### Why Classical Utilitarianism is the only (Archimedean) Ethic

Probably the most famous graph in ethics is this one of Parfit's:

He's constructing a series of worlds where each one has more people, but those people have a lower level of welfare. The question is whether the worlds are equivalent, i.e. whether it's equivalent to have a world with a huge number of barely happy people or a world with a small number of ecstatic individuals.

Classical utilitarianism answers "Yes", but some recent attempts to avoid unpleasant results (such as the "repugnant conclusion") have argued "No". For example, Parfit says:
Suppose that I can choose between two futures. I could live for another 100 years, all of an extremely high quality. Call this the Century of Ecstasy. I could instead live for ever, with a life that would always be barely worth living. Though there would be nothing bad in this life, the only good things would be muzak and potatoes. Call this the Drab Eternity. I believe that, of these two, the Century of Ecstasy would give me a better future.

The belief that the "Century of Ecstasy" is superior to the "Drab Eternity", no matter how long that eternity lasts, has been called "Non-Archimedean" by Arrhenius, in reference to the Archimedean Property of numbers, which says roughly that there are no "infinitely large" numbers.1 Specifically, a group is Archimedean if for any $x$ and $y$ there is some $n$ such that $$\underbrace{x+x+\dots+x}_{\text{n times}}>y$$
The following remarkable fact is true:
Classical Utilitarianism is the only Archimedean ethic.
This means that if we don't accept that the briefest instant of a "higher" pleasure is better than the longest eternity of a "lower" pleasure, then we must be classical utilitarians.

### Proof

First, define the terms. As always, we assume that there is some set $X$ which contains various welfare levels. There is an operation $\oplus$ which combines welfare levels; the statement $x\oplus y=z$ can be read as "A life with welfare $x$ and then welfare $y$ is equivalent to having a life with just welfare $z$."2 It is assumed that this constitutes a group, i.e. the operation is associative and inverses and an identity exist.

In order to make decisions, we need some ranking; the statement $x>y$ means "The welfare level $x$ is morally preferable to $y$." We require $>$ to agree with our operation, i.e. if $x>y$ then $x\oplus c > y\oplus c$ for all $c$.

With the stipulation that our group is Archimedean, this reduces to a theorem of Hölder's, which states that all Archimedean linearly ordered groups are isomorphic to a subgroup of the reals under addition, i.e. classical utilitarianism. The proof is rather involved, but a fairly readable version can be found here.∎

### Discussion

In order to be useful, non-Archimedean theories can't just say that there is some theoretical amount of welfare which is lexically superior - this level of welfare must exist in our day-to-day lives. Personally, when comparing a brief second of happiness on my happiest day to years of moderate happiness, I would choose the years. This leaves me with no choice but to accept classical utilitarianism.

Footnotes
1. Ethics with this property have also been called "discontinuous" or having a "lexical" priority.
2. Unlike in past blogs where I used $\oplus$ to be a population ethic, here I define it in terms of intra-personal welfare to fit more in line with Parfit's quote.

### Group Theory and the Repugnant Conclusion

A fundamental question in population ethics is the tradeoff between quantity and quality. The world has finite resources, so if we promote policies that increase the population, we do so at the risk of decreasing quality of life.

Derek Parfit is credited with popularizing the importance of this problem when he pointed out that any population ethic which obeys some seemingly reasonable constraints must end up with what he called "the repugnant conclusion" - the conclusion that a world full of miserable people is better than a sparsely-populated world full of happy people. Since Parfit, there have been a range of theories seeking to preserve our intuitions about ethics while still avoiding this conclusion.

One discovery of abstract algebra is that we can understand the limitations of systems based solely on the questions they are able to answer, even if we don't know what the answers are.

Here, I'll consider any system capable of answering a question like "Are two people who each live 50 years morally equivalent to one person who lives 100 years?" (Again, we don't require that the answer be "Yes" or "No", but merely that there be some answer.) For notational ease, I use the symbol $\oplus$ to be the "moral combination", e.g. the above question can be written $$(50\text{ years})\oplus(50\text{ years})=100\text{ years?}$$ Such a system I will call a "moral group" and require that it obey a few standard requirements. These are:

1. Any two people can be replaced with one who is (significantly) better off
2. There is some level of welfare which is "morally neutral", i.e. a person of that welfare neither increases nor decreases the overall moral desirability of the world.
3. For any level of welfare, no matter how high, there is some level of welfare which is so negative that the two cancel out

With this definition, we have an impossibility theorem:

Theorem: In any "moral group", the repugnant conclusion holds.

Proof: Suppose that $x$ is a welfare level that is better than "barely worth living". Formally, say that there must be some $y$ where $0 < y < x$, i.e. it's possible to be worse off than $x$ and still have a "life worth living". We'll show that a world with just $x$ is morally equivalent to a world with two people who are both worse off than $x$. Repeating this ad infinitum leads to the conclusion that a world with a few happy people is equivalent to a world with a large number of people whose lives are "barely worth living."

Choose some $y$ between $0$ and $x$ (one exists, by the definition of $x$). Note that $x=y\oplus z$ where $z=y^{-1}\oplus x$, so we just need to show that $z<x$. Since $y>0$, $y^{-1} < 0$ because if it weren't then we'd have $y^{-1} > 0$; adding $y$ to both sides results in $0>y$ which contradicts the assumption that $y>0$. Therefore $y^{-1} \oplus x < x$, or to write it another way: $z < x$. So $x=y\oplus z$, with $y$ and $z$ both worse than $x$.

This means that for any world with people $x_1,x_2,\dots$ of high welfare, there is an equivalent world $y_1,y_2,\dots$ with more people, each of whom have lower welfare. By adding some person of low (but still positive) welfare $y_{n+1}$ to the second world, it becomes better than the first, resulting in the repugnant conclusion.∎

### Algebra and Ethics

Symmetry is all around us. The kind of symmetry that most people think of is geometric symmetry, e.g. an equilateral triangle has rotational symmetry:

I've rotated the triangle by 1/3 of a rotation, but it remains the "same", just with a "relabeling" of the points. Hence this rotation is a symmetry of the triangle.

Ethical positions generally express another type of symmetry; when someone argues for "marriage equality" what they mean is that the gender of partners is merely a "relabeling" that keeps the important aspects like love and commitment the same. Symmetries in pain processing between humans and other animals has lead thinkers like Richard Dawkins to declare that species is merely a relabeling, and that causing pain to a cow is "morally equivalent" to causing pain to a human, calling our eating practices into question.

In 1854 Arthur Cayley gave the first modern definition of what mathematicians call a "group", and showed that groups are essentially permutations, thus establishing the theory of groups as the language of symmetry. Despite the importance of groups to symmetry and the importance of symmetry to ethics, I'm not able to find any ethical works based on group theory. So I hope to give what may be the first ever group-theoretical proof of ethics.

"Group-like" Ethics
I'm going to be concerned with questions like "is having two people, each of whom live 50 years, equivalent to having one person who lives 100 years?" I don't require that this question be answered either "yes" or "no", but only that the question has some answer.

So that this post doesn't take up a huge amount of space, I'm going to define the symbol $\oplus$ to mean "moral combination" and $=$ to mean moral equivalence, so the statement "two people, each of whom live fifty years, is equivalent to one person living 100 years" can be written as $$(50 \text{ years})\oplus(50 \text{ years})=100 \text{ years}$$ There are many different ways to define $\oplus$. For example, we might care only about the worst-off person - in this case $(50 \text{ years})\oplus(50 \text{ years})=50 \text{ years}$ as the worst-off person on the left-hand side of the equation has the same length of life as the worst-off person on the right. Alternatively, we might point out that quality of life degrades as you get older, so in fact maybe $(50 \text{ years})\oplus(50 \text{ years})=150 \text{ years}$ since the two young people get so much more joy out of their life. The World Health Organization follows this model and weights lives like this:

According to their formula, old age is so awful that $(40 \text{ years})\oplus(40 \text{ years})=125 \text{ years}$ and one person would have to live for thousands of years to be equivalent to two 50 year lifespans.

In addition to requiring that statements like $(50 \text{ years})\oplus(50 \text{ years})$ have some answer, I will also require that there is an "identity", i.e. there is some quality of life such that adding a person with that quality of life doesn't change the overall value of the world. This is a reasonable assumption because:
1. Sometimes increasing the population is a good idea, i.e. there is some $y$ such that $x\oplus y > x$
2. Sometimes increasing the population is a bad idea, i.e. there is some $z$ such that $x\oplus z < x$
3. By the intermediate value theorem, there must therefore be some value which I'll call $0$ such that $x\oplus 0 = x$

Any ethical system which has an operation like $\oplus$ I will call "group-like" (although observant readers will note that I'm making fewer assumptions than what groups require - technically this is a "unital magma").

"Utilitarian-like" Ethics
The classic definition of "utilitarianism" is to look only at happiness and to define $\oplus=+$, e.g. two people with five "units" of happiness is equivalent to one person with ten units of happiness.

There are a plethora of "utilitarian-like" ethical theories which define $\oplus$ as being sort of like addition, but not really. For example, negative utilitarians would first discard any pleasure, and look only at the pain of each individual before doing the addition. Prioritarians wouldn't completely disregard pleasure, but they would weight helping those in need more strongly. The Sen social welfare function weights income by inequality before doing the addition. And so on.

I will describe an ethical system as "utilitarian-like" if it is equivalent to doing addition with some appropriate transformation applied first. Formally, utilitarian-like operations are of the form $x\oplus y = f(x)+f(y)$.

The Theorem
With these definitions in mind, we can state our theorem:
The only ethical system which is both group-like and utilitarian-like is classical ("Benthamite") utilitarianism.
Observant readers will notice that my examples in the "group-like" section were different than the examples in the "utilitarian-like" section. This theorem proves that this is not an accident.

Proof: $x\oplus 0 = f(x)+f(0)$ so $x = f(x)+f(0)$ or to rewrite it another way, $f(x)=x - f(0)$ where $f(0)$ is some constant. This means that all group-like and utilitarian-like functions are equivalent, just shifted slightly. To use a formal definition of "equivalent", the homomorphism $\phi(x) = x + f(0)$ can be easily seen via the first isomorphism theorem to be an isomorphism $(\mathbb{R},\oplus)\to(\mathbb{R},+)$.

Discussion
The reason why Prioritarians et al. fail to be group-like is something I haven't seen discussed much in the literature: a lack of an identity element.

For example, suppose $x\oplus y = f(x)+f(y)$ where $$f(x) = \left\{ \begin{array}{lr} 2x & x < 0\\ x & else \end{array} \right.$$ This is a negative utilitarian-type ethics which weights suffering (i.e. negative experience) more strongly.

Consider a few possible worlds in which we add someone of utility 2:

1. $-1\oplus 2 = 0$
2. $-2\oplus 2 = -2$
3. $-3\oplus 2 = -4$

In the first case, adding someone of utility two improves the world. In the second, it keeps the world the same and in the third it makes the world worse.

That negative utilitarianism requires this isn't immediately obvious to me, and I believe it to be a non-trivial result of using group theory.

Conclusion
We might view negative utilitarianism or prioritarianism as a form of "pre-processing". For example, we might say that painful experiences affect utility more than positive ones. But when it comes to comparing utility to utility, it must be "each to count for one and none for more than one" with all the counter-intuitive results that implies.

### Poverty and Plant-Based Diets

Forty years ago, Frances Moore Lappe wrote Diet for a Small Planet, a combination cookbook and food industry critique. In it, she pointed out that the grain we feed to livestock animals could instead be fed to hungry people.

The recent shock in food prices has led to increased examination of food cost determinants, and the data provides interesting insights into how our diets can affect the lives of the world's poor.

### The Numbers

According to Counting Animals, a vegetarian saves 29 chickens, 1/2 of a pig and an eighth of a cow each year. Using the formula developed by Fortenbery and Park, 9 million such vegetarians would reduce the price of corn by $5/bushel. Using this as a proxy for soy, food prices of the ten staple foods would drop by 20%1. This corresponds2 to the central scenario of Dessus et al.,estimated to cause 233.2 million people to come out of absolute poverty (defined as living on less than$2/day). Using Goklany's estimates, this would avert 1.22 million deaths, and 42.7 million disability adjusted life-years.3

To put it in personal terms: one vegetarian saves one human for every eight years they're veg, and averts four DALYs per year of vegetarianism.

### Cost Effectiveness

EAA has previously estimated that the top charities create one vegetarian-year for around $11. This means that top veg charities save one person for$90, and spend around $2.75 to avert a DALY. For comparison, the Against Malaria Foundation, GiveWell's current top pick, spends$2,300 per life saved or between $29 and$169/DALY.

Even with the generous padding that these rough calculations deserve, veg charities may be competitive with other poverty-focused charities.

#### Footnotes

Code used to calculate these numbers can be found here.
1. This would cause a drop in soy and corn prices of 63%. However, these foods make up only a third of total global staples, meaning that aggregate staple price would drop by only ~20% (ceteris paribus). Note that Fortenbery and Park's model probably wouldn't handle such a large change well, so this should be considered a very rough estimate.
2. Dessus and Goklany both examined the other direction: how many more people would enter poverty as the result of increased food prices. I assume here that the change is symmetric, i.e. the badness caused by an increase of $x is the same as the goodness caused by a decrease of$y
3. Goklany separates DALYs meaning "disability with no death" from actual deaths, in contrast to places like GiveWell, which usually include premature death in their DALY calculation.

### Should Veg Advocates Use Health Arguments?

There are occasionally debates about how we can best advocate for veganism. Usually these debates take place between ethical vegans, so (unsurprisingly) the conclusion is usually that ethical arguments are the best approach.

For example, the Animal Activist's Handbook calls health-based arguments "problematic" and urges readers to focus on ethics-based approaches. No less an authority than Mahatma Gandhi said in his book The Moral Basis of Vegetarianism:
I notice also that it is those persons who become vegetarian because they are suffering from some disease or other - that is, from the purely health point of view - it is those persons who largely fall back. I discovered that for remaining staunch to vegetarianism a man requires a moral basis.
Like a lot of marketing advice, these theories are usually justified by an appeal to intuition, and like most such appeals I suspect that they aren't well supported by the facts.

A review of US meat consumption found that health information (as measured by the number of articles published in medical journals about the bad effects of cholesterol) had a stronger effect on demand than even price changes. A similar review of Canadian meat consumption found that government recommendations to eat less meat appear to have a significant impact. Concerns about cholesterol have sent the demand for butter and eggs plummeting. As Oprah fans know, information about the unhealthfulness of beef causes a huge drop in beef consumption - without increasing the consumption of pigs or chickens.

In a survey by the Vegetarian Journal, 82% of readers stated that they became vegetarian for health reasons, and among adolescents a vegetarian diet seems to be linked with a desire for weight control. This is confirmed by the Vegetarian Times' survey, which found that the majority of self-described vegetarians do it for health reasons. In a psychological survey of the origins of vegetarianism, the authors found that slightly less than half of vegetarians originally quit eating meat for health reasons. Vegetarians of all stripes are significantly more likely to be concerned about health aspects of their food.

And we shouldn't think that someone who becomes veg*n for health reasons will be less committed. An attempt to understand the process of becoming vegetarian found that slightly more than half the subjects were vegetarian for ethical reasons, but "health vegetarians became increasingly aware of animal welfare issues and this reaffirmed the transition." Indeed, the initial ethical/health distinction seems to fade over time as ethical vegetarians become more interested in health, and vice versa.

Unspeakably more depends on what things are called, than on what they are. - Friedrich Nietzsche

It's critically important to consider here too the benefit gained from advocacy that is not "vegan advocacy." You probably have heard of pink slime, a filler used in ground beef. The public outcry sent beef prices plummeting, causing at least one producer to declare bankruptcy. Several lawsuits regarding E. Coli-infected beef caused Topp's Meat Company to file for Chapter 11 a few years ago. The Hallmark/Westland Meat Packing Company went bankrupt after an investigation by the Humane Society of the US caused the largest beef recall in history - not because of animal cruelty violations (which were horrendous), but because of health concerns.

Bruce Schneier has said that no one should be concerned by what's on the news - if it's newsworthy, it's by definition unusual, hence it almost certainly won't affect you. This is a fact which a lot of advocates seem to forget. Pink slime is probably no worse than any other type of meat, yet some combination of branding, luck and timing caused tremendous economic damage to the beef industry. Similarly, your chance of dying from E. Coli even during an "outbreak" compares favorably with that of being struck by lightning, yet we find massively expensive recalls happening on an almost weekly basis.

So we have to be extremely careful when evaluating things like the evidence that vegan diets help with long-term weight loss. They stack up pretty well when compared to the competition, but the fact that they aren't overwhelmingly better than anything else doesn't necessarily mean that the health argument fails veganism.

Maybe health benefits aren't the best way to present veganism. Certainly there is a subgroup of people that is more responsive to ethical arguments than health ones, and we have to be careful about change which moves people from one type of animal consumption to another (although the evidence seems to indicate that this is less of a problem than one might think). But I hope I've convinced you that this is not something which can be decided by navel-gazing - it needs to be decided empirically, by doing surveys, handing out pamphlets and measuring what works.

If you are interested in learning more about health-based arguments for veganism, PCRM is a good place to start.

### Don't "Raise Awareness"

In an early analysis of efforts to convince people to act more pro-environmentally, Burgess et al. present the following flowchart on how people's minds change:

It seems pretty straightforward. It's also completely wrong.

In a later metasurvey, Kollmuss and Agyeman say:
These models from the early 1970s were soon proven to be wrong. Research showed that in most cases, increases in knowledge and awareness did not lead to pro-environmental behavior. Yet today, most environmental Non-governmental Organisations (NGOs) still base their communication campaigns and strategies on the simplistic assumption that more knowledge will lead to more enlightened behavior.
Problems go even further. Kollmuss and Agyeman add that "quantitative research has shown that there is a discrepancy between attitude and behavior." Wong and Sheth agree, saying that the relationship between beliefs and behavior is generally found to be "low and nonsignificant."

25% of Americans tell pollsters that "Animals deserve the same rights as people," yet only 2% are vegan. Unless a quarter of Americans believe it's ok to torture humans to death for their flesh, that's a pretty big gap between beliefs and behavior.

Henry Spira, one of the most effective animal advocates of all time, noted this problem in his list of tips for advocates when he disparaged "raising awareness". It's very easy to convince ourselves that we're building "mindshare" even if people's behaviors don't change, but without the explicit measurement of the sort that EAA's Top Charities do we're probably just building castles in the air.

### Kill the Young People

Suppose you were forced to choose between killing someone today, and killing someone a century from now. Which would you choose?

To be clear: the two people are exactly the same - equally happy, healthy, etc. And the effects on others are the same, and there is no uncertainty involved. The only difference between the two murders is when they occur.

It seems hard to give a justification for why one is better than the other, and in the landmark Stern Review on Climate Change the eponymous Nicholas Stern said as much. In his interrogation by Parliament, he stated that to choose one over the other is to "discriminate between people by date of birth," a position that is "extremely hard to defend".

There is a lot of controversy about whether he made the right decision, mostly motivated by the fact that even slightly different decisions on how we "discount" the future can cause huge differences in how we respond to threats which will kill people in the future, like climate change.

A remarkable proof by Peter Diamond shows that, under some reasonable assumptions, we should indeed "discriminate by date of birth," and choose to kill the person a century from now.

## Diamond's Proof

The full proof (and several others) can be found in his paper The Evaluation of Infinite Utility Streams, but I'll present a simplified version here.

First, some notation. We denote welfare over time as a list, e.g. $(1,2,3)$ indicates that at time 1 all sentient persons have utility 1, and time 2 they have utility 2 and so forth. Because time is infinite, these lists are infinitely long. We denote infinite repetition with "rep", e.g. $1_{rep}$ is the list $(1,1,1,\dots)$. These lists are given variable names - I use $u,v$ for finite lists and $X,Y$ for infinite lists - and they are compared with the standard inequality symbols ($>,\geq$).

There are four assumptions:

1. If $u\geq v$ then $u_{rep} \geq v_{rep}$. I.e. if some finite list of utilities $u$ is better than some other finite list $v$, then repeating $u$ for all of eternity is better than repeating $v$ for all of eternity.
2. If $u\geq v$ then $(u,X)\geq (v,X)$. I.e. if $u$ is better than $v$, starting off the world with $u$ is better than starting things off with $v$, given that the rest of time is equal.
3. If $X\geq Y$ then $(u,X)\geq (u,Y)$. I.e. if some infinite state of affairs $X$ is better than $Y$, starting them both off with $u$ won't change that.
4. If $u$ is the same as $v$ except some people are better off (and no one is worse off), then $u > v$. (This is sometimes known as Pareto efficiency.)
Proof: The proof actually isn't that complicated, but it looks intimidating because the notation is probably unfamiliar.

Suppose, for the sake of contradiction, that $(1,2)_{rep}\geq (2,1)_{rep}$. By (A4), $(2,2,(1,2)_{rep}) > (2,1)_{rep}$ since $(2,2,(1,2)_{rep})$ is the same as $(1,2)_{rep}$, except with people being better off at $t = 1$. By rearrangement, $(2,2,(1,2)_{rep})=(2,(2,1)_{rep})$ and $(2,1)_{rep}=(2,(1,2)_{rep})$ so $(2,(2,1)_{rep}) > (2,(1,2)_{rep})$. But we had assumed that $(1,2)_{rep}\geq (2,1)_{rep}$ which by (A3) means that $(2,(1,2)_{rep}) \geq (2,(2,1)_{rep})$. We've reached a contradiction, meaning that $(1,2)_{rep} < (2,1)_{rep}$.

This means that $(1,2) < (2,1)$, for if the opposite were true, $(1,2)_{rep} \geq (2,1)_{rep}$ by (A1). Therefore, by (A2), $(1,2,X) < (2,1,X)$, meaning that if we could shift happiness from year two to year one, we should.

We should value the happiness of those born earlier more than those born later, and kill the person living a century from now.

## Discussion

My girlfriend pointed out to me that the reason why we interpret this theorem to mean that people born earlier matter more is because we assume time has a beginning but no end. If we assumed the opposite, then people born later would matter more.