The "Concave" and "Convex" of the Blockchain World

Author: Vitalik Buterin

Original Title: 《Convex and Concave Dispositions》

Published on: November 8, 2020

I have noticed that a major philosophical difference in how people handle large-scale decisions in the world lies in how they deal with the trade-offs between "compromise" and "purity." If you make a choice between two options, often expressed in profound principled philosophical terms, you will naturally tend to believe that one of these paths is the right one and that we should stick to it, or you prefer to find a middle ground between the two extremes.

In mathematical terms, we can explain it this way: do you expect the world we live in, particularly in how it responds to our actions, to be fundamentally "concave" or "convex"?

Those leaning towards a "concave" position might say:

1. "Going to extremes has never benefited us; being too hot or too cold can lead to death. We need to find a balance between the two, and that is the right approach."
2. "If you implement just a little philosophy, you can choose the part with the highest returns and the lowest risks, avoiding the parts with greater risks. However, if you insist on going to extremes, once you pick the low-hanging fruit, you will be forced to work harder and harder to find smaller and smaller benefits, and before you realize it, the ever-increasing risks may outweigh the benefits of the whole endeavor."
3. "There may also be some value in opposing philosophies, so we should combine the merits of both and definitely avoid doing things that opposing philosophies consider extremely terrifying, just in case."

On the other hand, those leaning towards a "convex" position might say:

1. "We need to focus; otherwise, we risk becoming jack-of-all-trades and master of none."
2. "If we take a few steps down that path, it will become slippery, pulling us lower and lower until we fall into the abyss. There are only two stable positions on the slope: either we are at the bottom or we stay at the top."
3. "If you give an inch, they will take a mile."
4. "Whether we follow this philosophy or that philosophy, we should adhere to some philosophy and stick to it. Mixing everything together makes no sense."

Personally, I find that in a variety of contexts, I tend to agree with the "concave" approach more than the "convex" approach. If I had to choose (i) to flip a coin between anarcho-capitalism and Soviet communism, or (ii) a compromise with both occupying half, I would immediately choose the latter. I advocate for moderation in the debate over Bitcoin block size, opposing both 1-2MB small blocks and 128MB "super-sized blocks." I oppose the view that freedom and decentralization have no middle ground. I supported the DAO fork, but surprisingly to many, I have opposed hard forks similar to "state intervention" ever since. As I said in 2019, "Support for the Sabot Law (blockchain immutability) is a spectrum, not binary."

But as you might have noticed, not everyone shares the same intuition. I want to particularly point out that the Ethereum ecosystem overall has a fundamental "concave" temperament, while the Bitcoin ecosystem's temperament is essentially "convex." In the Bitcoin space, you often hear rhetoric like: either you have your sovereignty or you don't, or any system must have a fundamentally centralized or fundamentally decentralized tendency, with no possibility of being in between.

Occasionally, my half-joking support for Tron serves as a key example: from my own perspective, if you value decentralization and immutability, you should recognize that the Ethereum ecosystem sometimes does violate the purest concepts of these values, while Tron violates these values to a much greater extent and without any sense of remorse, which is why Ethereum remains the more popular of the two choices. But from a "convex" perspective, the extremity with which Tron violates these norms is a virtue: Ethereum pretends to be decentralized half-heartedly, while Tron is centralized, but at least it takes pride in that, which is honest.

The difference between "concave" and "convex" mindsets is not limited to the obscure points about efficiency and decentralization in cryptocurrency. It also applies to politics (guess which side has more thoroughgoing anarcho-capitalists), other technological choices, and even what food you eat.

But in all these matters, I personally find myself always standing on the side of balance.

The Fusion of "Concave" and "Convex"

It is worth noting that even at a meta level, a "concave" temperament is something one must be very careful to avoid in its extremes. Of course, there are situations where Policy A yields good results, Policy B yields worse but still tolerable results, but a careless mix between the two can lead to the worst outcomes. The coronavirus may be a good example: a 100% effective travel ban is more than twice as useful as a 50% effective travel ban. An effective lockdown can push the virus's R0 below 1, leading to a quick recovery, but a careless lockdown will only lower R0 to 1.3, resulting in months of suffering with little effect. This might explain why many Western countries have responded poorly: political systems designed for compromise can get stuck in the middle ground even when ineffective.

Another example is war: if you invade Country A, you conquer Country A; if you invade Country B, you conquer Country B. However, if you invade both countries simultaneously, sending your soldiers to each, the combined strength of those two countries will overwhelm you. Generally speaking, when the effects of a response are "convex," you will often find that some degree of centralization brings benefits.

But in many areas, mixing is clearly better than any extreme; a common example is the issue of setting tax rates. In economics, there is a general principle that deadweight loss is quadratic: that is, the harm caused by inefficient taxation is proportional to the square of the tax rate, for the following reason:

A 2% tax rate prevents very few transactions, and even the transactions it prevents are not very valuable—if a mere 2% tax rate is enough to deter participants from trading, how valuable can those transactions be? A 20% tax rate might prevent more than ten times as many transactions, but each transaction prevented is ten times more valuable than in the 2% case. Therefore, increasing the tax tenfold could result in a hundredfold economic loss. For this reason, low tax rates are generally better than high taxes or no taxes at all.

According to similar economic logic, a complete ban on certain behaviors can cause the greatest harm, while replacing existing bans with moderately high punitive taxes can improve efficiency, increase freedom, and provide valuable revenue to build public goods or help the poor.

The Laffer curve tells us: a zero tax rate will not increase revenue, nor will a 100% tax rate, because in such cases, no one would want to work, but some intermediate tax rate will maximize revenue. There is debate about what the tax rate for maximizing revenue is, but generally, people agree with the chart shown below:

If you had to choose the average of two proposed tax plans or flip a coin to choose between them, it is clear that the average is usually the best. Taxes are not the only things that exhibit this phenomenon; economics studies a range of broad "diminishing returns" phenomena that are prevalent in many other aspects of production, consumption, and everyday behavior. Ultimately, a common opposite of diminishing returns is accelerating costs: for a notable example, if you use the standard economic model of monetary utility, it directly implies that economic inequality squared will cause quadruple the harm.

The World is Not One-Dimensional

Another complex issue is that in the real world, policies are not merely one-dimensional numbers. There are many ways to average between two different policies or two different philosophies. A very easy-to-understand example is: suppose you and your friend want to live together, but you want to live in Toronto while your friend wants to live in New York; how would you compromise between these two choices?

Well, you could take a geographical compromise and enjoy your peaceful life at the arithmetic midpoint between the two lovely cities.

Image: This church is located about 29 kilometers southwest of Ithaca, New York.

Or you could be mathematically purer, taking the straight line midpoint between Toronto and New York without any fuss. Then, you would still be very close to that church, but 6 kilometers below it. Another way to compromise is to spend six months in Toronto and six months in New York each year—this might be a practically reasonable way for some.

The key is that when the options presented to you are more complex than simple one-dimensional numbers, figuring out how to compromise between the two options and truly extract the essence from both, rather than the worst parts of either, is an art and a challenge.

This is to be expected: "convex" and "concave" are terms best suited for mathematical functions where both input and output are one-dimensional. The real world is high-dimensional—just as machine learning researchers have now determined, in high-dimensional environments, the most common situation is that you may find yourself in an environment that is not universally "convex" or "concave," but rather a saddle point: locally "convex" in some directions and "concave" in others.

Saddle point: convex from left to right, concave from front to back.

This may be the best mathematical explanation for "why we need both tendencies to some extent": the world is not entirely "convex," but it is also not entirely "concave." However, between any two distant points A and B, there is a high likelihood of finding some concave path; if you can find that path, you can usually find a composite position between the two that is better than either.