In this issue:
Friends,
My second favorite thing after a piece of art, writing, music, movie, and sport that I love is watching someone else gush over things they love. It’s why the Lost in Vegas guys are so endearing, especially when the channel first started. You were watching hip-hop heads not only discover but unpeel the layers in Tool’s music.
One of my dad friends recently bought a 1979 Jeep to restore with his son and asked if my son Zak (13) would like to help. Zak’s “hell yea” beat the sound of my last syllable when I asked him. He started watching YouTube, and we landed on a channel where these obsessed details comb the U.S. looking for “barn finds” to clean for the owners. For free! It’s free because the channel has almost 2 million subscribers clamoring to watch an impossible cleaning.
I watched a few videos. I get it. Renewal of beauty, just like the objects of renewal never goes out of style. The process is as timelessly rewarding as the result.
Caitlin’s tweet is an appeal to our spirit of obsession and craft.
Which brings me to a luxurious reading experience:
I like ‘em thick (Adam Mastroianni)
This essay not only feels good but is an important lens on the fast-approaching experiment of whether infinite electric monkeys will pound out Shakespeare.
Normally I might apologize for extensive excerpting but they are the point. I can do no better than this.
It opens:
I owe an apology to every English teacher I ever had. I always assumed that so-called “great” literature was a hoax, a punishment inflicted upon adolescents for the crime of being young. These books did not have anything special about them, and covering up that fact was simply a make-work exercise for former English majors, a sort of “jobs for snobs” program.
I was wrong about this. There is such thing as greatness. More specifically, there is such thing as thickness. Great works of fiction—for that matter, great works of any art—unfurl in response to your attention. The more time you spend with them, the more you get out of them. That kind of responsiveness is so addicting that it can lead people to do crazy things, like try to teach literature to high schoolers.
But thickness is tricky, because rewarding the careful reader often means repelling the casual one. And this is where I would like an apology in return from my English teachers, because while this might have been obvious to them, they never made it obvious to me.
I was presented with art and literature as if it was self-explanatory, and that everything wonderful about it was plainly visible from the outside. But those works were much more like dark, winding caves with treasure stashed inside of them. My teachers were like, “Right, well, into the cave you go!” and I was like “But there’s nothing in there” and they were like “Entering the cave is 30% of your grade” and so I took a few steps into the darkness and I was like “Just as I suspected: an empty cave” and then I came trudging back out and pretended that I saw something.
It goes on to describe spectacular examples of “thickness”. You won’t want to miss the The Garden of Earthly Delights painting and its unintended invitation to hear “butt music”.
Adam presents 4 qualities that make something thick with examples from a children’s book, the “ears” in Hamlet, and Penn (of Penn and Teller fame) eating fire.
There’s an amazing takedown of what passes for popular non-fiction.:
Reading a book like this feels like wandering through a Potemkin village. Touch any of the ideas, and they tip over.
He contrasts gilded examples of non-fiction with solid gold:
Thickness comes from surfacing a few facts well, and in such a way that you realize the existence of entire universes of additional facts that could be known….[Jane Jacobs book] is pointing out a fact that millions of people observe every day, but almost none of them notice.
He addresses an easily anticipated objection, which you’ll be familiar with if you have read any of deBoer on “poptimism”:
If you allow for the existence of secret treasures that can only be accessed with effort and analysis, then you empower the elitists and the snobs. “Buddy, don’t even talk to me until you’ve been in the cave!”
But look around. The snobs are in full retreat. We have swung the pendulum so far toward poptimism, toward the blinkered idea that all art is equal because all humans are equal, toward the ethos that guilty pleasures are simply pleasures, that I’m not sure if we can ever swing it back.
And finally, Adam addresses the elephant:
Erasing the line between the thick and the thin has left us defenseless against slop at the exact moment of its onslaught. Everyone can sense there’s something amiss with the prose that comes out of the machines, but we lack the language to talk about it, and so we’ve converged on the idea that slop simply means using too many em dashes, bullet points, and line breaks.
What separates substance from slop is thickness. Slop holds no secrets; it signifies nothing. Under scrutiny, it evaporates. All it can offer is bottomlessness—sure, there’s nothing on, but at least there are infinite channels!
That’s why I’m neither surprised nor dismayed when studies find that people prefer AI art to human art. Of course they do! In the short term, thinness prevails. When people are making snap judgments, they want pretty flowers, poems that rhyme, pleasing pablum, the simulacrum of thought. But none of these last…
About once a week, I get a pitch from some AI startup that wants to automate some part of my writing. The most recent one says it’s “built for credible thinkers who have a book’s worth of ideas but not the time it typically takes to write one”.
I’m sorry, but if you’re building or using a tool like this, then you’ve got slop for brains. There is no such thing as having a “book’s worth of ideas” that are all ready to go except for the small matter of choosing the right words and putting them in the right order.
I know exactly the feeling that these slop-trepeneurs are preying on, because I feel it all the time: I’ve got these thoughts in my head, and boy oh boy they’re good ones, all-timers, really, and it’s so annoying that I have to spend all this time making the words sound good, when the ideas behind the words are already so good!
But this is an illusion. The ideas are not already good. They need to be thickened. I understand why it’s tempting to force a machine do the hard part for you, but it can’t, and the hard part is the only part worth doing anyway.
People seemed to like the post I wrote 2 weeks ago teaching readers how to compute Kelly optimal bet sizes in their heads. A lot of people reached out saying that despite learning Kelly in the past, this treatment not only made it clearer but also helped them appreciate how its approach informs risk-taking in wider contexts.
Panoptica graciously asked me to republish it under their own banner:
After this post you will be sizing bets in your head (Panoptica)
Just to put a bow on it, I’ll restate what I think are the most crucial lessons without dwelling on the formula:
If interested, Matt does this neat meta series “Notes on Notes”. It’s a short chat about how and why a particular article comes together in the first place. You can watch it here:
via Matt Zeigler@CultishCreative
To round out this Kelly sprint, I have 2 more bits that are again overtures to curious learners who may find math intimidating.
1) Slides
The first is a condensed slide version, which I hope makes this accessible. It was born out of teaching Kelly to my 8th grader at breakfast this past Tuesday. “Zak, I wanna see if I can teach you something neat in 5 minutes.” He rolled his eyes, but at least he humored me while scarfing down his cereal.
2) An empowering derivation
The derivation of the Kelly formula is so fun because as it rolls downhill, a number of concepts we talk about in this letter stick to it, so when we get to the end it feels like something grand, but it’s so damn compact.
Another teaching experiment. Let’s see if I can narrate the derivation in a way so that as you follow along it never feels “hard”. I want to prove that this is fun to do and while I don’t expect to convert everyone, I do think there’s a bunch of you who’d like to be able to learn this but feel blocked because you have gaps in your foundations or can’t remember HS math, or just lack some confidence.
Screw all that, I’ll lay my jacket over the puddles so you can see that it’s not so bad out here. Just come along.
Stating the question
You have a bet. You win with probability p and lose with probability q = 1 − p. If you win, you get paid B times what you risked. If you lose, you lose what you risked. B is also just a return. So if you double your money on a bet, B=1=100% return.
You’re going to bet the same fraction f of your bankroll every time and let it ride.
What’s the f that yields the highest compounded return?
If your wealth is W and you bet the fraction f:
Win: W → W(1 + Bf) Lose: W → W(1 − f)
Example:
Your starting wealth is $100 and you bet 50% of it on a coin flip. Remember B =1 because when you win you make 100%. When you lose you always lose f which is your bet size.
Win: 100 → 100(1 + 1*.50) = $150 Lose: 100 → 100(1 − .50) = $50
You keep the part you didn’t bet no matter what. A win adds B times your stake. A loss takes your stake away.
Play n bets. You win h of them and lose the other n − h. Each bet multiplies whatever the last one left you, so the multipliers stack:
Wₙ = W₀(1 + Bf)ʰ(1 − f)ⁿ⁻ʰ
The order of wins and losses doesn’t matter. Only the count does.
Example:
I bet 50% of my wealth each turn on the coin. I play 4 times and I win 3 of them.
100(1+1*.5)3(1-.5)1= 168.75
Order doesn’t matter. If I lost 50% on the first flip, then won 3 in a row:
Start: 100 Lose (×0.5): 50 Win (×1.5): 75 Win (×1.5): 112.50 Win (×1.5): 168.75
Wₙ is where you end up. We want a per-bet growth rate.
You already know how to do this. If an investment grew by a factor of (1 + 8%)¹⁰ over 10 years, you take the 10th root to get the CAGR back. Same move here: take the nth root of this equation: Wₙ = W₀(1 + Bf)ʰ(1 − f)ⁿ⁻ʰ
Note that W₀ can be divided out, just as if your starting identity was 150 = 100(1+8%)¹⁰ and you turned that into 1.5 = 1.08¹⁰ before taking the 10th root to get to annual growth rate.
This equation took our simple total growth equation and turned it into a growth rate equation:
We can clean up our new growth equation with several handy notation substitutions.
Ps and Qs
h/n is the share of bets won
(n-h)/n is the share of bets lost
Over a long run, the fraction of bets you win settles down to your win probability:
h/n → p and (n − h)/n → q
G
Wₙ/W₀ is just a wealth multiple. If you made 200% on your investment portfolio over a decade, your wealth multiple is 3 because you started with $1 and ended up with $3.
We’ll call that wealth multiple G (for “gross multiple”)
So the per-bet growth factor becomes:
G = (1 + Bf)ᵖ(1 − f)ᑫ
G = 1.05 means your bankroll grows 5% per bet on a compounded basis. In traditional investing, we’d substitute the word “year” for “bet”. Each flip is like a year.
Our job is to find the f that makes G (the gross multiple of our wealth) as big as possible.
Maximizing G means taking its derivative with respect to f and setting it to zero.
It’s 2026, you don’t need to know how to actually differentiate an equation. You just have to awaken that part of your brain that knows:
a) a derivative is the slope of a function at a given point on a curve
b) when the slope of a curve is 0 this is a maximum or minimum
c) intuitively, we can reason that this is a growth curve is a hill with no bumps. Its slope starts positive and only ever gets smaller as you bet more, so it can hit zero exactly once, and when it does, you’re at the top. (the jargon version: this growth curve is concave: it bends downward everywhere from the max, with no inflection points)
We are still here:
G = (1 + Bf)ᵖ(1 − f)ᑫ
But G is a product of powers, which is nasty to differentiate.
But remember from simple pleasures, logs fix this. They are the inverse of exponentiation, allowing them to turn multiplication into addition!
Push the cobwebs away to recall that this works:
log(100) = log(10 × 10) = log(10) + log(10) = 1 + 1 = 2
That’s not all logs do.
They pull exponents down in front: log(xᵖ) = p · log(x).
We can prep that equation for differentiation by taking the log of both sides and using both of those sexy log features:
log(G) = p · log(1 + Bf) + q · log(1 − f)
Maximizing log(G) gives the same f as maximizing G, because log always rises when its input rises.
There’s another little bonus.
Log(G) is the log return per bet, the same quantity as ln(S₁/S₀).
It re-expresses a compounded, multiplicative growth factor as a continuously compounded rate, and rates add cleanly.
Once you know this is a derivative problem because you are trying to maximize then we can rely on crutches (Claude) for thing that’s hard to remember.
Namely that:
So we have our equation:
log(G) = p · log(1 + Bf) + q · log(1 − f)
then we differentiate our terms to be added with respect to f:
Set the sum equal to zero, which is where the growth curve is flat at its peak:
pB/(1 + Bf) − q/(1 − f) = 0
Move the second term across and cross-multiply:
pB(1 − f) = q(1 + Bf)
pB − pBf = q + qBf
pB − q = pBf + qBf
pB − q = Bf(p + q)
But don’t turn your pattern recognition noggin off!
Since p + q = 1, this collapses to:
f = (pB − q) / B
f = p − q / B
Can you reproduce this right now on a blank sheet of paper?* Probably not. But go through it once the way you used to trace when you learned the motions for drawing comic book characters or flowers or in my case TMNT. After a single reproduction by hand, I assure you some neural pathways will re-open that have had construction signs in front of them for years.
And to tie this back to the beginning, that is the joy of thickness.
[Get your mind back over here, this is a family letter.]
*You will be able to reproduce it on your own after 2 or 3 attempts. I don’t know why the motion of the pencil is a 10x better instructor than reading it a bunch of times (generation effect maybe?). But this is one of those learning principles I take seriously and impress on my kids.
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