Dynamic Delegation & Reputation Feedback

• The paper introduces a recursive equilibrium model where experts delegate tasks and update reputation based on outcome feedback, linking risk recommendations to private signals.
• It employs a reputation‐dependent cutoff rule with Bayesian updating, ensuring that higher reputations lead to more cautious risk-taking due to the greater cost of failures.
• Comparative statics show that improvements in signal precision, higher prior success, and increased patience adjust risk thresholds, with practical implications such as success‐contingent bonus schemes.

Dynamic delegation with reputation feedback is a mechanism in which an experienced principal (expert) repeatedly advises or assigns tasks to a stream of self-interested implementers, with each implementer’s effort—and thus the informativeness of outcomes—being endogenous to the current public reputation of the expert. Reputation, in turn, is updated based on the results of delegated actions. The dynamic interplay between advice, reputation formation, agent effort, and outcome feedback leads to recursive, belief-based equilibria exhibiting sensitive dependence on the structure of outcome informativeness, private signal precision, prior beliefs, and incentives for experimentation.

1. Recursive Equilibrium Structure

The central model involves a long-lived expert interacting over time with a series of short-lived agents. The expert has access to a private signal $s$ about the relevant state, and each period must choose an action $a \in \{0, 1\}$ (e.g., recommend risk vs. recommend safety) for the implementer, who then exerts effort. The key state variable is the current public belief (reputation) $\pi \in (0,1)$, representing the probability that the expert is competent (high type).

The equilibrium is recursive and belief-based: the public reputation $\pi$ summarizes all payoff-relevant public history, and the value function for the expert admits a BeLLMan equation:

$$V(\pi) = \max_{a \in \{0, 1\}} \left\{ u(\pi) + \left(\phi \cdot \mathbb{1}_{\{a=1\}}\right) + \delta \cdot \mathbb{E}\left[ V(\pi') \mid a, \pi \right] \right\}$$

Here, $u(\pi)$ summarizes short-run returns from holding reputation $\pi$; $\phi$ is (optionally) a per-period bonus or fee for choosing the risk action; and $\delta$ is the discount rate. The conditional expectation calculates the continuation value after observing the outcome-dependent update of reputation $\pi'$ based on Bayes’ rule, with the distribution of outcomes (success/failure) and their informativeness being determined endogenously via the implementer’s effort, itself a function of $\pi$.

2. Reputation-Dependent Cutoffs and Reputational Conservatism

The equilibrium advice policy of the expert is described by a reputation-dependent cutoff rule. There exists a (weakly) increasing threshold $s^*(\pi)$ such that the expert recommends risk ($a=1$) if and only if $s \geq s^*(\pi)$. The threshold solves for indifference in the marginal continuation value of taking risk:

$$\Delta_H(s;\pi) \equiv \left\{ \phi + \delta \cdot \mathbb{E}[V(\pi') \mid a=1, s, \pi] \right\} - \{\delta V(\pi)\} = 0$$

A diagnosticity condition—specifically, that failures are at least as informative as successes—implies that the value loss from a failure at high reputation is greater than the incremental value of a success. As a result, the expert exhibits reputational conservatism: at higher public reputation, the expert selects a higher cutoff $s^*(\pi)$, taking risk less often and only when her private signal is especially favorable.

This mechanism is grounded in the distributional properties of the outcome signals: for monotone likelihood ratio properties (MLRP), the informativeness of failures induces highly nonlinear reputational risk, amplifying caution as reputation grows.

3. Comparative Statics and Calibration

The equilibrium cutoff $s^*(\pi)$ possesses transparent comparative statics:

• Private Signal Precision: Increasing expert information quality (e.g., in a Gaussian signal model via larger $\mu_1-\mu_0$ or smaller $\sigma_H$) decreases $s^*(\pi)$, increasing experimentation.
• Ex-ante Success Probability: A higher prior probability of a good (success-prone) state lowers the cutoff, making the expert less conservative.
• Discount Factor (Patience): Greater patience ($\delta$ closer to 1) raises $s^*(\pi)$, making the expert more conservative due to the increased value placed on future reputation.

These effects follow directly from first-order implicit differentiation of the BeLLMan recursion and are explicitly characterized in a Gaussian learning benchmark.

A practical calibration method is provided for incentivization—specifically, implementing a success-contingent bonus ($\beta$) that can be tuned to render any desired experimentation rate implementable, by ensuring:

$$\beta^{\min}(\pi, \tilde{s}) = -\frac{\Delta_H(\tilde{s}(\pi); \pi)}{P_S(\tilde{s}(\pi); \pi)}$$

where $P_S$ is the success rate conditional on $s$ and $\pi$.

4. Reputation Dynamics and Martingale Properties

Public reputation $\pi_t$ evolves via Bayesian updating on observed outcomes only when the expert recommends risk (i.e., when $a=1$). This updating process generates a reputation trajectory with the following key characteristics:

• If the expert is truly competent (high type), $(\pi_t)$ is a submartingale: $$\mathbb{E}[\pi_{t+1}|\mathcal{H}_t, \theta=H] \geq \pi_t$$.
• For less competent experts, $(\pi_t)$ is a supermartingale.

The process is jump-driven: only risky recommendations followed by observed outcomes produce belief updates; safe recommendations leave beliefs unchanged (no-news "absorption"). Thus, long safe streaks correspond to periods of reputation stasis, while experimentation (risk-taking) induces informative learning and possible boundary hitting (very high or very low $\pi$).

Boundary hitting distinguishes between two regimes:

• Learning: Repeated risky choices with outcomes. Eventually, $\pi_t$ converges to a high- or low-reputation absorbing state.
• No-news absorption: If the expert refrains from experimentation as $\pi$ rises, the process can stall in a region of stasis, with no further learning.

5. Implementation Mechanisms and Practical Predictions

The formalism yields testable predictions and practical design primitives. For example, in applications such as surgery (the operate vs. conservative care case):

• Prediction 1 (Risk Taking): Surgeons with higher reputation $\pi$ will recommend surgery less frequently, conditional on similar patient risk profiles.
• Prediction 2 (Conditional Success): When high-reputation surgeons do operate, observed success rates are higher, due to increased implementer (patient/team) effort, itself responding positively to reputation.
• Prediction 3 (Reputational Shock): Early successes that raise $\pi$ paradoxically lead to lower subsequent experimentation rates (higher cutoffs) but increase the stakes (adverse informational content) of future failures.

Empirical measurement strategies are outlined: reputation ($\pi$) can be proxied by historical risk-adjusted outcomes or performance scores, the private signal ($s$) by diagnostic reports, and the outcome ($y$) by clinical or quality-of-care indicators. Implementer effort may be inferred from adherence or engagement metrics.

6. Schematic Summary of Mechanism and Feedback Loop

Component	Functional Role	Endogenous Effect
Private signal $s$	Expert’s information on underlying state	Determines willingness to recommend risky action
Action $a$	Risk or safety advice choice	Triggers agent effort and outcome realization
Implementer effort	Level of exertion (monotonic in $\pi$)	Modulates outcome probabilities and informativeness
Outcome $y$	Success/failure	Updates credential via Bayes’ rule
Public reputation $\pi$	Sufficient statistic for beliefs	Governs future delegation and experimentation rates

7. Connection to Broader Reputation Mechanisms

The dynamic delegation with reputation feedback model presented in this work (Lukyanov et al., 27 Aug 2025) incorporates critical features of dynamic learning from feedback, endogenous task assignment, and reputation-driven effort incentives. It formalizes the way an expert’s advice policy adapts over time in response to both stochastic signals and evolving public beliefs about competence, highlighting structurally novel properties such as nonlinear reputational conservatism, implementability via calibrated bonuses, and recursive equilibria that deliver testable predictions for organizational, clinical, and market environments.

The model’s submartingale/supermartingale characterization of reputation dynamics, together with precise intervention guidance (such as the success-contingent bonus), positions it as a rigorous foundation for studies of dynamic agency, learning, and feedback in systems where trust, reputation, and recurrent delegation are pivotal.