📈 economics

Principal-agent problems with adverse selection: A stochastic target problem formulation

This paper reformulates a principal-agent problem with adverse selection, where the principal offers a unique contract, as a stochastic target problem to characterize the agent's optimization and solve the principal's objective as a stochastic optimal control problem with partial information and state constraints.

Original authors: Guillermo Alonso Alvarez, Ibrahim Ekren, Liwei Huang

Published 2026-05-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Guillermo Alonso Alvarez, Ibrahim Ekren, Liwei Huang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a boss (the Principal) trying to hire a worker (the Agent) to run a project. The project's success depends on how hard the worker tries, but you can't see their effort directly; you only see the final results (like sales numbers or output).

Usually, in these stories, the boss knows exactly what kind of worker they are hiring. But in this paper, the boss faces a mystery: The worker knows their own "cost" of effort, but the boss doesn't.

Type 0: A "Good" worker who finds it easy to work hard (low cost).
Type 1: A "Bad" worker who finds it very tiring to work hard (high cost).

The worker knows which type they are immediately. The boss only has a guess (a probability) about which type they are hiring.

The Big Problem: One Contract vs. A Menu

In classic business theory, if a boss suspects there are two types of workers, they offer a menu of two different contracts. The worker picks the one meant for them, effectively revealing their type.

This paper asks a different question: What if the boss is forced to offer only one single contract to everyone? They can't offer a menu. They have to design one deal that works for both types, hoping the "Good" worker doesn't get too lazy and the "Bad" worker doesn't quit.

The Solution: A "Trust Zone" (The Credible Set)

The authors realized that offering a single contract is like trying to walk a tightrope between two cliffs.

The Promise: The contract promises the worker a certain "future happiness" (utility) based on how well they do.
The Gap: Because the two types of workers have different costs, the "Good" worker and the "Bad" worker will naturally want different levels of future happiness to be motivated.
The Trap: If the boss promises too much happiness to the Bad worker, they waste money. If they promise too little to the Good worker, the Good worker quits.

The paper proves that there is a specific "Trust Zone" (mathematically called a credible set). This zone is a strip on a graph.

If the difference between the promised happiness for the Good worker and the Bad worker falls inside this strip, a single contract can successfully motivate both.
If the difference falls outside the strip, it's impossible to make a single contract work for both. One type will always be unhappy or unmotivated.

Think of this strip as a safety corridor. The boss must steer the contract so that the "gap" in promises stays inside this corridor. If the gap hits the wall of the corridor, the boss has to change the contract immediately to keep it from crashing.

The Boss's New Job: Filtering the Fog

Since the boss doesn't know the worker's type, they have to play detective.

Every time the worker produces a result, the boss updates their guess: "Hmm, that result was really good. Maybe I hired the Good worker after all?"
This is called filtering. The boss is constantly updating their belief (a probability score) about who they are dealing with.

The paper shows that the boss's problem becomes a complex game of Stochastic Control. The boss isn't just picking a salary; they are steering a ship through a foggy sea (the unknown type) while trying to keep the ship inside a narrow channel (the Trust Zone).

The "Magic" Formula

The authors developed a mathematical recipe (a set of equations called HJB equations) that tells the boss exactly how to adjust the contract in real-time.

Inside the channel: The boss has freedom to choose the best contract.
Hitting the wall: If the gap in promises gets too close to the edge of the Trust Zone, the boss is forced to make very specific, rigid adjustments to the contract to prevent it from breaking.

The Surprising Finding: One Contract vs. Two

The paper compares two scenarios:

The Menu (Screening): The boss offers two contracts, and the worker picks one. (The boss learns the type immediately).
The Single Contract: The boss offers one contract and has to guess the type over time.

The Result: It's not always better to offer a menu!

Sometimes, offering a single contract is actually more profitable for the boss.
Why? Because the "filtering" process (learning over time) allows the boss to extract more value from the worker than the rigid rules of a menu sometimes allow. It depends on the specific numbers (how different the workers are, how much the boss dislikes risk, etc.).

In a Nutshell

This paper solves a puzzle for bosses who can't offer a menu of choices. It shows that even without knowing who they are hiring, a boss can design a single, dynamic contract that works for everyone. The secret is to keep the "promise gap" between the two types of workers inside a safe, mathematical "Trust Zone," while constantly updating their guess about the worker's identity based on performance. Sometimes, this "one-size-fits-all" approach is actually smarter than trying to guess the worker's type upfront and offering a menu.

1. Problem Statement

The paper addresses a continuous-time Principal-Agent problem with adverse selection.

Setting: A Principal designs a contract $\xi$ (a terminal payment) based on an output process $X$ . An Agent accepts the contract and chooses an effort level (control) to maximize their expected utility.
Adverse Selection: The Agent possesses private information regarding their type $\Theta \in \{0, 1\}$ (e.g., cost of effort), known at time $t=0$ but unknown to the Principal. The Principal only knows the prior distribution of $\Theta$ .
Constraint: Unlike classical screening models where the Principal offers a menu of contracts to induce self-selection, this paper assumes the Principal can offer only a single unique contract. Consequently, the Principal cannot observe the Agent's type immediately upon acceptance and must infer it dynamically from the observed output path $X$ .
Objective: The Principal seeks to maximize their expected utility $E[Up(X_T - \xi)]$ subject to the Agent's incentive compatibility (optimal response) and participation (rationality) constraints.

2. Methodology

The authors develop a novel framework combining Stochastic Target Problems, Filtering Theory, and State-Constrained Stochastic Control.

A. Reparametrization via Stochastic Target

Following the Sannikov (2008) and Cvitanić et al. (2018) approach, the contract space is reparametrized. Instead of optimizing over the terminal payment $\xi$ , the Principal optimizes over:

Initial continuation utilities $(y_0, y_1)$ for each type.
Sensitivity processes $(Z_0, Z_1)$ encoding the contract's sensitivity to the output.

The contract is recovered as the terminal value of a forward SDE: $\xi = Y_T^{0, y_0, Z_0} = Y_T^{1, y_1, Z_1}$ .

The Target Constraint: Since the contract $\xi$ must be a single random variable independent of the type, the terminal values of the two type-specific continuation utility processes must coincide almost surely:
$Y_T^{0, y_0, Z_0} = Y_T^{1, y_1, Z_1}, \quad \text{a.s.}$
This is formulated as a Stochastic Target Problem (Soner & Touzi, 2002).

B. Characterization of the Credible Set

The set of feasible initial utility pairs $(y_0, y_1)$ that can be supported by a single contract is called the Credible Set $\mathcal{E}(X_0)$ .

The authors characterize $\mathcal{E}(X_0)$ explicitly as a strip in the $(y_0, y_1)$ plane defined by the gap $\Delta = y_0 - y_1$ .
The boundaries of this strip, $W(t, x)$ and $\bar{W}(t, x)$ , are solutions to a pair of fully nonlinear Partial Differential Equations (PDEs) (Hamilton-Jacobi-Bellman type).
State Constraint: The condition that the contract is valid translates to a state constraint on the gap process:
$W(t, X_t) \leq Y_t^0 - Y_t^1 \leq \bar{W}(t, X_t).$
If the gap hits a boundary, the controls $(Z_0, Z_1)$ are constrained to specific sets to keep the process within the strip.

C. Filtering and Augmented State

Because the Principal does not know $\Theta$ , they must maintain a belief $p_t = P(\Theta=0 | \mathcal{F}_t^X)$ .

Using the Wonham filter (Kushner-Stratonovich equation), the belief $p_t$ evolves based on the innovation of the output process $X$ .
The Principal's problem is reduced to a State-Constrained Stochastic Optimal Control Problem with an augmented state vector $(X_t, Y_t^0, Y_t^1, p_t)$ .

D. Handling State Constraints

Standard state-constrained control requires an "inward-pointing" condition, which fails here because controls on the boundary are pinned down by the credibility constraint.

The authors adapt the methodology of Bouchard, Élie, and Imbert (2010).
Under specific structural assumptions (data independent of $X$ , bounded effort, risk-neutral Principal), they show the problem reduces to a Dirichlet boundary value problem. The Principal receives a specific boundary utility when the state hits the lateral boundary of the credible set.

3. Key Contributions

Stochastic Target Formulation for Adverse Selection: The paper provides the first rigorous continuous-time formulation of a single-contract adverse selection problem using stochastic target constraints. It bridges the gap between moral hazard (Sannikov framework) and adverse selection.
Explicit Credible Set Characterization: The authors prove that the set of implementable utility pairs is a strip defined by the solutions to fully nonlinear PDEs. This provides a geometric interpretation of the feasibility of single contracts under asymmetric information.
Unified Framework for Screening vs. Single Contract: The paper recasts the classical screening problem (offering a menu) within the same stochastic target framework. This allows for a direct, apples-to-apples comparison between the value of offering a single contract (with filtering) versus a menu of contracts (with immediate revelation).
Verification and Structure of Optimal Contracts: The optimal contract is shown to depend on the output $X$ , the continuation utilities of both types, and the Principal's belief $p_t$ . It is a feedback rule in the augmented state space, proving that optimal contracts in this setting are non-trivial and cannot be expressed as functions of $X$ alone.

4. Key Results

Theorem 1 (Credible Set): The set of feasible initial utilities is $\mathcal{E}(X_0) = \{(y_0, y_1) : W(0, X_0) \leq y_0 - y_1 \leq \bar{W}(0, X_0)\}$ . The boundaries are determined by the gap PDEs.
Theorem 2 (Reduction): The Principal's value function is the supremum of a state-constrained control problem over the credible set.
Theorem 4 (Viscosity Solution): Under specific assumptions, the value function is the unique continuous viscosity solution to a Dirichlet problem, where the boundary conditions represent the utility obtained when the gap process hits the feasibility limits.
Numerical Findings (Section 9):
- Dominated Case: If one agent type is strictly "better" (lower cost), the Principal's value is convex in the prior belief. The Principal may offer negative utility to the "bad" type to extract surplus from the "good" type.
- Non-Dominated Case: When types are not uniformly ordered, the Principal's value is still convex in the prior.
- Single Contract vs. Screening:
  - In the Dominated Case, the Screening Menu (offering two contracts) generally yields a higher value than a single contract because it resolves information asymmetry immediately.
  - In the Non-Dominated Case, the Single Contract (with dynamic filtering) can sometimes outperform the Screening Menu. This counter-intuitive result suggests that the ability to learn the type dynamically from the output path can be more valuable than the immediate revelation provided by a menu, particularly when the types are not easily distinguishable by cost alone.

5. Significance

This paper makes a significant theoretical contribution to contract theory by:

Extending the Sannikov Framework: It successfully incorporates adverse selection into the continuous-time stochastic control framework, which was previously limited to moral hazard or discrete-time models.
Resolving the "Single Contract" Puzzle: It provides a tractable method to solve for optimal contracts when the Principal is restricted to a single offer, a scenario common in real-world applications (e.g., standardized insurance policies) but difficult to model mathematically.
Economic Insight: The comparison between single contracts and screening menus reveals that dynamic learning (filtering) can be a powerful tool for the Principal, potentially outweighing the benefits of immediate type revelation in complex environments where types are not strictly ordered.

The work opens new avenues for numerical analysis of dynamic contracts using Deep Galerkin methods and viscosity solution theory, offering a robust toolkit for analyzing principal-agent problems with hidden information.