A max-plus perspective on package relay and block assembly

The interconnectedness of package relay, Child Pays for Parent (CPFP), ancestor limits, and block assembly in Bitcoin's operation presents a complex interplay that influences transaction inclusion and block utilization. These elements interact in a manner that can lead to sharp thresholds for transaction inclusion, underutilized block space, and significant impacts from minor fee adjustments. A conceptual framework is proposed to navigate these intricacies without suggesting modifications to the protocol or policies but rather to elucidate the economic incentives and marginal trade-offs at play.

Miners are tasked with constructing blocks by selecting transactions within the confines of a block weight limit and dependency constraints, aiming to maximize fee collection. This selection process is modeled as an optimization problem on a directed acyclic graph (DAG), where both rewards (fees) and costs (weight) are additive. Despite miners adopting greedy heuristics for practicality, their goal aligns with maximizing total fees under these constraints.

Transaction selection evolves incrementally, with miners weighing the addition of transactions or their ancestor groups based on their marginal contributions, i.e., the additional fees versus weight consumed. This approach highlights the importance of considering transaction packages over individual transactions due to dependency-induced non-linearities in their marginal contributions. The decision-making process inherently involves a trade-off, governed by an implicit "shadow price" of weight, reflecting the minimum fee increase required to compensate for an additional unit of block weight. This shadow price, emerging from mempool competition, dictates the feerate cutoff for transaction inclusion at any given moment.

It's critical to recognize that optimal block construction does not necessarily equate to fully utilized block space. Optimal behavior from a miner’s perspective may involve leaving space unused if no transactions offer positive marginal benefits at the prevailing shadow price. Such occurrences are often attributed to mempool conditions or block template specifics, underscoring that leftover space should not immediately imply inefficiency. This insight challenges common interpretations and emphasizes the nuanced understanding required to assess miner behavior accurately.

The piecewise linear nature of the optimization problem underpinning block assembly manifests in threshold effects, where small changes in fees can abruptly alter the viability of transaction packages. This characteristic stems from the constrained linear optimization framework, highlighting structural rather than strategic implications.

Furthermore, the analytical lens provided by max-plus algebra and tropical geometry—emphasizing maximization and addition operations—offers a mathematical foundation for understanding the optimization logic inherent in block assembly and package relay processes. While the application of such advanced mathematical concepts provides clarity, it does not necessitate a deep mathematical proficiency to appreciate the underlying logic.

In summary, the coherent optimization logic guiding block assembly and package relay processes in Bitcoin is crucial for interpreting observed behaviors and avoiding misconceptions. This understanding is pivotal for navigating the Bitcoin protocol effectively, emphasizing the need for a refined comprehension of the economic and computational principles at play.