最优功能激励机制设计：具有双线性激励的线性二次情况
Optimal Functional Incentives for Control: The Linear-Quadratic Case with Bilinear Incentives
AI总结：
本文研究了动态系统功能激励机制的设计，提出在无实时调整的情况下，通过固定激励函数激励自利跟随者在长期内有益地操作系统。针对线性二次情况和双线性激励，推导了闭环系统的稳定性条件，并给出了激励参数的闭式表达。
摘要：
我们研究了动态系统功能激励机制的设计，其中领导者设计一个固定激励函数以激励自利的跟随者在扩展时间内有益地操作系统，而无需实时调整激励。这与适应性范式形成对比，后者中激励本身是持续更新的控制变量。我们将问题形式化为离散时间双层最优控制问题，并推导了线性二次情况下的分析结果，其中激励为双线性且跟随者具有短视行为。具体而言，我们建立了诱导闭环系统的必要且充分稳定性条件，推导了领导者成本对激励参数矩阵的梯度的闭式表达，并在标量情况下获得了完全闭式成本表达式。基于后者，在两个渐进行为中提供了激励参数的显式刻画：无限时间极限和高跟随者成本极限。对于长时间范围，最优激励被证明与跟随者的私人成本参数无关，这对在私人信息下的鲁棒机制设计有直接影响。
英文摘要：
We study the design of functional incentive mechanisms for dynamical systems, in which a leader designs a fixed incentive function to motivate a self-interested follower to actuate the system beneficially over an extended horizon, without real-time revision of the incentive. This stands in contrast to the adaptive paradigm, in which the incentive is itself a continuously updated control variable. We formalize the problem as a discrete-time bi-level optimal control problem and derive analytical results for the linear-quadratic case with bilinear incentives and a myopic follower. Specifically, we establish a necessary and sufficient stability condition for the induced closed-loop system, derive a closed-form expression for the gradient of the expected leader cost with respect to the incentive parameter matrix, and obtain a fully closed-form cost expression in the scalar setting. Based on the latter, explicit characterizations of the optimal incentive parameter are provided in two asymptotic regimes: the infinite-horizon limit and the limit of high follower cost. For long horizons, the optimal incentive is shown to become independent of the follower's private cost parameter, with direct implications for robust mechanism design under private information.
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