Blockchain and Algorithmic Stablecoins: Analyzing the Potential and Limitations of Recursive Operators

Algorithmic stablecoins have recently attracted widespread attention, with many believing they could achieve the goals that Bitcoin has failed to accomplish: a fully decentralized and self-regulating global currency. This idea arises not only from a limited understanding of blockchain and currency concepts but also because algorithmic stablecoins introduce novel recursive operators.

The recursive operator refers to an operation in continuous transformations of smart contracts, where the previous state is used as input to repeatedly generate the next state. This design utilizes the data transparency of the Blockchain and the serial characteristics of smart contracts to form a non-linear structure, which may even produce a geometric series effect. This strong positive feedback feature completely aligns with the self-reinforcing properties of on-chain games.

However, pure time series recursion is not an ideal choice, as it makes future states completely determined by the current state. What is truly worth noting is the combination of recursive operators with other elements, introducing new information between state changes. This new information reflects game-theoretic properties, possessing unpredictability, while also being influenced by recursive operators to form a certain common expectation. This complex interaction can be referred to as multiple recursive operators.

Taking common algorithmic stablecoins as an example, the pricing operator generates price Pt, and the total amount Mt is a function of Pt, while Pt+1 depends on Mt. In this way, Mt+1 and Mt form an indirect recursive relationship through the pricing operator, theoretically achieving price stability through periodic negative feedback. However, this design is based on the equilibrium of the supply and demand curve, and its game process occurs in the secondary market, resulting in insufficient accuracy, which may lead to a slow transmission process and difficulty in forming stable equilibrium.

In addition to negative feedback, recursive operators can also provide positive feedback. For example, the repurchase mechanism in certain systems raises prices by reducing market supply, thereby enhancing performance, meeting more demand, generating more revenue, and increasing repurchases, creating a virtuous cycle. This concise and powerful method with anti-Markov properties may gain more favor from future on-chain protocol developers.

From a mathematical perspective, it is still unclear whether recursive operators can construct stable short-cycle properties. Therefore, stablecoins that rely on recursive operators find it difficult to converge to a stable structure. This is especially true when algorithmic stablecoins indirectly influence supply and demand relationships by changing the total amount, as their transmission is slower, the constraints to reach stable equilibrium are greater, and the difficulty in achieving their own goals is higher.

In multiple recursive operators, the step of introducing new information is crucial. The general equilibrium properties of Blockchain indeed easily allow for the introduction of more information, which has a certain degree of uncertainty under specific game structures, yet follows a unified information framework. This characteristic, combined with recursive operators, can easily create an illusion of stability. Without a rigorous game theory analysis, it is difficult to fully grasp the overall equilibrium properties, which may lead to results contrary to expectations.

When designing recursive operators, it is important to pay attention to the steps of introducing information or independent operators, as too many can weaken the effect of the recursive operator. If the goal is to enhance positive and negative feedback, the frequency of introducing new information should be reduced; if the goal is long-cycle regression, the introduction of information flow itself should have a certain periodicity.

Most recursive operators in the DeFi space combine price sequences, as price games are the form of games where information is most concentrated and difficult for algorithms to predict or control. However, many current designs still rely on AMM mechanisms rather than effective decentralized oracles, which may lead the recursive process to become deterministic or controllable, contrary to the original intention of recursive operator design.

In addition, the recursive quantities designed by many projects are not directly linked to the supply and demand variables that determine the price sequence, but are related to the total asset volume. This may result in the operators being unable to directly affect the secondary market, and the transmission effect may be distorted.

In the future, we should explore the combination of more variables and recursive operators, especially the 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, thereby fully unleashing their potential in the Blockchain ecosystem.