Monitoring with Rich Data

TL;DR

This paper studies a static moral hazard with rich, observable monitoring data and shows that the principal's payoff converges to the first-best at an optimal exponential rate as data grows, regardless of contract sophistication. The key finding is that binary wage contracts—with a maximally lenient high/low cutoff—achieve this optimal rate, with the rate determined by the Kullback-Leibler divergence between signal distributions under the target action and the closest deviating action. In contrast, more nuanced contracts, such as linear designs, attain substantially slower, subexponential convergence, providing a principled rationale for the empirical prevalence of simple binary pay schemes in data-rich environments. The authors further show that the optimal convergence rate depends only on a simple statistic of the monitoring technology, enabling a detail-free ranking over monitoring designs; they extend the analysis to non-i.i.d. signals, severe limited liability, and settings with adjustable actions, maintaining the same convergence-rate framework. Altogether, the results offer a rigorous, information-theoretic lens on when simple contracts suffice and how to compare monitoring technologies in data-rich incentive problems.

Abstract

We consider moral hazard problems where a principal has access to rich monitoring data about an agent's action. Rather than focusing on optimal contracts (which are known to in general be complicated), we characterize the optimal rate at which the principal's payoffs can converge to the first-best payoff as the amount of data grows large. Our main result suggests a novel rationale for the widely observed binary wage schemes, by showing that such simple contracts achieve the optimal convergence rate. Notably, in order to attain the optimal convergence rate, the principal must set a lenient cutoff for when the agent receives a high vs. low wage. In contrast, we find that other common contracts where wages vary more finely with observed data (e.g., linear contracts) approximate the first-best at a highly suboptimal rate. Finally, we show that the optimal convergence rate depends only on a simple summary statistic of the monitoring technology. This yields a detail-free ranking over monitoring technologies that quantifies their value for incentive provision in data-rich settings and applies regardless of the agent's specific utility or cost functions.

Figures (3)

• Figure 1: Implementation costs in Example \ref{['ex:finite-n']} as a function of $n$ when $c =0.8$, $\eta=0.1$, so $C^{\rm FB} \approx 16.1$. The lenient (resp. symmetric) binary contract uses a cutoff $\gamma=0.1$ (resp. $\gamma=0.5$).
• Figure 2: Contour curves of ${\rm Var}_{1}[v_n(x^n)]$ under contract (\ref{['eq:test contract']}). Given $\gamma$, $n$ determines the probabilities of false positives and false negatives, which jointly determine the utility variance.
• Figure 3: Contract (\ref{['eq:test contract']}) with binary signals. The space of empirical signal frequencies $\Delta (X)$ is divided into two regions, "pass" ($L_n\geq \gamma$) and "fail" ($L_n< \gamma$). As $\gamma$ approaches the maximally lenient threshold ${\mathbb E}_{0}[L_n]$, the cutoff between the two regions approaches $\mu_0$.

Theorems & Definitions (26)

• Theorem 3.1
• Proposition 3.1
• Example 3.1
• Remark 1: Contrast with "shoot the agent" contracts
• Remark 2: Limit of optimal contracts
• Proposition 3.2
• Corollary 1
• Example 3.2: Precise bad vs. good news
• Theorem 4.1
• Remark 3: Risk-neutral case