Vitalik: “Concave” and “Convex” in the blockchain world


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Note: The original author is Vitalik Buterin, the co-founder of Ethereum.

I have noticed that a major philosophical difference in how people deal with large-scale decisions in the world lies in how they deal with the trade-off between “compromise” and “pure”. If you choose between two options, usually expressed in a deep principled philosophy, you will naturally tend to one of the two paths is correct, and think we should stick to it, or you are more Like to find a way between two extremes.

In mathematical terms, we can explain it this way: Do you expect the world we live in, especially its response to our actions, to be fundamentally “concave” or “convex”?

Those who prefer a “concave” position might say this:

  1. “Going to extremes has never been good for us, too hot or too cold will lead to death. We need to find a balance between the two, and this is the right thing.”
  2. “If you only implement a little philosophy, you can choose the part with the highest profit and the lowest risk and avoid the part with greater risk. However, if you insist on going to extremes, once you pick the low fruit, you will be forced Work harder and harder to find smaller and smaller benefits, and before you realize this, the growing risks may outweigh the benefits of the whole thing.”
  3. “The opposition to philosophy may also have a certain value, so we should combine the advantages of the two, and we must avoid doing things that the opposition to philosophy considers extremely terrible, just in case.”

Those who prefer a “convex” position might say:

  1. “We need to concentrate, otherwise, we may become knowledgeable but not precise people.”
  2. “If we take a few steps along that road, that road will become very slippery, and it will only pull us lower and lower until we fall into the abyss. There are only two stable positions on the slope: either we are in Below, or stay on top.”
  3. “If you give up an inch, they will ask for a mile.”
  4. “Whether we follow this philosophy or that philosophy, we should follow some philosophies and stick to it. There is no point in mixing everything together.”

I personally find that in a variety of environments, I myself always prefer the “convex” method. If I had to choose (I) toss a coin between anarchic capitalism and Soviet communism, or (ii) a compromise of half of the two, I would immediately choose the latter. I advocate that the Bitcoin block size debate should be moderate, and I oppose small blocks of 1-2MB and “super large blocks” of 128MB. I oppose the view that there is no middle ground between freedom and decentralization. I support the DAO fork, but to the surprise of many people, since then, I have been opposed to hard forks like “state interference”. As I said in 2019, “The support for Saab’s law (blockchain immutability) is a spectrum, not a binary one.”

But as you may see, not everyone has the same intuition. What I want to point out in particular is that the Ethereum ecosystem as a whole has a basic “concave” temperament, while the temperament of the Bitcoin ecosystem is essentially “convex” . In the Bitcoin field, you can often hear such arguments: either you have your own sovereignty, or you don’t, or any system must have a fundamentally centralized or a fundamentally decentralized tendency, impossible Somewhere in between.

Occasionally, my half-joking support for Tron is a key example: from my own point of view, if you value decentralization and immutability, you should realize that the Ethereum ecosystem does sometimes violate the purity of these values. The concept of ethics, and the degree to which Tron violates these values ​​is far beyond common sense, and there is no self-blame, so Ethereum is still the most popular of these two options. But from a “convex” perspective, the extreme nature of Tron’s violation of these specifications is an excellent quality: Ethereum half-heartedly pretends to be decentralized, and Tron is centralized, but at least it is proud of it. , This is honest.

The difference between “concave” and “convex” thinking modes is not limited to the obscure points of efficiency and decentralization in cryptocurrencies. It also applies to politics (guess which side has more downright anarchic capitalists), other technological choices, and even what food you eat.


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

The fusion of “concave” and “convex”

But it is worth noting that even at the meta-level, the “concave” temperament is an extreme thing that a person must be very careful to avoid. Of course, there are situations where policy A brings good results, policy B gives worse but tolerable results, but a careless mixture between the two is the worst result. Coronavirus may be a good example: a 100% effective travel ban is far more than twice as effective as a 50% effective travel ban. An effective blockade can lower the R0 of the virus below 1, leading to rapid recovery, but inadvertent blockade will only reduce R0 to 1.3. This will bring months of pain, but it has almost no effect. This is one possible explanation for why many Western countries have not responded well to this: political systems designed for compromise may fall into the middle course even when they are ineffective.

Another example is war: if you invade country A, you will conquer country A, if you invade country B, you will conquer country B, but if you invade these two countries at the same time, send your soldiers to each country, The combined strength of these two countries will crush you. Generally speaking, when the response effect is “convex”, you will usually find that a certain degree of centralization will bring benefits.

But in many places, mixing is obviously better than any extreme. A common example is the issue of 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. The reasons are as follows:

The 2% tax rate prevents very few transactions, even if the transactions it prevents are not very valuable-if only a 2% tax rate is enough to prevent participants from making transactions, how much value can the transaction be? A 20% tax rate may prevent more than 10 times of transactions, but each blocked transaction is 10 times more valuable to participants than in the case of 2%. Therefore, a tax increase of 10 times may cause 100 times economic losses. For this reason, low tax rates are generally better than high taxes and no taxation.

According to similar economic logic, the complete prohibition of certain behaviors may cause the greatest harm, and replacing the existing prohibition with a moderately high punitive tax can improve efficiency, increase freedom, and provide valuable To build public goods or help the poor.

The Laffer curve tells us that a zero tax rate will not increase income, and a 100% tax rate will not increase income, because in this case, no one will want to work, but a certain tax rate in the middle will increase the most income. There is controversy about the tax rate for maximizing returns, but in general, people generally agree with the following chart:


If you have to choose the average of the two proposed tax plans, or toss a coin between them, it is obvious that the average is usually the best. Taxation is not the only thing with this phenomenon. Economics studies a wide range of “diminishing returns” phenomena, which are common in many other aspects of production, consumption, and daily behavior. Finally, a common negative of diminishing returns is acceleration costs: to take a notable example, if you use the standard economic model of monetary utility, they directly imply that economic inequality is twice as much as it would cause four times the harm.

The world has more than one dimension

Another complicated issue is that in the real world, policies are more than just one-dimensional numbers. There are many ways to average between two different policies or two different philosophies. An easy-to-understand example is: Suppose you and your friend want to live together, but you want to live in Toronto, and your friend wants to live in New York, how would you compromise between these two options?

Well, you can take a geographical compromise and enjoy your peaceful life in the arithmetic between two lovely cities.


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

Or you can be more pure in mathematics and take the midpoint of the straight line between Toronto and New York without bothering. Well, you are still close to that church, but 6 kilometers below it. Another way to compromise is to stay in Toronto for 6 months each year and 6 months in New York-this may be a practical and reasonable way for some people.

The point is, when the options presented to you are more complex than simple one-dimensional numbers, figure out how to compromise between the two options and really take the best from the two, not the worst part of the two. This is an art and a challenge.

This is to be expected: “convex” and “concave” are the most suitable terms for mathematical functions, where both input and output are one-dimensional. The real world is high-dimensional—as machine learning researchers have now determined, in a high-dimensional environment, the most common situation is that you may find that your environment is not a universal “convex” or “convex” The “concave” environment is a saddle point: the local area is “convex” in some directions, and “concave” in other directions.


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

This may be the best mathematical explanation for “why these two tendencies are needed to some extent”: the world is not completely “convex”, but it is not completely “concave” either. But between any two distant locations A and B, there is some kind of concave path is very possible. If you can find that path, then you can usually find a path between these two locations that is better than both. Good comprehensive location.

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