### A Pure Math Argument for Total Utilitarianism

Addition is a very special operation. Despite the wide variety of esoteric mathematical objects known to us today, none of them have the basic desirable properties of grade-school arithmetic.

This fact was intuited by 19th century philosophers in the development of what we now call "total" utilitarianism. In this ethical system, we can assign each person a real number to indicate their welfare, and the value of an entire population is the sum of each individuals' welfare.

Using modern mathematics, we can now prove the intuition of Mills and Bentham: because addition is so special, any ethical system which is in a certain technical sense "reasonable" is equivalent to total utilitarianism.

### What do we mean by ethics?

The most basic premise is that we have some way of ordering individual lives.

We don't need to say how much better some life is than another, we just need to be able to put them in order. We might have some uncertainty as to which of two lives is better:

In this case, we aren't certain if "Medium" or "Medium 2" is better. However, we know they're both better than "Bad" and worse than "Good".

In the case when we always know which of two lives is better, we say that lives are totally ordered. If there is uncertainty, we say they are lattice ordered.

In either case, we require that the ranking remain consistent when we add people to the population. Here we add a person of "Medium" utility to each population:

The ranking on the right side of the figure above is legitimate because it keeps the order - if some life X is worse than Y, then (X + Medium) is still worse than (Y + Medium). This ranking below for example would fail that:

This ranking is inconsistent because it sometimes says that "Bad" is worse than "Medium" and other times says "Bad" is better than "Medium". A basic principle of ethics is that rankings should be consistent, and so rankings like the latter are excluded.

### Increasing population size

The most obvious way of defining an ethics of populations is to just take an ordering of individual lives and "glue them together" in an order-preserving way, like I did above. This generates what mathematicians would call the free group. (The only tricky part is that we need good and bad lives to "cancel out", something which I've talked about before.)

It turns out that merely gluing populations together in this way gives us a highly structured object known as a "lattice-ordered group". Here is a snippet of the resulting lattice:

This ranking is similar to what philosophers often call "Dominance" - if everyone in population P is better off than everyone in population Q, then P is better than Q. However, this is somewhat stronger - it allows us to compare populations of different sizes, something that the traditional dominance criterion doesn't let us do.

Let's take a minute to think about what we've done. Using only the fact that individuals' lives can be ordered and the requirement that population ethics respects this ordering in a certain technical sense, we've derived a robust population ethics, about which we can prove many interesting things.

### Getting to total utilitarianism

One obvious facet of the above ranking is that it's not total. For example, we don't know if "Very Good" is better than "Good, Good", i.e. if it's better to have welfare "spread out" across multiple people, or concentrated in one. This obviously prohibits us from claiming that we've derived total utilitarianism, because under that system we always know which is better.

However, we can still derive a form of total utilitarianism which is equivalent in a large set of scenarios. To do so, we need to use the idea of an embedding. This is merely a way of assigning each welfare level a number. Here is an example embedding:

• Medium = 1
• Good = 2
• Very Good = 3

Here's that same ordering, except I've tagged each population with the total "utility" resulting from that embedding:

This is clearly not identical to total utilitarianism - "Very Good" has a higher total utility than "Medium, Medium" but we don't know which is better, for example.

However, this ranking never disagrees with total utilitarianism - there is never a case where P is better than Q yet P has less total utility than Q.

Due to a surprising theorem of Holder which I have discussed before, as long as we disallow "infinitely good" populations, there is always some embedding like this. Thus, we can say that:
Total utilitarianism is the moral "baseline". There might be circumstances where we are uncertain whether or not P is better than Q, but if we are certain, then it must be that P has greater total utility than Q.

### An application

Here is one consequence of these results. Many people, including myself, have the intuition that inequality is bad. In fact, it is so bad that there are circumstances where increasing equality is good even if people are, on average, worse off.

If we accept the premises of this blog post, this intuition simply cannot be correct. If the inequitable society has greater total utility, it must be at least as good as the equitable one.

### Concluding remarks

There are certain restrictions we want the "addition" of a person to a population to obey. It turns out that there is only one way to obey them: by using grade school addition, i.e. total utilitarianism.

### Double Your Effectiveness with a Bunny Suit

I decided today that three leafletters were saturating the area, so after I ran out of my first stack I just started tallying the other two's success. One was in a bright blue bunny costume, and the other was more normally dressed.

The bunny won (p = .0008).

Accepted LeafletDeclined Leaflet
Bunny Suit2011
No suit1847
Contingency table of what fraction of people accepted a leaflet when offered.

Who would you take a leaflet from?

### A Graphical Introduction to Lattices

Here is my (extended) family tree:

Everyone in the tree shares at least one common ancestor and at least one common descendant. This makes my family tree a lattice, an important mathematical structure. While lattices are often presented in abstract algebraic form, they have a simple graphical representation called a Hasse diagram, which is similar to a family tree.

Because most lattice theory assumes a strong background in algebra, I think the results are not as well known as they should be. I hope to give a sampling of some lattices here, and a hint of their power.

### What are Lattices?

A lattice is a structure with two requirements:
1. Every two elements have a "least upper bound." In the example above, this is the "most recent common ancestor".
2. Every two elements have a "greatest lower bound." In the example above, this is the "oldest common descendant".
Note that the bound of some elements can be themselves; e.g. the most recent common ancestor of me and my mother is my mother.

Lattices are a natural way of describing partial orders, i.e. cases where we sometimes know which element came "first", but sometimes don't. For example, because the most recent common ancestor of my mother and myself is my mother, we know who came "first" - my mother must be older. Because the least upper bound of my mother and my father is some third person, we don't know which one is older.

### Shopping Carts

Here's an example of four different ways to fill your shopping cart:

The lines between two sets indicates preference: one apple is better than nothing, but one apple and one banana is even better than one apple. (Note that the arrows aren't directed, because every relation has a dual [e.g. the "better than" relation has a dual relation "worse than]. So whether you read the graph top-to-bottom or bottom-to-top, it doesn't really matter. By convention, things on the bottom are "less than" things on the top.)

Now, some people might prefer apples to bananas, and some might prefer bananas to apples, so we can't draw any lines between the "one apple" and the "one banana" situations. Nonetheless, we can still say that you prefer having both to just one, so this order is pretty universal.

The least upper bound in this case is "the worst shopping cart which is still preferred or equal to both things" (doesn't quite roll of the tongue, does it?), and the greatest lower bound is "the best shopping cart which is still worse than or equal to both things". Because these two operations exist, this means that shopping carts (or rather the goods that could be in shopping carts) make up a lattice.

A huge swath of economic and ethical problems deal with preferences which can be put into lattices like this, which makes lattice theory a powerful tool for solving these problems.

### Division

This is a more classical "math" lattice:

Here a line between two integers indicates that the lower one is a factor of the higher one. The least upper bound in this lattice is the least common multiple (lcm) and the greatest lower bound is the greatest common divisor (gcd, some people call this the "greatest common factor").

The greatest common divisor of 4 and 10 is 2, and the least common multiple of 2 and 3 is 6.

Again we don't have a total ordering - 2 isn't a factor of 3 or vice versa - but we can still say something about the order.

An important set of questions about lattices deal with operations which don't change the lattice structure. For example, $k\cdot\gcd(x,y)=\gcd(kx,ky)$, so multiplying by an integer "preserves" this lattice.

Multiplying the lattice by three still preserves the divisibility relation.

A lot of facts about gcd/lcm in integer lattices are true in all lattices; e.g. the fact that $x\cdot y=\gcd(x,y)\cdot \text{lcm}(x,y)$.

### Boolean Logic

Here is the simplest example of a lattice you'll probably ever see:

Suppose we describe this as saying "False is less than True". Then the operation AND becomes equivalent to the operation "min", and the operation OR becomes equivalent to the operation "max":
• A AND B = min{A, B}
• A OR B = max{A, B}
Note that this holds true of more elaborate equations, e.g. A AND (B OR C) = min{A, max{B, C}}. In fact, even more complicated Boolean algebras are lattices, so we can describe complex logical "gates" using the language of lattices.

I switch now from examples of lattices to a powerful theorem:
[Holder]: Every operation which preserves a lattice and doesn't use "incomparable" objects is equivalent to addition.1

The proof of this is fairly complicated, but there's a famous example which shows that multiplication is equivalent to addition: logarithms.

The relevant fact about logarithms is that $\log(x\cdot y)=\log(x)+\log(y)$, meaning that the problem of multiplying $x$ and $y$ can be reduced to the problem of adding their logarithms. Older readers will remember that this trick was used by slide rules before there were electronic calculators.

Holder's theorem shows that similar tricks exist for any lattice-preserving operation.

### Everything is a Set

Consider our division lattice from before (I've cut off a few numbers for simplicity):

Now replace each number with the set of all its factors:

We now have another lattice, where the relationship between each node is set inclusion. E.g. {2,1} is included in {4,2,1}, so there's a line between the two. You can see that we've made an equivalent lattice.

This holds true more generally: any lattice is equivalent to another lattice where the relationship is set inclusion.2

### Max and Min Revisited

Consider the following statements from various areas of math:
$$\begin{eqnarray} \max\{x,y\} & = & x & + & y & - & \min\{x,y\} &\text{ (Basic arithmetic)} \\ P(x\text{ OR } y) & = & P(x) & + & P(y) & - & P(x\text{ AND } y) & \text{ (Probability)} \\ I(x; y) & = & H(x) & + & H(y) & - & H(x,y) & \text{ (Information theory)} \\ \gcd(x,y) & = & x & \cdot & y & \div & \text{lcm}(x,y) & \text{ (Basic number theory)} \\ \end{eqnarray}$$When laid out like this, the similarities between these seemingly disconnected areas of math is obvious - these results all come from the basic lattice laws. It turns out that merely assuming a lattice-like structure for probability results in the sum, product and Bayes' rule of probability, giving an argument for the Bayesian interpretation of probability.

### Conclusion

The problem with abstract algebraic results is that they require an abstract algebraic explanation. I hope I've managed to give you a taste of how lattices can be used, without requiring too much background knowledge.

If you're interested in learning more: Most of what I know about lattices comes from Glass' Partially Ordered Groups, which is great if you're already familiar with group theory, but not so great otherwise. Rota's The Many Lives of Lattice Theory gives a more technical overview of lattices (as well as an overview of why everyone who doesn't like lattices is an idiot) and J.B. Nation has some good notes on lattice theory, both of which require slightly less background. Literature about specific uses of lattices, such as in computer science or logic, also exists.

Footnotes
1. Formally, every l-group with only trivial convex subgroups is l-isomorphic to a subgroup of the reals under addition. Holder technically proved this fact for ordered groups, not lattice-ordered groups, but it's an immediate consequence.
2. By "equivalent" I mean l-isomorphic.

### Why Inequality Can't Matter

A famous experiment of Hsee's asks people how much they would pay for two different sets of dishware:

Set ASet B
Dinner plates: 8, all in good condition 8, all in good condition
Soup/salad bowls: 8, all in good condition 8, all in good condition
Dessert plates: 8, all in good condition 8, all in good condition
Cups: 8, 2 of them are broken
Saucers: 8, 7 of them are broken

Note that Set A is a Pareto improvement over Set B - it has everything in Set B and some additional items as well. Therefore, people should be willing to pay at least as much for A as they are for B.

Nonetheless, people are willing to pay almost 50% more for B than for A. The explanation for this "less is better" result is that the "hard" question of finding the absolute value of the set is subconsciously replaced with the "easier" question of finding the relative value of each item in the set.

A similar phenomenon occurs in population ethics. Consider two populations:

Population APopulation B
Investment Bankers:100, very well off100, very well off
Secretaries:100, moderately well off

My guess is that Population A would raise more ire than Population B, even though A is a Pareto improvement over B. Suppose we require our population ethics to follow what is sometimes called "Dominance" or "Pareto Dominance":

If Population A and Population B differ by only one person, and that person is better off in A than in B, then A is better than B.

Note that this is a pretty weak condition: in real life, there will almost always be winners and losers to any policy change, so it's rare to be able to decide things based solely on the Pareto Dominance principle.

Despite being a weak condition, it rules out population ethics that value equality, diversity etc.

Consider an extreme example: we only care about inequality (as measured by say the Gini index). In the example above, Population A had more inequality (higher Gini index) and so it would be worse. But A was a Pareto improvement over B, so a contradiction arises; hence, the Gini index can't be the way we compare populations.

A more general version of this is true:

Suppose $(G,+)$ is a population ethics that obeys the group axioms and Pareto Dominance. Let's say there is also some function $f$ whereby if $pop_a$ and $pop_b$ differ by only one person $\Delta$ then $pop_a > pop_b$ if and only if $\Delta > f(pop_b)$, i.e. $f$ defines the minimum welfare needed for a person to "improve" the total value of the population.

Then $f$ is constant. Specifically, $f(x)=0$ for all $x$, where 0 is the identity of $G$.

In some ways, this is not a very surprising result - it just says that whether your life is good is independent of whether my life is good. But it seems to contradict a lot of things we believe as a society.

Proof: Arbitrarily choose some population $pop$ and consider $pop+f(pop)$, i.e. adding a person right on the "margin". There are two possibilities: $pop+f(pop) < pop$ (adding this person is a bad idea), or $pop+f(pop)=pop$ (adding the person doesn't matter).

Suppose that $pop+f(pop) < pop$. We know that there is some element $0$ such that $pop+0=pop$. If $0 < f(pop)$ then $pop+f(pop)$ is a Pareto improvement over $pop+0$, so $pop+0 < pop+f(pop) < pop$, which is a contradiction because $pop+0 = pop$. If $0 > f(pop)$ then by the definition of $f$, $pop+0 > pop$, another contradiction. Therefore $0=f(pop)$, proving the theorem in the first case.

Alternatively, suppose that $pop+f(pop)=pop$. This means that $f(pop)$ is an identity of $G$, and since identities in a group are unique, $f(pop)$ must be $0$.

Since $pop$ was chosen arbitrarily, we have shown this is true for all populations. QED.

### How to Create a Donor-Advised Fund

There are a lot of charities. So many, in fact, that some would-be altruists are struck with the alliterative analysis paralysis and end up not donating at all.

A tax vehicle known as a "Donor-advised fund" (DAF) allows you to get the best of both worlds: you can donate to charity, with all the psychological and tax benefits that go along with that decision, while still holding off on your decision as to which charity is best.

You can create your own DAF in about 15 minutes online. (Be sure to give it an awesome name like "The Jane Doe Fund for Paperclip Maximization" because how many chances do you have to name an organization after yourself?)  Once created, you can contribute money when you feel like it and deduct those contributions from your taxes. Your contributions will sit in an account accruing interest until you decide to write a grant to a specific charity.

There are interesting questions about when to put money in a DAF vs. donate directly, but if you are uncertain about the most effective charity, especially if you're so uncertain that you might not donate at all, you should create a DAF.