Incentivizing Information Acquisition

TL;DR

The paper tackles the problem of designing contracts to incentivize an agent to acquire precise information about an unknown state when effort and information are privately chosen. It develops a principal–agent model with a costly signal $s= heta+rac{1}{\
}\varepsilon$, where the agent picks precision $\\u0001$ at cost $c(\u0001)$ and reports a value $a$ after observing $s$, while the principal offers a transfer $t$ based on the state and report under a budget constraint. A central contribution is showing that, when the signal distribution has increasing elasticity above 1 (a mild condition satisfied by Gaussian, Laplace, logistic, and uniform densities), there exists an optimal transfer with a simple cutoff form: pay the entire budget if the report lies within a cutoff distance $d$ of the state and pay nothing otherwise; this result extends to multi-dimensional settings, Gaussian priors, and scenarios with unobserved states. The paper also provides a full characterization of the optimal cutoff, analyzes when cutoff transfers are not optimal, and derives comparative statics showing how budget, cost structure, and signal precision affect the optimal cutoff and induced precision. These insights imply that simple contracts are often optimal in information-acquisition contexts and offer a new perspective on classic principal–agent problems under budget constraints.

Abstract

I study a principal-agent model in which a principal hires an agent to collect information about an unknown continuous state. The agent acquires a signal whose distribution is centered around the state, controlling the signal's precision at a cost. The principal observes neither the precision nor the signal, but rather, using transfers that can depend on the state, incentivizes the agent to choose high precision and report the signal truthfully. I identify a sufficient and necessary condition on the agent's information structure which ensures that there exists an optimal transfer with a simple cutoff structure: the agent receives a fixed prize when his prediction is close enough to the state and receives nothing otherwise. This condition is mild and applies to all signal distributions commonly used in the literature.

Figures (9)

• Figure 1: Cutoff Transfer
• Figure 2: The Transfer $t$ and The Cutoff Transfer $d$. The given transfer $t$ and the matching cutoff transfer $d$ are shown in black. The wider distribution shown in blue is the signal distribution chosen by the agent under $t$, while the narrower (more precise) distribution in red is the one chosen under $d$.
• Figure 3: Expected Transfer
• Figure 4: Increment of Expected Transfer When Increasing $\lambda$ The agent slightly increases precision from $\lambda$ to $\lambda+\Delta\lambda$. The area of the red region is the expected transfer $E(\lambda;d)$. The area of two blue regions is the increment of probability that the signal falls into the cutoff.
• Figure 5: Expected Transfer

Theorems & Definitions (35)

• Definition 1
• Lemma 1
• Definition 2
• Theorem 1
• Corollary 1
• Theorem 2
• Lemma 2: Complements or Substitutes
• Example 1
• Proposition 1: Comparative Statics
• Corollary 2