  This paper studies the optimal transaction fee mechanisms for blockchains,
focusing on the distinction between price-based ($\mathcal{P}$) and
quantity-based ($\mathcal{Q}$) controls. By analyzing factors such as demand
uncertainty, validator costs, cryptocurrency price fluctuations, price
elasticity of demand, and levels of decentralization, we establish criteria
that determine the selection of transaction fee mechanisms. We present a model
framed around a Nash bargaining game, exploring how blockchain designers and
validators negotiate fee structures to balance network welfare with
profitability. Our findings suggest that the choice between $\mathcal{P}$ and
$\mathcal{Q}$ mechanisms depends critically on the blockchain's specific
technical and economic features. The study concludes that no single mechanism
suits all contexts and highlights the potential for hybrid approaches that
adaptively combine features of both $\mathcal{P}$ and $\mathcal{Q}$ to meet
varying demands and market conditions.
