---
tags: web3
---

# 翻譯共筆：Convex and Concave Dispositions

原文網址：
https://vitalik.ca/general/2020/11/08/concave.html

## 貢獻者：

## 翻譯：

### Convex and Concave Dispositions

凸凹配置

2020 Nov 08See all posts

2020 年 11 月 8 日查看所有帖子

One of the major philosophical differences that I have noticed in how people approach making large-scale decisions in the world is how they approach the age-old tradeoff of compromise versus purity. Given a choice between two alternatives, often both expressed as deep principled philosophies, do you naturally gravitate toward the idea that one of the two paths should be correct and we should stick to it, or do you prefer to find a way in the middle between the two extremes?

在世界上人們如何做出大規模決策時，我注意到的主要哲學差異之一是他們如何處理妥協與純潔之間古老的權衡。如果在兩種選擇之間做出選擇，通常都表達為深刻的原則哲學，你會自然而然地傾向於認為兩條道路中的一條應該是正確的並且我們應該堅持它，還是你更喜歡在兩者之間找到一條路兩個極端？

In mathematical terms, we can rephrase this as follows: do you expect the world that we are living in, and in particular the way that it responds to the actions that we take, to fundamentally be concave or convex?

用數學術語來說，我們可以將其改寫如下：您是否期望我們生活的世界，尤其是它對我們採取的行動做出反應的方式，從根本上是凹的還是凸的？


Someone with a concave disposition might say things like this:

性格內向的人可能會這樣說：

* "Going to the extremes has never been good for us; you can die from being too hot or too cold. We need to find the balance between the two that's just right"

* 「走極端對我們從來都沒有好處；你可能會死於太熱或太冷。我們需要在兩者之間找到恰到好處的平衡點」

* "If you implement only a little bit of a philosophy, you can pick the parts that have the highest benefits and the lowest risks, and avoid the parts that are more risky. But if you insist on going to the extremes, once you've picked the low-hanging fruit, you'll be forced to look harder and harder for smaller and smaller benefits, and before you know it the growing risks might outweigh the benefit of the whole thing"

* 「如果你只實施一點點哲學，你可以選擇收益最高和風險最低的部分，而避免風險較大的部分。但如果你堅持走極端，一旦你已經摘下了唾手可得的果實，你將被迫越來越努力地尋找越來越小的好處，在你意識到這一點之前，不斷增加的風險可能會超過整個事情的好處」

* "The opposing philosophy probably has some value too, so we should try to combine the best parts of both, and definitely avoid doing things that the opposing philosophy considers to be extremely terrible, just in case"

* 「對立的哲學可能也有一些價值，所以我們應該盡量把兩者最好的部分結合起來，絕對避免做對立哲學認為極其糟糕的事情，以防萬一」

Someone with a convex disposition might say things like this:

脾氣暴躁的人可能會這樣說：

* "We need to focus. Otherwise, we risk becoming a jack of all trades, master of none"

* 「我們需要集中註意力。否則，我們就有可能成為萬事通，一無是處」

* "If we take even a few steps down that road, it will become slippery slope and only pull us down ever further until we end up in the abyss. There's only two stable positions on the slope: either we're down there, or we stay up here"

* 「如果我們沿著那條路走幾步，它就會變成濕滑的斜坡，只會把我們拖得更遠，直到我們最終陷入深淵。斜坡上只有兩個穩定的位置：要么我們在下面，要么我們留在這裡」

* "If you give an inch, they will take a mile"

* 「如果你給一英寸，他們會採取一英里」

* "Whether we're following this philosophy or that philosophy, we should be following some philosophy and just stick to it. Making a wishy-washy mix of everything doesn't make sense"

* 「無論我們遵循這種哲學還是那種哲學，我們都應該遵循某種哲學並堅持下去。把所有東西都混為一談是沒有意義的」

I personally find myself perenially more sympathetic to the concave approach than the convex approach, across a wide variety of contexts. If I had to choose either (i) a coin-flip between anarcho-capitalism and Soviet communism or (ii) a 50/50 compromise between the two, I would pick the latter in a heartbeat. I argued for moderation in Bitcoin block size debates, arguing against both 1-2 MB small blocks and 128 MB "very big blocks". I've argued against the idea that freedom and decentralization are "you either have it or you don't" properties with no middle ground. I argued in favor of the DAO fork, but to many people's surprise I've argued since then against similar "state-intervention" hard forks that were proposed more recently. As I said in 2019, "support for Szabo's law [blockchain immutability] is a spectrum, not a binary".

我個人發現，在各種各樣的情況下，我自己對凹面法比凸面法更贊同。如果我不得不選擇 (i) 無政府資本主義和蘇聯共產主義之間的擲硬幣或 (ii) 兩者之間的 50/50 妥協，我會毫不猶豫地選擇後者。我主張在比特幣塊大小辯論中適度，反對 1-2 MB 的小塊和 128 MB 的「非常大的塊」。我反對這樣一種觀點，即自由和權力下放是「你要么擁有要么沒有」的屬性，沒有中間立場。我支持 DAO 分叉，但令許多人驚訝的是，從那時起我就反對最近提出的類似「國家干預」硬分叉。正如我在 2019 年所說，「對 Szabo 定律 [區塊鏈不變性] 的支持是一個範圍，而不是二進制」。

But as you can probably tell by the fact that I needed to make those statements at all, not everyone seems to share the same broad intuition. I would particularly argue that the Ethereum ecosystem in general has a fundamentally concave temperament, while the Bitcoin ecosystem's temperament is much more fundamentally convex. In Bitcoin land, you can frequently hear arguments that, for example, either you have self-sovereignty or you don't, or that any system must have either a fundamentally centralizing or a fundamentally decentralizing tendency, with no possibility halfway in between.

但正如您可能從我根本需要發表這些聲明這一事實中看出的那樣，似乎並不是每個人都有相同的廣泛直覺。我特別要指出的是，以太坊生態系統總體上具有基本的凹面氣質，而比特幣生態系統的氣質基本是凸面的。在比特幣領域，你經常會聽到這樣的爭論，例如，你要么擁有自主權，要么沒有，或者任何系統都必須具有根本上的中心化或根本上去中心化的趨勢，而不可能介於兩者之間。

The occasional half-joking support for Tron is a key example: from my own concave point of view, if you value decentralization and immutability, you should recognize that while the Ethereum ecosystem does sometimes violate purist conceptions of these values, Tron violates them far more egregiously and without remorse, and so Ethereum is still by far the more palatable of the two options. But from a convex point of view, the extremeness of Tron's violations of these norms is a virtue: while Ethereum half-heartedly pretends to be decentralized, Tron is centralized but at least it's proud and honest about it.

偶爾半開玩笑地支持 Tron 是一個關鍵的例子：從我自己的凹面角度來看，如果你重視去中心化和不變性，你應該認識到雖然以太坊生態系統有時確實違反了這些價值觀的純粹主義概念，但 Tron 更違反了它們令人震驚且毫無悔意，因此到目前為止，以太坊仍然是這兩種選擇中更受歡迎的一種。但從凸面的角度來看，Tron 違反這些規範的極端行為是一種美德：雖然以太坊半心半意地假裝是去中心化的，但 Tron 是中心化的，但至少它對此感到自豪和誠實。

This difference between concave and convex mindsets is not at all limited to arcane points about efficiency/decentralization tradeoffs in cryptocurrencies. It applies to politics (guess which side has more outright anarcho-capitalists), other choices in technology, and even what food you eat.

凹凸思維之間的這種差異並不局限於關於加密貨幣效率/去中心化權衡的神秘觀點。它適用於政治（猜猜哪一方有更多徹頭徹尾的無政府資本主義）、技術的其他選擇，甚至你吃的食物。


But in all of these questions too, I personally find myself fairly consistently coming out on the side of balance.

但在所有這些問題中，我個人也發現自己相當一致地站在平衡的一邊。

#### Being concave about concavity

關於凹凸

But it's worth noting that even on the meta-level, concave temperament is something that one must take great care to avoid being extreme about. There are certainly situations where policy A gives a good result, policy B gives a worse but still tolerable result, but a half-hearted mix between the two is worst of all. The coronavirus is perhaps an excellent example: a 100% effective travel ban is far more than twice as useful as a 50% effective travel ban. An effective lockdown that pushes the R0 of the virus down below 1 can eradicate the virus, leading to a quick recovery, but a half-hearted lockdown that only pushes the R0 down to 1.3 leads to months of agony with little to show for it. This is one possible explanation for why many Western countries responded poorly to it: political systems designed for compromise risk falling into middle approaches even when they are not effective.

但值得注意的是，即使在元層面上，凹氣質也是一個人必須非常小心避免極端的東西。當然，在某些情況下，策略 A 會產生良好的結果，而策略 B 會產生較差但仍然可以接受的結果，但兩者之間三心二意的混合是最糟糕的。冠狀病毒也許是一個很好的例子：100% 有效的旅行禁令比 50% 有效的旅行禁令有用得多。將病毒的 R0 降至 1 以下的有效封鎖可以根除病毒，從而導致快速恢復，但半心半意的封鎖只會將 R0 降至 1.3，這會導致數月的痛苦，但收效甚微。這是為什麼許多西方國家對此反應不佳的一種可能解釋：為妥協而設計的政治制度有可能陷入中間方法，即使它們並不有效。

Another example is a war: if you invade country A, you conquer country A, if you invade country B, you conquer country B, but if you invade both at the same time sending half your soldiers to each one, the power of the two combined will crush you. In general, problems where the effect of a response is convex are often places where you can find benefits of some degree of centralization.

另一個例子是一場戰爭：如果你入侵 A 國，你就征服了 A 國，如果你入侵了 B 國，你就征服了 B 國，但是如果你同時入侵兩個國家，就派一半的士兵去每個國家，兩個國家的力量結合起來會壓垮你。通常，在響應效果是凸的問題中，您通常可以找到某種程度的集中化的好處。

But there are also many places where a mix is clearly better than either extreme. A common example is the question of setting tax rates. In economics there is the general principle that deadweight loss is quadratic: that is, the harms from the inefficiency of a tax are proportional to the square of the tax rate. The reason why this is the case can be seen as follows. A tax rate of 2% deters very few transactions, and even the transactions it deters are not very valuable - how valuable can a transaction be if a mere 2% tax is enough to discourage the participants from making it? A tax rate of 20% would deter perhaps ten times more transactions, but each individual transaction that was deterred is itself ten times more valuable to its participants than in the 2% case. Hence, a 10x higher tax may cause 100x higher economic harm. And for this reason, a low tax is generally better than a coin flip between high tax and no tax.

但也有很多地方的混合明顯好於任何一個極端。一個常見的例子是設定稅率的問題。在經濟學中有一個普遍原則，即無謂損失是二次方的：也就是說，稅收效率低下造成的危害與稅率的平方成正比。出現這種情況的原因如下。 2% 的稅率阻止的交易很少，甚至它阻止的交易也不是很有價值——如果僅 2% 的稅率就足以阻止參與者進行交易，那麼交易能有多大價值？ 20% 的稅率可能會阻止十倍的交易，但每筆被阻止的交易本身對其參與者的價值是 2% 情況下的十倍。因此，10 倍的高稅收可能會導致 100 倍的經濟損失。出於這個原因，低稅通常比在高稅和不稅之間擲硬幣要好。

By similar economic logic, an outright prohibition on some behavior may cause more than twice as much harm as a tax set high enough to only deter half of people from participating. Replacing existing prohibitions with medium-high punitive taxes (a very concave-temperamental thing to do) could increase efficiency, increase freedom and provide valuable revenue to build public goods or help the impoverished.

根據類似的經濟邏輯，完全禁止某些行為可能造成的危害是設置高到足以阻止一半人參與的稅收的兩倍多。用中等高的懲罰性稅收取代現有的禁令（一種非常凹的氣質的事情）可以提高效率，增加自由並提供寶貴的收入來建設公共產品或幫助窮人。

Another example of effects like this in Laffer curve: a tax rate of zero raises no revenue, a tax rate of 100% raises no revenue because no one bothers to work, but some tax rate in the middle raises the most revenue. There are debates about what that revenue-maximizing rate is, but in general there's broad agreement that the chart looks something like this:

拉弗曲線中類似效果的另一個例子：零稅率不會增加收入，100% 的稅率不會增加收入，因為沒有人願意工作，但中間的一些稅率會增加最多的收入。關於收入最大化率是多少存在爭論，但一般來說，人們普遍認為圖表看起來像這樣：


If you had to pick either the average of two proposed tax plans, or a coin-flip between them, it's obvious that the average is usually best. And taxes are not the only phenomenon that are like this; economics studies a wide array of "diminishing returns" phenomena which occur everywhere in production, consumption and many other aspects of regular day-to-day behavior. Finally, a common flip-side of diminishing returns is accelerating costs: to give one notable example, if you take standard economic models of utility of money, they directly imply that double the economic inequality can cause four times the harm.

如果你不得不選擇兩個提議的稅收計劃的平均值，或者在它們之間擲硬幣，很明顯平均值通常是最好的。稅收並不是唯一這樣的現象；經濟學研究廣泛的「收益遞減」現象，這些現像在生產、消費和日常行為的許多其他方面無處不在。最後，收益遞減的一個常見反面是加速成本：舉一個值得注意的例子，如果你採用貨幣效用的標準經濟模型，它們直接意味著雙倍的經濟不平等會造成四倍的傷害。

#### The world has more than one dimension

世界不止一個維度

Another point of complexity is that in the real world, policies are not just single-dimensional numbers. There are many ways to average between two different policies, or two different philosophies. One easy example to see this is: suppose that 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?

另一個複雜點是，在現實世界中，策略不僅僅是一維數字。有很多方法可以在兩種不同的政策或兩種不同的哲學之間進行平均。一個簡單的例子是：假設你和你的朋友想住在一起，但你想住在多倫多而你的朋友想住在紐約。你會如何在這兩個選項之間做出妥協？

Well, you could take the geographic compromise, and enjoy your peaceful existence at the arithmetic midpoint between the two lovely cities at....

好吧，你可以採取地理上的妥協，在兩個可愛城市之間的算術中點享受你的和平生活……


This Assembly of God church about 29km southwest of Ithaca, NY.

這座神召會教堂位於紐約州伊薩卡西南約 29 公里處。

Or you could be even more mathematically pure, and take the straight-line midpoint between Toronto and New York without even bothering to stay on the Earth's surface. Then, you're still pretty close to that church, but you're six kilometers under it. A different way to compromise is spending six months every year in Toronto and six months in New York - and this may well be an actually reasonable path for some people to take.

或者你可以在數學上更加純粹，甚至不必費心留在地球表面就可以取多倫多和紐約之間的直線中點。然後，你仍然非常靠近那個教堂，但你在它下面六公里處。另一種妥協方式是每年在多倫多呆六個月，在紐約呆六個月——這對某些人來說可能是一條真正合理的道路。

The point is, when the options being presented to you are more complicated than simple single-dimensional numbers, figuring out how to compromise between the options well, and really take the best parts of both and not the worst parts of both, is an art, and a challenging one.

關鍵是，當呈現給你的選項比簡單的一維數字更複雜時，弄清楚如何在選項之間很好地妥協，並真正利用兩者最好的部分而不是兩者最壞的部分，是一門藝術, 和一個具有挑戰性的。

And this is to be expected: "convex" and "concave" are terms best suited to mathematical functions where the input and the output are both one-dimensional. The real world is high-dimensional - and as machine-learning researchers have now well established, in high-dimensional environments the most common setting that you can expect to find yourself in is not a universally convex or universally concave one, but rather a saddle point: a point where the local region is convex in some directions but concave in other directions.

這是意料之中的：「凸」和「凹」是最適合輸入和輸出均為一維的數學函數的術語。現實世界是高維的 - 正如機器學習研究人員現在已經確定的那樣，在高維環境中，您可以期望發現自己所處的最常見環境不是普遍凸起或普遍凹陷的環境，而是馬鞍點：局部區域在某些方向上凸出但在其他方向上凹入的點。


A saddle point. Convex left-to-right, concave forward-to-backward.

一個鞍點。從左到右凸，從前到後凹。

This is probably the best mathematical explanation for why both of these dispositions are to some extent necessary: the world is not entirely convex, but it is not entirely concave either. But the existence of some concave path between any two distant positions A and B is very likely, and if you can find that path then you can often find a synthesis between the two positions that is better than both.

對於為什麼這兩種傾向在某種程度上都是必要的，這可能是最好的數學解釋：世界不完全是凸的，但也不完全是凹的。但是在任何兩個遠距離位置 A 和 B 之間很可能存在一些凹路徑，如果你能找到那條路徑，那麼你通常可以在兩個位置之間找到比兩者都更好的綜合。