The Application and Limitations of Recursive Operators in Decentralized Finance

Algorithmic stablecoins have sparked the interest of many, partly due to their introduction of novel recursive operators. These operators take the previous state as input and repeatedly loop to produce the next state during continuous transformations of smart contracts. Due to the transparency of blockchain and the serial design of smart contracts, recursive processing can lead to nonlinear structures or geometric series effects, forming strong positive feedback characteristics.

However, simple time series recursion is not ideal. More attention should be paid to multiple recursive operators, which introduce new information between state changes, reflect game attributes, and create unpredictability. This unpredictability is influenced by recursive operators, forming common expectations that feed back into other operators, resulting in controllable expectation attributes.

Taking algorithmic stablecoins as an example, the pricing operator generates price Pt, and the total supply Mt acts as a multiple recursive operator. Mt is a function of Pt, and Pt+1 depends on Mt, forming an indirect recursive relationship. Under the coordination of the pricing operator, periodic negative feedback is generated, gradually approaching price stability. However, this concept is based on the equilibrium of supply and demand curves, and the actual transmission process is slow, making it difficult to form a stable equilibrium.

Recursive operators can also provide positive feedback, such as the buyback mechanisms in certain systems. Buybacks reduce market supply, drive up prices, enhance performance, satisfy more demand, and generate more revenue, which further increases buybacks, creating a virtuous cycle.

From a mathematical perspective, it is still unclear whether recursive operators can construct stable short-term attributes. Algorithmic stablecoins indirectly influence supply and demand relationships by changing the total amount, with slow transmission and numerous constraints to achieve stable equilibrium, making it difficult to realize their own goals.

In multiple recursive operators, introducing new information is crucial. The general equilibrium properties of blockchain easily incorporate more information, presenting uncertainties under game structures. This information, combined with recursive operators, establishes overall expectations, which may create an illusion of stability. Without a rigorous game-theoretic analysis, it is difficult to grasp the overall equilibrium properties.

When designing DeFi, one should carefully analyze the information transmission mechanism of recursive operators to avoid being predicted and controlled. In the future, there may be more variables combined with recursive operators, especially parameters reflecting the difficulty of the overall market game, which is a nonlinear operator series worth exploring.