## 10 December, 2018

### Fastly Fast Growing Functions

In a previous post, I discussed Really Big Numbers, moving from many children's example of a big number, a million, up past what most people I meet would think of as a huge number, a googol, and ultimately going through Graham's number, TREE(3), the busy beaver function, infinities and beyond. I wasn't aware of it at the time, but a much better version of that post already existed: Who Can Name the Bigger Number?, by Scott Aaronson.

In my original post, I made a few errors in the section about fast growing functions. Some kind commentors helped correct the most egregious errors, but the ensuing corrections littered that entire section of the post with strikethrough text that I was never really happy with. Now, six years later, I'd like to finally make up for my mistakes.

### The Goal

I'd like to name some really, really big numbers. I'm not going to talk about the smaller ones, nor the ones that delve into infinities; you can read the previous post for that. Here I just want to point toward some really big finite numbers. The numbers I'm aiming for are counting numbers, like 1, 2, or a billion. They're not infinite in size. These are numbers where, if someone asked you to write a really, really big number, these would be way beyond what the questioner was thinking of, and yet still wouldn't be infinite in extent.

### Why Functions?

We always use functions when writing numbers. It's just that most of the time, it's invisible to us. If we're counting apples, we might make a hatch mark (or tally mark) for the first apple, another hatch for the second ("‖"), and so on. This works fine for up to a dozen apples or so, but it starts to get pretty difficult to understand at a glance. You might fix this by making every fifth hatch cross over the previous four ("卌"), but you quickly run into a problem again if you get too many sets of five hatch marks.

It's easier to come up with a better notation, like using numerals. Now we can use "1" or "5", rather than actually write out all those hatch marks. Then we can use a simple function to make our notation easier to follow. The rightmost numeral is the ones place, then next to the left is the tens place, and the next to the left is the hundreds place, and so on. So "123" means (1*100)+(2*10)+(3*1). Of course, I'm being loose with definitions here, as I've written "100" and "10" using the very system I'm trying to define. Feel to replace with tally marks: 2*10 is ‖*卌卌.

As you can see, functions are integral parts of any notation. So when I start turning to new notations by using functions to describe them, you shouldn't act as though this is somehow fundamentally different from the notations that you likely already use in everyday life. Using Knuth arrow notation is no less valid for saying a number's name than writing "123". They're both just names that point at a specific number of tally marks.

### Defining Operations

Let's start with addition. Addition is an operation, not a number. But it's easier to talk in terms of operations when you get to really big numbers, so I want to start here. We'll begin with a first approximation of a really big number: 123. In terms of addition, you might say it is 100+23, or maybe 61+62. Or you may want to break it down to its tally marks: 卌卌卌…卌⦀. This is all quite unwieldy, though. I'd prefer to save space when typing all this out. So let's instead use the relatively small example of 9, not 123. You might not think of 9 as a really big number, but we've only just started. The first function, F₁(x,y), involves taking the numeral X and doing whatever operation it is Y times. In this series of functions, I'm always going to use 3 for both x and y to make things as simple as possible. F₁ is addition, so F₁(3,3)=3+3+3=9.

Each subsequent function Fₓ is just a repetition of the previous function. Addition is repeated counting, but when you repeat addition, that's just multiplication. So our second operation, multiplication, can be looked at as F₂=3*3*3=27.

(As an aside, a similar function to Fₓ(3,2) can be seen at the On-Line Encyclopedia of Integer Sequences. Their a(n) is equivalent to our Fₓ(3,2), where x=n-1. So their a(2) is our F₁(3,2). You may also notice that F₂(3,2)=F₁(3,3),  so although the OEIS sequence A054871 is out of sync on the inputs, the series nevertheless matches what we're discussing here.)

I want to pause here to point out that multiplication grows more quickly than addition. Look at the first few terms of F₁:
• F₁(3,1)=3
• F₁(3,2)=3+3=6
• F₁(3,3)=3+3+3=9
Then compare to the first few terms of F₂:
• F₂(3,1)=3
• F₂(3,2)=3*3=9
• F₂(3,3)=3*3*3=27
What's important here isn't that 27>9. What's important is that the latter function is growing more quickly than the previous one.
We can keep going to F₃, which uses the exponentiation operation. This is as high as most high school math classes go. F₃=3^3^3=19683. The first few terms of F₃ are:
• F₃(3,1)=3
• F₃(3,2)=3^3=27
• F₃(3,3)=3^3^3=19683
You can see that each subsequent function is growing more and more quickly, such that the only the third term, Fₓ(3,3), is fast approaching really big numbers.

Next in the series is F₄, which uses tetration. F₄=3⇈3⇈3=7,625,597,484,987. Here I am using Knuth arrow notation for the operator symbol, but the idea is the same as all the previous operations. Addition is repeated counting. Multiplication is repeated addition. Exponentiation is repeated multiplication. Tetration is repeated exponentiation. In other words:
X*Y = X+X+…+X, where there are Y instances of X in this series.
In the case of F₂(3,2), 3*3=3+3+3.
• Exponentiation is repeated multiplication:
X^Y = X*X*…*X, where there are Y Xs.
3^3=3*3*3
• Tetration is repeated exponentiation:
X⇈Y = X^X^…^X, where there are Y Xs.
3⇈3=3^3^3
Pentation is next: F₅=3↑↑↑3↑↑↑3. It takes a bit of work to figure out this value in simpler terms.
• F₅=3↑↑↑3↑↑↑3
=3↑↑↑(3↑↑↑3)
=3↑↑↑(3⇈3⇈3)
=3↑↑↑(3⇈(3⇈3))
=3↑↑↑(3⇈(7,625,597,484,987))
Remember that tetration is repeated exponentiation, so the part in the parentheses there (3⇈7,625,597,484,987) is 3 raised to the 3 raised to the 3 raised to the 3…raised to the 3, where there are 7,625,597,484,987 instances of 3 in this power tower. The image to the right shows what I mean by a power tower: it's a^a^…^a. In our example, it's 3^3^…^3, with 7,625,597,484,987 threes. And this is just the part in the parentheses. You still have to take 3↑↑↑(N), where N is the huge power tower of threes. It's truly difficult to accurately describe just how big this number truly is.

### Fastly Fast

So far I've described the first few functions, F₁, F₂, F₃, F₄, and F₅. These are each associated with an operation. I could go on from pentation to hexation, but instead I want to focus on these increasingly fast growing functions. F₅(3,3) is already mindboggingly huge, so it's difficult to get across how huge F₆(3,3) is in comparison. Think about the speed at which we get to huge numbers from F₁ to F₂ to F₃, and then realize that this is nothing compared to where you get when you move to F₄. And again how this is absolutely and completely dwarfed by F₅. This happens yet again at F₆. It's not just much bigger. It's not just bigger than F₅ by the hugeness of F₅. It's not twice as big, or 100 times as big, nor even F₅ times as big. (After all, the word "times" denotes puny multiplication.) It's not F₅^F₅ even. Nor F₅⇈F₅. Nor even F₅↑↑↑F₅. No, F₆=3⇈⇈3⇈⇈3=3⇈⇈(F₅(3,3)). I literally cannot stress how freakishly massive this number is. And yet: it is just F₆.

This is why I wanted to focus on fast growing functions. Each subsequent function is MUCH bigger than the last, in such a way that the previous number basically approximates to zero. So imagine the size of the numbers as we move along to faster and faster growing functions.

These functions grow fast because they use recursion. Each subsequent function is doing what the last function did, but does it repeatedly. In our case, Fₓ(3,3) is just taking the previous value and using the next highest operator on it. F₂(3,3)=3*F₁(3,3). F₃(3,3)=3^F₂(3,3). F₄(3,3)=3⇈F₃(3,3). F₅(3,3)=3↑↑↑F₄(3,3). And as we saw two paragraphs ago, F₆(3,3)=3⇈⇈F₅(3,3).

I chose this recursive series of functions because I wanted to match up with the examples I used in my previous discussion of really big numbers. But most mathematicians use the fast growing hierarchy to describe this kind of thing. Think of it as a yardstick against which we can compare other fast growing functions.

### Fast Growing Hierarchy

We start with F₀(n)=n+1. This is a new function, unrelated to the multiple input function we've used earlier in this blog post. F₀(n) is the first rung of the fast growing hierarchy. If you want to consider a specific number associated with each rung of the hierarchy, we might choose n=3. So F₀(3)=3+1=4.

We then use recursion to define each subsequent function in the hierarchy. Fₓ₊₁(n)=Fₓ(Fₓ(…Fₓ(n)…)), where there are n instances of Fₓ.

So F₁(n)=F₀(F₀(…F₀(n)…)), with n F₀s. This is equivalent to n+1+1+…+1, where there are n 1s. This means F₁(n)=n+n=2n. In our example, F₁(3)=6.

Next is F₂(n)=F₁(F₁(…F₁(n)…)), with n F₁s. This is just 2*2*…*2*n, with n 2s. So F₂(n)=n2^n. In our example, F₂(3)=3*(2^3)=24.

At each step in the hierarchy, we roughly increase to the next level of operation each time. F₀ is basically addition; F₁ is multiplication; F₂ is exponentiation. It's not exact, but it's in the same ballpark. This corresponds closely to the function I defined earlier in this blog post. Mathematicians use the fast growing hierarchy to give an estimate of how big other functions are. My F₂(3,3) from earlier is roughly F₂(n) in the FGH. (F₂(3,3)=27, while F₂(3)=24.) (Egads, do I regret using F for both functions, even though it should be clear since one has multiple inputs.)

### Diagonalization

So at this point you probably get the gist of the fast growing hierarchy for F₂, F₃, F₆, etc. Even though they are mind-boggingly large numbers, you may be able to grasp what we mean we talk about F₉, or F₉₉. These functions grow faster and faster as you go along the series of functions, and there's an infinite number of functions in the list. We can talk about Fₓ with the subscript x being a googol, or 3↑↑↑3↑↑↑3. These functions grow fast. But we can do even better.

Let's define F𝜔(n) as Fn(n). (Forgive the lack of subscripts here; we're about to get complex on what's down there.) Now our input n is going to be used not just as the input in the function, but also as the FGH rank of a function that we already defined above. So, in our example, F𝜔(3)=F₃(3)=F₂(F₂(F₂(3)))=F₂(F₂(24))=F₂(24(2^24))=F₂(24(16777216))=F₂(402653184)= 402653184*(2^402653184)≈10^120000000.

As you can see, F𝜔(n) grows incredibly quickly. More quickly, in fact, than any integer value of Fₓ(n). This means that the sequence of functions I've been talking about previously in this blog post can't even get close to the fast growing F𝜔(n), even though there are infinite integer values you could plug in for Fₓ. An example of a famous function that grows at this level would be the Ackermann function.

But we can keep going. Consider F𝜔₊₁(n), which is defined exactly as we defined the FGH earlier. F𝜔₊₁(n)=F𝜔(F𝜔(…F𝜔(n)…)), where there are n F𝜔s. This grows faster than F𝜔(n) in a way that is exceedingly difficult to describe. Remember that each function in this sequence grows so much faster than the previous function so as to make it approximate zero for a given input. An example of a famous function that grows at this level would be Graham's function, of which Graham's number is oft cited as a particularly large number. In particular, F𝜔₊₁(64)>G₆₄.

There's no reason to stop now. We can do F𝜔₊₂(n) or F𝜔₊₆(n) or, in general, F𝜔₊ₐ(n), where a can be any natural number, as high as you might please. You can use a=googol or a=3↑↑↑3↑↑↑3 or even a=F𝜔(3↑↑↑3↑↑↑3). But none of these would be as large as if we introduced a new definition: F𝜔*₂(n)=F𝜔₊n(n). This is defined in exactly the same way that we originally defined F𝜔(n), where the input not only goes into the function, but also into the FGH rank of the function itself. F𝜔*₂(n) grows even faster than any F𝜔₊ₐ(n), regardless of what value you enter in as a.

I'm sure you see by now where this is going. We have F𝜔*₂₊₁(n) next, and so on and so forth, until we get F𝜔*₂₊ₐ(n), with an arbitrarily large a. Then we diagonalize again to get F𝜔*₃(n), and then the family of F𝜔*₃₊ₐ(n). This can on indefinitely, until we get to F𝜔*ₑ₊ₐ(n), where e can be arbitrarily large. A further diagonalization can then be used to create F𝜔*𝜔(n)=F𝜔²(n), which grows faster than F𝜔*ₑ₊ₐ(n) for any combination of e and a.

Yet F𝜔²(n) isn't a stopping point for us. Beyond F𝜔²₊₁(n) lies F𝜔²₊ₐ(n), beyond which is F𝜔²₊𝜔(n), beyond which is the F𝜔²₊𝜔₊ₐ(n) family, and so on, and so forth, past 𝜔²₊𝜔*₂(n), beyond 𝜔²₊𝜔*ₑ₊ₐ(n), all the way to F𝜔³(n). At each step, the functions grow so fast that they completely and utterly dwarf the function before it, and yet we've counted up several times to infinity in this sequence, an infinite number of times, and then did this three times in order to get to F𝜔³(n). These functions grow fast.

Still, there's more to consider. F𝜔³(n) is followed by F𝜔(n), all the way up to F𝜔(n), beyond which lies yet another digonalization to get to F𝜔^𝜔(n). From here, you can just redo all the above: F𝜔^𝜔₊ₐ(n) to F𝜔^𝜔₊𝜔₊ₐ(n) to F𝜔^𝜔₊₂𝜔₊ₐ(n) to F𝜔^𝜔₊ₑ𝜔₊ₐ(n) until we have to rediagonalize to F𝜔⇈𝜔(n), which we set equal to Fₑ₀(n) just for the purpose of making it easier to read. There are two famous examples of functions that grow at this level of the FGH: the function G(n) = "the length of the Goodstein sequence starting from n" and the function H(n) = "the maximum length of any Kirby-Paris hydra game starting from a hydra with n heads" are both at the FGH rank of Fₑ₀(n).

You can keep going, obviously. Tetration isn't the end for 𝜔. We can do Fₑ₀₊₁(n), then the whole family of Fₑ₀₊ₐ(n), followed by Fₑ₁(n). And we can keep going, to Fₑ₂(n) and beyond, increasing the exponent arbitrarily large, followed by Fₑ𝜔(n). And this ride just doesn't stop, because you go through the whole infinite sequence of infinite sequences of infinite sequences of infinite sequences of infinite sequences yet again, increasing the subscript of e to the absurd point of ε₀. And then we can repeat that, and repeat again, and again, infinitely many times, creating a subscript tower where ε has a subscript of ε to the subscript of ε to the subscript of ε to the suscript of… -- infinitely many times. At this point the notation gets too unwieldy yet again, so we move on to using another greek letter: 𝛇, where it starts all over again. And we can do this infinite recursion infinitely yet again, until we have a subscript tower of 𝛇s, after which we can call the next function in the series η.

Each Greek letter represents an absolutely humongous jump, from 𝜔 to ε to 𝛇 to η. But as you can see it gets increasingly complicated to talk about these FGH functions. Enter the Veblen Hierarchy.

### Veblen Hierarchy

The Veblen Hierarchy starts with 𝜙₀(a)=𝜔a, then increases with each subscript to a new greek letter from before. So:

• 𝜙₀(a)=𝜔a
• 𝜙₁(a)= εa
• 𝜙₂(a)= 𝛇a
• 𝜙₃(a)= ηa
This FGH grows much faster than the previous one, because it skips over all the infinite recursions to the final tetration of each greek letter, which it defines as the next greek letter in the series. The Veblen Hierarchy grows fast.

The subscript can get bigger and bigger, reaching 𝜙ₑ(a), where e is arbitrarily large. You can follow this by making 𝜔 the next subscript in the series, then follow the same recursive expansion as before until you get to 𝜔⇈𝜔, which we'd define as ε. And go through the greek letters, one by one, until you've gone through an infinite number of them, after which we can use 𝜙 as the subscript for 𝜙. Then do this again and again, nesting additional 𝜙 as the subscript for each 𝜙, until you have an infinite subscript tower of 𝜙, after which you have to substitute a new notation: Γ₀.

Here we finally reach a new limit. Γ₀ is as far as you can go by using recursion and diagonalization. It's the point at which we've recursed as much as we can recurse, and diagonalized as much as we can diagonalize.

But we can go further.

We can already see Γ₀ as 𝜙(a,0)=a. Let's extend Veblen function notation by defining 𝜙(1,0,0)=γ₀. Adding this extra variable let's us go beyond all the recursion and diagonalization we could do previously. Now we have all of that, and can just add 1.

Let's explore this sequence:
• γ₀=𝜙(1,0,0) Start here.
• γ₁=𝜙(1,0,1) Increment the last digit repeatedly.
• γ𝜔=𝜙(1,0,𝜔) Eventually you reach 𝜔.
After this, the next ordinal is 𝜙(1,1,0). As you can see, we have a new variable to work with. We can keep incrementing the right digit until we get to 𝜔 again, after which we reach 𝜙(1,2,0). And we can do this again and again, until we reach 𝜙(1,𝜔,0). Then the next ordinal would be 𝜙(2,0,0). And we can keep going, more and more until we get to 𝜙(𝜔,𝜔,𝜔). At this point, we're stuck again.

So now we have 𝜙(1,0,0,0) as the next ordinal. And we can max this out again until we need to add yet another variable, and then yet another variable, and so on, until we have infinite variables. This is called the Small Veblen Ordinal.

ψ(ΩΩω)=φ(1,0,,0ω)

Among FGH functions, the Small Veblen Ordinal ranks in just the lower attic of Cantor's Attic. It's not even the fastest growing function on the page it's listed on. We're nowhere near the top, despite all this work. Of course, there isn't a top -- not really. But what I mean is that we're nowhere near the top of what mathematicians talk about when they work with really large ordinals.

### …and Beyond!

You might notice that at no point did I mention TREE(3), which was one of the numbers I brought up in my last blog post. That's because the TREE() function is way beyond what I've written here. You have to keep climbing, adding new ways of getting to faster and faster growing functions before you reach anything like TREE(3). And beyond that to the point of absurdity is SSCG(3). And these are all still vastly beneath the Church Kleene Ordinal, which (despite being countable) is uncomputable. This is where you finally run into the Busy Beaver function. The distances between each of these functions that I've mentioned in this paragraph are absurdly long. It took this long to explain up to the Small Veblen Ordinal, and yet it would take equally long to get up to the TREE() function. And then just as long to get to SSCG(). And just as long to Busy Beaver.

I want to be clear: I'm not saying they are equal distances from each other. I'm saying that it would take an equal amount of time to explain them. At each step of my explanation, I've gotten to absurdly faster and faster growing functions, leaping from concept to concept more quickly than I had any right to. And I would explain that much faster if I kept going, using shorthand to handwave away huge jumps in logic. And yet it would still take that long to explain up to these points.

And I still wouldn't even be out of the lower attic, with the Church Kleene Ordinal.

If you want to keep going, you may be interested in this readable medium post by Josh Kerr, the absolutely beautifully written Who Can Name the Bigger Number? by Scott Aaronson, or the wiki at Cantor's Attic. Parts of this post were inspired by my own previous post on large numbers and a reddit post by PersonUsingAComputer. I'd also like to thank professor Edgar Bering and grad students Bo Waggoner and Charlie Cunningham for helping to correct errors in this essay.

### Slow Growing Functions

I'm a terrible amateur mathematician.  Sure, I watch Grant Sanderson's 3Blue1Brown videos for fun, but I never pause them to work out the math on my own. I participate in math forums occasionally, and every once in a while something I play around with gets some press (e.g., when I helped in a thread with Ed Pegg, Jr., and Laura Taalman with determining that the scutoid shape always has non-planar faces, for which Taalman's 3d print model was later popularized in a Matt Parker video), but to be honest, these are just nothing more than weird flexes. Beyond the thesis I wrote back in my school days about applying Gödel numbering to Aristotelian logic (which had no discernable practical applications), I haven't added anything novel to the field of mathematics at all.

Nevertheless, I love math. There's something about the way you can navigate its simple rules and come up with surprising results that makes me feel excited and full of genuine wonder. I enjoy board games and video games for much the same reason: I like to play around with rulesets and see what comes out. But mathematics has an unreasonable effectiveness when it comes to reality that few other invented systems have, so it occupies a special place in my heart.

Six years ago, I found myself talking with my friend Dale about extraordinarily large numbers. The conversation prompted me to write a short blog post on the topic. It was written just for my own enjoyment, but a number of better mathematicians than I got their hand on it and wrote a few discouraging words. One commenter in particular pointed out a few errors in the last few paragraphs of my post, and then, after I replied and edited my post, they wrote: "I'm sorry if I came off a little brusk and harsh. It's good that you're interested in this stuff and trying to learn more!" As a layperson, it felt simultaneously good and bad to read their comment. Good, because they're right: it is good that I'm trying to improve on this stuff. But also: Bad, because they're right: I'm just a nonmathematician writing another poorly written post on mathematics.

Anyway, the part of that past blog post where I was most confused was on fast growing functions. I not only explained what I knew poorly, but I also didn't fully understand the concepts behind those ideas. I really should not have included fast growing functions in that post, since it was not something I fully understood at the time, but it fit thematically and I really wanted to make the post thorough.

Now, I realize how much more important it is that all portions of a blog post are researched well enough to pass for at least acceptable to experts in whatever field it is. I've striven to ensure that even reddit posts I make in specialized subreddits are suitable enough so that experts in those fields wouldn't downvote me. It's a weird goal to have, not wanting experts to downvote me, but it's the best a layperson can strive for, I think. My contributions to r/philosophyofscience, r/boardgames, r/startrek, r/philosophy, amongst others, are examplars of what I aim to do in my everyday life: to know enough in each facet of life to not be a total idiot in it. My eventual aim of competence starts with a desire to function adequately, and slowly grow to more knowledge in each field as I can.

It sounds a bit silly when I put it this way. There are areas where I have a great deal of competence: effective animal advocacy, communications data analysis for organizations, knowing every nook and cranny of the worlds of balance and ruin in Final Fantasy VI. But for everything else, I just want to do well enough so that an expert in that field wouldn't laugh at me, and then I want to slowly build from there.

It's in that vein that I'd like to make up for the mistake I made six years ago. And so I present a short essay on fastly fast growing functions, written for a lay audience that's moderately comfortable with high school level mathematics.

## 12 November, 2018

### Great Harms

If I had to figure out the greatest harm I'd ever caused in my life, my initial thought would be from my first 23 years of life when I ate meat every single day. Cumulatively, this likely caused the direct suffering of several animals.

But I get strange looks for saying so. "Eric," my well-meaning adjudicator may say, "the animal is already dead by the time you eat it. How can buying one hamburger have any causal effect on how many cows die?"

It's true that it is unlikely that my purchasing a single hamburger from a restaurant will cause more animals to die. But that chance is not zero. Let's say that the restaurant buys hamburgers in lots of 100. Each week, they purchase as many lots as they expect to sell. This week, they purchased 10 lots, expecting to sell between 901 and 1000 hamburgers. They ended up selling 957; had I not eaten there that week, they would have instead sold 956. This is a clear example where my purchase affected nothing in terms of how many lots they buy each week.

But 1% of the time, I will be the purchase that goes over some threshold. In this naive example, let's say they sold 1000 burgers without me, and I was purchaser number 1001. In this case, in order to sell to me, they will need to purchase an additional lot of 100 hamburgers. In other words, 1 out of every 100 times that I buy a hamburger, it will cause the restaurant to purchase an additional lot of 100 hamburgers. On average, buying one hamburger causes one additional hamburger to be purchased by the store.

Of course, this simplified description isn't exactly what happens. Lots are bigger than this, and the threshold number of sales that causes them to purchase an additional lot is far less than the last hamburger they have for sale. Buying hamburger #957 may in fact be the trigger that causes them to keep 1100 hamburgers on hand rather than 1000. And we still have to consider who they're buying from; one additional lot may not necessarily cause the upstream company to then buy more meat -- you have to do a similar calculation there, and repeat up the chain until you get to the farmer who makes the choice of whether or not an additional cow should be raised and killed. And there's additional concerns of giving the restaurant some miniscule more buying power by making a small hamburger purchase, and elasticity also comes into play if you're really trying to figure out the truth. But, in the end, the math still works out the same. For every burger I eat, I cause approximately one burger worth of harm down the chain.

So, although it is unlikely that any individual meat purchase made on my behalf in my first 23 years caused any additional animals to die, it's probably the case that I was the threshold buyer at least a few times, causing many more lots to be purchased. In terms of expected value, I definitely caused a lot of harm.

That's a lot of suffering. But maybe not as much as I've caused from wasting energy. Sure, having the a/c on constantly doesn't seem like it could cause any great harm. But, in combination with every other wasteful American, the massive power waste has hastened us to a potential major climate change scenario. It's not likely that my actions caused any of what may potentially occur, but there is still a small chance that the overriding factor was me. To calculate expected value, I need to multiply this exceedingly small percentage by whatever harm comes about due to climate change. If it isn't more than moderately bad, then maybe this is not a big deal, since I'm multiplying by such a very, very small percentage. But if catastrophic climate change occurs, with massive amounts of suffering on par with the wildest predictions of video games or movies, then even my very small percentage will end up being the single largest contribution of harm that I've ever done in my life.

"But, Eric," my adjudicator exclaims, "how can you think of expected value calculations when you yourself have caused such direct harm so many times in your life!" If I'm judging the greatest harm I've caused in my life, then the adjudicator is correct: I shouldn't be too quick to dismiss direct harms.

As a young child, between ages 8 and 12, I would play games with my friends. In the course of physical play, I acted as a storyteller, allowing me to pause actions and rearrange conditions at will. So even though everyone used long thick staffs we found in the woods to hit each other as hard as we could, only I had the ability to escape injury by modifying the battle verbally. Those rocks that I threw as hard as I could: is there any word better suited to describe my actions than bully? It was consensual, but it was also direct physical harm to fellow children.

What about when I was mowing a yard, oblivious to my surroundings, only to suddenly see a squirt of blood -- a rabbit, dying in front of me, suffering due to my carelessness. Through the crying and terror I felt for making such a horrible mistake, I nevertheless was able enough to grab a shovel and end the poor creature's misery. Unintended harm is still harm, is it not?

In an early relationship at fifteen years old, feeling trapped, with anger for how I perceived my life to be ruined, I turned violent with my partner. What a stupid idiot I was back then for making such a choice. But I did. A part of me wants to clarify: it's nothing like what you see on tv. I never was violent enough to leave a mark. I never beat up, or slapped, or used some implement to cause harm. But I was emotionally abusive, and I used physical force to overpower. Just because abusers on tv appear worse does not make what I did okay. I was a stupid child.

In a few relationships after that, I was less harsh, but still not anywhere near being a nice guy. It took several years before I got over this terrible habit of my youth. Slowly, I learned to be better, but the learning came at the expense of those I dated between the ages of 15 and 23.

Later, as an adult, I learned how to have positive, happy relationships.  Occasionally, I still caused harm, but not of a violent type. One partner's father died. She had always spoke of him negatively, without love. I took her at her word, so when I had an existing trip planned and he passed away, it did not occur to me that I should have canceled the trip to be there for her. Instead, I left her alone. Another partner told me in advance what was acceptable and what was not, and mentioned that she had low willpower, so I should only do what was acceptable, even if in the moment she said otherwise. Later, in the moment, she said otherwise, and my willpower broke just as surely as hers did. Afterward, she considered it a violation, because she had verbally set boundaries in advance that both of us had broken. Shared causation of harm doesn't negate the harm that is caused.

But none of the above is why I now consider it my greatest harm. Instead, it is the fact that this particular person was especially affected by my actions. She ended up taking these events and magnifying them to the point where the harms negatively affected her life in major ways even ten years after they occurred. She has PTSD. She is unable to work. She has bouts of depression and has suicidal ideations stemming directly from the incidents that occurred ten years earlier with me. What seemed like a relatively minor example of overpowering to me was, to her, an event so intense that entire swaths of her current life revolve around memories of these abuses.

What I did to her was unacceptable. It was abusive and wrong. I should not have held her down in the way that I did, even though the alternative would have been to allow her to scratch me. I should not have stayed in a relationship with her for two years, knowing that this kind of thing was happening, even though it only came to a head every six months or so. I do not excuse any of it today, even though, at the time, I think I was trying to excuse it by contrasting how minor the events seemed compared to the actions I took when I was much younger.

In the end, it doesn't matter how 'terrible' what I did was, when judged by an outside observer. Rather, what matters is the level of suffering I inflicted by doing so. And in this particular case, the amount of suffering is overwhelming. Regardless of my intent, the harm suffered by her was (and still is) undeniably intense. Her everyday life is much, much worse than it would have been had she not been in a relationship with me. The only good to come out of it has been the lesson it taught me: to never allow any situation to even get close to a point where harm could occur again. I honestly believe that over the past decade, I've become a much better, kinder, and more fair person directly because of the experience I had with her.

Maybe it's just moral luck that the other people in my life that I harmed were strong enough to bounce back and lead happy lives. But when I evaluate the greatest harm I've caused in my life, I cannot help but to think of this one person, who suffers even today, ten years after the fact, from things I am now far removed from.

Expected value calculations just don't work in the face of utility monsters. My greatest harm is determined more by the person that was harmed than by the actions I used to cause the harm in the first place. This makes me feel conflicted, but, most of all, I just feel shitty.

## 18 October, 2018

### Enjoyment Through Attention and Meaning

I've been making some errors lately.

I valued content consumption too highly. Listened to too many podcasts, read too many articles, browsed reddit for too many hours. Describing me as time-poor would have been an understatement. It's not that I didn't find the activities worthwhile; to the contrary, I love board game night. Playing my Switch is always a highlight of my day. The new tastes I experience after trolling veg*n cooking videos on youtube are definitely worth it. But when you put it all together, it's just too much.

This week, I started walking between 2–4 a.m. every other day, and one key decision has made it more worthwhile than I could have expected: I left behind my headphones.

I've walked for years. That part isn't new. It was always between 1–2 hours each weekday in half hour increments in downtown DC, or VA, or wherever I happened to be working that day. But I always would listen to podcasts. I felt like I was wasting my time if I wasn't learning something new. I subscribed to podcasts on philosophy, news, economics, effective altruism, rationalism, gaming, history, etc. The more content I consumed, the better I felt. And on the rare occasions that I didn't feel like focusing on something deep, I would put on a lighter podcast or listen to orchestral music. If I was walking, my headphones were on.

But I no longer work in DC. I left my position at ACE last month and I have nowhere regular that I need to walk to anymore. I'd gone from time-poor to time-sufficient. So when I decided to start walking in my local area earlier this week, I didn't feel the pressure to bring my headphones. Instead, I merely listened to the rustling of the leaves for a couple of hours.

Tonight, I trekked to Little Seneca Creek in silence. It was glorious. The night air was chill, but invigorating. The deer I saw today disappeared between the trees when I stepped on a twig. The fish weren't visible, but I could hear them playing as they leapt from the water's surface. It felt good.

Black Hill Park is beautiful, and I feel so lucky to live within walking distance.

I must say that I've been thinking about this for a while. Too often I've found myself watching a sub-top-tier tv show while playing on my phone. Or reading about politics while StarCraft matches are going on in the background. Others have also brought this up. CGP Grey posted a video about this a few weeks back that's well-worth watching. (Seriously: if you only click one link in this post, it should either be CGP Grey's video or John Michael Greer's blog post in the penultimate paragraph.Day[9] talked about related issues on stream a few days ago. And when Dorek was still trying to convince me to visit Europe with him, he pounded on this point repeatedly:

 At 36, I finally learned what a glacier was.Hint: It's not the equivalent of an iceberg but on land.
"There's something different about just going out in nature and experiencing it. You get something out of climbing the parapets of a thousand year old wall that isn't he same as just looking up the wall on wikipedia." Dorek described the feeling of backpacking across Australia, looking up at a multitude of stars when no one else was around for many dozens of miles. He reminded me of the peace we felt when we would go to Dauphin Island, walk through the bird sanctuary, and relax on the deserted beach just next door to the sea lab. He'd walk west; I'd walk east. Then we'd sit, too far away from each other to talk or see one another, and wait for the sunrise.

Last year, I went with him to Europe. We saw castles and towns and people and food. But the best part was nature. He took me to a mountain in the Swiss alps overlooking the Aletsch glacier. He pointed me toward an easy peak to climb, then left to climb a tougher peak just nearby. Making it to the top of that peak by myself felt really, really good.

I think the disconnect here has to do with a difference between the way I think and the way some others that I know think. There's a feeling that people sometimes get where the essence of an event or storied thing somehow imbues the physicality of an object. It's the reason why houses where murders take place are avoided by some people. The Smithsonian castle has (what I think is) a really weak exhibit of just bits of random stuff that Americans donated from world events. Like a scrap of cloth from the bedsheet Lincoln died upon. Or a miniscule piece of rock from the Berlin wall. (I was going to link a partially examined life episode on this kind of thing, but I can't recall when it was discussed there.) These things bore me. I don't feel anything special by actually being in a place that I wouldn't otherwise feel from learning about the place. I don't have a feeling that events imbue something extra into physical locations or objects.

Don't get me wrong; I want those places to exist. I want historical locations preserved. I want to be able to visit Auschwitz-Birkenau. But I feel like a proper meditation on the events from reading in a classroom or home setting can be just as powerful an experience. Others disagree strongly with this sentiment.

John Michael Greer wrote a great piece on the next ten billion years. It's worth reading, and I won't spoil the story. But when you finish it, you should also definitely read his next post, which points out a great divide between two camps of readers. You may know what side I fell on after having read this blog post.

I really want to recreate the experience I had at the top of that mountain more often. Walking without headphones at 3 a.m. to the local creek is not nearly as provocative, but it is similar enough to make me feel good. Now I just need to look into purchasing a foldable kayak.

## 16 October, 2018

### Night Walks

2 a.m.
The night air relaxes me as I walk through my neighborhood. I always thought of suburban areas like this as still being more city than country, yet I am accompanied on my sidewalk trek by two wild deer, who pause every dozen feet to turn and look at me. Eventually they continue forward while I make a left turn. It's calm. My neighbors are mostly asleep.

Two days ago, we had to switch from using the air conditioner to the heater; I'm not quite comfortable with the change yet. The house smells different when heated. I miss the cold air. I will always prefer cold surroundings, so long as I can warm up in my own way, using electric blankets or exerted energy in a small space. My favorite was when I lived in Colorado: anytime it would snow, my first instinct would be to go outside and sit in the hot tub. Something about having a toasty warm body under the water while snowflakes gathered on my eyelashes and glasses always made me feel comforted.

This is why the walks outside feel so good this late at night. The air is cool. The breeze occasionally makes me shudder. But it is such a positive feeling to get my warmth from exercise while the air cools my body.

I also like the feeling of being alone. Walking without a phone. No podcast on. Just the deer to accompany me. It's peaceful.

## 14 September, 2018

### Signaling Versus Habit

If Robin Hanson is to be believed, then most everything we do relates back to signaling in one form or another. Whether it's the clothes I choose to wear in public, the way I speak to others, or even the rationalist essays I read to supposedly better myself, if how I proceed carries a cost, then signaling must be a part of it.

Yesterday, I carried a backpack from the car into my house. To my knowledge, no one was looking in my direction, nor was even present to look. I used one shoulder strap, not two.

I find this interesting because when I was a kid, using only a single shoulder strap on your backpack was considered the "cool" thing to do. I don't think anyone really thought of it in those terms, of course; you weren't exactly called out for being uncool if you used both straps. But in southern Alabama where I grew up as a teeenager, it was just known that you only used one shoulder strap when wearing a backpack.

I haven't used a backpack in years. When I do see backpacks these days, they're generally carried by children in school. Here, in the Maryland suburbs of DC, all the kids know you're supposed to use both straps when wearing a backpack. It's not an explicit knowledge, mind you. When I query kids between the ages of 10 and 18 about this sort of thing, they claim to have never really thought about it until I asked them directly. Yet it's nevertheless a shared knowledge between all that using both straps is the way you're supposed to do it.

So yesterday, I have the task of taking a backpack from the car to the house. Without thinking, and without any intended reason, I used a single strap while carrying it. As I was walking, I had the thought to myself: This backpack is a little heavy; it would have been better for me to use both straps. But I was halfway to the door by then and didn't bother changing how I carried it.

I think that I used only a single strap out of habit, and not for any signaling reason. Except it can't really be habitual, since I haven't carried a backpack in years. Could it be a remembrance, then, of an old established signaling guide that no longer functions? After all, current people who use backpacks think of using two straps as the cool thing to do, even if they aren't able to explicitly say this without being guided first. Am I signaling my tribe unintentionally -- a tribe of mid-eighties era southern Alabama kids that have long since grown up and none of which could possibly have seen me signal?

I don't like the signaling model because it makes me feel like shit. When I needed high quality headphones that could block out ambient sound while I took the metro, I bought Bose QC35 noise-canceling bluetooth headphones. My intention was to buy something which sounded good for podcasts (my main use of the headphones), which didn't have tangling cords, and which would allow me to understand the spoken word even when going through the noisiest portions of the DC metro. Bose was a relatively expensive brand, but the headphones hit all the notes I cared about, so I purchased them.

Before purchasing, I thought about two signaling issues. First, among non-audiophiles, Bose is seen as an expensive high-status brand, especially when the headphones are wireless. Second, among people who actually use high-quality headphones, Bose is seen an overly-expensive rip-off brand with only medium-quality sound, especially when the headphones are wireless. I remember thinking: Do I care that most people will fall into one of these two groups and judge me by my wearing of this product? My conclusion at the time was that it didn't matter, thinking about signaling was a waste of time, and I just wanted the best product for the features I desired, which ended up being the QC35s. I bought them and was very happy with them for years, even though I think it increased the rate at which people asked me for money during my commute. The use of them was great, but the fact that I had to think about signaling was not.

I have a partner, Katherine, who has an unconventionally large body size. When we first met, I connected with her online and found myself fascinated by her personality, charm, wit, and humor; we've since grown much closer together. My honest inner thoughts are that signaling has nothing whatsoever to do with how our friendship has grown, but, just as with the example of using a single shoulder strap on that backpack, it doesn't have to be consciously intentional in order for it to be true.

Size acceptance isn't exactly popular in society today, so obviously I can't be signaling for status among the general public by associating myself with Katherine. And I don't really associate with a specific tribe that specifically extols size diversity, so I can't be signaling to them. But what if I'm countersignaling, showing that I'm beyond base prejudice by associating with someone who is fat? But this leaves out counter-countersignaling, where you need not bother showing that you're beyond that base prejudice by needlessly associating with someone of size.

I'm sure you can see now why thinking about signaling makes me feel like shit. Katherine is a person, to think of her in this way just feels wrong. And, to be fair, I'm not thinking of her in this way at all -- to the extent that there may be signaling going on with her, it is all unconscious, undesired, and unnoticed. On the inside, it feels like I just found someone I clicked with (or is it "cliqued"?) and we grew closer over time. But taking the Hansonian view makes me have to think about signaling (or countersignaling) or the only real alternative that explains my costly actions: habit.

What I want is to not even care about this sort of thing. But that's a kind of signaling all in itself, so it helps not at all to think of it that way. I'm left, then, with habit -- but that sounds almost as bad as signaling. I may not be a deontologist, but it still makes me feel like shit to think about how I may be using people as a means, rather than as an end. I'm reminded of how easily I continue to fall into the tropes of "helping" women, despite my best efforts. In the poly community, it's not even standard for the male to pay for meals at dates anymore, yet anytime I've dated, I end up doing just that. In my mind, it isn't sexist; it's just a factor of my earning more money than the usual person I'd date. But the end result is the same: I pay for meals. Combine this with the habit (or is signal?) from my southern upbringing of holding open doors, using overly polite language, etc., and I can't help but to feel like I'm just not acting correctly in these types of situations. I feel like I need to manually insert a bias against "helping" in order to correct for my maybe-prejudiced norms (read: habits).

The main takeaway that I have from all this is that if I treat how I act as a series of habits, then I feel like I can correct for it through inserting manual bias. But if it's all really signaling, then I don't really understand how I could even start to correct for it. This is because habits deal with the actions you take, whereas signaling deals with the desires behind the actions you take. So I'm going to treat how I act as the results of a series of habits, rather than as signals -- regardless of whether the Hansonian view is accurate. (Which means, perhaps ironically, that I'm not valuing love strongly enough through the reality of signaling being a prime motivator. As Hanson tries to relate: "I love people, even if I don’t think they are as good as they like to let on." Whether you view this as me being a failure or of me countersignaling, I don't really care.)

The above paragraphs meander a bit too much for this to really be a publishable essay, but in the interest of trying to avoid making decisions due to signaling, I'm going to publish anyway.

## 04 September, 2018

### A Vacation Review

Following my final day at Animal Charity Evaluators back in August, I enjoyed a much-needed three-week-long vacation. I house-sat with Osi (the family dog) for a week, taking him to a local golf course for fun; we went out to two plays at the Kennedy Center; I got a chance to play way too many board games and video games with people I love and care about; and I even went on a trip to New York City, the highlight of which was the Nintendo New York store. Plus, my sister was with me for part of my vacation, which is a big deal since I only get to see her twice each year.

Admittedly, it was a little strange taking a vacation so soon after having taken another vacation (back in July for three weeks), but it just works out that way sometimes. It's not like I could choose when Dice Tower Con is held, nor could I really choose which date I could see Hamilton, due to the difficulty of obtaining tickets.

### Dice Tower Con

 Dice Tower Con in Orlando, Florida.
In July, I and several of my friends went to Dice Tower Con in Orlando, Florida. It was, as usual, extremely awesome. Conventions in general are awesome for introverts like me. Sure, each night I always need to go be alone for several hours in order to recharge for the massive influx of interacting with people each day, but other people do something similar, too (when they sleep), so it's not that different from most other peoples' experiences. At Dice Tower Con, the great part is the ease with which you can get games going with strangers. They have signs you can put up at your table to invite others to join, and it tends to work pretty well. The only poor experience I had there was a poorly run "escape room", which was inaccessible to a handicapped person in our party and for which the main devices had been broken by earlier participants. Also, despite being advertised as an escape room, it was more of an escape box, containing electronics that did not work. I especially hate the fact that in a paragraph about an otherwise awesome Dice Tower Con experience, the 'escape room' was so bad that my complaining about it takes up half the paragraph.

### The Color Purple

 The Color Purple in the Eisenhower Theater on August 21.
The Kennedy Center production of The Color Purple was great. As usual, we sat in what I consider to be the best possible seats (box seats, two to the side from the center, in the very front), and I did my usual (which my partners sometimes chastise me for) of relaxing with shoes off and a drink at my side. I never drink during the play, mind you; but it's nice to have a water bottle at hand during intermission. But I still get quite a bit of flak for walking outside the box seat without shoes. I'm told that it's "inappropriate", but honestly: when you pay extra for nice seats, doesn't that at least warrant an allowance for feet comfort on the carpeted floor?

 The Color Purple. Photo by Matthew Murphy.
Beforehand, I had a wonderful salad at the KC Café; it's in all seriousness usually the best place to get salad that I go to all year, despite the fact that it's a café in the Kennedy Center itself. But despite being awesomely good, it's still massively overpriced. And this year it took second place to a salad I had at Sweet Tomatoes in Orlando back in July for something like a third of the price. Nevertheless, one thing I really like about play days is the salad beforehand. (c:

As plays go, I'd seen better. Maybe I'm biased, but it seems like the productions done in the Opera House at the Kennedy Center are almost always more enjoyable than the ones that are shown in the Eisenhower Theater. Nevertheless, I really enjoyed it because I went in blind. Watching the story unfold in person was a great experience, and the actors and musicians both did an excellent job. The production was too sparse for my tastes, though. It was literally a 50 foot backdrop of nothing but a wall from chairs were suspended; the actors then used these chairs as props for every action they took. Time to stand on a hill: stand on a chair. Time to harvest the crops: rotate the chair sideways by 90°. Time to crack a whip? Okay, they used a real whip that time. But still: nothing but chairs and a whip for props.

I was told afterward that John Doyle (the director/designer) deliberately chose to do a minimalist production. While I'm sure that more cultured audience-goers than I fully enjoyed the minimalism, to someone like me who is relatively new to theater productions, I really tend to appreciate the costume changes and set designs much more.

 From The Partially Examined Life.
While we're on the topic, I'd like to bring up Pierre Bourdieu's Distinction: A Social Critique of the Judgment of Taste. I've never directly read Bourdieu's work, but I did go moderately deep into his ideas through discussions with friends during a listen of The Partially Examined Life podcast episode on Bourdieu. The idea is that our tastes in art reflects our social position. In other words, my negative personal experience of John Doyle’s minimalist production is reflective of my 'middle-brow' vantage point. Not because my social position fails to allow me to experience the 'true' version of art on display here, but because humans' need for social strata creates an unconscious difference in how we perceive art. I'm not sure I agree with him (I'm generally skeptical of "just so" arguments that attempt to explain already existing data without presenting falsifiable claims), but I do find the ideas fascinating, especially after I found out that others who saw the play with me actually enjoyed the minimalist production unironically. For me, it was clearly the worst part of the show.

### Hamilton

The very next day, we saw Hamilton. It was absolutely mind-blowing.

I'm not sure how exactly to put this. Generally when people review a play, they use words that talk about how good that play was in comparison to the usual reference class of other shows that are also called great. So earlier in this blog post, when I wrote that the actors and the musicians in The Color Purple both did an excellent job, what I meant was that, in a field of other high-quality plays, where the usual compliment is that they are excellent, this, too, should be considered excellent. In other words, it was expectedly good. The best expectedly good play I've seen was The Book of Mormon, and it was heads and shoulders above any other that I'd seen before it. It was so good that I actively sought to have others see it, despite its media being an art form that requires consumers to go through several hoops just to see it: you have to spend a relatively expensive amount for tickets, actually drive to a theater in a city big enough to get the play to show, and you have to do this during a time period not necessarily of your choosing; and then to top it off you have to make it a big part of your evening in order to fully appreciate it. That's a lot to ask for when your alternatives are movies that you can just watch at home while eating dinner, or reading a book that you can literally pick up anytime to enjoy, or games which allow you to interact with friends while enjoying the media. A play has to overcome a lot in order to get me to endorse it, like The Book of Mormon did.

But Hamilton was quite different. It wasn't just a head and shoulders above the rest. Rather, tt may very likely have literally ruined all future play performances for me. It was utterly enjoyable on all kinds of levels.

First: the instrumental music. This music is good. Themes are introduced and reinserted at appropriate times, pushing the story forward appropriately. It isn't just a catchy tune that a play was then written to make sense of. It isn't just a well orchestrated song that sits beside the story. Instead, the music pushes the story forward in the same way that good rpg video game music does. In early video games, voice acting wasn't a thing. So the music had to be a prime actor that informed the player of what was going on. In Final Fantasy VI, Nobuo Uematsu's songs had to immediately tell you the prime characters in a scene, firstly by reusing their theme; but then they also had to combine the themes of disparate characters that interact in a single scene to indicate where the story was headed. Doing this kind of work was horribly difficult in those days. Yasunori Mitsuda literally worked himself into the hospital during his composition work for Chrono Trigger. So to hear this style of story-driven instrumental music in a musical for the first time was absolutely amazing to me.

Add to this the lyrics. Lin-Manuel Miranda used various styles of rap for each character to tell their story, and did so in ways that I've never seen in any previous musical. (See the genius lyrics annotations for details.) Phrases came into play early with one meaning, then get re-used later (alongside their musical accompaniment) with a different meaning for the phrase. This then happens several times, which reminds me heavily of reference-laden novels (in the James Joyce style, not the Ernest Cline or Cory Doctorow style of heavy referencing). For example: "not throwing away one's shot" refers not just to social-climbing, but also drinking alcohol in small glasses, telling his soldiers to literally empty their guns so as to be able to walk more quietly through the night, getting the woman he likes to be with him, remaining quiet about revolution (and later, about politics) in public so you can achieve what you want in public, and, most famously, about throwing away one's shot in a duel. This is just one example of wordplay that goes on throughout the musical. There are also themes on (for one example) 'being satisfied' that are woven throughout, like when Laurens' shot satisfies him, where the music interweaves the themes for "throwing away a shot" and "being satisfied" in a jerky 7/8 time signature. The way that this musical is written makes me repeatedly feel as though I need to watch it again and again to catch more, in an Arrested Development sort of way.

The dance is perhaps my least favorite portion, because it sprinkles in references that are impossible to understand without reading background material. For example, "throwing away one's shot" uses a baseball pitching motion immediately before hand, then a gun pointed upward afterward. One of these I could get from watching; the other I can't imagine anyone getting just by watching them dance. This motif is then repeated when combining references, like "being satisfied" with a baseball pitch motion. Keep in mind that the dance moves are too quick to really see them like this, but if you watch supplementary material, they show the moves in slow motion and explicitly call out what each is meant to represent. So although I like the dancing the least, it's because it uses Cline/Doctorow style referencing, which (to be honest) isn't that bad. It's just not as good as the lyrics using Joyce-style referencing.

The production is *amazing*. Sound is recorded in real time during the show and then played back to make an echo effect. Costumes are elaborate when they need to be and sparse when it makes more sense for the story. The set moves and flows with the story; people move into positions that make sense story-wise (not just for dance purposes), and the props are excellent. The floor moves to allow stationary acting to appear dynamic (it is a show about someone famous for being a writer, after all). And the lighting is on point. Seriously, some of the lighting patterns made the floor look as though a carpet had been laid down, and this one scene where the actors freeze time, then rewind it -- that was breathtakingly choreographed between the dancers, the sound designer, and the lighting person in a way that makes it just feel right. While the lyrics were the best part of the show for me, I nevertheless really need to call out the production team for doing a far better job at lighting, sound, set, and costumes than literally any other play or musical I've ever seen. It just worked.

Hamilton was so good that I want to encourage anyone reading this to take the time and expense required to go see it. It's definitely worth it.

### New York City

 Natalia and Eric at Vegetarian Dim Sum.
Appropriately, not long after seeing Hamilton we went to visit Manhattan. It was my sister's first time, so we did some of the touristy type stuff, including visiting the 9/11 memorial and taking a ferry to see the Statue of Liberty. But in between walking through Times Square and seeing bull statues and too old graveyards, we also got a chance to eat at an all-vegetarian restaurant in Chinatown that Robin recommended to us: Vegetarian Dim Sum. And we were able to go shopping!

This was exciting to me because my previous visits to NYC have all been about going to places that you see on tv. In other words: boring. I don't really get much out of being at the top of the Empire State building, other than perhaps it being an opening for me to talk about the concept of Schelling points to my traveling companions. And seeing the Rockefeller Center in person is just utterly wasted on me, despite having moderately enjoyed 30 Rock. Places just don't interest me as much as ideas. Touching a piece of the Berlin wall doesn't make me feel any differently from touching a random piece of rock, if you take away the story aspect of it. So NYC, for me, would be an exceptionally boring place if not for the chance to go to retail stores that have stuff I care about. And so my mini-shopping tour of NYC commenced.

First: the Disney Store. Two floors of Disney paraphilia. It was…interesting, I guess. There were all kinds of shirts, but all of them were kid-sized, and none had anything I might ever want to wear on it. I like The Nightmare Before Christmas, but not enough to buy anything associated with it. Keep in mind that my office is full of plushes from My Little Pony, Kiki's Delivery Service, Final Fantasy, etc., so it's not as though I wouldn't want to buy something if it caught my eye. But all the Disney characters I saw were just too much aimed at children. Upstairs, the Star Wars section sold nothing but toys for kids and clothes for kids. The same was true for Marvel.  I really thought I'd find something, but instead the only thing that held my attention for more than ten seconds in the entire store was a too expensive and too large Star Wars Lego set. I left disappointed.

 Nintendo NY. Photo by Gustavo Camargo.
Next was the Nintendo New York store. I'd seen video of it before, so I knew what to expect in advance. With expectations set, it was a great experience. I loved seeing the shirts (though I wasn't as happy with the limited size options), and the museum pieces were pretty cool. I guess it was a little weird, as everything behind the glass was just stuff that I'm already quite familiar with. I played with the Virtual Boy when it came out, I owned all the old systems, my collection of amiibo is just about as comprehensive as theirs -- even the oldest thing on display, hanafuda cards from before Nintendo made video games, wasn't that impressive given that I had literally handled a deck of those cards a month earlier in Orlando, when my friend showed them off after purchasing a deck in Japan. It felt weird to see all this stuff behind glass that just literally sits in boxes in the corner of my house. But I thoroughly enjoyed myself anyway! Lots of great plush characters, nice journals for sale, and great shirts of all types. I came away with only two purchases, but this was after lots of internal debate about getting more. (No jewelry, unfortunately. My gift plans were ruined due to no jewelry.)

 Tree of Creativity in Denmark Lego House.
The final store I visited was the Lego Store. I've watched videos of the LEGO House in Billund, Denmark, and have been fascinated by stories from the architect, tours by fans, and the large variety of stuff on display there. So when I saw a two story Lego store in the Rockefeller Center, I anticipated seeing great things. But from the moment I entered, my excitement was tempered way down. Sure, they had a lot of bricks and sets for sale, and there were two or three moderately sized creations on display, using 30,000 or so bricks. They even had a dragon that snaked through the ceiling that used maybe double that number. But my youtube experience of the Billund house had me expecting things like their three Dinosaurs (~253,274 pieces each) or their Tree of Creativity (6+ million bricks), so I came away quite underwhelmed. In retrospect, I should have expected this. I was expecting New York City to be the site of each company's grandest store, where they could really show off all that they have available. Instead, they all had to limit their ability to showcase impressive stuff due to the high cost of space in Manhattan proper. If all I wanted was to just buy something, I could have used Amazon instead. I should have realized: physical stores just aren't my thing.

 From left to right: duplo, standard, technic. They're each roaring because they've stepped on a lego brick. Both photos are by Lego; I can't find the photographer's name.

### In Conclusion

There were several ups and downs during my vacation(s) this year. I didn't even get a chance to talk about playing six player rocket league, nor doing several board game nights (one of which had 8+ hours of straight board games), nor how I was able to dogsit Osi while binge-watching foreigners finally win in StarCraft for the first time, nor even just the pleasant time of being able to hang out with my sister and do nothing much whatsoever. We had friends come in from Malaysia that I hadn't seen for three years. We stayed in an awesome hotel in Savannah. We even went to Myrtle Beach. There were just too many good times to really name during July and August.

But now I'm ready to move on from vacationing and toward the next phase of my career. My time at Animal Charity Evaluators was great, and taking two months to travel around the country and do fun stuff was enjoyable, but I really do look forward to seeing what happens next with my career in effective altruism. (c:

## 13 June, 2018

### Effective Advertising and Animal Charity Evaluators

This entry was originally posted on Effective-Altruism.com. It is reposted here for reference only.

[Summary: Animal Charity Evaluators wants to address feedback that we've heard from the effective altruism community regarding our online marketing practices. Although we follow best practices in the advertising industry, some EAs feel that we sometimes use advertising which glosses over details and is potentially misleading to the public. As the communications data scientist for ACE, I explain why we feel these advertising practices are not only net beneficial to the EA community, but also should not be considered to be misleading.]
Our mission at ACE is to find and promote the most effective ways to help animals. One of the ways in which the promotion part of our mission is fulfilled most effectively is through reaching those who are most passionate about helping animals but who have not yet been introduced to the concept of effective altruism.
While some of our ads are aimed at existing EAs, the majority of our advertising efforts focus on the above audience for three reasons: (1) the large scale of this audience compared with the size of the EA audience, (2) our belief that they have the most potential for change once they learn about effective animal advocacy (EAA), and (3) the fact that their counterfactual donations seem likely to be less impactful than those of EAs.
We have received some feedback on a few of these marketing practices—specifically, we have received feedback suggesting that we might not be advertising in a way that the EA community would most like to see. I will go over a few such examples before sharing why we use our current methods of advertising.

## Double Donations

In late 2017, we held a donation-matching campaign. A generous donor offered to match any donations we received to the ACE Recommended Charity Fund. This Fund is not used for ACE operations; the entirety of this Fund goes toward our recommended charities. (We now have a checkbox indicating that users may choose to donate 10% of their gift directly to ACE, but at the time we had already reached our funding cap for the year and were not accepting new donations.)
We know that the EA community is generally not in favor of donation-matching campaigns. There are four main reasons why this is so:
1. Most donation-matching campaigns are illusory
2. Non-illusory campaigns may do harm via double-counting
3. Matching campaigns encourage dishonesty
4. There may be better ways for large donors to leverage their money

### Most donation-matching campaigns are illusory

At ACE, we are careful to ensure that any matching campaigns are entirely non-illusory. GiveWell has pointed out several problems with illusory matching campaigns. We agree with their reasoning. For example, last year a donor approached us asking about doing a matching campaign to benefit ACE; we declined it because their donation was going to happen whether we set up a campaign or not.
However, not all matching campaigns are illusory. For our year-end donation-matching campaign in 2017, we had a donor who would not otherwise have given to our Recommended Charity Fund but who was interested in doing a donation-matching campaign with us. We discussed how a non-illusory campaign might very likely reach a broader audience, inspiring hundreds of new people to participate in effective giving for the first time. We believe this type of influence matching is especially effective with non-EAs. However, as Karnofsky points out, it may be better for existing EAs to ignore the influence matching aspect when making decisions on where to donate—even if we prefer non-EAs to be influenced by them.
ACE’s commitment to non-illusory matching campaigns alleviates the concern raised by Jeff Kaufman about counterfactual trust and contradicts the assumptions made by most polled EAs on Facebook. Sometimes even illusory donation-matching campaigns can be beneficial (as Avi Norowitz points out with the Facebook #GivingTuesday retrospective) but this requires special circumstances.
We are careful to explicitly note our position on these influence-matching campaigns in our FAQ, and we believe that using them has been a net positive for our non-EA audience.

### Non-illusory campaigns may cause harm via double-counting.

Ben Hoffman has rightfully pointed out that overassignment of credit can obscure opportunity costs with donation-matching campaigns. Given two rational EA actors, believing that each is causing the other to donate may result in a scenario where each is giving less optimally than they’d otherwise choose. However, this example only causes harm if both sides are changing their mind on where to donate due to the existence of the donation match.
At ACE, we only accept matching campaigns where this does not occur. While we don’t have full certainty of their counterfactual actions, we believe our matching donor not only would likely have counterfactually given a smaller amount elsewhere had we not done the campaign (thus making our campaign non-illusory), but also that they would have counterfactually funded a different non-EAA donation-matching campaign. This ensures that, at most, only one side is influenced by the matching campaign—and so evades the situation Hoffman describes where double-counting causes harm.¹

### Matching campaigns encourage dishonesty

Overassigning credit isn’t the only way that matching campaign incentives reward dishonesty. In 2016 Benjamin Todd reported that 80,000 Hours had previously run partially counterfactually valid donation matching campaigns, by allowing the donor to commit to delaying funding rather than ensuring that the funding is completely counterfactually valid. Ben Kuhn ran a survey that found that most donors expect matching campaigns to be entirely counterfactually valid, so this remains an issue.
At ACE, we’ve tried to maintain a balance of being intellectually honest and using copywriting that isn’t overly wordy. During our year-end donation-matching campaign, we initially used advertisements that contained the phrase “Double Your Impact”, but then switched to “Double Your Donation” after receiving feedback about where that balance made the most sense.²
We believe using “Double Your Donation” as an advertisement headline strikes that balance well. It is an appropriate shorthand for what is actually occuring: the amount of money that would go to this fund was indeed doubled, and, had the donation match not occurred, our matching donor may have given to a different non-EA cause.
As further evidence that these are real counterfactual matches where influenced donations are legitimately doubled, in 2016 and 2017 we surpassed the agreement of what the donor had originally offered to match, after which the donor followed up by continuing the match through these additional donations to more than double what they originally offered toward the matching challenge.

### There may be better ways for large donors to leverage their money

Ben Kuhn performed a survey of research on the effectiveness of matching campaigns, concluding that matching campaign effects are generally smaller than we might at first expect, and suggesting that there are more effective ways for large donors to leverage their money. We have no reason to doubt the validity of his analysis, though we do have limited conflicting anecdotal data from our matching campaigns.
When we are approached by large donors, we generally try to steer them away from the donation-matching campaigns they are ordinarily used to, advising them to fund general unrestricted programs and administration instead. However, last year we set a funding cap for our own operations; once met, there are only so many other ways that large donors can leverage their money. When a legitimate influence-matching campaign opportunity arises, we don’t think it is inappropriate to take advantage of it at the 1:1 rate, even if the returns may not be as much as you might expect.
Anecdotally, we’ve found that our matching campaigns have brought in a disproportionately large number of new donors—the majority of whom were not previously involved with effective giving. While we did not set up a control group, we can report that 73% of the donors to our 2017 matching campaign were first-time donors with ACE, and our post-donation survey showed that over 80% of respondents reported being motivated to give specifically due to the matching opportunity. In addition to the hundreds of thousands of dollars raised by this new audience,³ we were able to teach them about effective animal advocacy and to support them in effective giving elsewhere in the EA movement. The amount that these donors will give to effective charities during their lifetime is significantly higher than the donation-matching campaign that attracted them; we continue to build relationships with these new donors. So while we concur with Kuhn that the raw donation amounts might not be as influenced by a donation match as we may at first think, in our case the flow-through effects seem to more than make up for this difference.

### Utilons Versus Fuzzies

Another advertising example that has received some limited criticism from the EA community is our tendency to use cute pictures of animals and catchy messaging in many of our ads. We’ve heard this critique several times in person at EA conferences, although it’s rare to see the argument explicitly laid out online. The idea is that advertising in a way that tugs at people’s fuzzies may be appropriate for direct-level EAA organizations such as The Humane League, but that a meta-charity like ACE should focus solely on arguments for maximizing utility. By posting cute pictures or clever one-liners, we may be misrepresenting the type of work we do. After all, GiveWell’s Facebook feed rarely uses pictures other than to show graphs or their logo; and MIRI’s feed focuses on showcasing data, not cute pictures.
We feel that these lines of argumentation misunderstand the role that ACE plays in the EA movement. Yes, we have a core audience of EAs who use our charity recommendations—and yes, those recommendations are based on what we find to be most effective. We fully identify as an effective altruist organization. However, a large portion of our intended audience is comprised of animal lovers who are not yet aware of EA principles. We strongly believe that this is the audience that is capable of making the largest positive change once they learn about effective animal advocacy. Catching their attention via cute animal pictures is the best way we’ve found so far to get them to read more about why effectiveness is important.
We have data to support this. We’ve experimented with pushing out various types of Facebook posts to both our current audience (to increase engagement) and to potential new followers. Although we make a point to post a combination of EA-oriented messaging alongside posts that showcase cute animals, the only posts that receive traction significant enough to generate engagement and reach a larger audience are the ones that use fuzzies, not utilons, as the main hook.
Facebook in particular has a positive feedback loop where the posts which garner the most reactions and engagement tend to be shown to more followers, whereas the lower-performing posts only get shown to a subset of our Facebook audience. This results in the average Facebook user getting the impression that the majority of our posts are fuzzies-centric on trending topics, when in reality a considerable amount of our posts are utility-centric on data-oriented news.
This feedback loop works to our advantage. Over time, the best performing posts reach non-EA audiences who are likely to be sympathetic to the cause of effective animal advocacy. This allows our brand to grow consistently, introducing new people to the ideas of the EA movement.

We’re currently in the process of improving some of our graphs in order to make them more easily shareable. For example, our donation impact page includes a violin plot that shows how many animals we estimate are spared by a $1,000 donation. While the data is accurate, the resulting graph is very difficult to interpret—and is therefore not ideal for sharing at animal rights conferences or on social or journalistic media. The amount of time it takes to explain probability density to passersby at non-EA conferences or to general online audiences looking for a quick picture that explains ACE’s thoughts usually exceeds their (understandably limited) attention span. If we want to reach these audiences, we need to produce graphs that illustrate the point much more efficiently. However, sharing just the average estimates (seven animals spared for shelters and 4,056 for ACE recommended charities) could be seen as deceitful, as it doesn’t take into account the uncertainty involved in making these calculations. Clearly, there is some compromise between a full violin plot and sharing just the mean as the best single estimate for each category. We are still working out where that balance lies. A similar issue comes up with our donation allocation chart. The data on the left omits wild fishes, who receive almost no portion of animal charity donations. The data on the left also includes animals used for clothing, but the data on the right replaces that category with “mixed or other activities.” On the page, we explicitly point out how the “mixed or other” category was put together, giving an example of guide dog training as one component. This matters because some advocates may consider subcategories like guide dog training as a primarily human-centric charity, not an animal charity, and this might affect the relative size of the chart regions. However, when this chart does gets shared online, it’s generally just the image portion that people will copy and paste. Although these shared images will not include the surrounding text, we believe it is sufficient to include the extra information on our website. We feel similarly with regard to the analogous donation impact chart; including the probability densities as additional information on our site is sufficient, so long as the chart itself includes ranges for our cost-effectiveness estimates. ## Marketing Best Practices The above marketing practices are specific to EA organizations. Most non-EA organizations do not object to setting up illusory donation-matching campaigns, nor reaching out based on fuzzies, nor ensuring that graphics are as accurate as possible. We care about these issues because we are committed to the norm of being extraordinarily honest. As Ben Hoffman rightly points out, “When the activity of extracting money from donors is abstracted away from…other core activities of an organization…, best practices tend towards distorting the truth.” This doesn’t mean we should shy away from all nonprofit marketing best practices. To the extent that we can reach out to new audiences, increase the number of donors who are giving effectively, and grow our brand without compromising what we believe in, we feel that using best practices is both acceptable and desirable. One of our goals is to continually be better able to introduce non-EA animal advocates to effective animal advocacy ideas. We accomplish this through several methods: • We try to speak in language that non-EA animal advocates already understand when in an appropriate context. This not only means using cute pictures in social media posts, but also talking about projects and interventions that may not be a currently understood top intervention, so long as it reasonably could apply. • We use multivariate testing on social media adverts and email campaigns, but we are careful to only test messaging that matches our brand and to only use subject lines that accurately convey the content of each email. • We segment our audience, showing different information to different audiences, while being careful not to cut information sent to various audiences in a way that might be seen as deceitful. Specifically, we take care to only show different content based on the different interests of our audience; we do not alter the meaning behind our messages when segmenting our audience. • We use a Google ad grant and SEO efforts to gain traffic from audiences not yet familiar with effective animal advocacy, without deceiving visitors as to what they will see once they visit. • We track key performance indicators to judge how effective our communications strategies perform, but are careful not to focus on quantity at the expense of quality. • We create videos that appeal to a general audience through fuzzies in addition to posting videos of webinars and symposium talks. • We take advantage of opportunities to direct more funding to effective animal charities, such as running a non-illusory matching campaign that is very likely to inspire new people to give effectively. While we are careful not to blindly follow marketing best practices, we nevertheless utilize them when they don’t interfere with our values. We prize integrity and take care to exhibit norms of honesty when following accepted marketing principles. We do not use “rhetorical tricks” nor “sales techniques” to convince others; our use of imagery is solely used to gain attention. Arguments for EAA on our site are fully transparent, and we both accept and encourage feedback on the research we perform. I’m proud to report that we are continuing to grow our brand, increase the number of donors who are giving effectively, and introduce new audiences to effective animal advocacy. In December last year, we raised over$1.26 million for our Recommended Charity Fund—a significant portion of which came from extremely generous first-time donors not already identifying themselves as EAs. Of course, we couldn’t achieve these results for our recommended charities without the support of EAs, who so generously helped to fully fund ACE directly last year, but the larger point is that these strategies are encouraging non-EAAs to donate to more effective causes and are making a subset more aware of effective altruism in general.
If you work with an EA organization, we would love to hear about the marketing/communications techniques and/or successes that you’ve had when promoting EA organizations. We’d also like to hear if anyone has any concerns about ACE or any other EA charity using these kinds of marketing techniques. Are you comfortable with ACE’s methods of using donation-matching campaigns? Do you agree that marketing with fuzzies is acceptable even for a charity evaluator like ACE? What’s the minimal amount of information our shareable images should convey? Do you feel that the way ACE follows nonprofit marketing best practices is appropriate?
We look forward to hearing your thoughts.

¹ This donor decided to give to our Recommended Charity Fund on the basis of our recommendation, not by calculating the additional effects of the matching campaign, because they counterfactually would have done a different donation-matching campaign anyway. This means that while an overassignment of credit may have occurred, this overassignment did not change how the donor would have counterfactually acted.
² In November 2017 we initially used the phrase “This means that you can double the impact of your donation from now through the end of the year by donating to our Recommended Charity Fund” in at least one marketing material, but after receiving feedback from both ACE staff members and outside EAs (including Remmelt Ellen and Marianne van der Werf—thank you to both!) we standardized to the “double your donation” language instead across all marketing materials.
In previous years we were less strict about our language, using “Double Your Impact” in several advertisements during the 2015 and 2016 giving seasons. We did not make this choice blindly; at the time, we felt that “double your impact” and “double your donation” were different mostly in what kinds of audiences they attract, and that this overrode any concerns about one being more strictly accurate than the other. “Double your impact” emphasizes the effect of helping others, whereas “double your donation” emphasizes what one can personally accomplish.
Note that these terms are regularly used interchangeably in fundraising campaigns outside the EA community, and we then felt that most people wouldn’t take campaign headlines literally. In 2017, we updated to believing that it was not enough for our FAQ to describe our commitment to non-illusory matching campaigns, and we standardized to only use “double your donation” language going forward.
³ New donors made up 56% of the pre-matched amount raised during our 2017 matching campaign.
⁴ This positive feedback loop means that the majority of our Facebook post views are of fuzzy-style content. However, this is only true for posts that get pushed out to users; for those who actively come to the ACE Facebook page for content, the ratio of fuzzy-style to utilon-style is roughly one-to-one.
⁵ This is also explicitly why we do not publish cost-per-animal-spared numbers in the same way that you may see for QALYs or DALYs elsewhere.
⁶ We acknowledge that there are limitations to this chart and are working on an updated version that will include both data on wild animal suffering and more clear proportions that indicate how limited animal charity donations are overall.
⁷ ACE has publicly endorsed and acts in accordance with CEA’s guiding principles of effective altruism.
⁸ In total, we influenced over $6 million to our recommended charities in 2017, including the$1.26 million that flowed through our Recommended Charity Fund.
⁹ We are currently working on creating several more charts for non-EAs that are more shareable, and we would especially appreciate any remarks that express what level of simplification the EA community would be comfortable with.