Moral Hazard in LTI Dynamics: A Hypothesis Testing Approach

TL;DR

This paper models moral hazard in control systems with linear dynamics, proposing an optimal payment scheme based on hypothesis testing to incentivize agents to select cost-effective controllers.

Contribution

It introduces a hypothesis testing approach to design optimal payments in control systems under moral hazard with linear dynamics and risk-averse agents.

Findings

Optimal payment scheme uses likelihood ratio hypothesis testing.

The approach is demonstrated in power systems load frequency control.

Application to wellness interventions for weight loss.

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.