Thoughts on the Design of DEX Trading Operators

The core of DEX development lies in designing a trading operator, which can be linear or nonlinear. Similarly, when designing interest rate operators, there is also a distinction between linear and nonlinear, but this distinction is often not easy to understand.

Linear operators use equilibrium prices, and trading is merely a simple linear transformation of asset portfolios. Using equilibrium prices means accepting the no-arbitrage assumption, under which reasonable financial transactions should be linear. If non-linear results occur, it will lead to asset portfolios that are either unpriceable or that present arbitrage opportunities. In principle, trading models that use oracle should adopt linear operators; otherwise, they are prone to arbitrage. In other words, only linear operators can achieve no-arbitrage in a complete market with effective pricing.

However, linear operators also have limitations: any liquidity pool is treated equally, and this operator cannot achieve tokenization. This is because linear transformations are equivalent in any contract and cannot capture value in specific contracts. Non-linear operators, on the other hand, attempt to simultaneously achieve three goals: pricing, trading, and value accumulation ( tokenization ).

Non-linear operators can be designed with scale-related self-enhancing properties, thereby accumulating value. However, this also brings some issues: when the market gradually becomes complete, non-linear operators are essentially fitting linear operators within extremely small trading scales; when the market is incomplete, whether the design cost and efficiency of non-linear operators are sufficient; and who provides the value input of non-linearity, whether this value input will gradually diminish under the competition of linear operators.

When the market is complete, arbitrage trading is linear. Therefore, the validity of nonlinear operators depends on market efficiency. Once the market is sufficiently complete, contracts using nonlinear operators are essentially fitting linear operators within a very small range. Many AMMs adopt a fixed product trading model (, such as XY=K), which is a typical scale-related nonlinear operator. Only when the liquidity provider's pool is sufficiently large can linear trading be simulated locally.

Some people hope to place pricing power on the chain, but this may be an illusion. In a complete market, the advantages of centralized exchanges are more pronounced. Every on-chain action is a product of an auction, which creates a huge gap with the demand for pricing trading services. For incomplete markets ( such as new projects ), the key demand is to quickly form prices at low cost and complete large-volume transactions, with the main constraints being the cost of quickly forming prices and the cost of completing large-scale transactions.

Non-linear trading operators handle pricing and trading simultaneously but must compete with linear trading models that accept oracle inputs. In terms of trading efficiency, linear operators under oracles far outperform non-linear operators. The remaining comparable advantages mainly lie in pricing cost and efficiency, but intuitively, linear operators also hold an advantage.

The value input problem of nonlinear trading operators is also crucial. In a complete market, a large number of small transactions are needed to compensate for the arbitrage losses of nonlinear operators during equilibrium price fluctuations, which is a stringent condition. In a highly incomplete market, any nonlinear operator can meet trading demands, with the focus on completing as many transactions as possible, which then transforms into a quasi-linear model.

In summary, the nonlinearization of trading operators is not a valuable direction. In protocols that settle decentralized value on-chain, nonlinear trading operators may not be the type of nonlinear operators we need to seek.

The interest rate operator, as a special trading operator, is slightly different due to the difficulties of interest rate arbitrage. Currently, the interest rate market on the blockchain is still very thin, lacking good interest rate oracles. Therefore, using nonlinear operators for interest rate pricing has certain value, but this is more of a stopgap measure.

Non-linear trading operators can also be improved, for example by introducing recursive information to reduce arbitrage risk. Research in this area is currently scarce, but some have realized the potential to combine recursive operators and non-linear trading operators to mitigate issues such as impermanent loss in DEX. The key lies in a deep analysis of the core risks behind each operator and clear modeling of trading objectives. This is precisely the direction some communities are committed to: unifying all financial services under operator theory, obtaining more effective mathematical equations, making product design more efficient and complete, and promoting the development of the on-chain financial world.