Recursive Operators in the Blockchain World: Opportunities and Challenges

The recursive operator has attracted widespread attention in the blockchain field, especially in the design of algorithmic stablecoins. This novel concept has led many to fantasize that it could achieve a goal that Bitcoin failed to accomplish: a fully decentralized and self-adjusting global currency system. However, the emergence of this idea is not only due to a lack of understanding of the nature of blockchain and currency but also relates to the characteristics of the recursive operator itself.

The recursive operator refers to a calculation that takes the previous state as input and repeatedly loops to generate the next state in continuous smart contract transformations. This design is quite natural in a blockchain environment, as the openness of on-chain data and the serial nature of smart contracts create a time series. Recursive processing of similar operations can produce non-linear structures and even geometric series effects, which align with the self-enhancing characteristics of on-chain games.

However, simple time series recursion is not an ideal solution. What is truly worth noting is the multiple recursive operators, which introduce new information between two state changes. This information reflects game attributes and possesses unpredictability. This unpredictability is also influenced by the recursive operators, forming certain common expectations, thereby generating controllable expectation attributes.

Taking algorithmic stablecoins as an example, the pricing operator generates a price P, and the total supply M is a multiple recursive operator. M is a function of P, while the next moment's P depends on M, forming an indirect recursive relationship. This design attempts to achieve price stability through periodic negative feedback, but due to the slow transmission process, it is difficult to form a stable equilibrium.

Recursive operators can provide not only negative feedback but also construct positive feedback mechanisms. For example, the buyback mechanism in certain systems is a typical positive feedback recursive operator, which can lead to a decrease in market supply, an increase in prices, an improvement in system performance, an increase in demand, an increase in revenue, and an increase in buybacks, thereby forming a positive cycle.

From a mathematical perspective, it remains unclear whether recursive operators can construct stable short-cycle properties. Especially in algorithmic stablecoins, since what is changed is the total supply rather than the direct market supply and demand relationship, the transmission is slower, and the conditions for achieving stable equilibrium are more stringent.

In multiple recursive operators, the step of introducing new information is crucial. The general equilibrium properties of blockchain do indeed make it easy to introduce more information, which has a certain degree of uncertainty under specific game structures, yet follows a unified information structure. These characteristics, when combined with recursive operators, may create an illusion of stability. Without a rigorous game theory analysis, it is difficult to fully grasp the overall equilibrium properties.

When designing recursive operators, it is important to pay attention to the frequency and manner of information introduction. Too many steps or independent operators in introducing information can lead to a gradual weakening of the effects of the recursive operator. Therefore, there is an indicator of feedback strength for recursive operators. If the goal is to strengthen positive and negative feedback, the frequency of introducing new information needs to be reduced; if the goal is to pursue long-cycle regression, the information flow introduction itself should also have certain cyclical attributes.

In most DeFi projects, recursive operators tend to combine with price series, as price games are the most concentrated form of information and are difficult for algorithms to predict or control. However, many projects currently rely on AMM mechanisms instead of decentralized oracles, which may lead the recursive process to become deterministic or controllable, contrary to the original intention of recursive operator design.

Moreover, the recursive quantities designed by many projects are not directly tied to the supply and demand variables that determine the price sequence, but rather are related to the total asset amount. This may lead to an inability to directly influence the secondary market, causing deviations in the propagation effects of the operators.

In the future, more combinations of variables and recursive operators should be explored, especially those parameters that reflect the difficulty of the overall market game. When designing DeFi projects, a detailed analysis of the information transmission mechanism of recursive operators should be conducted to avoid being predicted and controlled, thereby achieving true innovation and breakthroughs.