In-depth Discussion on Recursive Operators in Blockchain

The recursive operator has attracted widespread attention in the blockchain field, especially in its application in the design of algorithmic stablecoins. Many people have high expectations for it, even believing that it could realize a fully automated decentralized global currency system. The emergence of this idea is related not only to a lack of understanding of the essence of blockchain and currency but also to the novelty of the recursive operator.

The recursive operator refers to an operation that takes the previous state as input and repeatedly generates the next state in continuous smart contract state transformations. In a blockchain environment, the openness of data and the serial design of smart contracts naturally form a time series, and recursive processing of similar operations can produce nonlinear structures, even geometric series effects. This strong positive feedback characteristic highly aligns with the self-reinforcing attributes of on-chain games.

However, simple time series recursion is not ideal due to the lack of new information input. What truly deserves attention is combining recursive operators with other elements to introduce new game information during state changes. This unpredictability is influenced by the recursive operators, forming certain common expectations, which in turn affect other operators, resulting in controllable expectation attributes. We refer to these types of operators as multiple recursive operators.

Taking algorithmic stablecoins as an example, the pricing operator generates price Pt, and the total supply Mt is a function of Pt, while Pt+1 depends on Mt, forming an indirect recursive relationship. With the cooperation of the pricing operator, it gradually approaches price stability. However, this design is based on the assumption of supply and demand equilibrium, and the actual transmission process may be slow, making it difficult to quickly achieve stable equilibrium.

The recursive operator can provide not only negative feedback but also positive feedback. For example, a system's buyback mechanism can increase prices by reducing supply, thereby enhancing performance, increasing demand, and revenue, creating a virtuous cycle. This straightforward and clear method with anti-Markov properties may attract more protocol developers' attention in the future.

From a purely mathematical perspective, it is unclear whether recursive operators can construct stable short-cycle attributes. In particular, algorithmic stablecoins indirectly affect supply and demand relationships by changing the total amount, have slower transmission, and face more constraints on stable equilibrium, making it more difficult to achieve the target.

In multiple recursive operators, introducing new information is crucial. The general equilibrium properties of the Blockchain help to bring in more information, which has certain uncertainties and a framework under specific game structures. The combination of recursive operators may generate overall expectations, easily leading to an illusion of stability. Without rigorous game theory analysis, it is difficult to fully grasp the equilibrium properties, which may result in outcomes contrary to expectations.

In certain designs, randomness is introduced, assuming zero dependency on information. This randomness, combined with recursive operators, may more easily lead to stable traits and is a direction worth exploring for future algorithmic stablecoins.

In design, excessive introduction of information or independent operators can weaken the effect of recursive operators. If the goal is to strengthen positive and negative feedback, new information should be minimized; if the aim is long-cycle regression, the information flow introduced should have cyclical properties. Proving that random operators can achieve regression under specific recursive designs is not an easy task.

Currently, recursive operators in the DeFi space are often combined with price sequences. However, relying on the AMM mechanism rather than decentralized oracles can lead to predictability and controllability, turning the recursive process into a deterministic or controlled process. Attacks are directly reflected in the AMM price sequences, making them difficult to eliminate algorithmically, which contradicts the uncertainty required by recursive operators.

In addition, the recursive quantity designed by many projects is not directly related to the supply and demand variables that determine the price, but rather related to the total asset amount, which may lead to transmission deviations.

In the future, more combinations of variables and recursive operators should be explored, especially 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.