Transaction fee mechanism for Proof-of-Stake protocol

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 $\texttt{BSP}(γ)$ proposed in Chung and Shi (where $γ$ is a key parameter in the strict $γ$-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 $\texttt{BSP}(γ)$ mechanism will fail a key desired property in TFM, $c$-side contract proofness ($c$-SCP). As a remedy, we introduce another parameter $θ$, and propose a new $\texttt{BSP}(θ)$ 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 $θ$ falls into a specific range, along with a proper tick size imposed on user bids.