Multidimensional Blockchain Fees are (Essentially) Optimal (2402.08661v2)

Abstract: In this paper we show that, using only mild assumptions, previously proposed multidimensional blockchain fee markets are essentially optimal, even against worst-case adversaries. In particular, we show that the average welfare gap between the following two scenarios is at most $O(1/\sqrt{T})$, where $T$ is the length of the time horizon considered. In the first scenario, the designer knows all future actions by users and is allowed to fix the optimal prices of resources ahead of time, based on the designer's oracular knowledge of those actions. In the second, the prices are updated by a very simple algorithm that does not have this oracular knowledge, a special case of which is similar to EIP-1559, the base fee mechanism used by the Ethereum blockchain. Roughly speaking, this means that, on average, over a reasonable timescale, there is no difference in welfare between 'correctly' fixing the prices, with oracular knowledge of the future, when compared to the proposed algorithm. We show a matching lower bound of $\Omega(1/\sqrt{T})$ for any implementable algorithm and also separately consider the case where the adversary is known to be stochastic.

• The paper demonstrates that multidimensional fee mechanisms achieve optimal welfare outcomes with the welfare gap shrinking as O(1/sqrt(T)).
• It employs regret analysis via online convex optimization to compare optimal foresight pricing with algorithmic strategies under adversarial conditions.
• The study models both stochastic and worst-case scenarios, providing practical insights for designing resilient blockchain fee systems.

Overview of "Multidimensional Blockchain Fees are (Essentially) Optimal"

This paper presents a comprehensive analysis of the optimality of multidimensional blockchain fee markets. The authors focus on establishing the efficiency of previously proposed models that incorporate multiple resource pricing functionalities within blockchains. Their results are grounded in the context of adversarial environments where adversaries aim to manipulate the fee structures. The authors introduce a framework that characterizes and proves the effectiveness of these multidimensional fee mechanisms. Two fundamental scenarios are investigated: one where prices are set optimally with complete foresight and another where pricing evolves based on a simple algorithm without such foresight.

Key Contributions

1. Optimality of Multidimensional Fees: The core assertion of this research is that multidimensional fee mechanisms are proven to deliver optimal welfare outcomes, with the welfare gap between foresight and non-foresight scenarios decreasing at a rate of $O(1/\sqrt{T})$. This rate illustrates a minor difference as the number of observed time periods $T$ grows large, implying robustness over extensive timescales.
2. Analytical Approach: The authors utilize a regret analysis foundation, borrowing from concepts in online convex optimization. Regret, defined as the difference in performance between the proposed algorithms and the optimal price setting, is minimized under adversarial conditions. This analysis highlights that no significant advantage is gained from knowing future transactions over the proposed pricing algorithms.
3. Stochastic and Adversarial Models: The paper further analyses scenarios in both stochastic adversary environments and deterministic worst-case settings. A clear distinction emerges, indicating that in stochastic settings, prices trend towards a market-clearing state with explicit rates, while in worst-case scenarios, regret is bounded in a manner similar to that of competitive online algorithms.
4. Practical Implications: The authors apply this optimality framework to real-world scenarios, such as Ethereum's impending EIP-4844 that proposes two-dimensional fee strategies. Here, they illustrate that these blockchain upgrades are potentially optimal, echoing their mathematical findings.

Strong Numerical Results and Claims

The research provides compelling numerical results, with rigorous proof that the welfare gap scales as $O(1/\sqrt{T})$. The authors also establish a matching lower bound of $\Omega(1/\sqrt{T})$ for any implementable algorithm, emphasizing the rarity of such optimality results in adversarial-defending mechanisms. Their comprehensive exploration of multiplicative and gradient descent-based update rules confirms the findings, affirming that the convergence of prices, under specific conditions, reliably reflects optimal resource allocation.

Implications for Future Developments

This paper's findings have significant theoretical implications, as they shed light on the broader design space of blockchain fee mechanisms, suggesting models that resist adversarial interference while sustaining efficient market operations. Practical impacts are substantial, as blockchain developers might leverage these results to craft more resilient fee algorithms that optimize resource distribution and network stability.

Avenues for Future Work

While the authors propose a fine-tuned adversarial model, they acknowledge potential extensions in defining weaker risk models that could mimic realistic conditions more accurately. Additionally, examining more varied fee adjustment strategies and understanding the interplay between multiple dimensions of blockchain resources could illuminate further the optimization phenomena explored herein.

In conclusion, "Multidimensional Blockchain Fees are (Essentially) Optimal" elevates the discourse on blockchain fee design, equipping blockchain architects with the analytical foundation necessary to navigate the complexities of multi-resource environments. The theoretical advancements and rigorous empirical validation provided in this paper serve as a cornerstone for both the existing structure and future evolution of blockchain economic models.