Algorithmic Stablecoins and Recursive Operators: A New Direction in Blockchain Finance

Many practitioners in the blockchain field have developed a strong interest in algorithmic stablecoins. Compared to traditional collateralized stablecoins or automated market maker (AMM) mechanisms, algorithmic stablecoins seem to bring new possibilities. Some even fantasize that it can achieve the mission that Bitcoin failed to complete: to create a fully decentralized and self-adjusting global currency system. The emergence of this idea stems not only from the limitations of understanding the essence of blockchain and currency, but more importantly, because algorithmic stablecoins introduce a novel concept – recursive operators.

The recursive operator refers to a method of computation in the continuous transformation process of smart contracts, where the previous state is used as input, and the next state is generated through repeated cycles. The emergence of such operators is not surprising, as the openness of blockchain data and the serial design of smart contracts naturally create a time series. Recursively processing the same type of operation can produce non-linear structures and even exhibit geometric series effects. This strong positive feedback feature completely aligns with the self-enhancing properties of on-chain games, making it one of the preferred solutions for developers seeking new possibilities in non-cooperative games.

However, simple time series recursion does not bring much innovation. What is truly noteworthy is the multiple recursive operator: it introduces new information between two state changes, reflecting game properties and possessing unpredictability. This unpredictability is influenced by the recursive operator, forming a certain degree of common expectation, which in turn reacts to other operators, creating a resonance effect, ultimately forming controllable expectation attributes.

Taking a simple algorithmic stablecoin as an example, the pricing operator generates price P(t), while the expansion total M(t) is a multiple recursive operator. M(t) is a function of P(t), while P(t+1) relies on M(t), thus establishing an indirect recursive relationship between M(t+1) and M(t). With the cooperation of the pricing operator, a periodic negative feedback is formed, gradually approaching price stability. This concept is based on the equilibrium of the supply and demand curve, and its game process occurs in the secondary market, so it is not very precise, which may lead to a slow price transmission process, making it difficult to form a stable equilibrium.

In addition to providing operators that offer negative feedback, there are also recursive operators that provide positive feedback. The goal of such operators is to achieve self-enhancement rather than price stability. For example, buyback mechanisms in certain systems belong to this type: buybacks lead to a reduction in market supply, causing prices to rise, which in turn enhances system performance, meets more demand, brings in more revenue, increases buybacks, and forms a virtuous cycle.

From a purely mathematical perspective, it is still unclear whether a recursive operator can construct a stable short-term attribute. Therefore, stablecoins that rely on recursive operators find it difficult to converge to a stable structure. This is especially true considering that algorithmic stablecoins do not change the direct supply and demand relationship in the secondary market, but instead indirectly affect supply and demand by changing the total amount. This leads to slower transmission and more constraints to reach a stable equilibrium, making it even more challenging to achieve their own goals.

In multiple recursive operators, the step of introducing new information is crucial. The general equilibrium properties of Blockchain do indeed facilitate the introduction of more information, which has a certain degree of uncertainty under specific game structures, yet there exists a framework of unified information structure. This information, combined with recursive operators, establishes a holistic expectation, which easily creates an illusion of stability. Without a rigorous game-theoretic analysis as a basis, it is difficult to fully grasp the overall equilibrium properties, which may be contrary to expectations.

When designing recursive operators, it is important to note that as the number of steps or independent operators for introducing information increases, the effect of the recursive operator will gradually weaken, and its positive and negative feedback properties will gradually dissipate. Therefore, there exists an indicator of feedback strength for recursive operators. If one wants to strengthen positive and negative feedback when designing decentralized finance (DeFi) systems, it is necessary to reduce the frequency of introducing new information; if the goal is long-cycle regression, the introduction of information flow itself should also have certain cyclical properties.

In the DeFi space, most recursive operators combine with price sequences, as price games are the form of game with the most concentrated information and are difficult to predict or control by algorithms. However, currently when using price sequences, there is often a reliance on the AMM mechanism rather than an effective decentralized oracle, which may lead the entire recursive process to become a deterministic or controllable process, contrary to the original intention behind the design of recursive operators.

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 amount of assets. This may lead to an inability to directly reach the core of the game in the secondary market, causing deviations in the transmission of the algorithm.

In the future, there should be more combinations of variables and recursive operators, especially those parameters that reflect the difficulty of the overall market game. This is a series of nonlinear operators worth exploring in depth. When designing DeFi systems, a detailed analysis of the information transmission mechanisms of recursive operators should be conducted to avoid being predicted and controlled, thus truly unleashing the potential of recursive operators in Blockchain financial innovation.