What is the essence of DeFi? In-depth understanding of the mathematical principles of DeFi economics

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DeFi is not completing the linear transformation of the underlying securities, but looking for a new operator to capitalize it.

Written by: Banach

The value of blockchain

“What do you want to do on the chain? What can you do?” This is an essential and core issue that is often downplayed.

Assuming that the positive feedback structure of the blockchain remains unchanged, then the marginal cost of each behavior that occurs on the chain should increase relative to the legal currency, which is a basic constraint. On the contrary, if it decreases relative to the legal currency, it will enter the negative cycle that everyone often says, which is very different from the traditional world.

The reason why the general service needs to be completed in a centralized way is because its marginal cost is diminishing, otherwise it should not be centralized to solve it, or there is no advantage. The cost here includes not only explicit, but also implicit, such as risk. Of course, the cost can also remain unchanged on an expected basis.

It is almost impossible for the price of the blockchain system to achieve long-term stability (for example, the long-term inflation rate of legal currency and the inflation rate of ETH remain the same). Unless fiat currency is issued at Friedman’s fixed growth rate, coincidence is a small probability event. The growth rate of legal currency must exceed ETH, or the growth rate of its demand needs to exceed the issuance speed of ETH. The demand here not only refers to the chain, but also includes financial needs, such as secondary and mortgage lending. This has nothing to do with performance, it is determined by the blockchain mechanism, and this mechanism creates more certainty. In essence, the certainty of blockchain is higher than that of legal currency, and this certainty even exists in a relatively deflationary way. Performance is determined by the technical basis, inflation is determined by the economic basis, and certainty is determined by the game basis. The first two determine the degree, and the last one determines the essence. This kind of game structure determines the certainty of certain performances in equilibrium, such as the non-tamperable data or the cost of tampering. It can be said that for this certainty, people are willing to pay higher costs. But it should be the other way around. The unit cost of this deterministic payment is higher than any centralization. Such a mechanism automatically forms this structure. This is the charm of game equilibrium. Participants did not aim for this result, but in the end they achieved this result without the participants knowing it. This game structure is like using a function to approach an absolute certainty infinitely. Data on the chain is limited, and the cost per unit of data is getting more and more expensive, which makes change almost impossible. The core of centralization is its scale effect, the unit cost is increasing, and the intrinsic value is also increasing.

The realization of equilibrium is a dynamic process. Here is a question worth thinking about, why would anyone want to buy in the secondary market? If there is no secondary market, then BTC is also closed loop. Every miner has a judgment on the intrinsic value of BTC, and then chooses a computing power to play with everyone. The game here is not the kind of chess, but refers to everyone’s behavior affecting each other. One of the most important factors is something called a random operator . Without this random operator, the game is difficult to unfold. Because the data cannot be tampered with, in addition to the miners themselves who feel that BTC has a certain value, buyers will also choose BTC based on this performance (including easy storage and transfer). In addition to this, a deterministic value anchor is needed, that is, the game between pricing in the secondary market.

Assuming that buyers buy BTC based on the certainty of value, this demand can directly push up the price of BTC, and then a steady stream of people enter, BTC enters a positive cycle , which strengthens its intrinsic value . This form has never appeared before, and is not a simple so-called exponential thing. Many value structures used to be linear, but this is a non-linear structure . So I think BTC is at the same level as the market mechanism, not at the same level as the enterprise. There is no counterpart yet, and it may be more like a chain reaction . Most of the research on game theory is split around market mechanisms. Random operators are not difficult, but it is more difficult to combine them to form meaningful things, and they can easily become gambling. Therefore, the value lies in the blockchain, not the random operators. Random operators can generate a gamble, but the gamble has no value and cannot attract risk aversion. The idea of ​​blockchain is meaningful. The data is processed by algorithms to ensure the coupling between data (through HASH to ensure the consistency between data and data), so that the cost of tampering increases with the depth of the game .

In the past, data was difficult to automatically confirm rights, and it was easy to copy. Only by controlling the use environment of the data can it be ensured that the data is not copied, and asymmetric encryption also ensures the decentralized confirmation of the data (the higher the attack cost, the higher its value) . BTC has solved many problems, and these problems are also realized through this decentralized GAME. BTC has built such a foundation. On this basis, ETH provides a more open data structure, and smart contracts make transfers a state function.

What else can we do on top of the functions and balance that ETH has achieved?

The essence of DeFi

I used to think that ordinary programmers can make BTC when they get a basic idea, but now it is impossible. Only very deep thinking can find these components. The following changes are not known for their value, but under the structure of Satoshi Nakamoto, this foundation is not easy to obtain. In addition to implementing simple calculation functions, it is a new data structure. I prefer to think of BTC as an advancement in storage technology. AI and quantum computing are advancements in computing technology. 5G currently feels that the theory is not good enough, so quantum communication must be developed. Computing, storage and communication are probably the basic elements of the universe.

Converging to DeFi, blockchain finance, what are they doing? Finance is essentially a linear transformation of the underlying securities portfolio. There are two goals to be achieved here:

  1. Linear transformation of underlying securities;
  2. Form more basic securities.

If the underlying security has been determined, the nonlinear transformation becomes meaningless. Here we will begin to describe it in mathematics.

Let me explain a few terms first. Basic securities: represent the basic risks of a given return space. Each basic risk corresponds to a basic security. If there is risk, there is a corresponding return. To be precise, each basic risk and return constitutes a basic security, and we only divide it according to risk rather than income. There are only several types of returns in a given income space, and there are also several corresponding risks. Then a two-dimensional matrix is ​​formed, and this matrix is ​​the matrix of the underlying securities. If a point is used to describe a basic security, then all the elements of this matrix form a vector.

For example, there are 12 elements, representing the basic securities, forming a basic securities vector S, financial services is STP=Y, or change S to X, X is the vector you trade; X=SK, K is the coefficient matrix. Any X can be expressed as SK, which is called a complete market . T is a linear transformation and P is a price vector. Each risk-return portfolio represents a basic security, and as many risk-return possibilities there are, how many return structures can be combined. The real world is infinite dimensional , so simplify the processing here. In infinite dimensionality, it is generally described by an operator, and T is an operator. X=SK, any vector can be expressed as a linear combination of vector space basis . Linear algebra generally studies linear transformations. When it comes to abstract linear spaces, they start to study operators, because the dimensionality may be infinite, and infinite dimensional spaces cannot be simply represented by matrices. They are generally called operators. Simply put, how to compress an infinite dimension to a finite dimension without losing a lot of information is somewhat similar to the eigenvalues ​​of the eigenvectors in the line algebra. This is called the spectral theory in operator algebra. Because there are many representations, just like spectra.

The most important problem is the problem of Y and T. The exact value should be SKTP (K is the coefficient matrix of any security expressed as the underlying security). Because what you are trading is not necessarily the underlying securities, but may be arbitrary securities. Any securities are represented as basic securities, transactions or financial services, not necessarily triggered from the underlying securities. But it is ultimately equivalent to the combination of the underlying securities and the coefficient matrix. And this combination is linear. It is assumed that P is effectively priced, or called the equilibrium price. Assuming effective pricing, why is T linear? If it is non-linear, what Y is called, in fact, the truth is very simple.

A stock is sold for money at the price, XTP=Y. Where X is the stock, T is the trading operator, P is the price, and Y is the money. In the case of effective pricing, T must be linear, otherwise 1+1 will be greater than or less than 2. If the transaction is reversible, that is, T is a reversible matrix, no matter it is greater or less than, there will be: risk-free arbitrage . This is the law of linear pricing . If T is non-linear, then Y is called an unpriceable asset. To give a simple example, you can get a WETH by hitting an ETH, and then hitting a WETH in exchange for an ETH. This is the simplest linear transformation. WETH must be priced and equal to 1ETH. If you get 1 ETH if you hit one ETH, and 4 get 4 if you hit 2 ETH, square production (non-linear pricing), then this WETH is not priceable. All derivatives are priced linearly, otherwise it would be impossible to price assets. The meaning of linear operator, a_T+b_T=(a+b)*T. Non-linearity cannot be priced. Linear pricing is a proper term. Non-linearity must be arbitrage.

Let’s simplify it and say STP=Y. The interesting part here is how P guarantees that it is a decentralized oracle, which is what NEST does. Secondly, if T is linear, how to avoid being copied? Either there is a perfect pricing operator on the chain to generate P, or there is a decentralized oracle. The first one does not exist, because the characteristics of the blockchain determine that the pricing power must be off-chain. The pricing here refers to effective pricing, there is no such problems as arbitrage or information advantage, incomplete market, etc. This oracle machine can generate a new asset, which may not be linearly synthesized from the underlying securities of the originally assumed securities space. That is, NEST has created a new basic security. When we borrowed, that is, NEST1.0 tried to capitalize the loan, and finally we had to create a non-linear structure, deliberately convergent compression (deliberately compressed the relationship between transaction volume and mining into a non-linear structure). Because it is linear, it is immediately equal to the linear relationship of ETH. It is the attenuation coefficients at that time. The handling fee and mining output you pay are related to the output of 300 blocks. This deliberately formed nonlinearity is problematic because the operator is linear. Simply put, the linear operator cannot be GAME. Because only GAME can make the linear operator equivalent to an asset, this asset is not a combination of underlying securities, this asset does not exist, it will be copied, and there is no return.

What are you doing on the chain? It is not completing the linear transformation of the basic securities, but looking for a new operator to capitalize it and form a new basic securities to create new original assets. Because linear transformations can be simply copied, they do not have any uniqueness, cannot capture value, and therefore have no revenue structure. Linear transformation cannot generate new assets. Linear transformation can be a meaningful general agreement. In the real world, an operator is not an asset, and an intermediary guarantees the feasibility of the operator, thereby charging fees. On-chain operators are completely open and credible and can be copied at any time, so they cannot capture value. STP=Y, in fact, there are some structures, such as SO=Y, which means that there is no equilibrium price, so there is no need for a linear transformation. O is a non-linear operator, which directly combines the original securities and a non-linear operator. SO=Y has barriers, because in principle: SO=Y, this Y is a non-priceable asset and will definitely be arbitrage. But if you consider the infinitesimal part, O may be close to linear. This is the design idea of ​​Uniswap. In fact, there is no need to consider XY=K, but to see that most simple functions are close to linear in an infinitesimal part. This infinitesimal is relative. Therefore, the larger the pool, the closer the local small transactions are to linear pricing, which constitutes a competitive advantage. That is, non-linear operators automatically have a GAME structure (in popular terms: 1, free entry and exit 2, strategies are related to others, 3 can be accumulated to form equilibrium characteristics ), the reason is actually partial linearization. If you use the operator’s thinking XY=K, it can be optimized a lot. After the nonlinear operator forms an internal correlation, the larger the scale, the stronger the linear characteristic of the local transaction, the stronger the linear characteristic, the higher the value, and the more attractive. This kind of nonlinear operator must automatically make a market to form a pool, so that the transaction and the pool have correlation.

Let’s talk about Compound. Mortgage operator is a linear operator, but the reason Compound can form GAME is because of the lack of interest rate oracle, so it must use the non-linear operator of interest rate. If Compound is to form an asset, it must not be related to mortgages, but to interest rates. But everyone is not sensitive to interest rates, so it is difficult to find a good asset structure from Compound. Interest rates are related to scale, and interest rates should be generated by oracles and cannot be related to scale. Just like Uniswap, the price should be generated by an external pricing machine, not related to scale.

to sum up

What DeFi has to do is to find more non-linear operators, and to classify these non-linear operators, find the basic structure, and find the basic securities of the operator. Except for intermediate transition type linear operators (such as transaction operators), linear operators must not be combined with nonlinear operators.

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