Many proof-of-stake protocols finance validator rewards from two sources: transaction fees and a finite reserve of tokens. This creates a dynamic hand-off problem. Early in the life of the system, fees may be too small to fund the target level of security; later, fees may become sufficient. The central question is whether the reserve provides enough runway for the protocol to remain secure until this fee-only region is reached. We study this problem in a discrete-time stochastic model of validator participation. Token price and transaction demand fluctuate over time, while validators choose participation strategically. We solve the validator entry game and derive an exact state-dependent reserve threshold, i.e., the minimal reserve stock necessary and sufficient to sustain a target security level. This threshold separates three regions: infeasibility, reserve-dependent security, and fee-only security. Security fails if the reserve first falls below the state-dependent threshold, and a successful hand-off occurs exactly if the fee-only region is reached before that failure time. We derive stress-test guarantees that convert lower confidence bands for token price and demand into reserve requirements, and obtain explicit failure-probability and expected hand-off-time bounds. Finally, we extend the model to forward-looking validators and derive the Markov participation condition that captures how current participation affects future reserve-funded rewards. The main implication is that reserve policy should not be evaluated by nominal depletion dates or steady-state reward ratios alone. A protocol can have a large nominal reserve and still be close to security failure after adverse price or demand shocks. Conversely, once demand crosses the fee-only threshold, the reserve becomes redundant for security. This paper provides a tractable equilibrium framework for stress-testing this transition.

翻译：许多权益证明协议通过两种来源为验证者奖励提供资金：交易费用和有限的代币储备。这形成了一个动态的衔接问题。在系统运行初期，交易费用可能过低，无法支撑目标安全水平；而随着系统发展，费用可能变得充足。核心问题在于：储备能否为协议提供足够的运行窗口，使其在达到仅依赖费用的安全阶段之前保持安全。我们在一个验证者参与的离散时间随机模型中研究此问题。代币价格和交易需求随时间波动，而验证者则策略性地选择是否参与。我们求解验证者入场博弈，并推导出一个精确的状态依赖储备阈值，即维持目标安全水平所需的最小储备存量。该阈值划分出三个区域：不可行区、储备依赖安全区和仅费用安全区。当储备首次低于状态依赖阈值时，安全失效；而恰好在失效时间之前到达仅费用安全区时，即为成功衔接。我们推导出压力测试保证，将代币价格和需求的置信下界转化为储备要求，并获得明确的失效概率和期望衔接时间界限。最后，我们将模型扩展至前瞻性验证者，并推导出马尔可夫参与条件，该条件刻画了当前参与如何影响未来由储备资助的奖励。其主要含义是：储备政策不应仅通过名义消耗日期或稳态奖励比率来评估。一个协议可能拥有庞大的名义储备，但在经历不利的价格或需求冲击后仍可能接近安全失效。反之，一旦需求超过仅费用安全阈值，储备对于安全而言便成为冗余。本文为压力测试这一过渡阶段提供了一个易于处理的均衡分析框架。