Recursive Operator: The Mathematical Magic and Challenges Behind Algorithmic Stablecoin

The appeal of Algorithmic Stablecoins lies in their novelty and the promise of Decentralization, but this appeal often stems from a misunderstanding of Blockchain and the nature of currency. Among them, the introduction of recursive operators is a key factor, as it brings new possibilities for on-chain operations.

The recursive operator, in the continuous transformation of smart contracts, takes the previous state as input and loops to generate the next state. This structure naturally forms in a blockchain environment due to the openness of on-chain data and the serial design of smart contracts, creating a time series. Recursive processing can produce nonlinear structures and even geometric series effects, resulting in strong positive feedback characteristics.

However, simple time series recursion is not ideal because it lacks uncertainty. What truly deserves attention is the multiple recursive operators, which introduce new, unpredictable game information between state changes. This unpredictability interacts with the recursive operators to form a controllable expectation attribute.

Taking algorithmic stablecoins as an example, the pricing operator generates the price Pt, and the expansion total Mt serves as a multiple recursive operator. Mt is a function of Pt, while Pt+1 depends on Mt, forming an indirect recursive relationship. This design aims to achieve price stability through negative feedback, but its accuracy and efficiency are limited due to reliance on the supply and demand relationship of the secondary market.

Recursive operators can also provide positive feedback, such as the repurchase mechanism in certain systems. This mechanism can lead to a reduction in market supply, an increase in prices, improved performance, an increase in demand, an increase in revenue, and an increase in repurchases, creating a self-reinforcing cycle.

From a mathematical perspective, it is still unclear whether recursive operators can construct stable short-cycle properties. Especially for algorithmic stablecoins, due to their indirect influence on supply and demand relationships by changing the total amount, the transmission process is relatively slow, making it even more difficult to achieve stable equilibrium.

In multiple recursive operators, the step of introducing new information is crucial. The general equilibrium properties of Blockchain help to introduce more information, which has a certain degree of uncertainty under specific game structures. However, this uncertainty exists within a unified framework, which can easily create an illusion of stability. Without a rigorous game theory analysis, it is difficult to accurately grasp the overall equilibrium properties.

When designing recursive operators, it is important to pay attention to the frequency of information introduction. Excessive introduction of information can weaken the effect of the recursive operator. If the goal is to enhance positive and negative feedback, the introduction of new information should be reduced; if the goal is long-cycle regression, the information flow itself should also have periodicity.

Most recursive operators in the DeFi space are combined with price sequences, as price games concentrate a large amount of information and are difficult to predict or control. However, there is currently a reliance on AMM mechanisms rather than decentralized oracles, which may lead to the recursive process becoming predictable or controllable, contradicting the original design intent.

In the future, the application of recursive operators may expand to more variables, particularly parameters that reflect the difficulty of the overall market game. When designing DeFi, a detailed analysis of the information transmission mechanism of recursive operators should be conducted to avoid being predicted and controlled. There remains a large series of nonlinear operators in this field that are worth exploring in depth.