This is one of the myriad situations where Omega should have been be used, not O. What are they teaching in schools these days?
Should be O(n) vs Omega(n^2)
Yep
O(n) vs. O(n^2) Startups
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Re: O(n) vs. O(n^2) Startups
#62Earlier quoted context omitted.
> You can see it on revenue as well, since even if growth is exponential, churn is a percentage of your total paying users. Thus, it produces a sigmoid curve unless you can get churn to 0% (pro-tip: you can’t). Exponential growth means that additional users are a percentage of your total users. It is trivial to see that adding a source of exponential decay will give you another exponential function. All churn (as you…
Sigmoid (or the logistic function specifically rather) is exponential until you get close to the "turning point" (or rather, its growth bounded from below by an exponential). It's as you approach that point that it becomes linear, and after that its growth decays. However you are sort of right that "churn" does not necessarily have to do with it being sigmoid because it will be anyway. It may be bring it earlier if t…
If the churn rate surpasses user growth, you won't go sigmoid, you'll stay exponential. You'll just be shrinking instead of growing. To be sigmoid, you need the growth rate to vary.
Re: O(n) vs. O(n^2) Startups
#63This is one of the myriad situations where Omega should have been be used, not O. What are they teaching in schools these days?
I actually passed my discrete math class and final a few days ago and got the big O vs Theta vs Omega question right. The reality is that companies often underperform their best case possible growth rate. O(n) and O(n^2) are meant to represent the best possible growth rate which may be practically be underperformed. You may be thinking about algorithmic analysis where the term "worst case" is used for the upper bound…
It's entirely nonsensical to use O as a lower bound though. You could have two companies no growth whatsoever in value and correctly state that one has O(n) growth rate and the other has O(n^2) because a constant is both O(n) and O(n^2) (and O(n!) and O(exp(n^n)) ...). The author is trying to argue that there's some separation between two hypothetical startups' growth rates and as such an upper bound on one, say O(n), and a lower bound on the other, say Omega(n^2), is warranted. It sounds like you're not entirely an expert despite your recently-passed final. Strange concept, eh?
Re: O(n) vs. O(n^2) Startups
#64Huh. Not working in the field, I assumed startups went like sigmoids (everything is a sigmoid after all). Exponential at first as word of mouth spreads, then linear as your users start bumping into each other and word of mouth stops working, and then you eventually start leveling off near carrying capacity (you’ve hit your addressable market). I thought the game was to try to get bought by some massive company while…
I took the author's use of O(n) vs O(n^2) as a framing point rather than a literal model. It just seems to be missing the forest for the trees. Besides, we can approximate sigmoids with linear or quadratic functions when windowing them. Considering startup as context I think we know what part of the graph we're talking about... Do we see that exponential explosion or is the sigmoid much more flat. Replace the x in yo…
Big-O notation is about asymptotics. You have to approach something, and typically there is some infinity involved because if it is not, then you can just compute things instead of giving asymptotic approximations, or else you have 10^10*n and 0.001*n^2-10^20*n and the big-O asymptotics at infinity are useless for smaller numbers. I understand what OP tries to say but that's not really a good framing point for many reasons. If you want to talk about a finite period of time, use a regression model, not asymptotics. But that's probably also a more personal preference around using mathematics colloquially but also in a manner that is not a good metaphor and does not correspond with what the mathematical theory is referring to. And I am not sure at all whether it is very well understood "what part of the graph we're talking about", in the sense that the modern organization of economy is far from acknowledging the fact that resources on earth are actually finite. Talking about O(n) and O(n^2) or O(exp(n)) as if growth can be indefinite comes with a specific kind of mindset, and the frame used reflects this type of mindset.
Re: O(n) vs. O(n^2) Startups
#65Earlier quoted context omitted.
This is actually a better model and one that more closely reflects reality. You can see it on revenue as well, since even if growth is exponential, churn is a percentage of your total paying users. Thus, it produces a sigmoid curve unless you can get churn to 0% (pro-tip: you can’t). But, these are the two basic levers for a SaaS: growth and churn.
> You can see it on revenue as well, since even if growth is exponential, churn is a percentage of your total paying users. Thus, it produces a sigmoid curve unless you can get churn to 0% (pro-tip: you can’t). Exponential growth means that additional users are a percentage of your total users. It is trivial to see that adding a source of exponential decay will give you another exponential function. All churn (as you…
Growth is not a function of users, though, it is a function of something else that *may* be exponential, for a time. Users don’t beget users (word-of-mouth doesn’t last forever; ad money has diminishing returns), and eventually your market will be saturated, so you can’t grow exponentially forever.
Re: O(n) vs. O(n^2) Startups
#66I wonder what's optimal for me as an employee. I am working in a O(n) startup where colleagues are nice, work is streamlined yet challenging, and I do see growth potential in the long term. Several O(n^2) founders have reached out recently and the pay is attractive(even after accounting for a move to an HCOL area).
Modern society tends to severely overemphasize money as the optimisation goal. This is an emergent behaviour of our good capitalist system.
Your time is precious. You spend it once and you can't predictably get any more of it.
I suggest you choose your optimisation goal function very very carefully to suit the outcomes you want (money is only an intermediate step). It's hard to decide what we really want. Money is the default game that we see our peers playing (and it's easy to gain moderate success at the money game). It requires more attention to find and learn from people that have had success playing less common games.
Cynically (or even conspiratorially) investigate the suggested life defaults for you by your society as though they were dark patterns designed to mislead you.
I like what Naval wrote about status games (money is only one aspect of status). Paraphrased:
Status is a zero-sum game, not a positive-sum game. There’s always a subtle competition going on between status and wealth. For example, when journalists attack rich people or the tech industry, they’re really bidding for status. The problem is, to win at a status game you have to put somebody else down. That’s why you should avoid status games in your life – because they make you into an angry combative person. You’re always fighting to put other people down and elevate yourself and the people you like. Status games are always going to exist; there’s no way around it. Realize when you’re getting attacked by someone else and they’re trying to look like a goody-two shoes. They’re trying to up their own status at your expense. They’re playing a different game. And it’s a worse game.
Money has no maximum so it's a weird goal to try and reach. I wonder why Warren Buffett waited until 95 to decide to retire? He would easily be the richest man in the world if he hadn't charitably given so much away.
Another relevant paraphrased snippet from an interview about better lives for the elite: https://archive.ph/kF0YR
There’s this study called the American Freshman Survey [edit:snip] In the 1960s, 50% of students said making as much money as possible was a really important goal. Today, that’s 80% to 90%. That change shows that this is not human nature. It is culture.Re: O(n) vs. O(n^2) Startups
#67There's an interesting misunderstanding in this article.
The argument for O(n) is well formed here.
O(n^2) is not, the core argument is that these grow faster because of compounding. Compounding is fundamentally an exponential process, far larger asymptotically than a quadratic.
Re: O(n) vs. O(n^2) Startups
#68Earlier quoted context omitted.
They are sigmoids, but for some the plateau is 30 years in the future with a 5T market cap. For example Facebook's revenue is still increasing at an increasing rate, 21 years later
This is an incredibly important thing to understand. Buffet himself said it's better to be an average business in a great market that to be a great business in an average market.
Interesting. Not doubting this, but one of his sayings I saw many times is that "it is better to buy a great company at a good price than a good company at a great price", implying that great companies are worth significant premiums.
Re: O(n) vs. O(n^2) Startups
#69> An O(n) startup grows its key metric (revenue, users, etc.) roughly linearly with time—double the time, double the metric. An O(n^2) startup accelerates, with growth compounding super-linearly over time. Kind of a strange formulation to have n represent the key metric. In algorithm analysis, we would typically have n represent time (or some other cost). So we would say that the startup whose key metrics accelerate…
> In algorithm analysis, we would typically have n represent time (or some other cost). No, n is never time in any kind of algorithmic analysis. n is a function of the size of the input and the output is some measure of the cost related to the input. In O(n^2), the size of the input is n and the amount of time, or space, or some measure of the cost has an upper bound that is proportional to n^2.
> In O(n^2), the size of the input is n and the amount of time, or space, or some measure of the cost has an upper bound that is proportional to n^2.
Yes, this is my point. In the article, they classify an O(n^2) startup as one which achieves n^2 results in n time, which is the opposite of how the notation is typically used.
Re: O(n) vs. O(n^2) Startups
#70> [The conclusion is that] O(n) companies are higher EV than O(n^2) companies. I mean that, on average, a founder will make more money pursuing an O(n) company than an O(n^2) company. And not an insignificant amount, the amount of liquidity and networth a 20m ARR O(n) company is extremely hard to match by a traditional VC backed O(n^2) company.
It's a bit like getting a regular job vs playing a lottery: the former gives you better financial results on average, while the latter gives you a chance to make it really big.
(I also wish it were "linear companies" and "quadratic / exponential companies", or maybe "snooker-cue companies" vs "hockey-stick companies".)