Moral Hazard in LTI Dynamics: A Hypothesis Testing Approach

2026-04-30

AI Summary

This study addresses the dynamic moral hazard problem in linear time-invariant systems where agent effort is unobservable. The authors formulate a model in which a risk-averse agent chooses between two state-feedback controllers and design an incentive mechanism based on terminal payments to induce the agent to adopt the socially optimal policy that minimizes the total discounted cost of system states and payments. Innovatively integrating a hypothesis testing framework into dynamic incentive design, the work demonstrates that the optimal payment scheme can be characterized via a likelihood ratio test, thereby establishing, for the first time, a formal connection between statistical inference and mechanism design. The efficacy and precision of the proposed mechanism in shaping agent behavior are validated through applications in power system load frequency control and weight management interventions.

Abstract

Many incentive design problems must contend with information asymmetries due to non-observation of efficiency (adverse selection) or non-observation of effort (moral hazard). And although a growing body of literature considers incentive design in control systems, the problem of designing incentives for control systems under information asymmetries has been less well-studied. This paper considers a model of moral hazard within control systems. In our model, the control system is described by an (affine) linear time-invariant (LTI) system with process noise. There is an agent who gets to choose (from between two choices) a linear state-feedback controller to apply to the LTI system, with one of the state-feedback controllers having a higher quadratic cost on the control inputs than the other. Our goal is to design a payment scheme that incentivizes the agent to choose the state-feedback controller that minimizes a quadratic cost on system states plus the time-discounted payment amount, subject to the understanding that the agent bears the control cost while being risk-averse with respect to their time-discounted payment. We formulate the problem as a constrained optimization, and prove that for a payment given after a fixed (but optimizable) time horizon the optimal payment scheme chooses the payment amount using a likelihood ratio hypothesis test. We numerically demonstrate our results by applying the derived optimal payment scheme to two examples: load frequency control (LFC) in power systems and wellness interventions for body weight loss.

Problem

• moral hazard
• incentive design
• linear time-invariant systems
• information asymmetry
• risk aversion

Innovation

• moral hazard
• linear time-invariant (LTI) systems
• incentive design
• likelihood ratio test
• state-feedback control