Mere Addition Paradox Resolved

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David Cooper
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Mere Addition Paradox Resolved

Post by David Cooper » April 30th, 2018, 4:57 pm

Diagram:

See https://en.wikipedia.org/wiki/Mere_addition_paradox for the diagram, though in case the key parts of it are edited into a different form in the future, I'll provide a description here (adding appropriate numbers of my own invention, chosen to reflect the height of the bars in the diagram)

Description of diagram:

Population A might have 1000 people in it with a quality of life of 8, which we'll call Q8.

Population A+ is a combination of 1000 people at Q8 (population A) plus another 1000 people at Q4 (population A', though this population is not normally named).

Population B- is a combination of two lots of 1000 people which are both at Q7.

Population B has 2000 people at Q7.

The distinction between group B and B- is that B- keeps the two lots of 1000 people apart, which should reduce their happiness a bit as they have fewer options for friends, but we're supposed to imagine that they're equally happy whether they're kept apart (as in B-) or merged (in B).

Parfit's argument (to illustrate the paradox):

"Parfit observes that i) A+ seems no worse than A. This is because the people in A are no worse-off in A+, while the additional people who exist in A+ are better off in A+ compared to A [where they simply wouldn't exist] (if it is stipulated that their lives are good enough that living them is better than not existing)."

"Next, Parfit suggests that ii) B− seems better than A+. This is because B− has greater total and average happiness than A+."

"Then, he notes that iii) B seems equally as good as B−, as the only difference between B− and B is that the two groups in B− are merged to form one group in B."

"Together, these three comparisons entail that B is better than A. However, Parfit observes that when we directly compare A (a population with high average happiness) and B (a population with lower average happiness, but more total happiness because of its larger population), it may seem that B can be worse than A."

Paradox Lost:

First of all, we need to understand why there should be an optimal population size for a given amount of available resources, and if the population grows too high, total happiness goes down rather than up. This must be the case because the happiness of a population falls to zero long before the resources per person approach zero, and if you drag people out of poverty by giving them a modest increase in resources, their happiness shoots up, so it isn't a linear relationship either. The paradox superficially appears to deny this, but it only does so by introducing a fundamental error.

The error in the argument is hidden in the allocation of resources for A. Initially, A has access only to the resources of A and not to the resources of A'. When A' is added to A to make A+, new resources are brought in at the same time.

We can see now that A with access to all the resources of A+ (but without the population A') is inferior to A+ in terms of happiness because it's failing to use all the resources available to it, whereas A with access only to the resources of A is superior to A+ in terms of happiness per unit of resources. This is the key difference which Parfit missed.

When we look at A+, we see an unfair distribution of resources, and if we fixed that by sharing things evenly for all members of A+, A+ might well end up looking like B- because so many people would be lifted out of poverty and gain greatly in happiness without dragging A down very far.

We can thus see that A+ is inferior to an adjusted A+ with a redistribution of resources to even them out, and we can see that A+ is inferior to B- and B if it lacks that even distribution of resources, or it might be on a level with B- and B if it has redistributed of resources to even them out.

B is superior to A if A has access to the resources of A' while population A'=0, but A is superior to B if it doesn't have access to those extra resources of A' when you compare A and B per unit of available resources (which B has a lot more of).

So, the paradox evaporates: A is better than B if A only has the resources of A; but B is better than A if A has access to the full resources of A+ while it fails to use the component of those resources relating to A'.

(Note: If A was to use the resources of A' as well, it might go up to Q9 and have happier people in it, but the optimum population would then be higher and so it would have to grow to maximise total happiness per unit of resources.)

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Re: Mere Addition Paradox Resolved

Post by David Cooper » May 2nd, 2018, 8:40 pm

I should add a little more explanation for those who aren't familiar with this issue. This is seen as being an important paradox which undermines utilitarianism on the basis that A+ is better than A, B is better than A+, and yet B is also inferior to A. It may not be obvious that B is inferior to A, but if you keep adding more populations of people at lesser levels of happiness, you can create C, D, E ... etc. in the same manner with the total amount of happiness going up each time, and by the time you get to Z you have enormous amounts of people who are barely happy at all, but where the total happiness is higher than for A. This is argued to mean that utilitarianism says we should choose to have that massive population Z which is reduced to bare subsistence rather than the tiny population A where everyone is very happy, but it's regarded as a paradox because it's obvious that A is actually better than Z where everyone is reduced to a level where it's hardly worth them existing.

So, it is claimed to be a paradox on the basis that A < A+ < B < A. Crucially though, that is not correct mathematics because the last of these is worked out on a different basis from the others. If you stick with the same basis throughout, you either get A < A+ < B > A, or you have A > A+ < B < A. The latter applies if you're consistently using the basis that maximum happiness per unit of available resources is what matters (or on an alternative basis that maximum average happiness is what matters regardless of resources). What makes us see A as being better than B (or Z) is that we are applying different rules from the stated ones - we are making a judgement based on our knowledge that we can adjust population to maximise total happiness for a given amount of resources and that there is actually a point beyond which happiness must plunge to zero and accelerate on into the negative, more than cancelling out any imagined extra happiness provided by the greater number of people. We can take Z and reduce the population over a long length of time until it is of optimal size, and then it will have much greater total happiness than the original Z, and after we've done that, it is clearly better than A because it has more happiness in total and the same average happiness per person.

There is a way to break the link to resources though and still show that the paradox is resolved. If we replace the people with unintelligent sentient devices, we can have A populated by devices which are always happy at the level Q8, while we can have Z populated by trillions of similar devices which are always happy at the level Q0.001 and which are incapable of being set to a higher happiness level than that. Is it better for a universe to contain A or Z? The answer is that Z is better because that introduces much greater total happiness. By switching to devices of this kind instead of using humans, we eliminate all the undeclared factors which make this answer seem so wrong for humans: we now have no means to optimise the population sizes for their available resources, but we also don't imagine the trillion devices being deeply discontented about not being very happy at all in the way that humans would be, so it becomes much clearer that it is correct to go by total happiness. With all thought experiments in moral philosophy, it's essential either to eliminate all the undeclared factors or to expose them so that they can be accounted for in full. People's intuitions are based on these undeclared factors which they may not consciously recognise, but they can be identified and made clear, and they must be if arguments about morality are to be correct.

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Re: Mere Addition Paradox Resolved

Post by LuckyR » May 3rd, 2018, 12:51 pm

The flaw of this entire issue is that happiness (like pain) is not additive. Thus having ten Q8 individuals is not inherently worse than twenty Q8 individuals.

Once this observation is factored in, the whole house of cards collapses.
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Re: Mere Addition Paradox Resolved

Post by David Cooper » May 3rd, 2018, 6:29 pm

Utilitarianism says that a universe with twenty Q8 individuals in it is better than a universe with only ten Q8 individuals in it, and I think it's got that right, because the first ten of them aren't harmed in any way by the addition of the second ten. The real difficulty is with how you judge when a population is optimal. There comes a point where adding more members to a population reduces total happiness rather than increasing or maintaining it, and that's where utilitarianism says it should stop growing, but in the run up to that point, quality of life is already going down for everyone as the population goes up (while total happiness is still rising).

There will be a point previous to that before which the addition of new members to the population is still improving quality of life for everyone else (on average) in addition to adding the new happiness of the new members, and where we reach the point where quality of life for existing members starts to decline when new members are added, that is arguably the moral place to stop the population growing rather than the point where total happiness is maximised. I'm now looking for thought experiments that might help to reveal which of these two approaches is actually better, of if there's some rational way of picking a position somewhere in between the two. Of course, by the time when average quality of life is clearly falling as the population goes up, the resentment at that loss will quickly lead to total happiness falling, so the distance between the two points where the population might be considered optimal may be very small.

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Re: Mere Addition Paradox Resolved

Post by LuckyR » May 4th, 2018, 5:51 pm

David Cooper wrote:
May 3rd, 2018, 6:29 pm
Utilitarianism says that a universe with twenty Q8 individuals in it is better than a universe with only ten Q8 individuals in it, and I think it's got that right, because the first ten of them aren't harmed in any way by the addition of the second ten. The real difficulty is with how you judge when a population is optimal. There comes a point where adding more members to a population reduces total happiness rather than increasing or maintaining it, and that's where utilitarianism says it should stop growing, but in the run up to that point, quality of life is already going down for everyone as the population goes up (while total happiness is still rising).

There will be a point previous to that before which the addition of new members to the population is still improving quality of life for everyone else (on average) in addition to adding the new happiness of the new members, and where we reach the point where quality of life for existing members starts to decline when new members are added, that is arguably the moral place to stop the population growing rather than the point where total happiness is maximised. I'm now looking for thought experiments that might help to reveal which of these two approaches is actually better, of if there's some rational way of picking a position somewhere in between the two. Of course, by the time when average quality of life is clearly falling as the population goes up, the resentment at that loss will quickly lead to total happiness falling, so the distance between the two points where the population might be considered optimal may be very small.
Your post is mixing topics hither and yon, thus muddying the waters such that little if any sense can be made.

For example, for simplicity's sake let's suppose that there are 10 individuals on earth (with Q8 happiness) and they live on 10 separate islands and don't even know of one another. Does adding another person on an 11th island at Q1 raise earth inhabitant happiness? According to the OP the answer would be yes (10 x 8 ) + 1 increases from total from 80 to 81. But for most folks (especially economists and statisticians) the answer would be no (average of 8 drops to average of 7.4

This doesn't even step into the mess of interactions between the individuals such that adding people adds (or subtracts) to the original's happiness, which of course adds many more variables and makes the computation significantly more difficult to understand.
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Re: Mere Addition Paradox Resolved

Post by David Cooper » May 4th, 2018, 7:39 pm

LuckyR wrote:
May 4th, 2018, 5:51 pm
Your post is mixing topics hither and yon, thus muddying the waters such that little if any sense can be made.
It's all the same topic. When you said "Thus having ten Q8 individuals is not inherently worse than twenty Q8 individuals", that depends on lots of unstated factors. If there are no unstated factors, having twenty Q8 individuals is twice as good as having ten Q8 individuals because there's twice as much happiness present.
For example, for simplicity's sake let's suppose that there are 10 individuals on earth (with Q8 happiness) and they live on 10 separate islands and don't even know of one another. Does adding another person on an 11th island at Q1 raise earth inhabitant happiness? According to the OP the answer would be yes (10 x 8 ) + 1 increases from total from 80 to 81. But for most folks (especially economists and statisticians) the answer would be no (average of 8 drops to average of 7.4
But if you use that maths, let's suppose you have nine individuals at Q8 and one at Q8.001. If one of the nine dies, you would have a happier world, and it would continue to get happier as the rest of the nine die until the only person left is the one at Q8.001. That is why the addition of someone at Q1 doesn't make things worse, and so long as the Q level of this individual is > 0, (s)he is within the range of contentment.

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Re: Mere Addition Paradox Resolved

Post by LuckyR » May 5th, 2018, 1:07 am

David Cooper wrote:
May 4th, 2018, 7:39 pm
LuckyR wrote:
May 4th, 2018, 5:51 pm
Your post is mixing topics hither and yon, thus muddying the waters such that little if any sense can be made.
It's all the same topic. When you said "Thus having ten Q8 individuals is not inherently worse than twenty Q8 individuals", that depends on lots of unstated factors. If there are no unstated factors, having twenty Q8 individuals is twice as good as having ten Q8 individuals because there's twice as much happiness present.
For example, for simplicity's sake let's suppose that there are 10 individuals on earth (with Q8 happiness) and they live on 10 separate islands and don't even know of one another. Does adding another person on an 11th island at Q1 raise earth inhabitant happiness? According to the OP the answer would be yes (10 x 8 ) + 1 increases from total from 80 to 81. But for most folks (especially economists and statisticians) the answer would be no (average of 8 drops to average of 7.4
But if you use that maths, let's suppose you have nine individuals at Q8 and one at Q8.001. If one of the nine dies, you would have a happier world, and it would continue to get happier as the rest of the nine die until the only person left is the one at Q8.001. That is why the addition of someone at Q1 doesn't make things worse, and so long as the Q level of this individual is > 0, (s)he is within the range of contentment.
That is a very unpopular opinion. Most would describe the life of someone in the lowest 10th of the world population as dreadful, and adding more sufferers lowers the world happiness level (as I demonstrated above).

Of course you are free to have your opinion, I'm just saying that it is an unusual one.
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Re: Mere Addition Paradox Resolved

Post by David Cooper » May 5th, 2018, 7:22 pm

LuckyR wrote:
May 5th, 2018, 1:07 am
That is a very unpopular opinion. Most would describe the life of someone in the lowest 10th of the world population as dreadful, and adding more sufferers lowers the world happiness level (as I demonstrated above).
Ah, but you're equating Q1 with someone in the lowest 10th of the world population, which you describe as dreadful. I would suggest that Q1 is not dreadful. The lowest 10th of the population are likely at highly negative levels of Q, whereas Q1 people are quite contented with their lives. The "paradox" misleads people by implying that levels of Q between 1 and 0 are descriptions of grinding poverty, but that's not the case at all.

Also, when it suggests that a trillion people at Q0.001 don't outweigh the happiness of a thousand people at Q8, they're cheating with the numbers in another way. They might equally say that two people at Q4.1 aren't better than one person at Q8, but what they're doing here is smuggling in a hidden factor where they're considering the happier person to be further happier because (s)he is a lot happier than the other two people, which means they're adding to the happiness of the Q8 person without declaring it in the score (which they're imagining to be more like Q12 while still labelling it as Q8). If they have actually set the Q values correctly, adding them all up for two cases will produce a correct answer as to which case is better, so if they want amplification of happiness caused by happiness levels being higher, they need to add that in clearly and turn a Q8 into an explicit value like Q12 instead of continuing to call it Q8.

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Re: Mere Addition Paradox Resolved

Post by LuckyR » May 6th, 2018, 1:34 am

David Cooper wrote:
May 5th, 2018, 7:22 pm
LuckyR wrote:
May 5th, 2018, 1:07 am
That is a very unpopular opinion. Most would describe the life of someone in the lowest 10th of the world population as dreadful, and adding more sufferers lowers the world happiness level (as I demonstrated above).
Ah, but you're equating Q1 with someone in the lowest 10th of the world population, which you describe as dreadful. I would suggest that Q1 is not dreadful. The lowest 10th of the population are likely at highly negative levels of Q, whereas Q1 people are quite contented with their lives. The "paradox" misleads people by implying that levels of Q between 1 and 0 are descriptions of grinding poverty, but that's not the case at all.

Also, when it suggests that a trillion people at Q0.001 don't outweigh the happiness of a thousand people at Q8, they're cheating with the numbers in another way. They might equally say that two people at Q4.1 aren't better than one person at Q8, but what they're doing here is smuggling in a hidden factor where they're considering the happier person to be further happier because (s)he is a lot happier than the other two people, which means they're adding to the happiness of the Q8 person without declaring it in the score (which they're imagining to be more like Q12 while still labelling it as Q8). If they have actually set the Q values correctly, adding them all up for two cases will produce a correct answer as to which case is better, so if they want amplification of happiness caused by happiness levels being higher, they need to add that in clearly and turn a Q8 into an explicit value like Q12 instead of continuing to call it Q8.
Well, as I said before, until you make the case that happiness is additive (which many if not most disagree with) you are wasting your time with the rest of your commentary. First things first.
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Re: Mere Addition Paradox Resolved

Post by David Cooper » May 6th, 2018, 5:14 pm

LuckyR wrote:
May 6th, 2018, 1:34 am
Well, as I said before, until you make the case that happiness is additive (which many if not most disagree with) you are wasting your time with the rest of your commentary. First things first.
Clearly it is additive for one person: one big chunk of happiness can be traded for a series of smaller chunks of happiness which add up to a greater total. A book with one big laugh in it can be outdone by a book with a lesser laugh on every page. However, the examples in the "paradox" are based on the idea of small amounts of happiness for many people adding up to a greater amount of happiness than greater amounts of happiness for a few people, and we need to know how to do this kind of maths because we will soon have a world run by artificial intelligence which makes moral judgements about all issues and which thereby dictates political policies.

If these machines judge things on the basis that happiness is additive and that more people of lesser happiness can be better than fewer people of greater happiness, that may lead to policies that grow the population. If they judge on some other basis, it may lead to them introducing policies to reduce the population in order to drive up average happiness, but it's possible that if they work on that basis, they will reduce the population over time to a tiny size in order to maximise happiness of the people who still exist, and if the greatest average happiness can be achieved by having just a few selfish people exist who keep all the world's resources for themselves, that would be the natural destination of such an approach (where the role of happiness addition is rejected).

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Re: Mere Addition Paradox Resolved

Post by LuckyR » May 7th, 2018, 4:17 am

David Cooper wrote:
May 6th, 2018, 5:14 pm
LuckyR wrote:
May 6th, 2018, 1:34 am
Well, as I said before, until you make the case that happiness is additive (which many if not most disagree with) you are wasting your time with the rest of your commentary. First things first.
Clearly it is additive for one person: one big chunk of happiness can be traded for a series of smaller chunks of happiness which add up to a greater total. A book with one big laugh in it can be outdone by a book with a lesser laugh on every page. However, the examples in the "paradox" are based on the idea of small amounts of happiness for many people adding up to a greater amount of happiness than greater amounts of happiness for a few people, and we need to know how to do this kind of maths because we will soon have a world run by artificial intelligence which makes moral judgements about all issues and which thereby dictates political policies.

If these machines judge things on the basis that happiness is additive and that more people of lesser happiness can be better than fewer people of greater happiness, that may lead to policies that grow the population. If they judge on some other basis, it may lead to them introducing policies to reduce the population in order to drive up average happiness, but it's possible that if they work on that basis, they will reduce the population over time to a tiny size in order to maximise happiness of the people who still exist, and if the greatest average happiness can be achieved by having just a few selfish people exist who keep all the world's resources for themselves, that would be the natural destination of such an approach (where the role of happiness addition is rejected).
Perhaps it is clear to you (maybe based on your personal experience) but in my experience since happiness (like pain) can neither be seen, felt, or experienced by another person, does not lend itself to being quantified, let alone summed.
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Re: Mere Addition Paradox Resolved

Post by David Cooper » May 7th, 2018, 1:33 pm

LuckyR wrote:
May 7th, 2018, 4:17 am
Perhaps it is clear to you (maybe based on your personal experience) but in my experience since happiness (like pain) can neither be seen, felt, or experienced by another person, does not lend itself to being quantified, let alone summed.
How do judges work out how much compensation to give people if these things can't be quantified? (Not that they're always good at it, but they can get it reasonably right when they aren't handing out six-figure sums to already-rich people who've had their phones tapped).

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Re: Mere Addition Paradox Resolved

Post by LuckyR » May 8th, 2018, 2:17 am

David Cooper wrote:
May 7th, 2018, 1:33 pm
LuckyR wrote:
May 7th, 2018, 4:17 am
Perhaps it is clear to you (maybe based on your personal experience) but in my experience since happiness (like pain) can neither be seen, felt, or experienced by another person, does not lend itself to being quantified, let alone summed.
How do judges work out how much compensation to give people if these things can't be quantified? (Not that they're always good at it, but they can get it reasonably right when they aren't handing out six-figure sums to already-rich people who've had their phones tapped).
I guess we agree, even "professionals" cannot reliably quantify the unquantifiable. The thread evaporates in a puff of magic mist.
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Re: Mere Addition Paradox Resolved

Post by David Cooper » May 8th, 2018, 5:39 pm

The incompetence of many judges doesn't destroy the ability of more competent people (or machines) making good judgements about balancing things up. If a company's reckless behaviour has led to someone's dogs being poisoned (to death) and someone else's child being poisoned (to death), which do you think should get more compensation? Or how about an explosion which tears off one person's leg and another person's ear. Which one deserves more compensation, and how big a difference in compensation should there be? Having banned any calculations to determine the answer, you're in an awkward position.

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Re: Mere Addition Paradox Resolved

Post by LuckyR » May 9th, 2018, 1:40 am

David Cooper wrote:
May 8th, 2018, 5:39 pm
The incompetence of many judges doesn't destroy the ability of more competent people (or machines) making good judgements about balancing things up. If a company's reckless behaviour has led to someone's dogs being poisoned (to death) and someone else's child being poisoned (to death), which do you think should get more compensation? Or how about an explosion which tears off one person's leg and another person's ear. Which one deserves more compensation, and how big a difference in compensation should there be? Having banned any calculations to determine the answer, you're in an awkward position.
Well, in the examples you cite, if 10 "experts" give 10 identical answers, they know something (or there is something to be known). If you get 10 different answers you are dealing with opinions, not knowledge.
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