On the interpersonal comparison of well-being, part 2

In a previous article I discussed the possibility of comparing the levels of well-being between different individuals. This is a crucial problem if we want to compare levels of suffering and happiness between different animals such as humans, insects, fish or birds. I demonstrated a general method to put everyone’s utilities (evaluations of well-being) on the same scale. But I ended that article with an open question about the existence of an objective utility measure. Here I discuss a hypothetical possibility for the such an objective measure of well-being that can always be compared with other individuals.

The discreteness of subjective experiences

The crucial idea for an objective measure of well-being, is the finiteness and discreteness of our subjective experiences. From a neurobiological point of view, we can expect that our experiences are fundamentally discrete. Experiences such as pain are generated by a brain, and a brain consists of a discrete, countable number of neurons (about 100 billion in a human brain), which consist of a discrete number of atoms following the laws of quantummechanics where fundamental properties are quantized. Furthermore, each neuron has a discrete number of connections with other neurons (about 1000), and a neuron has a threshold potential, a critical level of polarization of the neuron membrane that initiates an action potential (a firing of the neuron). This means that a neuron fires a discrete number of times per second (about 200 times per second). As a result, a brain can process a discrete, finite amount of information bits per second (about 20 million billion bits per second for the human brain).

The discreteness of brain information processing is confirmed in experimental psychological, in particular psychophysics: the quantitative investigation of the relation between objective external stimuli and subjective internal experiences or conscious perceptions and sensations. Our perceptions appear to be discrete, with a so called just-noticeable difference or JND: the minimum amount an objective stimulus must be changed before the corresponding subjective experience changes. For example, I can increase room temperature from 20°C to 20,1°C and I don’t notice the difference, but when I enter a room at 21°C, I notice the temperature increase.

Our brains are also finite, which means they cannot generate infinite experiences. Hence, there must also be a maximum noticeable difference or MND. For example, above a certain temperature, say 1000°C, I’m no longer able to feel any differences. A thousand or a billion degrees feels the same.

In summary: there is plenty of evidence that our subjective experiences are finite and discrete in nature. This discrete nature of our experiences is crucial for an objective, interpersonal comparison of experiences. Without this discreteness, we have to rely on the general method described in the previous article to interpersonally compare well-being.

The staircase model of experiences

If a stimulus (for example room temperature) is increased, our perception (for example the sensation of heat) increases with many small discrete steps, just like a staircase.The staircase has two important properties. First, and most importantly in this context, each step has a minimum height. No step is infinitesimally high. In other words, the staircase does not look like an inclined plane or slide. Second, the staircase has a minimum and a maximum level. The minimum level is the ground floor, the value 0. And when you reach the maximum level, there is no further step going up.

When there are several stimuli, the staircase can have multiple directions or dimensions. Each stimulus corresponds with a direction. The stimuli include pinpricks, room temperatures, taste sensations, and all other possible perceptions, including the flow of time. So we have a just-noticeable difference for a time interval. We may experience a difference between seconds, but not between milliseconds. There is a maximum frequency, and changes of stimuli at a rate above this frequency are not noticed. Due to the finite neuronal firing rate, we do not experience changes faster than a fraction of a second. If your brain is faster in information processing, you can experience more in one second. For example the visual perception part of a fly’s brain is four times faster than ours, so when a fly looks at something for one second, it is comparable to us looking at it for four seconds. It is as if the fly sees everything in slow motion.

The Weber-Fechner law says that the steps of the staircase can have different widths. In particular: the width of the steps increase when you are higher on the staircase. For example, I can notice a difference between 20°C and 21°C, but when I’m in a sauna, I do not notice the difference between 90°C and 91°C, even though the absolute temperature difference is the same. However, I can notice a difference between say 90°C and 100°C. So the step at 90°C is much wider than the step at 20°C. Around 20°C, the staircase of my heat perception is much steeper.

You can say that evolutionary pressures determined the steepness of the staircase. It is useful to experience a difference between 20°C and 21°C, because these temperatures are experienced in daily life, whereas it becomes less useful to be able to experience the difference between 90°C and 91°C. It would require too much brain capacity and information processing energy to be as highly sensitive at all temperatures as we are at 20°C. So we became most sensitive in the stimulus ranges that we encountered mostly in our evolutionary history. If I burn my hand, it doesn’t matter if the water is 90°C or 100°C, so I don’t need to be able to feel that temperature difference. But when I go for a swim, it matters if the water is 10°C or 20°C.

However, our sensitivity, i.e. the steepness of the staircase and the widths of the steps, can change according to circumstances. For example, when I hold my hand in warm water for a while, my heat sensation becomes adapted to that temperature, and that means I become more sensitive when I put my hand in colder water. Habituation and drowsiness can decrease sensitivity and make the staircase less steep. Also some drugs such as analgesics and anesthetics can make the staircase flatter: a much stronger external stimulus is required before the threshold for pain is reached. When you are in an unconscious state such as a coma or deep sleep, the steps are basically infinitely wide and the staircase is completely flat. When you are slowly waking up after sleep, the staircase becomes steeper again. Expectations, anticipations, attention or focus, and some stimulant drugs can also increase the sensitivity and the steepness of the staircase. For example when you expect a temperature increase, you may notice an increase in temperature sooner.

Due to these circumstantial factors, it is difficult to precisely measure the width of the steps, i.e. the just-noticeable differences (that is why the JND is measured with a detection rate of more than 50%, i.e. when you notice the difference more than 50% of the times). But the crucial point is that the height of the steps is indeterminate, and therefore we can assume that all the steps have the same height. If I can as barely feel the difference between water at 50°C and 51°C as I can feel the difference between 90°C and 100°C, the pain increase when water temperature is raised from 50°C to 51°C is equal to the pain increase when water temperature raises from 90°C and 100°C.

The experiential (hedonic) utility function

A crucial notion of well-being is the utility function. One sentient being can be defined by having one integrated utility function that allows for trade-offs between different costs and benefits, i.e. different positive and negative experiences. Here I focus on the experiential or hedonic utility function, i.e. a function of how good or bad an experience is, where all possible subjective perceptions are the inputs of the function. This utility function is used in hedonistic utilitarianism that aims at maximizing the sum of everyone’s (selfish) happiness minus suffering. For example I have perceptions of room temperature, fear, joint pain, taste pleasure, musical enjoyment, income,… My utility function is integrated, because I can trade-off these perceptions: for example going to a colder, less comfortable room to get a delicious ice cream. Or working less to have more leisure time to listen to music, at a cost of decreasing my income.

The experiential utility function can also include expectations of future experiences. For example I may experience fear of getting toothache when I eat sweet ice creams, and if this fear is worse than my enjoyment of the ice cream, I decide not to eat the ice cream.

The whole input space of the utility function can be divided in three parts: the inputs that generate a positive, a zero and a negative utility. For example, when I’m very rich but have pain from a disease, my overall utility or well-being can be zero. But when a friend comes to visit me, my utility can become positive. The set of inputs that generate a zero utility cuts the input space in two regions. However, this zero utility set can also shift, for example due to adaptive preferences. When I win the lottery and become very happy, after a while I can adapt to my new situation and the richness no longer makes me happy. When I encounter a richer person, I may become jealous and suffer from frustration that I do not have even more money. Or when I have a disability, I can adapt and learn to live with it, such that feelings of discomfort decrease.

Also the steepness of the utility function can change due to circumstances (expectations, drugs, alertness, habituation…). For example consider the pain from pinpricks. I may be able to feel a difference between 2 and 3 needles in my arm, but with some drugs, it is possible that adding an extra needle doesn’t bother me so much anymore (even if I’m still able to feel the difference if the extra needle). In that case, the utility function in that dimension becomes very flat. There are people with pain asymbolia or pain dissociation, who can feel pain, but who do not have any negative evaluation or feeling of unpleasantness of that pain. For them, their utility function in the pain dimension is flat. On the other hand, when you become more and more sensitive to pain from pinpricks, and when you negatively evaluate that pain, your utility function becomes steep. Suppose you are being tortured and you have 100 needles in your arm. The torturer adds a few extra needles, but you won’t feel extra pain. Only after 10 needles have been added, you start to feel a pain increase and your utility level decreases with 1 utility unit due to the extra pain. Your just-noticeable difference in this case is 10 needles, i.e. the width of the pain steps can be measured as 10 input units (10 needles). But the torturer gives you a drug such that you become ten times more sensitive. You can start to feel pain from the slightest tough of an extra needle. You can now feel a difference in pain between 100 and 101 needles. Adding one extra needle becomes as painful as adding 10 needles used to be. So your utility function decreases with one utility unit when only one needle is added. This means your utility function becomes ten times as steep.

 

Comparing experiential utilities

There are two crucial properties of the inputs of a utility function that allows us to compare utilities of different sentient beings. First, all the inputs are measured on the positive real line segment, i.e. they have positive values. Second, the inputs are discrete and finite.

Concerning the first property, consider for example your income, your fever when you are ill, your desire for food, your stress level, your number of itchy mosquito bites, your number of friends, the number of times you laugh when watching a comedy, your amount of leisure time,… None of these values can be a negative number. These values can be inputs of your utility function, and there is a unique corner point where all the values are zero. In that case, you have zero income, zero friends, zero pain from cold, zero pain from heat, zero suffering from disease, zero enjoyment from food, zero hunger, zero minutes of relaxation with music, zero minutes of stressful work. This corner point is comparable to non-experience or non-existence, so we can give it a utility of zero. Every sentient being has such a unique corner point with utility zero, with zero positive and zero negative experiences. So the corner point always belongs to the zero utility set. Starting from this corner point, we can derive the zero utility set as an indifference curve by trading of inputs. For example increase the fever of a disease and then increase the intensity of laughter with a friend visiting you in the hospital, such that the positive experience of laughter exactly cancels out the negative experience of fever. Increase pleasure that cancels out pain. Increase income that cancels out the loss of leisure time. Decrease income to obtain perfect health, as in the notion of equivalent income for health.

With a corner point of utility zero, and a well-defined utility function that allows to derive an indifference curve starting at this corner point, we can interpersonally compare zero well-being. But we can not yet compare different positive or negative levels of well-being. For this, we need the second crucial property of the inputs: the finiteness and discreteness. We have seen that the inputs of a utility function are discrete, and that means that the levels that the utility function can take are discrete as well. The utility function becomes like a multidimensional staircase. Consider a room with walls in the north-south and west-east directions. In this room, there is a two-dimensional staircase. The south-western corner of the room is at the ground floor level (i.e. zero utility). Moving north corresponds with increased happiness, so the stairs go up. Moving east corresponds with increased suffering, so the stairs go down to the basement. Moving in a north-eastern direction keeps you at the ground floor level. The stairs cannot only go up or down, but might also have different steepness levels. Moving from south to north might be steep, with high steps, moving from west to east might be relatively flat, with small steps.

Furthermore, because of the finiteness of our brains, the inputs (stimuli) as well as the utility function itself are always finite: they cannot take a value of plus or minus infinity. The utility function has horizontal asymptotes. That means our multidemensional staircase has a highest and a lowest level. Also, the finiteness of the inputs imply that the steps cannot go infinitesimally small (low or flat). That is crucial, because now we can go looking for the smallest step in this multidimensional staircase: the step with the least height. There is a point and a direction, such that if you are at that point and move in that direction, then you rise the least, because that step has the smallest height. You can define its height to be one utility unit. So now we have our utility scale that allows for interpersonal comparisons.

For each sentient being, there is such a smallest difference in experienced utility, a smallest step. That step can be defined to have a height of one utility unit. That means that next to the zero level we can also interpersonally compare utility units or utility differences. Each sentient being has its own utility function that can be represented as a multidimensional staircase in a multidimensional room. Some stairs go up, some stairs go down. Two persons have two rooms that each contain one staircase, and the challenge is to derive whether a point in the first room is at the same height as another point in the second room. This can be done: every staircase has a unique ground floor (level zero), and the corner of the room is at this level. And the height of the smallest step of each staircase is the same for each room. With these two properties, we can compare the heights between different rooms.

 

The full (preference) utility function

For most sentient beings, their utility function has only subjective perceptions or experiences as inputs. However, some sentient beings with the capacity for abstract, rational thought, can also include non-experiential aspects in their utility function. For example you might have a preference for the truth, even if it hurts when you know the truth. You might have a preference that your spouse doesn’t cheat on you, even if you will never know about the adultery and never experience bad emotions. You might also have an altruistic preference that other people are happy, even if you will never know those other people or will never see their happiness. You might have a preference for fairness and justice, even when you are not the victim of injustice. You might even have a preference for your own non-existence.

The full utility function takes into account our hedonic experiences plus all other things that we want, all our other preferences. So our complete utility function can be very complex and broad. It contains literally everything that we value and prefer, even if we don’t experience it. This full utility or preference utility is used in preference utilitarianism.  The non-experiential aspects might raise or lower the full utility function. This property is useful in population ethical theories such as (variable) critical level utilitarianism where the experiential utility function gets subtracted with a constant.

The interpersonal comparison of utility can easily be generalized to the full utilities, because we can make trade-offs between a perceptual input and a non-perceptual input: how much are you willing to pay (experience a lower income) or suffer to know the truth? How much are you willing to be tortured in order for there to be world peace after you die? How much are you willing to sacrifice yourself in order to conserve nature or beautifull works of art after you (and all other sentient beings) die? The point is that we can compare the minimum difference in experiential utility with a difference in non-experiential utility, even if the utility of the non-experiential preferences would be continuous instead of discrete, and even if we do not subjectivelly feel those more abstract non-experiential preferences like we feel pain from a needle.

Comparing full utilities

There are some issues with comparing full utilities. First is the issue of multiple simultaneous utilities. It is possible that the brains of one sentient being generate two or more separate utility functions at the same time, as if that person has a multiple personality disorder. Think for example of a split-brain patient whose right hemisphere has different preferences than his left hemisphere. Or a sentient being who can make a trade-off between food and safety (e.g. moving to an unsafe area to obtain food and avoid a feeling of hunger), and between mating and pain relief (e.g. choosing for painful mating), but not between food and pain relief. Each utility function counts as a separate person.

Second is the issue of counting different utilities through time. The evaluation of time introduces two problems. First is the question how much we experience in a time interval, and second is the question how we value the different experiences in a time interval.

We can directly experience the flow of time, but not time intervals (except indirectly through memory, but then the time interval is already in the no longer experienced past). However, as mentioned above, our experiences have a maximum frequency due to the finite information processing rate of our brains. This means your utility function is evaluated a finite number of times per second. After your just-noticeable time difference, you reevaluate your utility function. Someone with a faster brain has more experiences and more evaluations of its utility function in a time interval. In interpersonal comparisons, every separate evaluation of the utility function counts. It is as if each instantaneous experience counts as an experience of a separate person. In other words, for an interpersonal comparison of well-being, the well-being experienced in a second for a sentient being whose brains are twice as fast as yours, counts twice as much as your well-being experienced in that second.

But we also have to make a distinction between the remembering self and the experiencing self. The experiencing self lives in the present moment and evaluates well-being or utility after each just noticeable time period. The remembering self evaluates utility for a past episode, based on memory of the experiences in that episode. The utility function of the remembering self can be different from the (sum of) utility functions at each of the experienced instantaneous moments, i.e. for each of the experiencing selves during a time interval. Suppose for example you can choose between two painful episodes. The first has a longer time interval with a steady pain slowly decreasing to zero. The second has a shorter time interval, with a very brief high peak of pain and an abrupt ending where pain suddenly drops to zero. Combining your experiencing selves, they choose the second episode, because there are fewer moments of experienced pain. But the remembering self uses a peak-end rule: in the second painful episode, the pain at its peak and the pain at the end are both higher than in the first episode, so the second episode is considered worse. Also, the remembering self evaluates life satisfaction, whereas the experiencing self evaluates momentaneous happiness, which is different from life satisfaction.

When comparing well-being between persons, we have to make sure that we either compare the utilities evaluated by their remembering selves, or the utilities evaluated by their experiencing selves, but not the evaluation of one person’s remembering self with the other person’s experiencing self.

An example

Assume, as an example, that a caterpillar is a sentient being and is on fire. How bad are the burns for the caterpillar? According to the discrete input model discussed above, we can measure the just-noticeable differences of pain from the burns. Going from zero burns to a just-noticeable burn decreases the utility with one unit. An extra burning sensation decreases the utility with another unit, and so on. Suppose the caterpillar on fire experiences 100 negative utility units, which means 100 just-noticeable differences of pain are exceeded. Now I want to compare this with me being on fire. Perhaps, hypothetically, when my little finger is on fire, I also experience 100 negative utility units, as much as when the whole caterpillar is on fire (because my little finger is as big as the caterpillar’s body). The surface area of my whole body is say 1000 times the surface area of my little finger, but that does not yet mean that my whole body being on fire is 1000 times worse than my finger being on fire. Due to the Weber-Fechner law, the more I am on fire, the less extra pain I experience when an extra square centimeter of skin is on fire. That means when my body is fully on fire, I experience say only 10.000 negative utility units instead of 100.000. In other words, for a caterpillar, being completely on fire feels the same as my little finger being on fire feels to me. When I’m completely on fire, it feels 100 times worse than what the caterpillar can experience.

But we also need to take into account the brain processing speed. Perhaps my brain is ten times faster, which means ten times more pain evaluations per second. In that case, the caterpillar being on fire for ten seconds feels like my little finger being on fire for one second. That does not yet mean that my finger being on fire for one second is 10 times worse than a caterpillar being on fire for one second, because I might have different preferences for time intervals. We have to consider the utility function of my remembering self and the remembering self of the caterpillar. Perhaps my remembering self doesn’t care that much about the number of experiences per second or the length of a painful episode. That means the length of time gets discounted. If my remembering self mainly cares about the peak pain experience, then the length of time is less important and one second of having a finger on fire is say only twice as bad as a caterpillar being on fire for one second. When my whole body is on fire for one second, it becomes 200 times as bad as what the caterpillar experiences.

What if experiences are not discrete?

Suppose that our assumption of discreteness of inputs of the utility function is invalid: the inputs as well as the utility function are continuous, which means that the utility function can take any real value and there is no smallest utility difference (minimal step height). In that case, we can no longer compare the differences in levels of utility, and we have to revert to the method described in the previous article, namely normalization (e.g. dividing your utility level by the spread or standard deviation of all possible utility levels that you can get for all possible choices that we can make). This can have drastically different results when comparing well-being and suffering.

In the above example with the caterpillar, I argued that my pain could be 100 times worse than the pain of the caterpillar. Now suppose that pain experiences were continuous instead of discrete. With a continuous input model, we have to use the utility normalization procedure. Suppose there are only two possible situations for the caterpillar: either the caterpillar dies by fire, or lives by avoiding the flames. Suppose the first situation corresponds with 100 negative utility units, the second with 0. The spread of utilities between those two situations is also 100, so dividing the utility levels by the spread gives a normalized utility of minus 1 for being on fire. Now we do the same for me: there are two possible situations: being completely on fire, with 10.000 negative utility units, and not being on fire, with utility 0. Dividing by the spread (10.000) gives a normalized utility of minus 1 when I’m on fire. In that case, being on fire for the caterpillar is as bad as being completely on fire for me, and 1000 times worse than my little finger being on fire. If insects can suffer, and if suffering is continuous instead of discrete, insect suffering might vastly trump human suffering.