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Motivating Effort with Information about Future Rewards

Chang Liu

TL;DR

This paper develops a unified framework to optimally design dynamic information disclosure in a dynamic principal–agent model without transfers, where task quality is private and evolves via Poisson arrivals. By converting the principal’s problem into an optimization over time lotteries and leveraging exponential discounting, it derives closed-form optimal policies that vary with stationarity and nonstationarity of the environment. The authors identify two precise channels where dynamic disclosure is strictly valuable: when the principal is more impatient than the agent, which favors front-loaded effort via maximal delay, and when the agent would become pessimistic over time without information, which dynamic disclosure can counteract. Across binary and general reward distributions, stationary settings yield static disclosure (KG) for patient principals or dynamic cutoff cascades for impatient ones, while nonstationary/pessimistic environments favor Poisson or gradual revelation to stabilize beliefs. The results illuminate how time-preference and information drift shape robust information-design policies and point to richer nonstationary extensions as a fruitful direction for future work.

Abstract

This paper studies the optimal mechanism to motivate effort in a dynamic principal-agent model without transfers. An agent is engaged in a task with uncertain future rewards and can quit at any time. The principal knows the reward and provides information over time to motivate effort. We provide a unified framework and derive the optimal information policy in closed form across stationary and nonstationary environments. Within this framework, we identify two precise conditions, each of which guarantees that dynamic disclosure is strictly valuable. First, if the principal is impatient compared to the agent, she prefers the front-loaded effort schedule induced by dynamic disclosure; in a stationary environment, dynamic disclosure is beneficial if and only if the principal is less patient. Second, in an environment where the agent would become pessimistic over time absent any disclosure, dynamic information provision can counteract this downward trend and encourage longer effort. Notably, patience remains a crucial determinant of the structure of the optimal policy.

Motivating Effort with Information about Future Rewards

TL;DR

This paper develops a unified framework to optimally design dynamic information disclosure in a dynamic principal–agent model without transfers, where task quality is private and evolves via Poisson arrivals. By converting the principal’s problem into an optimization over time lotteries and leveraging exponential discounting, it derives closed-form optimal policies that vary with stationarity and nonstationarity of the environment. The authors identify two precise channels where dynamic disclosure is strictly valuable: when the principal is more impatient than the agent, which favors front-loaded effort via maximal delay, and when the agent would become pessimistic over time without information, which dynamic disclosure can counteract. Across binary and general reward distributions, stationary settings yield static disclosure (KG) for patient principals or dynamic cutoff cascades for impatient ones, while nonstationary/pessimistic environments favor Poisson or gradual revelation to stabilize beliefs. The results illuminate how time-preference and information drift shape robust information-design policies and point to richer nonstationary extensions as a fruitful direction for future work.

Abstract

This paper studies the optimal mechanism to motivate effort in a dynamic principal-agent model without transfers. An agent is engaged in a task with uncertain future rewards and can quit at any time. The principal knows the reward and provides information over time to motivate effort. We provide a unified framework and derive the optimal information policy in closed form across stationary and nonstationary environments. Within this framework, we identify two precise conditions, each of which guarantees that dynamic disclosure is strictly valuable. First, if the principal is impatient compared to the agent, she prefers the front-loaded effort schedule induced by dynamic disclosure; in a stationary environment, dynamic disclosure is beneficial if and only if the principal is less patient. Second, in an environment where the agent would become pessimistic over time absent any disclosure, dynamic information provision can counteract this downward trend and encourage longer effort. Notably, patience remains a crucial determinant of the structure of the optimal policy.

Paper Structure

This paper contains 22 sections, 9 theorems, 75 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Model
  4. Binary-Stationary Case
  5. Static Disclosure
  6. Delayed Disclosure
  7. Optimal Information Policy
  8. General Stationary Case
  9. Static Disclosure
  10. Cutoff Cascades
  11. Immediate and Gradual Disclosure
  12. Delayed Gradual Disclosure
  13. Binary-Nonstationary Case
  14. Agent Behavior and the Critical Belief Threshold
  15. The Free-Riding Lemma
  16. ...and 7 more sections

Key Result

Lemma 1

Fix a time $t>0$. $\operatorname{DD}(t)$ increases the principal's payoff relative to $\operatorname{KG}$ for some range of priors if and only if the principal is less patient than the agent, i.e., $r_p>r$.

Figures (7)

  • Figure 1: Optimal static information policy $\operatorname{KG}$ corresponds to a concavification.
  • Figure 2: The principal's payoff comparison between $\operatorname{DD}(t)$ and $\operatorname{KG}$.
  • Figure 3: The principal's payoff comparison among $\operatorname{DD}(t)$, $\operatorname{MDD}$ and $\operatorname{KG}$.
  • Figure 4: Examples of cutoff cascades.
  • Figure 5: Relationship among $\operatorname{KG}$, $\operatorname{OIGD}$ and $\operatorname{MDGD}$.
  • ...and 2 more figures

Theorems & Definitions (31)

  • Definition 1: Information policy
  • Definition 2
  • Definition 3: Delayed disclosure
  • Lemma 1
  • proof
  • Definition 4: Maximally delayed disclosure
  • Proposition 1
  • Definition 5
  • Proposition 2
  • Definition 6: Cutoff cascade
  • ...and 21 more