Weighing Criteria

Original text Luciano Carlos Cunha · Volume XI, p. 107 of the PDF edition · English translation

7.1. Introduction

As we saw in Chapter 1, a situation can be good/better in one respect and bad/worse in another. How, then, can we assess how good/bad a situation is all things considered¹⁰²? In section 1.6 we briefly explored 13 possibilities. In this chapter we will expand the discussion of these 13 possibilities by adding examples, lists of problems/difficulties, and counterexamples.

In discussing many of these possibilities, we will look at two types of example, meant to illustrate that the better/worse all things considered assessment can be used both to assess the well-being of each individual (individual well-being) and to assess how good/bad a situation containing several individuals is (collective well-being). The counterexamples will also refer to these two types of scenario.

Examples of the first type involve weighing the aspects of the intensity and the duration of pain in a single individual. All of these examples involve deciding which of two situations is worse: (1) a more intense pain for a shorter time, or (2) a less intense pain for a longer time. The importance of knowing how intense the pain will be and how long it will last will be addressed separately for each method.

Examples of the second type involve several individuals, and weigh the aspects of the number of victims and the suffering per victim. All of these examples involve deciding which of two situations is worse: (1) fewer victims suffering more, or (2) more victims suffering less. The importance of knowing how much more each member of the minority suffers, and how much larger the number of victims who suffer less is, will be addressed separately for each method.

Although all the examples compare these aspects (intensity vs. duration, and severity vs. number), it is important to remember that the whole discussion here is not restricted to

comparing these aspects, since it applies equally to any other aspects being compared, whether they are negative or positive.

All the examples we will see concern value theory (that is, assessments of which situations are better/worse). However, the same methods apply equally to weighings in normative ethics (for example, to establish what weight each aspect taken into account should have when determining what we ought to do).

Let us turn, then, to the discussion of the methods.

7.2. Total incommensurability

What it consists of

To hold total incommensurability is to recognize that there are several aspects that are good/bad in themselves, but to maintain that it makes no sense to compare them (nor even to multiply or add them). According to this approach, the only circumstances in which it is possible to say which of several options is better all things considered are those in which one option is better in every respect. In circumstances in which each option is better in some respect, this approach will say that it is not possible to say which of them is better all things considered¹⁰³.

Example 1

If in one situation the pain is more intense but lasts a shorter time, and in another it is less intense but lasts longer, according to this view there is no way to assess which of them is worse all things considered, whatever the duration and intensity of each.

Example 2

If one situation contains fewer victims suffering more and another contains more victims suffering less, according to this view there is no way to assess which of them is worse all things considered, whatever

the intensity of each victim's suffering and the number of victims in each situation.

Problems/difficulties

This method does not help us choose when the various options before us are each good/better in one respect and bad/worse in another. It would help us choose only when, among the various options, there is one that is good/better in every relevant respect.

Counterexamples

If we do in fact make choices in situations where the various options are better/worse in different respects, then it is because we agree that it is possible to commensurate the different aspects. Otherwise, in cases like these we would always flip a coin to decide. It is one thing to say that there may be some aspects that are incommensurable with one another. It is another to say that there is no aspect at all that is commensurable with any other. We will return to this point in section 7.13.

7.3. Monism (general)

What it consists of

According to monist approaches, there is only one aspect that is good/bad in itself. All the others are seen as good/bad only instrumentally, depending on whether they increase or decrease what is seen as good/bad in itself. Thus, in monist approaches, how good/bad a situation is all things considered is determined solely by how good/bad it is in terms of a single aspect (namely, the one seen as having value/disvalue in itself). Although they share this common structure, the various monist approaches differ from one another according to which aspect is seen as good/bad in itself.

Example 1

Suppose a monist theory holds that only the intensity of suffering is bad in itself. According to this view, when two situations are compared, the one in which the suffering is more intense will be considered worse all things considered (no matter how much longer the less intense suffering lasts) – and vice versa if it held that only duration is bad in itself.

Example 2

Suppose a monist theory holds that only the number of victims is bad in itself. According to this view, when two situations are compared, the one containing the greater number of victims will be worse all things considered (no matter how much more the victims in the minority are suffering) – and vice versa if it held that only the severity of each victim's situation is bad in itself.

Problems/difficulties

Being able to explain why only one of the aspects is good/bad in itself. If it is the number of victims, why not the severity of the victims' situation, and vice versa? If it is the intensity of the pain, why not its duration, and vice versa? This does not mean that it is impossible to ground this choice, but it is clearly something that needs to be explained.

Counterexamples

There are both situations in which it seems preferable to help a much larger number of victims (even if they are not in the worst situation) and situations in which it seems preferable to help victims who are in a much worse situation (even if they are not the majority). Similarly, there are both situations in which it seems preferable to undergo a more intense pain for a shorter time and situations in which it seems preferable to undergo a less intense pain for a longer time.

7.4. Aggregative monism (a specific application of general monism)

What it consists of

A common form of monism consists in determining how good/bad a situation is all things considered by aggregating what is seen as good/bad in itself¹⁰⁴. One example is the approach adopted in utilitarianism. In the classical utilitarianism of Bentham (1996 [1789]) and Mill (1969 [1861]), what is good in itself is positive experiences and what is bad in itself is negative experiences. In the utilitarianism proposed by, for example, Singer (1993 [1979]) and Hare (1981), by contrast, what is good in itself is the satisfaction of preferences (desires) and what is bad in itself is their frustration. In either version, however, how good/bad a situation is all things considered is obtained through total aggregation.

Example 1

Suppose a monist theory holds that neither the intensity nor the duration of pain is bad in itself, but that each is bad only instrumentally, because it involves suffering. According to this view, how bad a situation is all things considered would be obtained by adding up the intensity of the suffering at each moment (thereby obtaining the total aggregate of suffering).

Imagine that in situation S1 we have 20 minutes of pain of intensity -5. In situation S2, by contrast, we have 10 minutes of pain of intensity -10. According to this approach, the total disvalue of S1 is -100 (since it is -5x20). The total disvalue of S2 is also -100 (since it is - 10x10). According to this approach, S1 and S2 are equally bad (the fact that in S1 the pain lasts longer, and that in S2 the pain is more intense, is not seen as relevant in itself).

Example 2

Suppose a monist theory holds that neither the severity of each victim's situation nor the number of victims is bad in itself, but that each is bad only instrumentally, because

it involves suffering. In that case, what is bad in itself would be the total aggregate of suffering. Thus, according to this view, how bad a situation is all things considered would be obtained by adding up each victim's level of suffering (thereby obtaining the total aggregate of suffering).

Imagine that in situation S1 we have 20 victims, each with suffering of -5. In situation S2, by contrast, we have 10 victims, each with suffering of -10. According to this approach, the total disvalue of S1 is -100 (since it is -5x20). The total disvalue of S2, in turn, is also -100 (since it is -10x10). That is, in this approach, neither the number of victims nor the severity of each victim's situation is seen as bad in itself (but only as bad instrumentally, insofar as they contribute to making the total aggregate negative). Thus, according to this approach, S1 and S2 are equally bad (the fact that S1 has more victims, and that in S2 each victim is suffering more, is not seen as relevant in itself).

Problems/difficulties

This method could say that a situation in which a much larger number of victims have minor scratches is worse than one in which a much smaller number of victims suffer excruciatingly (provided that the result of the aggregation in the first situation is greater). Similarly, this method could say that a situation involving a great amount of time of very slight discomfort (again, minor scratches) is worse than one involving excruciating torment that lasts a much shorter time (provided that the result of the aggregation in the first situation is greater).

Counterexamples

There are both situations in which it seems preferable to help a much larger number of victims, even if they are not in the worst situation, and situations in which it seems preferable to help victims who are in a much worse situation, even if they are not the majority (even when the total aggregate suffering is the same in all these situations). Similarly, there are both situations in which it seems preferable to undergo a more intense pain for a shorter time and situations in which it seems preferable to undergo a less intense pain that lasts longer (even when the total aggregate suffering would be the same in all these situations). This suggests that these other aspects matter in themselves, and not merely instrumentally.

7.5. Equal-weight pluralism

What it consists of

In this approach two or more aspects are recognized as good/bad in themselves, and how good/bad the situation is all things considered is determined either by multiplying these aspects (the multiplicative approach) or by adding them (the additive approach). What is characteristic of this approach is that each aspect is seen as equally important (that is why they are multiplied or added, but both have the same weight).

Note that, in a multiplicative variant of this approach, the result will be identical to the result obtained by an aggregative monist approach (which we saw in the previous section). Only the grounding will be different (aggregative monism sees only one thing as good/bad in itself, while multiplicative equal-weight pluralism sees more than one thing as good/bad in itself). An additive variant, on the other hand, may yield results different from those generated by aggregative monism.

Example 1

How bad all things considered a situation involving pain for a single individual is would be determined by multiplying the average intensity of the pain by its duration (in the multiplicative approach) or by adding the average intensity of the pain to its duration (the additive approach).

Example 2

How bad all things considered a situation involving several victims is would be determined by multiplying the average severity of each victim's situation by the number of victims (the multiplicative approach) or by adding the average severity of each victim's situation to the number of victims (the additive approach).

Problems/difficulties

The multiplicative variant will have the same problems/difficulties as aggregative monism (which we saw in the previous section). The additive variant, on the other hand, avoids some problems faced by the multiplicative variant and by aggregative monism, but at the same time has to face other, additional problems. The problem it avoids is the following: in the additive approach an intense, short pain can be considered worse than a mild, long one, even if the total aggregate is smaller in the intense, short pain. It could also consider a scenario with fewer victims who are suffering terribly worse than a scenario with many more victims with mild discomfort, even if the aggregate is smaller in the scenario with fewer victims. In this way, some counterexamples that could be raised against aggregative monism and against the multiplicative variant do not apply to the additive variant. However, the additive variant runs into new problems, since it can be accused of comparing units in a very strange way (for example, adding “20 of pain intensity” to “3 hours of duration”). We will explore this problem below.

Counterexamples

With respect to the multiplicative variant, the counterexamples are the same as those raised against aggregative monism (which we saw in the previous section). However, there are other counterexamples that illustrate different problems, some of which apply to both the multiplicative and the additive variant, while some apply to only one of the two. We will look at some of them below.

The problem of adding heterogeneous quantities

Counterexample 1. One individual (intensity vs. duration)

Multiplicative variant

S1: pain of intensity 9 for 3 minutes → 9 x 3 = 27 S2: pain of intensity 4 for 7 minutes → 4 x 7 = 28

The multiplicative variant concludes that S2 is worse than S1, but it may be that pain of intensity 9 for 3 minutes is much harder to bear than pain of intensity 4 for 7 minutes.

Additive variant

S1: pain of intensity 9 for 3 minutes → 9 + 3 = 12 S2: pain of intensity 4 for 7 minutes → 4 + 7 = 11

The additive variant concludes that S1 is worse (and this agrees with the criticism made of the multiplicative variant), but here the strangeness comes from the fact that “9 points of pain” were added directly to “3 minutes,” as if they were quantities of the same kind. Moreover, the result of the additive model depends on the scale we choose for measuring intensity and duration. For example, if intensity is measured on a scale from 0 to 10, and duration in hours, we will get one result. But if, to assess the same phenomenon, intensity is measured on a scale from 0 to 100, the sum changes radically, and with it the evaluative/moral assessment of the situation also changes. In short, one problem with this model is that the result of the assessment depends arbitrarily on the unit of measurement chosen.

Counterexample 2. Several victims (average severity vs. number of victims)

Multiplicative variant

S3: 100 victims with extreme suffering (100 each) → 100 x 100 = 10,000 S4: 10,001 victims with mild discomfort (1 each) → 1 × 10,001 = 10,001

The multiplicative variant concludes that S4 is worse, but this seems mistaken: there being more people with mild discomfort does not seem worse than there being fewer people with extreme suffering.

Additive variant

S3: 100 victims with extreme suffering (100 each) → 100 + 100 = 200 S4: 10,001 victims with mild discomfort (1 each) → 1 + 10,001 = 10,002

The additive variant concludes that S4 is much worse. But this is far from seeming true. Moreover, the additive variant has an additional problem: it adds “suffering per victim” to “number of victims” as if they were quantities of the same kind, but they obviously are not.

The central problem with both the multiplicative and the additive approach is that the aspect of the number of victims “dilutes” the aspect of suffering per victim, in such a way that atrocious situations for a few individuals would have to be considered better than trivial discomforts for many individuals.

7.6. Pluralism with an absolute trump

What it consists of

In this approach two or more aspects are recognized as good/bad in themselves, but a hierarchy is constructed in which one factor is an absolute trump over those that come after it in the hierarchy¹⁰⁵. According to this approach, no increase in the secondary aspects, however large, can tip the balance so as to make the situation better all things considered if it represents a decrease in the main aspect, however small. That is, in this approach how good/bad a situation is all things considered is determined almost exclusively by the trump aspect (the secondary aspects serve only as a tiebreaker).

Example 1

Suppose the aspect of pain intensity is an absolute trump over the aspect of pain duration. On this view, when several situations are compared, the one in which the pain is more intense is worse all things considered, no matter how much longer the pain lasts in other situations. Or, conversely, if the absolute trump is duration, when two situations are compared, the one in which the pain lasts longer is worse all things

considered, no matter how much more intense the pain is in other situations.

Example 2

Suppose the aspect of the severity of each victim's situation is an absolute trump over the aspect of the number of victims. On this view, when several situations are compared, the one in which the victims are in the worst situation is worse all things considered, no matter how much larger the number of victims in other situations. Or, conversely, if the absolute trump is the number of victims, when two situations are compared, the one with the greater number of victims is worse all things considered, no matter how much worse the condition of the victims in other situations.

Problems/difficulties

Any approach of this type needs to explain why the chosen aspect is the trump, and not any other. In addition, it needs to explain why the aspect in question is an absolute trump (since it could be a trump, but not an absolute one, or it could have greater weight without being a trump, as we will see in the following sections.

Counterexamples

There are both situations in which it seems preferable to help a much larger number of victims, even if they are not in the worst situation, and situations in which it seems preferable to help victims who are in a much worse situation, even if they are not the majority. Similarly, there are both situations in which it seems preferable to undergo a more intense pain for a shorter time and situations in which it seems preferable to undergo a less intense pain that lasts longer. If these judgments are correct, then neither of these aspects is an absolute trump over the other.

7.7. Pluralism with a trump up to a certain point

What it consists of

In this approach two or more aspects are recognized as good/bad in themselves; a hierarchy is constructed in which one factor is a trump over those that come after it in the hierarchy, but this trump is not absolute. Beyond a certain point (called the cutoff point or threshold), the hierarchy is reversed¹⁰⁶. Hence the name “pluralism with a trump up to a certain point.”

In this approach, the relationship between the aspects is neither continuous nor gradual (as is the one we will see in the next section), but stepwise (that is, it reverses completely and instantly once the cutoff point is reached, but until the cutoff point is reached, the weight of each aspect remains completely unchanged). This method is therefore said to involve a discontinuous relationship, that is, there is an abrupt jump when the cutoff point is reached, rather than a gradual change¹⁰⁷.

Example 1

Suppose the aspect of pain intensity is a trump up to a certain point over the aspect of pain duration. This means that there will be a point beyond which the aspect of pain duration becomes the trump (for example, lasting 10 times longer than a pre-established “baseline time”). Conversely, imagine that the duration of the pain is a trump up to a certain point over its intensity. This means that there will be a point beyond which the aspect of intensity becomes the trump (for example, being 10 times more intense).

Example 2

Suppose the aspect of the severity of each victim's situation is a trump up to a certain point over the aspect of the number of victims. This means that there will be a point beyond which the aspect of number becomes the trump (for example, there being 10 times more

victims). Conversely, imagine that the aspect of the number of victims is a trump up to a certain point over the aspect of the severity of each victim's situation. This means that there will be a point beyond which the aspect of the severity of each victim's situation becomes the trump (for example, being 10 times worse than the situation of any other victim).

Problems/difficulties

Any approach that uses this method needs to explain:

(1) Why one of the aspects is the main trump. (2) Why the relationship is one of a trump up to a certain point (and not, for example, greater weight without being a trump, or no fixed weight, and so on). (3) Why the cutoff point is set where it was, and not elsewhere (otherwise, the cutoff point seems arbitrary).

Counterexamples

Wherever we set the cutoff point, it will seem arbitrary. This is illustrated by the following examples.

Imagine that we adopt the following rules:

• The intensity of the pain is the trump, unless the duration is 10 times greater than a previously defined baseline time (say, 1 minute of pain) – if that happens, duration becomes the trump.
• The severity of each victim's situation is a trump over the number of victims, unless the situation containing the larger number of victims who are not in the most severe condition is 10 times larger than the situation in which the victims are in the most severe situation of all – if that happens, the number of victims becomes the trump.

There are two problems here:

The first is that it could be pointed out that the reference baseline value (in this case, a pain duration 10 times longer than the baseline time, and a number of victims ten times larger) is arbitrary. Why 10 times? Why not 9 or 11 times? Why not 5 or 20 times, or any other value? Moreover, regarding the first rule, one could ask: “why is the baseline time 1 minute? Why not 30 seconds? Why not 2 minutes? Why not any other value?

The second is that it could be pointed out that, even if we arrived at a non-arbitrary reference baseline value, measurement based on that value would still necessarily be arbitrary. For example, imagine that the values serving as the reference baseline in the examples above were demonstrably non-arbitrary. We could still ask: “if a pain 10.1 times longer becomes the trump, why not a pain 9.9 times longer? What is so ‘magical’ about the point of being 10 times greater?” Yet the same criticism could be directed at any other reference value. For example, if, with respect to the second example, the reference value were “there being 20 times more victims,” one could ask: “why not 19.9 times more victims? Why not 20.1 times more victims? What is so special about 20 times more?”, and so on.

Regarding these problems, proponents of the trump-up-to-a-certain-point approach could reply that, although it is true that wherever we draw the line it will seem arbitrary, drawing the line somewhere is better than not drawing it anywhere. In addition, they could reply that, although it is hard to find a precise non-arbitrary threshold, we can detect thresholds at both extremes that are more arbitrary than others. For example, requiring that the pain last 10 minutes for duration to become the trump seems reasonable, but it does not seem reasonable to require that it last 10 years, nor that it last 1 second.

Another way to get around both problems is to abandon an approach centered on trumps up to a certain point, and adopt an approach in which the various aspects have different weights, but there are no abrupt cutoffs at which the hierarchy is reversed: instead, the hierarchy gradually and continuously reverses. We will look at two approaches of this kind in sections 7.8 and 7.9, respectively.

7.8. Pluralism with fixed-weight aspects (linear)

What it consists of

In this approach two or more aspects are seen as good/bad in themselves, and they have different weights in the calculation of how good/bad the situation is all things considered – but without using the idea of trumps. The reasoning here is similar to what we saw in section 7.5 (equal-weight pluralism), except that here the weight of each factor is different. For example, suppose factor A has twice the weight of factor B. In that case, the total value/disvalue of the situation will be determined by (2x.A) + (1x.B), if an additive method is used, or by (2x.A) x (1x.B), if a multiplicative method is used¹⁰⁸. The difference from the use of trumps is that here the influence of the factors is continuous and linear (with the use of trumps, it is discontinuous and non-linear).

Example 1

Imagine that the intensity of the pain has twice the weight of the duration of the pain. In that case, if the approach is additive, the calculation of total disvalue will look like this: total disvalue = (2x pain intensity) + (1x pain duration). If the approach is multiplicative, on the other hand, the calculation will look like this: total disvalue = (2x pain intensity) x (1x pain duration).

Example 2

Imagine that the number of victims has twice the weight of the severity of the situation of each victim. In that case, if the approach is additive, the calculation of total disvalue will look like this: total disvalue = (2x number of victims) + (1x severity of each victim's situation). If the approach is multiplicative, on the other hand, the calculation will look like this: total disvalue = (2x number of victims) x (1x severity of each victim's situation).

What are the advantages and disadvantages of additive and multiplicative models?

The advantage of additive models is that they allow a situation to be considered very serious because it is very bad in one respect, even if it is not very bad in another (for example, a single victim, but one suffering to the extreme). If the model were multiplicative, such a situation would have to be considered not very serious.

However, multiplicative models also have some advantages. The first is that they capture the idea that, for example, a 1-second pain does not matter much, even if it is very intense (because it lasts only an instant), and that a very slight discomfort does not matter much either, even if it lasts a long time. That is, the multiplicative model captures the idea that a situation is truly bad only when the values are high in each respect. The second advantage is that it avoids excessive trade-offs. For example, in the additive model, a very slight discomfort that lasts a billion seconds can be considered worse than an unbearable pain that lasts 10 minutes. The multiplicative model makes this kind of trade-off harder, because a low value in intensity or in duration greatly reduces the total value. In short, the multiplicative model reinforces that both aspects matter to the total severity of the scenario.

In the following examples, we will show the results according to each of the models.

Problems/difficulties

Any approach that uses this method needs to explain: (1) why it is a certain aspect that weighs more than another, and not the reverse, and (2) why it weighs N times more (and not any other amount). Moreover, this method generates strange conclusions, illustrated below.

Counterexample 1

Suppose the intensity of pain were given a weight 5 times greater than its duration. Now imagine that we need to say which of two situations is worse:

S1: An extreme pain (of intensity 100) for 500 seconds. S2: A moderate pain (of intensity 20) for 3000 seconds. So we have:

Additive approach:

S1: (5x100) + (1x500) = 1000 S2: (5x20) + (1x3000) = 3100

Multiplicative approach:

S1: (5x100) x (1x500) = 250,000 S2: (5x20) x (1x3000) = 300,000

According to the linear model (whether additive or multiplicative), S2 is worse all things considered (that is, in this model, 3000 seconds of moderate pain is considered worse than 500 seconds of extreme pain). The point is this: however much greater the weight assigned to intensity, according to the linear approach (whether additive or multiplicative) there will always be a sufficiently long time of moderate (or even very mild) pain that would outweigh the peaks of extreme suffering. For example, this model would say that “little pinches,” provided they lasted long enough, would be worse than extreme torments lasting much less time (even if greater weight is given to intensity).

Counterexample 2

Suppose the severity of each victim's situation were given a weight 5 times greater than the number of victims. Now imagine that we need to say which of two situations is worse:

S3: 1000 individuals with extreme suffering of intensity 100 each. S4: 200,000 individuals with mild discomfort of intensity 1 each.

So we have:

Additive approach:

S3: (5x100) + (1x1000) = 1500 S4: (5x1) + (1x200,000) = 200,005

Multiplicative approach:

S3: (5x100) x (1x1000) = 500,000 S4: (5x1) x (1x200,000) = 1,000,000

According to the linear model (whether additive or multiplicative), S4 is (much) worse all things considered (that is, the mild discomfort of 200,000 individuals is considered much worse than the extreme suffering of 1000 individuals).

In short, these two counterexamples show that the linear approach (whether additive or multiplicative) allows a large number of trivial harms to outweigh extreme harms that are fewer in number, even if we assign greater weight to the extreme harms. Even if we try to compensate for this by increasing the weight of extreme harms still further, there will always be a sufficiently large number of trivial harms to outweigh the extreme harms. Assigning different weights to the various aspects, as is done in the linear model, mitigates this problem – compared to how the problem appears in multiplicative pluralism and in aggregative monism – but it does not eliminate it.

7.9. Pluralism with variable-weight aspects (diminishing returns)

What it consists of

In this approach two or more aspects are seen as good/bad in themselves, but there is no fixed weight between them: their weight is variable, decreasing proportionally as they are “filled” (which is why this approach is also called pluralism with diminishing returns)¹⁰⁹. If we were to represent it on a graph, this would amount to each aspect having a concave value/disvalue function.

This approach is different from pluralism with a trump up to a certain point. Although they are similar in that, in both approaches, one factor may come to be more important than the other beyond a certain level (both are non-linear), here the relationship is continuous, with each factor having its importance gradually reduced with each new unit filled (with a trump up to a certain point, there is a discontinuous relationship, since the trump is reversed abruptly if a cutoff line is crossed).

This approach is also different from the method we saw in the previous section (linear fixed weights). Although neither approach uses the idea of trumps, there are crucial differences. In the linear approach, each aspect always has a fixed weight, regardless of how much of it has already been filled. For example, if the weight of the number of victims is twice the weight of the severity of each victim's situation, it remains twice as much whether we are talking about ten, a hundred, or a million victims: there is no proportional change. In the diminishing-returns approach, by contrast, a given aspect has greater weight while not much of it has yet been filled, but this weight declines as it is filled.

Let us take an example isolating the factor of the severity of each victim's situation. In this method, moving a victim from -100 to -99 has more weight than moving a victim from -98 to -97, and so on. That is, as the victim's situation improves, adding one more point to their well-being has proportionally less and less value. That is why, according to this method, it is much more important to move one victim from -100 to -99 than to move another from -2 to -1 (since the first is in excruciating suffering, while the second is in moderate suffering). That is, in this approach the well-being of those who are worse off matters more, but as it improves, gaining one more point of well-being matters less and less.

Example 1. Intensity vs. duration of pain in a single individual

Imagine that we measure the intensity and duration of pain as follows

• The intensity of the pain ranges from -100 (maximum pain) to 0 (absence of pain).
• The duration of the pain is measured in seconds.

In this diminishing-returns model, the relative importance of alleviating each aspect is not linear:

• Each point less in pain intensity matters more the more intense the pain is (for example, moving a pain from -100 to -99 is more important than moving a pain from -20 to -19).
• Each second less in pain duration matters more the greater the remaining total of pain (for example, if someone would suffer for 500 seconds and we have the opportunity to reduce this to 499 seconds, that matters more than going from 300 seconds to 299 seconds).

Now let us compare two possible interventions:

• A: reduce the intensity of the pain from -100 to -99 (keeping a duration of 500 seconds).
• B: reduce the duration of the pain from -500 to -499 seconds (keeping an intensity of -100).

It seems that the gain from relieving one point of extreme pain intensity (going from -100 to -99) across all 500 seconds is more important than eliminating just 1 second of a pain that will still be very extreme and will still last a long time.

However, suppose we had already reduced the intensity of the pain substantially, for example, from -100 to -10. Now, moving the intensity from -10 to -9 represents a very small marginal gain. Meanwhile, the individual still faces hundreds of seconds of intense pain. In this scenario, we have the following options:

• C: reduce the intensity of the pain from -10 to -9 (keeping a duration of 500 seconds).
• D: reduce the duration of the pain from -500 to -451 seconds (keeping an intensity of -10).

Given the circumstances, according to the diminishing-returns approach, scenario D comes to matter more than scenario C, because the marginal weight of the time aspect is still high, while that of the intensity aspect (already greatly reduced) has become small.

Note that these results are different from those we would get if we gave equal weight to each factor (since, if we did, at least according to a multiplicative approach we would have to prefer scenario C):

• C: (9) x (500) = 4,500
• D: (10) x (451) = 4,510

Thus, in the diminishing-returns model, the hierarchy between the aspects can reverse depending on how “filled” each aspect already is.

In short, reducing by one point an extreme pain that lasts a long time is more significant than cutting just 1 second off that pain, but when the pain is already much less intense, cutting a few seconds off it may be more important than reducing its intensity by one point. Example 2. Severity per victim vs. number of victims

Imagine that in the diminishing-returns model we have the following:

• Severity per victim: moving a victim's situation from -100 to -99 matters more than moving it from -99 to -98, and so on progressively. Thus, moving a victim's situation from -100 to -99 matters much more than moving it from -2 to -1.
• Number of victims: given 100 victims, leaving 99 suffering (saving only one) is worse than leaving 98 suffering (saving two), and so on progressively. Thus, given 100 victims, leaving 99 suffering (saving only one) is much worse than leaving 1 suffering (saving 99). Now let us compare the following options:
• E: 100 victims in extreme suffering (-100 per victim)
• F: 10,000 victims with mild discomfort (-1 per victim).

In this case, it seems that scenario E is worse, since, despite having far fewer victims, the suffering per victim is extreme, whereas in F, despite there being many more victims, each one has only very mild discomfort. That is, it seems that in this case the aspect of suffering per victim weighs more.

However, with diminishing returns, the weight of both aspects progressively decreases depending on how filled they already are, and there will be a point at which the hierarchy reverses. For example, consider the following scenarios:

• E: 100 victims with suffering of -100 each.
• G: 120 victims with suffering of -80 each.

Comparing E with G, the fact that there are 20 more victims in G with suffering that, although somewhat less than the suffering of the victims in E, is still very severe suffering, seems to make situation G worse all things considered (even though the aggregate of suffering in E is greater, that is, -10,000, compared to -9,600 in G). This can be explained by the diminishing-returns model: in G the aspect of severity has had its weight reduced because it is already more filled (-80, compared to -100 in E), and the goal of reducing the number of victims has had its weight increased because it is less filled (in G there are more victims, 120, compared to E's 100 victims).

Problems/difficulties

Any position that uses this method needs to justify the shape of the curve (how much and how the weight decreases as the aspect is filled). It also needs to explain whether the shape of the curve should be the same for each aspect or not, and why.

Counterexample 1

In S1 we have a pain of twice the intensity, but very brief. In S2 we have a much more moderate pain, but a much longer one. The diminishing-returns method may end up assessing both situations as equally bad. However, many people would say that one of these two situations is worse than the other.

Counterexample 2

Situation Number of victims Suffering per victim S3 1000 10 S4 100 100

S3 has many more victims, but each with relatively much less suffering. S4, on the other hand, has far fewer victims, but each with extreme suffering. In the method we are analyzing, because of the variable weights, increases in the number of victims or in the extreme severity of each victim's suffering do not contribute linearly, and this may generate equal or similar final values. However, once again, many people would say that one of the situations above is worse than the other (and not that they are equally bad).

7.10. Maximin/leximin of priorities

What it consists of

In this method the idea is always to prioritize the worst individual case, then the second-worst case, then the third, and so on. This method resembles the idea of trumps, but whereas trumps are used to rank the different aspects, in this method the goal is to rank the priority of individuals within one and the same aspect.

However, this method is also used to compare situations with several individuals. For example, let us compare the following situations, where the letters represent individuals and the negative numbers their level of suffering (the greater the number, the more suffering).

According to this method, situation S1 would receive priority, since it contains the individual in the worst condition of all (A = -50). This is where maximin comes in, which prescribes maximizing the minimum level (and is, so to speak, the trump criterion). Now, imagine we had to compare S2 and S3 to determine which of them is more urgent. In that case, maximin does not help, since the victims in the worst condition in each of these situations are equally badly off (C = -49, E = -49). In that case, leximin comes in as a tiebreaker, prescribing that we prioritize the second-worst case. Here, situation S2 would be given priority, since the second-worst case of all is D = -40.

Problems

The main problems with this approach are set out below, with a counterexample presented for each problem.

A minimal gain for the worst case outweighs any benefit

Counterexample 1:

Situation Well-being of each individual in the situation S1 -100 -1 -1 -1 -1 -1 S2 -99 -99 -99 -99 -99 -99

The approach in question would say that S2 is better than S1, since its minimum level is higher. However, this ignores the fact that in S2 there are 6 individuals suffering terribly, whereas in S1, although the individual who is suffering the most suffers slightly more than any of the individuals in S2, there are very large improvements in the well-being of the other 5 individuals.

A minimal gain for the worst case outweighs any number of victims

Counterexample 2:

S3: 1 victim at -100 each S4: 1000 victims at -99 each

The approach in question would say that S4 is better than S3, but in S4 a much larger number of victims suffer almost as much as (if slightly less than) the victim in S3 and, moreover, there is a much larger number of victims.

In short, the maximin/leximin approach to priorities has all the problems of the absolute-trump approach, which we saw in section 7.6.

7.11. Sufficiency

What it consists of

In this method an aspect matters in itself until it reaches a certain threshold of fulfillment established as sufficient. After that it ceases to count. In an approach of this kind, if there are two or more aspects seen as good/bad in themselves, it can adopt any of the previous methods to commensurate them. What is distinctive about this kind of approach, however, is that a point comes at which the aspect is already sufficiently filled, and it then ceases to matter in itself.

One example of an approach that adopts this method, as we saw in section 6.11, is sufficientarianism. As we saw, it establishes a threshold line of well-being, above which life is considered sufficiently good¹¹⁰. With respect to those above the threshold line, sufficientarianism establishes no criterion of priority¹¹¹. With respect to those who

are below the line, on the other hand, sufficientarianism establishes priority according to three criteria: (a) how far each individual is from this threshold (the farther, the higher the priority); (b) how many individuals are benefited; and (c) the size of the benefit that can be brought about¹¹².

Problems

The main problems and difficulties for this approach are:

1) Explaining why, above a certain level, there should be no criterion of priority among individuals. 2) In sufficientarianism, for individuals who are already above the line there is no criterion of priority, even if there are enormous inequalities among these individuals. 3) The difficulty of defining the sufficiency line. 4) Small changes in the sufficiency line can drastically change who receives priority.

7.12. Hybrid approaches

What they consist of

These approaches combine one or more of the previous approaches. For example, they may use trumps (absolute or up to a certain point) up to a cutoff line, and from there on use a continuous relationship (with fixed weight or variable weight).

Problems

Among the main problems this kind of approach faces are:

1) Excessive complexity. It can generate excessively complex rules that are hard to apply. The more approaches are combined, the more complex the reasoning becomes.

2) Arbitrariness in choosing the cutoff line. Small changes in this line can drastically alter priorities. 3) The risk of suboptimal decisions. There could be a better overall impact if the continuous relationship were applied in all circumstances than if trumps were used, for example.

7.13. Partial indeterminacy

Unlike total incommensurability, which we saw in section 7.2, this method holds that, in some cases, there is clearly a way to assess which situations are better/worse all things considered, but that, in other cases, there is no correct answer (and so, in those cases, this method operates like the approach that defends incommensurability, with the judgment remaining indeterminate)¹¹³.

Problems

1) It has to offer a criterion for saying which cases have a correct answer and which are indeterminate (otherwise, one and the same case may be treated as determinable by one agent and indeterminable by another). 2) In indeterminate cases, the approach offers no criteria for choosing between the alternatives, which may lead to inaction or arbitrary decisions.

7.14. Probabilistic aggregation based on uncertainty

In this method several different approaches (or even all of those we saw earlier) are weighed on the basis of uncertainty about which of them is correct. In this method, one assesses how good/bad the situation would be all things considered according to each method taken into account, and multiplies each result by the degree to which one believes each method is correct. Then everything is added up, yielding probabilistic aggregation based on uncertainty. In short, in this method a probabilistic mixture of several approaches is made, weighting how much one believes each of them is correct and aggregating these results¹¹⁴.

Problems

1. It opens the door to arbitrary assignments of probability. 2. Weighing probabilistically requires many calculations, which makes the method hard to apply (and the more approaches are involved, the harder it becomes). 3. The final result is a kind of “weighted average” of the various approaches included, but this does not guarantee that the result will be considered justified from the standpoint of any of the approaches adopted. 4. Small changes in the assigned probabilities can produce large changes in the final decision.