shifts the ranges by changing which probability functions set the endpoints, a shift that independence says should be irrelevant. So the flip may be less a defect of graded views than a reflection of a potential substantive disagreement about what the comparison should be sensitive to. But again, there’s a broad range of graded views, and this simple form of midpoint ordering may not be among the most plausible such views.
Note that neither act dominates the other here, before or after Z is added, so maximality returns “both permissible” in each case. The sign-flip therefore occurs, and can only occur, entirely within the region where maximality delivers no ranking.