  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.
