In-depth Exploration of the Linear and Non-linear Properties of Decentralization Trading Operators

When developing the decentralized exchange (DEX), the core task is to design a trading operator. This operator can be linear or nonlinear. Similarly, when designing an interest rate operator, it is essentially also designing a trading operator, with the same distinction between linear and nonlinear. However, this distinction may not be easy for most people to understand.

The characteristic of linear trading operators is that they trade using equilibrium prices, and the asset portfolio only undergoes simple linear transformations at this price. The reason for using linear operators is the acceptance of the no-arbitrage assumption. In this case, rational financial trades should be linear. If non-linear results occur, then the asset portfolio may present arbitrage opportunities or be unpriced. In principle, trading models using oracle should adopt linear trading operators; otherwise, arbitrage may occur. From another perspective, in a complete market with effective pricing, only linear trading operators can achieve no arbitrage.

However, linear operators also have their limitations. It implies that any trading pool is equal, and this operator cannot achieve tokenization. This is because linear operators are completely identical after being replicated and cannot capture unique value on the chain. Imagine that when every on-chain asset accepts a given equilibrium price, these assets are equivalent in any contract. Therefore, any trading contract or operator struggles to capture additional value and achieve tokenization.

In contrast, non-linear trading operators attempt to simultaneously achieve three goals: pricing, trading, and value accumulation (tokenization). The design of non-linear operators is more open and can theoretically be designed with scale-related self-enhancing properties to accumulate value. However, this approach also faces some challenges:

1. As the market gradually matures, non-linear trading operators essentially fit linear operators within a very small trading scale.
2. When the market is incomplete, is the design cost and efficiency of nonlinear trading operators sufficiently excellent?
3. Who will provide the value input for nonlinear operators? Will this value input gradually diminish under the competition of linear trading operators?

Many existing automated market makers (AMM) adopt a fixed product trading model (such as XY=K), which is a typical scale-related nonlinear trading operator. Linear trading can only be locally simulated when the market maker's pool is large enough. However, if the trading objects of the AMM are a complete market, its core meaning lies only in the fitting effectiveness after scale effects, which is not very essential.

A common misconception is the hope of placing pricing power on-chain. However, when the market is complete (supply and demand are extremely large and difficult to manipulate), the advantages of centralized exchanges become very obvious. Every on-chain action is the result of an auction, which creates a huge gap with the demand for pricing trading services. Pricing trading is an extreme activity; even ordinary centralized exchanges have the highest requirements for computation, storage, and communication, let alone the discreteness and auction attributes on-chain.

For incomplete markets (such as tail assets or new projects), the core requirement should be to form prices quickly and at a low cost, while completing a relatively large number of transactions. The main constraints are the costs of quickly forming prices and the costs of completing large-scale transactions. Here, costs do not refer to marketing or traffic costs, but to the costs generated within the transaction operator.

Non-linear trading operators handle pricing and trading simultaneously, while also facing competition from the linear trading models that accept oracle (price operator). In this competition, at least in terms of trading efficiency, the linear trading operators under oracles far surpass non-linear trading operators.

Non-linear trading operators also face the issue of value input. In a complete market, a large number of small transactions are needed to input value, compensating for the arbitrage losses of non-linear operators during equilibrium price fluctuations. This condition is very stringent, as the increasing marginal cost of on-chain transactions may eliminate a large amount of small demand. In a highly incomplete market, although there may be a large number of traders indifferent to price slippage, any non-linear operator can meet the trading demand, and it is important to complete as many transactions as possible.

Based on the above analysis, the non-linearization of trading operators is not a particularly valuable direction. Among the protocols that embed decentralized value on-chain, non-linear trading operators may not be the type of non-linear operators we should pursue.

It is worth noting that the interest rate operator, as a special trading operator, is slightly different from pure secondary market buying and selling transactions. This difference arises from the difficulty of interest rate arbitrage due to the lack of sufficient term structure trading markets to achieve arbitrage. Currently, the interest rate market on the blockchain is still very thin and has not reached a level of effective trading. In the absence of good interest rate oracles, using nonlinear operators for interest rate pricing holds some value, but this is more of a stopgap measure rather than an essential innovation.

Non-linear trading operators can also be improved, for example by introducing recursive information, which captures valuable components from historical transaction data to reduce arbitrage risk. Currently, there is little market research in this area, but some have realized that combining recursive operators with non-linear trading operators can help mitigate issues such as impermanent loss in current DEX.

The future challenge lies in conducting in-depth analysis of the core risks behind each operator and clearly modeling the trading objectives. This requires unifying all financial services under operator theory to obtain more effective mathematical equations, thereby improving the effectiveness and completeness of product design and promoting the development of the on-chain financial world.