A Framework for Combined Transaction Posting and Pricing for Layer 2 Blockchains

TL;DR

This work addresses the intertwined challenges of posting costs on L1 and congestion on L2 by presenting a joint dynamic model that links Layer 1 gas prices, Layer 2 transaction queues, and user demand through L2 fees. It establishes a threshold-based optimal posting policy and a rigorous dynamic L2 pricing mechanism that integrates budget balance and congestion control, including existence and uniqueness results for the optimal fees $f^*$ and $p^*$ and an adaptive update scheme. The framework is validated via simulations under both i.i.d. and mean-reverting L1 gas-fee regimes, demonstrating convergence of fees to their targets and stable throughput under realistic network conditions. The results offer a principled, implementable approach for scalable, financially sustainable L2 rollups, with practical pathways to extend to more complex transaction types and dynamic demand patterns.

Abstract

This paper presents a comprehensive framework for transaction posting and pricing in Layer 2 (L2) blockchain systems, focusing on challenges stemming from fluctuating Layer 1 (L1) gas fees and the congestion issues within L2 networks. Existing methods have focused on the problem of optimal posting strategies to L1 in isolation, without simultaneously considering the L2 fee mechanism. In contrast, our work offers a unified approach that addresses the complex interplay between transaction queue dynamics, L1 cost variability, and user responses to L2 fees. We contribute by (1) formulating a dynamic model that integrates both posting and pricing strategies, capturing the interplay between L1 gas price fluctuations and L2 queue management, (2) deriving an optimal threshold-based posting policy that guides L2 sequencers in managing transactions based on queue length and current L1 conditions, and (3) establishing theoretical foundations for a dynamic L2 fee mechanism that balances cost recovery with congestion control. We validate our framework through simulations.

Figures (3)

• Figure 1: Illustration of the fee update mechanism under decreasing step size and i.i.d. L1 fees. The top row displays the evolution of the posting fees over time, with red dashed lines marking the theoretical optimal levels $f^*$ and $p^*$. In the bottom row, the plots show the proportion of times each type of fee update mechanism (budget balance vs. congestion control) is selected. All four trajectories exhibit convergence behavior in line with the theoretical results.
• Figure 2: Illustration of the fee update mechanism under decreasing step size and non-i.i.d. L1 fees. The top row displays the evolution of the posting fees over time, with red dashed lines marking the theoretical optimal levels $f^*$ and $p^*$. In the bottom row, the plots show the proportion of times each type of fee update mechanism (budget balance vs. congestion control) is selected. All four trajectories exhibit convergence behavior in line with the theoretical results.
• Figure 3: Illustration of the fee update mechanism under constant step size and non‐i.i.d. L1 fees. The top row shows time‐series of the fee updates $f_t$ and $p_t$ under a mean‐reverting L1 fee process. In the middle row, the plots track the long‐run fraction of times each update rule is chosen, indicating how often the budget‐balance versus congestion‐control mechanism is active. The bottom row presents empirical histograms of $f_t$ and $p_t$. Although no formal convergence theorem applies in this setting, the simulation demonstrates that both fees remain near the targets, and that the selection frequencies also stabilize, highlighting the mechanism's robustness under realistic market dynamics.

Theorems & Definitions (24)

• theorem 1
• proof
• proof
• corollary 1
• proof
• proof
• theorem 2
• proof
• theorem 3
• proof