Feedback Design in Dynamic Moral Hazard: Ely, Georgiadis & Rayo (2025)

What this is. The paper’s core theoretical results, the model equations (principal-agent setup with binary success signal), and the solution method (minimal reward characterization, bang-bang relaxed problem): enough to know what it found and how, without reading all 25 pages. To replicate or extend it, read the full source at the original.

TL;DR

The paper studies the optimal joint design of performance feedback and monetary rewards when the only available performance signal is coarse: a binary “success” that arrives stochastically as a function of the agent’s accumulated effort. A “backward compounding effect” governs the analysis: promising greater future rewards to prevent pausing raises the cost of all earlier rewards as well, and this cost is larger the further into the future the information is hidden. As a result, the optimal baseline contract front-loads the agent’s ignorance. The optimal contract has at most two phases (Theorem 1, p. 606): when c < 1/2, a fully silent phase in which the principal says nothing and the agent works regardless of success, followed by a full-transparency pronto phase in which the agent is immediately informed and stops upon success; otherwise, the silent phase has zero length. The two-phase structure remains optimal with continuation payoffs, a more informed agent, and upfront effort investment. With costly direct monitoring, Proposition 4 shows that a sufficiently short early-silence policy dominates always-pronto when m > c; it does not characterize the fully optimal contract in that extension.

Core results

All results are theoretical; locators point into the source PDF.

#	Result	Locator	Stated form
R1	Local incentive constraint (LIC): necessary condition for incentive compatibility via deterring instantaneous pauses	Prop. 1, p. 602	r(t)R(t)f(t) - cp(t) ≥ cr(t)Q(t)f(t) + ∫ r(s)[R(s)-cQ(s)]|f’(s)| ds - c∫ r(s)f(s) ds
R2	Minimal implementing reward schedule: unique least-cost reward satisfying LIC with equality at every t	Prop. 2, p. 603	r(t)R(t) = c[p(t)/f(t) - ∫ f’(s)/f(s)² p(s) ds - ∫ (r(s)-r(t)q(s|t)) ds] (eq. 4)
R3	Principal’s objective simplifies to (Obj); relaxed problem (P) admits bang-bang solution with cutoffs t* and T*	Lemma 1 + Prop. 3, pp. 605-606	t* = min{t : 1-2c-cΦ(t) ≤ 0}; p = 1 on [0,t*], 1-F(t) on (t*,T*], 0 thereafter; r = 1 on [0,T*]
R4	When T > t*, every optimal policy has at most two phases: silent (no disclosure) then pronto (immediate disclosure)	Thm. 1, p. 606	Silent reward: c/λ(T*) + cF(t*)/f(t*); pronto reward: c/λ(T*); silent phase has positive length iff c < 1/2
R5	With direct monitoring cost m > c, early silence dominates always-pronto when the silent phase is sufficiently short	Prop. 4, p. 612; proof, p. 620	Expected cost ratio Δ(t) < 1 for sufficiently small t iff m > c
R6	With continuation payoffs for both parties, the silent cutoff is unchanged and the principal extends the pronto phase	§5.1, text p. 609	The objective gains (v + π)F(T̃); t* is unchanged and T* weakly increases
R7	With probability h of privately learning success, the two-phase policy remains optimal with a lower silent-phase reward	Cor. 1, p. 610	Silent-phase reward: c/λ(T̃) + c(1-h)F(t*)/f(t*), below the baseline reward; cutoff t* unchanged
R8	An upfront effort investment preserves the silent-then-pronto structure and expands the range supporting a silent phase	§5.3, text p. 611	Silent phase iff c < 1/(2-μ), compared with c < 1/2 in baseline; its length grows with μ

Overall (paper’s conclusion). The backward compounding effect unifies the analysis: silence grants the agent rents, and these rents compound the longer they are deferred, so the optimal baseline contract front-loads ignorance. The two-phase structure remains in the continuation-payoff, more-informed-agent, and upfront-investment extensions; the direct-monitoring result compares early silence with always-pronto rather than establishing a fully optimal contract. The key quantity is Φ(t) = F(t)(d/dt)(1/f(t)), which captures the compounding effect.

Theory / model

The baseline model (Section 2, pp. 599-601) is a continuous-time game on [0, T] with T large and finite. At each instant, the agent privately chooses to work (incurring flow cost c < 1 per unit time) or wait. A success occurs at most once and is observed only by the principal. The agent’s cumulative effort e maps to success via a CDF $F : [0, T] \to [0, 1]$ with F(0) = 0 and F(T) ≤ 1. Both the density $f = F'$ and the hazard rate $\lambda(e) = f(e)/[1-F(e)]$ are weakly decreasing in e. The key auxiliary function (p. 600, eq. 1) is:

$$\Phi(t) := F(t)\,\frac{d}{dt}\frac{1}{f(t)}, \quad \Phi(0) = 0, \quad \Phi \text{ weakly increasing.} \tag{1}$$

A larger $\Phi(t)$ means the backward compounding effect is more severe: the density is declining faster in inverse-density terms (large $-f'/f^2$), and more cumulative probability mass is available for past rewards to compound (large $F(t)$).

The principal designs (i) a reward schedule $R(t) \geq 0$ paid upon success at t and (ii) a feedback policy. By a direct-mechanism argument it suffices to consider direct policies: $q(s|t)$, the probability the agent is asked to continue working at s conditional on having succeeded at $t \leq s$, and $r(s)$, the probability the agent is asked to continue conditional on no success yet. The total probability the agent is asked to work at least until s is (p. 600, eq. 2):

$$p(s) = r(s)\bigl[1 - F(s)\bigr] + \int_0^s r(u)\,f(u)\,q(s|u)\,du. \tag{2}$$

The agent’s continuation payoff from obeying recommendations from t onward is (p. 601, eq. 3):

$$U(t) := \int_t^T r(s)\,R(s)\,f(s)\,ds - c\int_t^T p(s)\,ds. \tag{3}$$

The principal maximizes total expected effort net of rewards: $\int_0^T p(s)\,ds - \int_0^T r(s)R(s)f(s)\,ds$, subject to incentive compatibility. The agent is risk neutral and cash-constrained (no punishment below zero). Both players are assumed not to discount time, though the results are qualitatively unchanged with a common discount rate.

The paper builds on the continuous-time principal-agent approach of Sannikov (2008) and the information-design frameworks of Kamenica and Gentzkow (2011) and Rayo and Segal (2010), extending them to a setting where the principal jointly controls both the feedback policy and the monetary rewards.

Method

Step 1 - Local incentive constraint (Proposition 1, p. 602). It suffices to focus on instantaneous deviations (brief pauses). Dividing the no-pause condition by $\Delta t$ and letting $\Delta t \to 0$ establishes the local incentive constraint (LIC): for all t,

$$r(t)R(t)f(t) - c\,p(t) \geq c\,r(t)Q(t)f(t) + \int_t^T r(s)\bigl[R(s) - cQ(s)\bigr]\lvert f'(s)\rvert\,ds - c\int_t^T r(s)\,f(s)\,ds, \tag{LIC}$$

where $Q(t) = \int_t^T q(u|t)\,du$ is the agent’s expected future work conditional on succeeding at t. The three right-hand terms represent: (i) the effort savings (the agent avoids cost c for $Q(t)$ future periods), (ii) the higher density of future success because accumulated effort is lower (the backward compounding integral), and (iii) the higher probability that future recommendation-to-work periods are reached.

Step 2 - Minimal reward schedule (Proposition 2, p. 603). For any given recommendation policy, there is a unique least-expensive reward schedule satisfying (LIC) with equality at every t. Equation (4) gives it:

$$r(t)R(t) = c\!\left[\frac{p(t)}{f(t)} - \int_t^T \frac{f'(s)}{f(s)^2}\,p(s)\,ds - \int_t^T \bigl(r(s) - r(t)\,q(s|t)\bigr)\,ds\right]. \tag{4}$$

The first term is the zero-rent reward (just enough to cover current expected cost). The second term is the backward compounding penalty: since $f' \leq 0$, this term is positive and requires raising current rewards to compensate for high future rewards. The third term is the information rebate: when the principal reveals a success at t (low $q(s|t)$), the agent works less thereafter, reducing future reward obligations, so current rewards can be lower.

For deterministic-deadline contracts, the minimal reward schedule in equation (4) simplifies to equation (5) (p. 603):

$$R(t) = c\left[\frac{p(t)}{f(t)} - \int_t^{\tilde T} \frac{f'(s)}{f(s)^2}p(s)\,ds - \int_t^{\tilde T} \bigl(1-q(s|t)\bigr)\,ds\right]. \tag{5}$$

Step 3 - Simplified objective and bang-bang relaxed problem (Lemma 1 + Proposition 3, pp. 605-606). Substituting (4) into the objective, Lemma 1 shows it simplifies to:

$$\int_0^T p(t)\,dt - c\int_0^T \bigl[p(t)(1 + \Phi(t)) - (r(t) - p(t))\bigr]\,dt. \tag{Obj}$$

The integrand $r(t) - p(t)$ is the information rebate (positive only when a successful agent is still asked to work). Rearranging and collecting terms, the relaxed problem (P) that ignores global deviations is:

$$\sup_{r(\cdot),\,p(\cdot)} \int_0^T p(t)\{1 - 2c - c\Phi(t)\}\,dt + c\int_0^T r(t)\,dt \tag{P}$$ $$\text{s.t.} \quad r(t)[1-F(t)] \leq p(t) \leq 1, \quad 0 \leq r(t) \leq 1 \text{ non-increasing.} \tag{Feas}$$

The numbered feasibility restriction in the relaxed problem is equation (6) (p. 605):

$$r(t)[1-F(t)] \leq p(t) \leq 1. \tag{6}$$

Since $\Phi$ is weakly increasing, the coefficient $1-2c-c\Phi(t)$ is weakly decreasing in t: positive at t = 0 when c < 1/2 and eventually non-positive. Because the objective is linear in $p(t)$ and $r(t)$, Proposition 3 establishes a bang-bang solution: define $t^* = \min\{t : 1-2c-c\Phi(t) \leq 0\}$ and let $T^* \in (t^*, T]$ be the optimal terminal date. Then:

$$p(t) = \begin{cases} 1 & t \in [0, t^*], \\ 1-F(t) & t \in (t^*, T^*], \\ 0 & t > T^*, \end{cases} \qquad r(t) = \begin{cases} 1 & t \leq T^*, \\ 0 & t > T^*. \end{cases}$$

Step 4 - Optimal contract with at most two phases (Theorem 1, p. 606). When $T > t^*$, the bang-bang solution is implemented by a deterministic-deadline contract with at most two phases. In Phase 1 (silent phase, $t \in [0, t^*]$): the principal is silent; $q(s|t) \equiv r(t) \equiv 1$; the agent works continuously regardless of success. In Phase 2 (pronto phase, $t \in (t^*, T^*]$): $r(t) = 1$ and $q(s|t) \equiv 0$; the agent quits immediately upon success. Substituting into (4) and simplifying using $\lambda(T^*) = f(T^*)/[1-F(T^*)]$, the reward for each phase is (Appendix A.5, pp. 618-619):

$$R(t) = \frac{c}{\lambda(T^*)} + \frac{cF(t^*)}{f(t^*)} \quad \text{for } t \in [0, t^*], \qquad R(t) = \frac{c}{\lambda(T^*)} \quad \text{for } t \in (t^*, T^*].$$

The phase 1 reward exceeds the phase 2 reward by $cF(t^*)/f(t^*)$: the “silent premium” that compensates for the rents the agent would earn during the pronto phase, compounded backward into the silent phase. The cutoff $t^* > 0$ (positive silent phase) if and only if c < 1/2. Global incentive compatibility is verified by showing that the non-increasing reward profile makes any global deviation unprofitable (Appendix A.5). The result contrasts with Ely and Szydlowski (2020), where quitting is irreversible and rewards are exogenous, and with Halac, Kartik, and Liu (2016) and Mason and Välimäki (2015), neither of which allows the principal to strategically withhold performance feedback.

Empirical specifications

This is a pure theory paper and has no empirical estimating specifications, fixed effects, standard-error treatment, or estimation sample.

Datasets used

Dataset	Role in paper	Wiki page
None	Pure theory paper; all results are analytical propositions and theorems	n/a

When to read the full paper

Read the original if you are: designing performance evaluation or promotion systems with coarse binary signals in employment or professional services settings; extending the two-phase framework to multiple agents or contest environments (the paper’s own suggested direction); studying how information design interacts with monetary incentives under dynamic moral hazard; or working through the formal proofs, which are detailed and constructive (Appendix A, pp. 613-620). Section 5 (pp. 608-613) covers four extensions: continuation payoffs, a more informed agent, an upfront learning-by-doing investment phase, and costly direct monitoring.

Attribution and rights

Source: peer-reviewed, Econometrica 93(2), March 2025. This distillation was updated by an LLM on 2026-10-04 and is not human-verified or independently reproduced. The CC BY-NC 4.0 licence permits noncommercial sharing; the verbatim PDF is not hosted in this batch.

Attribution (CC BY-NC 4.0). Ely, Jeffrey C., George Georgiadis, and Luis Rayo. “Feedback Design in Dynamic Moral Hazard.” Econometrica 93, no. 2 (March 2025): 597-621. DOI: 10.3982/ECTA21871. © 2025 The Authors. The Econometric Society. Licensed under CC BY-NC 4.0. This page is an adaptation by the Institute for Automated Research: core results extracted and re-expressed; changes were made.