Population ethics and the welfare function
Population ethics studies the optimal welfare distributions in situations where the population size is variable or where our choices influence who will exist in the future. The problem of population ethics is that we always encounter counter-intuitive results. Population ethics is is very important in the effective altruist movement (see for example the problem of saving lives versus preventing unwanted pregnancies). It is a part of consequentialist welfare ethics that looks at the consequences of our actions in order to maximize welfare. The starting point of welfare ethics is the welfare function that we should maximize. This welfare function is a function of the utilities of all individuals. For example total utilitarianism corresponds with a welfare function that takes the sum of everyone’s utilities. Here I define utility as a function of lifetime well-being which says how much an individual prefers that level of lifetime well-being.
If the lifetime well-being is positive, the utility is positive, which means that the life is worth living, that someone who lives that life prefers that life above non-existence. If the lifetime well-being is negative, utility becomes negative, which means the life is not worth living. Someone who has a negative utility prefers not being alive (not having been born) above living that life. A neutral lifetime well-being corresponds with a zero utility, which means someone who lives that life is indifferent between having that life versus non existence. Someone who does not exist also has a zero utility, i.e. no preference at all.
Suppose there are two individuals, the first one has lifetime well-being 0 (neutral), the second has a very high lifetime well-being at level 10. Suppose we can shift one unit of lifetime well-being from the best-off person (at level 10) to the worst-off person (at level 0). The first person gets lifetime well-being 1, the second person decreases to level 9. It is possible that the first person prefers 1 above 0 to a stronger degree than that the second person prefers 10 above 9. In other words, a increase from 0 to 1 is more important for the first person than an increase from 9 to 10 is important for the second person. The utility increases stronger when the lifetime well-being of the first person is increased from 0 to 1 than when the lifetime well-being of the second person is increased from 9 to 10. So the total utility (the sum of the utilities of both individuals) increases when a unit of lifetime well-being is shifted from the better-off to the worse-off person, even if total lifetime well-being remains constant. Here we see an egalitarian or prioritarian effect: more egalitarian distributions of lifetime well-being are better, and increasing the lifetime well-being of the worst-off persons should get a priority. If two persons have lifetime well-being of respectively 0 and 9, and we can add one unit of lifetime well-being, we should give it to the worst-off person at level 0.
This egalitarian or prioritarian effect corresponds with utilities that are concave (humped) functions of lifetime well-being. It also means that a person is risk averse in situations of uncertainty. Suppose that a person has a choice between getting a lifetime well-being of 5 with certainty, or having a probability ½ of getting a lifetime well-being 0 and a probability ½ of getting a lifetime well-being 10. The average lifetime well-being in this second option equals 5, as high as the first option. But if a person is risk averse, that person prefers the first option, because then there is no risk of getting a low lifetime well-being 0.
Now we enter population ethics. Let us start with the most simple expression of the welfare function: the sum of everyone’s utilities. This is total utilitarianism, which says that we should maximize total utility in the world (including the future). What could happen if we maximize total utility, given as the sum of all utilities of all individuals? Derek Parfit demonstrated that we face the so called repugnant conclusion.
Start with a population of very happy individuals with very high levels of lifetime well-being equal to 10. Now we can double the population by adding a second group with rather high lifetime well-being equal to 8. No-one is made worse off, happy people are added, the total utility almost doubles, so this second situation with a double population size is better than the starting situation. Now we can move to a third situation where we equalize everyone’s lifetime well-being. Everyone gets level 9. If the utility functions of the individuals are concave, this equalization increases total utility, so this third situation is better than the second. This process of doubling the population and equalizing well-being can be repeated ten times, so in the end we end up with a population more than 500 times larger than the first situation, where everyone has a very low (but still positive) lifetime well-being at level 1. But everyone in the original situation would strongly prefer having lifetime well-being 10 above 1.
The figure below illustrates the repugnant conclusion. The height of the bar corresponds with the lifetime-well-being, the width with the size of a population. The dotted lines corresponds with all the individuals who do not exist.
Total utilitarianism faces an even worse problem: the very repugnant conclusion. Suppose you can choose between two situations. In the first situation, 10 people have a high utility level 10. In the second situation, those same 10 people have very negative utilities -10, and there are 1000 people with a low utility 1, i.e. lives barely worth living. The total utility is 900 in the second situation (-10×10+1×1000), 9 times higher than the total utility in the first situation (10×10). It is very counter-intuïtive to say that the second situation is 9 times better.
If total utilitarianism results in these repugnant conclusions, perhaps we can modify the welfare function?
Total utilitarianism values the quantity of well-being in the world. Instead of taking the sum of all utilities, we can take the average utility of a population, by dividing the sum of utilities with the population size. Then we arrive at average utilitarianism, which values the quality of a life instead of the quantity of lives and well-being. Taking the average utility avoids the repugnant conclusion. However, this average utilitarianism also has a lot of counter-intuitive results.
First, we have the reverse repugnant conclusion. If adding people with lower lifetime well-being is bad because it lowers average utility, only persons with lifetime well-being higher than the average should be born. If those individuals are born, it raises average utility even further. Fewer and fewer people are allowed to be born.
Second, average utilitarianism faces the population dependence problem. Adding happy people with lifetime well-being 8 is bad if the rest of the population has average well-being 10, but if the rest of the population has well-being 6, adding those same people would be good. In the figure below, the situation on the top left is worse than the situation on the bottom left, but the situation on the top right is better than the situation on the bottom right. This is strange, because the difference between the top and bottom cases is the same: in the two cases on the top, the same population is added.
This has far reaching consequences. If we discover that in the past or on another planet there was or exist a huge population of people happier than us, it is not good to add people like us who have a lower lifetime well-being.
Third, we have what Gustaf Arrhenius calls the dominance addition problem. In the figure below, the situation at the top is worse than the situation at the bottom, because in the top situation the average well-being is 9, which is lower than 10. Yet, everyone who exists in the bottom situation is better off in the top situation, so they would prefer the top situation. And the extra people at level 7 in the top situation cannot complain either, because they still have a positive well-being. Their alternative was non-existence.
It is getting worse with the first sadistic conclusion. Suppose we have a population with a small minority of happy individuals (level 10) and a huge majority of individuals with very negative utilities (level -10). This situation might be the case in nature, where most animals might have miserable short lives with bad experiences due to hunger, diseases, parasites, predation and so on. As a result, the average utility is also negative. We can increase this average utility by adding a population with negative utilities (level -8) that are a bit higher than the average negative utility of the already existing population. Hence, adding people with negative utilities (lives not worth living) would be good, if that would be the only option to increase average utility.
A final problem of average utilitarianism is the second sadistic conclusion. Suppose we have a huge population of very happy people at utility level 10, except for one individual. This individual can have utility levels 5 (a life worth living) or -10 (very miserable). Giving that person utility 5 is only possible if we add a huge population of happy people at utility level 8. See the top situation of the figure below. If we choose not to add this extra population, the one individual will get a miserable life (utility level -10), which corresponds with the bottom situation of the figure below. In this latter situation, average utility is a tiny bit lower than 10. In the former situation, average utility is a tiny bit lower than 9. So if we increase the lifetime well-being of the miserable person to a utility level 5 by adding the happy people, average utility decreases. Yet, in the latter situation, no-one can complain: the people at level 10 still get level 10, the miserable person is better off at level 5, and the extra people have lives worth living.
Some of the above problems of average utilitarianism can be avoided by taking complex combinations of total and average utilitarianism. But we cannot avoid all problems, and sometimes we get the repugnant conclusion again. We can propose new welfare functions that are mixtures of the welfare functions of average and total utilitarianism, to arrive at new theories such as so called critical level utilitarianism, negative utilitarianism and number-dampened utilitarianism. But I will not explain those theories, because they always contain many of the above problems.
So the basic choice seems to be between total and average utilitarianism, both of which have counter-intuitive results. Which one is correct? We cannot tell, because there is no-one who experiences neither total nor average well-being or utility. Each individual in a population has its own lifetime well-being and utility. Each individual has subjective experiences and preferences. But there is no ‘collective individual’ who represents the total population and has the experiences of the total population. The welfare functions of total and average utilitarianism do not correspond with the utility (preferences) of an existing individual.
In this sense, both total and average utilitarianism are impersonal theories. As a moral being, I can give a values to total utility, or average utility, or biodiversity or whatever other quantity I want to see maximized. But these values are given by me and are not the utilities of an existing sentient being who has preferences. Looking at the above second sadistic conclusion, average utility is decreased, even if no-one is harmed (no-one experiences a lower lifetime well-being). And looking at the repugnant conclusion, total utility increases a lot, even if a lot of people are harmed in the sense that they can complain that their well-being is decreased.
Person affecting utilitarianism
Next to the above impersonal utilitarian theories, there are more personal utilitarian theories. The so called person affecting view looks at how people who exist (or will exist independent of our choices, i.e. in all possible futures that we can choose) are affected by our choices.
Person affecting utilitarianism says that we should minimize the harms done to existing people. One version of person affecting utilitarianism is the minimum complaint theory. It counts the total complaint of all individuals and this total complaint should be minimized. A complaint indicates how strong a person chooses a situation relative to other possible situations. Suppose you can choose between different situations. Your choice affects the persons who are involved and those persons may or may not complain about your choice. The less they prefer your choice, the more they complain. All the complaints from all individuals can be add up to a total complaint. Your moral duty is to choose the situation that results in the least total complaint.
How do we calculate the total complaint? Take a situation S. This situation corresponds with a world history, where people lived, live and will live. Take all the individuals who existed, exist or will exist in situation S. All these individuals have their own lifetime well-beings and corresponding utilities or welfare preferences. A welfare preference of a person is a function of the well-being of that person and indicates how much that person wants a life (or situation) with such a well-being. The higher your well-being, the more you want that situation or that life with such a well-being. Now we can compare these utilities with the utilities that those individuals have in all other situations. If an individual does not exist in another situation, it gets utility 0 in that situation (because in that situation the non-existing individual has no preferences).
With the utilities or welfare preferences we can calculate frustrations. Take a person P who exists in situation S. In that situation S, that person has a welfare preference w(P,S,S) for that situation and a welfare preference w(P,S,X) for any other possible situation X that we can choose. Suppose that B is the best situation for persons P who sits in situation S. P would prefer situation B. Thus, w(P,S,B) is a maximum. In this situation S, person P has a maximum frustration w(P,S,B)-w(P,S,S). This frustration measures how strong person P in situation S has a preference for the best possible situation B above his or her actual situation S. (Note that if a person does not exist in situation Q, he or she cannot experience any well-being and does not have any welfare preferences, so for this non-existent person w(Q,S,X) is zero for all situations X.)
Usually, the complaint of a person is equal to its maximum frustration. Then you could say that we just need to take the sum of the maximum frustrations of all people involved and choose the situation in which the sum is minimized. But then we still encounter a small problem. Suppose you have the choice between three situations. In the first situation, no additional people are born. In the second situation, two people are born, the first having a utility 9 and the second a utility 10. In the third situation, the same two people are born, but the first has a utility 10 and the second a utility 9. If you choose the third situation, the second person has a frustration equal to 1, because in the second situation his or her utility was 10 and in the third it is 9. If you choose the second situation, then the first person has a frustration, because that person has a utility 9 instead of 10. So in the second and third situations, there is always someone with frustration.
The first situation is the only situation where there is no frustration because there is simply no-one in that situation. If you want to minimize the amount of frustration, it is better that no extra persons are born. In the second and third situations, there is more frustration than in the first. We can look at the first person in the second situation. That person has a frustration, but would it be wise for that person to complain to you that you ought to have chosen the third situation? No, because in the third situation, the second person can complain, and if you want to minimize the complaints, you will go for the first situation in which no-one is born and hence no-one complains. The first person in situation 2 still prefers that second situation above the first, because a utility 9 is higher than 0. Conclusion: no-one of the two persons can complain in the second and third situations. In terms of complaints, those situations are not worse than the first situation.
For each situation S we can calculate the total complaint. The complaint minimization version of person affecting utilitarianism says that we should look for the situation S’ that minimizes this total complaint. This total complaint counts as a negative welfare function, a function that we should minimize. (We can add a minus sign to arrive at a welfare function that we should maximize.)
As with the case of total and average utilitarianism, this negative welfare function of person affecting utilitarianism is not the utility of a ‘collective individual’. But it is less impersonal than total and average utilitarianism, because it is completely based on the utilities of existing individuals. If every existing individual takes an impartial perspective, every individual can want to minimize the total maximum complaint. Almost by definition, no-one can give valid (impartial) reasons to complain against this choice of welfare function that minimizes complaints. In contrast, some people have reasons to complain in total utilitarianism, if they become victim of the repugnant conclusion where they will end up with a much lower well-being. And some people – in particular the one person in the second sadistic conclusion – have reasons to complain in average utilitarianism, where they will end up with much lower (even negative) utilities.
According to person affecting utilitarianism, the top situations in the above illustrations of the repugnant conclusion, the dominance addition problem and the first and second sadistic conclusions are the best. And the population dependence problem is avoided as well, so this is all in line with the moral intuitions of many people.
Let us return to the example of the repugnant conclusion. Suppose we have a choice between only two situations: the first situation contains a population at utility level 10. The second situation contains two populations: the same population as in the first situation, at level 10, plus an additional population at level 8. If these are the only two options available, they are equally good according to complaint minimizing person affecting utilitarianism, because in both situations there are no complaints.
But what if the third situation in the argument towards the repugnant conclusion becomes possible? In this third situation the utilities of both populations are equal at level 9. If this third situation is possible, then the total frustration changes in the second situation. The second population in the second situation had utility 8 and in the third situation they get utility 9. Those people in the second population in the second situation are frustrated, because they then have preferred the third situation. They have a frustration 9-8 = 1 in the second situation.
Suppose those people in the second population go complaining because of their frustration. What if we then choose the third situation? In that situation, the first population may complain, because their utility is 9 in the third situation, while they had a utility 10 in the first situation. That first population prefers the first situation above the third situation. The first situation is the only situation in which nobody has a frustration and so no-one can complain. The first population has a maximum utility level and the second population does not even exist in the first situation, and remember that non-existing persons cannot complain.
There are now two options: either the people in the second population go complaining in the second situation. In that case, the first situation is the best, because that is the only situation without complaints. Or the people in the second population are not complaining in the second situation because they realize that if they complain, the first situation is the best, and in the first situation those people in the second population were not even born. If the people do not complain in the second situation, then the first and second situations are equally good, because in both cases there are no complaints. The third situation is clearly the least good, because there is always at least one person (from the first population) complaining. That is why the repugnant conclusion is avoided.
Unfortunately, person affecting utilitarianism faces its own counter-intuitive problems. A first, minor problem is the choice dependence problem. Consider again the first two situations we saw in the repugnant conclusion example. If these were the only options available to us, we could choose the second situation (doubling the population by adding people at level 8), because no-one in this second situation can complain. No-one is worse off than in any other available situation. But now suppose the third situation becomes an option as well. If we would choose the second situation, there is a group of people who can complain: those people at level 8 in the second situation could have a well-being at level 9 in the third situation. So the third situation is better for those people. But if we pick the third situation, we again have a group of people who can complain, because the first half of the population has level 9 whereas they could have level 10 if we chose the first situation. In situations two and three there are always people who can complain, if those two situations are possible. The first situation is the only situation where no-one can complain. But if the third situation is no longer possible, situations one and two become equal, because in both situations no-one can complain. It can seem counter-intuitive to say that the betterness of situation one relative to situation two depends on whether or not another situation three is possible.
Person affecting utilitarianism has a more serious counter-intuitive implication: the intergenerational justice problem. Consider environmental problems such as global warming and fossil fuel depletion. If the current generation decreases its carbon footprint, for example by using bikes instead of driving cars and doing research for alternative renewable energy sources, we avoid future problems of global warming and we give future people new opportunities with new energy sources. If we invest a lot in research, future generations can even be made better off than the current generation. So, as in the figure below, the current generation might have utility level 9 whereas future generations can have high levels as well, for example level 12. However, the utility of the current generation can increase to level 10 if we do not lower our carbon footprint, if we do what we most strongly prefer, such as driving cars and produce consumer goods for ourselves instead of doing more research. As a consequence, future generations will have a lower utility, because they face the problems of global warming and resource scarcities. We might end up with a small future population because the earth is no longer capable of supporting a larger population, and this remaining population might have a low utility level 1.
Now here is the tricky part: If we (the current generation) lower our carbon footprint, we will change our behavior. We will take the bike instead of the car and we will do other kinds of work. As a result people arrive a little bit later at home and go a little bit later to bed. If they want to make children, the timing becomes very important. Male sperm determines the sex of the future children. Multiple sperm cells compete with each other, at one moment, the sperm cell with a female X chromosome is ahead, and the future baby would be female, but a fraction of a second later, another sperm cell with a male Y chromosome might have the lead and fertilize the egg cell. The exact moment of intercourse determines whether the future child will be male or female. And it determines many other properties of the child. Completely different individuals can be born. This generates a butterfly effect: very small changes in our behavior will have very big consequences as they determine who will be born in the future. (The butterfly effect says that the flapping of a butterfly can determine whether or not there will be a hurricane in another country a year later.) In other words: the small future population at level 1 is not the same population as the larger population at level 12. These populations contain different individuals.
So if we choose not to decrease our carbon footprint and future generations will get a low level 1, can they complain? Can they say that for their benefit we should have taken some actions to avoid global warming? No, because in the other situation, they would no longer exist. As long as their utilities are positive, they cannot complain. And if we take the first situation where we lower our carbon footprint and the current generation ends up with utility level 9, is there someone who can complain? Yes, the current generation can complain, because in the alternative situation they would have a higher level 10. So for a person affecting utilitarian, the first situation is worse than the second, even if in the first situation total and average utilities are higher and there is more equality between the generations.
Of course, once global warming becomes so bad that new people will be born who will have negative utilities, those future people can complain, because they would prefer not being born above having a negative lifetime well-being. Hence we should avoid serious environmental problems and promote some level of sustainability such that future generations have positive utilities. We have to be very careful with environmental problems that decrease future well-being, because we risk that some future people will be born who will have lives not worth living. For example, suppose an additional 1% of future generations will get negative utilities due to environmental problems. Those extra 1% of people can complain. They have valid reasons to accuse us that we have done too little against the environmental problems. An additional 1% of people with a negative welfare seems negligible, but because there will be so many future generations – literally hundreds of generations – there may well be many future people who can complain. Much more than all the people of the current generation.
A related problem of person affecting utilitarianism is what Gustav Arrhenius calls the extreme priority problem. Suppose we have to choose between two situations as in the figure below. In the first situation we have a population at level 10, a larger population at level 15 and one person at level -1. The second situation contains the first population again at level 10, and a new population at level 1. This second situation is better than the first according to person affecting utilitarianism, because in the first situation there is one person who can complain: the person with the negative utility. This may seem counter-intuitive, because the first situation has a much higher total and average utility. There is only one person who can complain a little, but this one persons complaint dominates and as a consequence the very big population of very happy people at level 15 should not exist. Only a small group of future people who will have a low utility level 1 can ben born.
These problems of the person affecting view determine how we should treat for example the economic costs of future environmental problems such as climate change. In environmental economics that studies cost benefit analyses involving future generations, the discount rate is a very important parameter that determines which policy choices are good in terms of economic costs and benefits. An average or total utilitarian would choose a zero discount rate: the utilities of future generations are as important as those of the current generation. But a person affecting utilitarian would choose a higher discount rate. The more different the future populations are (the less overlap they have) between different situations, the higher the discount rate, and the less the utilities of the future generations matter (except if they have negative utilities). In other words: an average or total utilitarian would prefer higher levels of environmental sustainability compared to a person affecting utilitarian. A strong interpretation of sustainable development, in terms of ensuring that future utility levels are not lower than current levels, is less important for the person affecting utilitarian.
Summary and concluding remarks
There are basically three different population ethics: total utilitarianism, average utilitarianism and person affecting utilitarianism. These can be combined and slightly modified to generate other population ethics, but they all face at least one counter intuitive result. Hence we have a trilemma: which of the three utilitarian theories is the best? I think person affecting utilitarianism is the strongest candidate, because it is the only theory that minimizes complaints of real individuals (who exist now or will definitely exist in the future). Therefore it avoids the repugnant conclusion, the reverse repugnant conclusion, the two sadistic conclusions, the dominance addition problem and the population dependence problem.
However, person affecting utilitarianism faces other problems. One minor problem is the choice dependence problem. But this has no important practical implications, because we don’t have to consider non-existing options anyway. More serious problems it faces are the intergenerational justice problem and the extreme priority problem. These problems may seem counter-intuitive in terms of justice (intergenerational equality) and effectiveness (improving total well-being), but at least no-one can complain against it.
I would personally give some value to impersonal properties, such as total utility, average utility and biodiversity, and these imply stronger notions of sustainability than are implied by person affecting utilitarianism. However, I think we should not give too much value to those properties, because if we maximize them, there can be people who have reason to complain. And these properties are values that I give, they are not utilities of a sentient being such as a ‘collective individual’. Populations or ecosystems are not sentient beings, so they do not have preferences (utilities), so they don’t care whether they have high or low levels of total or average utility or biodiversity. If I value total utility or biodiversity and if those properties increase, my individual utility increases. This increase contributes to an increase of total utility, but the increase of total utility will be very small if I am the only person who values those things.
I also think that sustainability remains important in a person affecting view, because unsustainable development involves the risk that future generations with negative utilities will be born, and they have a reason to complain. And the degree to which future populations are different between different choices, is uncertain. If there are future people that exist in both the situation with a high future utility and a low future utility, those people in the situation where they have the low utility have very strong complaints.
For a mathematical formulation of the utilitarian population ethics, see here.
 Derek Parfit raised some important problems in population ethics. Gustaf Arrhenius demonstrated that we cannot avoid counter-intuitive results in population ethics. Some important works in population ethics are:
Parfit, D. (1984). Reasons and Persons, Oxford: Clarendon Press.
Arrhenius, G. (2000). Future Generations: A Challenge for Moral Theory, PhD dissertation, Uppsala University.
Blackorby, C., Bossert W. & Donaldson, D. (2005). Population Issues in Social Choice Theory, Welfare Economics, and Ethics. Cambridge University Press.