We study a mechanism design problem in the blockchain proof-of-stake (PoS) protocol. Our main objective is to extend the transaction fee mechanism (TFM) recently proposed in Chung and Shi (SODA, p.3856-3899, 2023), so as to incorporate a long-run utility model for the miner into the burning second-price auction mechanism $\texttt{BSP}(\gamma)$ proposed in Chung and Shi (where $\gamma$ is a key parameter in the strict $\gamma$-utility model that is applied to both miners and users). First, we derive an explicit functional form for the long-run utility of the miner using a martingale approach, and reveal a critical discontinuity of the utility function, namely a small deviation from being truthful will yield a discrete jump (up or down) in the miner's utility. We show that because of this discontinuity the $\texttt{BSP}(\gamma)$ mechanism will fail a key desired property in TFM, $c$-side contract proofness ($c$-SCP). As a remedy, we introduce another parameter $\theta$, and propose a new $\texttt{BSP}(\theta)$ mechanism, and prove that it satisfies all three desired properties of TFM: user- and miner-incentive compatibility (UIC and MIC) as well as $c$-SCP, provided the parameter $\theta$ falls into a specific range, along with a proper tick size imposed on user bids.

翻译：我们研究区块链权益证明（PoS）协议中的机制设计问题。主要目标是将Chung和Shi（SODA，第3856-3899页，2023年）近期提出的交易费用机制（TFM）进行扩展，以将矿工的长期效用模型纳入其提出的燃烧第二价格拍卖机制$\texttt{BSP}(\gamma)$中（其中$\gamma$是严格$\gamma$-效用模型中的关键参数，该模型同时适用于矿工和用户）。首先，我们利用鞅方法推导出矿工长期效用的显式函数形式，并揭示效用函数的一个关键不连续性：即偏离诚实行为的小偏差将导致矿工效用发生离散跳跃（上升或下降）。我们证明，由于这一不连续性，$\texttt{BSP}(\gamma)$机制将无法满足TFM中一个关键期望性质——c-侧合约抵抗性（c-SCP）。作为补救措施，我们引入另一个参数$\theta$，提出新的$\texttt{BSP}(\theta)$机制，并证明：当参数$\theta$落在特定范围内，且对用户出价施加适当的报价步长时，该机制能满足TFM的所有三个期望性质，即用户激励相容性（UIC）、矿工激励相容性（MIC）以及c-SCP。