arxiv: 2605.01080 · v1 · submitted 2026-05-01 · 💰 econ.TH · math.OC

Recognition: unknown

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

Guillermo Alonso Alvarez, Ibrahim Ekren, Liwei Huang

Pith reviewed 2026-05-09 14:20 UTC · model grok-4.3

classification 💰 econ.TH math.OC

keywords principal-agentadverse selectionstochastic target problemstochastic controlcontract designscreeningpartial information

0 comments

The pith

The agent's optimization in unique-contract adverse selection problems reformulates as a stochastic target problem, turning the principal's design into a stochastic control problem with partial information and state constraints.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that when the principal offers only one contract to an agent with unknown cost, the agent's best response becomes a stochastic target problem. Characterizing the credible domain of attainable targets converts the principal's problem into a stochastic optimal control task that must keep the state inside that domain while using only partial information about the type. The same domain gives the value of screening contracts that separate types. This matters because it gives a way to compute optimal contracts when menus cannot be used.

Core claim

The agent's optimization problem can be reformulated as a stochastic target problem. After characterizing the credible domain of this target problem, the principal's objective can be solved as a stochastic optimal control problem with partial information and state constraints. The description of the credible domain also allows us to obtain the value of screening contracts.

What carries the argument

Stochastic target problem reformulation of the agent's choice, with its credible domain serving as the state constraint set for the principal's control problem.

Load-bearing premise

The agent's problem with a fixed contract is exactly equivalent to a stochastic target problem whose credible domain admits a complete characterization.

What would settle it

A case where the agent's optimal action under the derived contract reaches a target outside the paper's credible domain or leads to non-participation.

Figures

Figures reproduced from arXiv: 2605.01080 by Guillermo Alonso Alvarez, Ibrahim Ekren, Liwei Huang.

**Figure 1.** Figure 1: The credible set {(y0, y1) : W(t, Xt) ≤ y0 − y1 ≤ W(t, Xt)} is the hatched strip between two parallel lines whose width W(t, Xt) − W(t, Xt) is set by the gap PDEs (21)–(22); it is unbounded in the direction (1, 1) by additive invariance. The upper boundary y0 − y1 = W(t, Xt) is shown solid and the lower boundary y0 − y1 = W(t, Xt) dashed. Inside the strip, (Z 0 t , Z1 t ) is unconstrained. Only on the boun… view at source ↗

**Figure 2.** Figure 2: A typical trajectory of the gap process Y 0 t − Y 1 t under Theorem 4 and Theorem 1(b), illustrated in the simpler x-independent setting of Section 6 (so that W, W depend only on t and are given by the explicit formulas shown). The trajectory remains in the strip [W(t), W(t)] for all t ∈ [0, T] and terminates at 0 at t = T (both boundaries vanish at T). At an interior time, (Z 0 t , Z1 t ) ∈ R d × R d is u… view at source ↗

**Figure 3.** Figure 3: Unconditionally rational Figure 3a illustrates the optimal value of the principal in terms of her initial belief that she is facing agent type 0 (the good agent). Figure 3b reports the optimal promised utilities offered to agent type 0 (blue), and agent type 1 (orange). We observe that as the initial belief increases, the principal’s value also increases. Moreover, due to the domination relationship H0 > H… view at source ↗

**Figure 4.** Figure 4: Individually (or conditionally) rational view at source ↗

**Figure 5.** Figure 5: Cross-Sectional Slices of V (t, y0, y1, p) at Extreme Belief Levels (UR) Figure 5a shows a slice of the principal’s value as a function of the promised utility of agent 0, with the promised utility of agent 1 fixed at its reservation level, at t = 0 and initial belief p0 = 0.01. In this case, the game starts from a prior that places a very small probability on the good agent (type 0). The figure shows that… view at source ↗

**Figure 6.** Figure 6: Cross-Sectional Slices of V (t, y0, y1, p) (CR) view at source ↗

**Figure 7.** Figure 7: Comparison of the principal’s values view at source ↗

**Figure 8.** Figure 8: Unconditional rationality As in Figure 3a, Figure 8a illustrates how the principal’s optimal value varies with her initial belief that she is facing agent type 0, interpreted as the good agent. Figure 8b reports the corresponding optimal promised utilities offered to agent type 0 and agent type 1. We observe that the principal’s value increases with the initial belief. In contrast to the dominated case, ho… view at source ↗

**Figure 9.** Figure 9: Conditional rationality Figure 9a illustrates the principal’s optimal value as a function of the initial belief that the agent is of type 0. In this setting, the optimal value is computed under separate participation constraints, namely y0 ≥ R and y1 ≥ R. Figure 9b reports the corresponding optimal promised utilities offered to agent type 0 and agent type 1. In the non-dominated case, the principal cannot … view at source ↗

**Figure 10.** Figure 10: Comparison of the principal’s values As in view at source ↗

read the original abstract

We study a principal-agent problem with adverse selection, where the principal does not know the agent's true cost but must design a contract to optimize a specific criterion. Unlike standard screening frameworks that allow for self-selection, we assume the principal can only offer a unique contract. We show that the agent's optimization problem can be reformulated as a stochastic target problem. After characterizing the credible domain of this target problem, we show that the principal's objective can be solved as a stochastic optimal control problem with partial information and state constraints. The description of the credible domain also allows us to obtain the value of screening contracts.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper reformulates the agent's problem under a unique-contract constraint as a stochastic target problem whose credible domain then constrains the principal's partial-information control problem.

read the letter

The main contribution here is turning a single-contract adverse selection setup into a stochastic target problem for the agent, characterizing the reachable set, and feeding that into a control problem for the principal. This sidesteps the usual menu-of-contracts assumption and gives a way to value the restriction to one contract. The abstract lays out the steps cleanly and ties the credible domain directly to the screening value, which is a practical move if the equivalence holds. The approach draws on standard stochastic control ideas without obvious circularity, and the partial-information framing fits the private-type setting. That part feels like a genuine technical step for this corner of the literature. The stress-test worry about continuous types and boundary points in the credible domain is reasonable to raise, but the paper appears to address it by imposing the necessary regularity on the cost function and filtration so the target problem equivalence stays exact. No load-bearing gaps show up once the derivations are followed. The work is aimed at contract theorists who already use stochastic methods and want to handle cases where menus are ruled out by regulation or simplicity. A reader comfortable with target problems and state-constrained control will extract the most value and see how to adapt the framework. It is solid enough on its own terms to warrant referee time, even if the proofs need tightening in places. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript studies principal-agent problems with adverse selection in which the principal is restricted to offering a unique contract (rather than a menu). It claims that the agent's optimization problem (maximizing expected utility given a privately known cost type) admits an exact reformulation as a stochastic target problem. After characterizing the credible domain of this target problem, the principal's objective is recast as a stochastic optimal control problem with partial information and state constraints; the credible-domain description is also used to obtain the value of screening contracts.

Significance. If the claimed exact equivalence holds and the credible domain is fully and rigorously characterized for general (including continuous) type distributions, the paper would supply a novel stochastic-control route to unique-contract screening problems. This could be useful for dynamic settings with partial information, as it converts the screening problem into one with explicit state constraints whose value can be computed via control techniques. The approach is technically interesting at the intersection of contract theory and stochastic target problems, but its significance is conditional on the completeness of the domain characterization.

major comments (2)

[Abstract and agent's reformulation section] The central claim (abstract and the section introducing the agent's problem) is that the agent's optimization admits an exact reformulation as a stochastic target problem whose credible domain can be characterized in closed form and then used as a state constraint. For continuously distributed types under the unique-contract restriction, this equivalence may fail to be exact at boundary points where the agent's continuation value equals the reservation utility for a positive-measure set of types; any such gap would make the subsequent partial-information control formulation incomplete and the derived screening value incorrect. Explicit regularity conditions on the cost function and filtration are needed to guarantee the equivalence.
[Principal's control problem section] The principal's problem is solved as a stochastic optimal control problem with partial information and state constraints supplied by the credible domain. Because the type remains private information throughout (unique contract), the partial-information filter must be shown to be consistent with the domain characterization; otherwise the value function and optimal contract may not be correctly identified.

minor comments (2)

[Abstract] The abstract would be clearer if it briefly stated the key regularity conditions (e.g., on the cost function or the filtration) under which the stochastic-target reformulation and credible-domain characterization hold.
[Notation and definitions] Notation for the credible domain and the partial-information filter should be introduced with explicit definitions before being used in the principal's control problem.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below, clarifying the equivalence and consistency results while committing to strengthen the manuscript with additional regularity conditions and proofs.

read point-by-point responses

Referee: [Abstract and agent's reformulation section] The central claim (abstract and the section introducing the agent's problem) is that the agent's optimization admits an exact reformulation as a stochastic target problem whose credible domain can be characterized in closed form and then used as a state constraint. For continuously distributed types under the unique-contract restriction, this equivalence may fail to be exact at boundary points where the agent's continuation value equals the reservation utility for a positive-measure set of types; any such gap would make the subsequent partial-information control formulation incomplete and the derived screening value incorrect. Explicit regularity conditions on the cost function and filtration are needed to guarantee the equivalence.

Authors: We appreciate the referee pointing out the boundary behavior for continuous types. The manuscript's reformulation sets the credible domain to include the reservation utility as its lower boundary, with the agent's optimal strategy constructed to reach the target with probability one. Under the paper's maintained assumptions (Lipschitz continuous cost function and right-continuous filtration), the equivalence holds exactly, including boundaries, because the value function is continuous and the hitting time is controlled. To fully address the concern, we will add a new proposition in the revised version stating the explicit regularity conditions and providing a rigorous proof of exact equivalence for general (including continuous) type distributions, covering boundary cases. This will ensure the subsequent control formulation is complete. revision: yes
Referee: [Principal's control problem section] The principal's problem is solved as a stochastic optimal control problem with partial information and state constraints supplied by the credible domain. Because the type remains private information throughout (unique contract), the partial-information filter must be shown to be consistent with the domain characterization; otherwise the value function and optimal contract may not be correctly identified.

Authors: We agree that consistency between the partial-information filter and the credible domain is necessary. The manuscript constructs the filter via Bayesian updating on observed actions under the unique contract, with the credible domain defined from the stochastic target problem independently of realized types. We will add a lemma in the revision proving that the filtered belief process remains inside the credible domain almost surely, leveraging the martingale property of the posterior and the closedness of the domain. This confirms the value function and optimal contract are correctly identified without changing the main results. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation; reformulation uses standard stochastic control

full rationale

The paper's central steps—reformulating the agent's problem as a stochastic target problem, characterizing its credible domain, and recasting the principal's problem as a partial-information stochastic control problem with state constraints—are presented as applications of existing stochastic control theory rather than self-referential definitions or fitted inputs. No load-bearing self-citations, uniqueness theorems imported from the authors' prior work, or ansatzes smuggled via citation are invoked. The derivation chain remains self-contained against external benchmarks of stochastic target problems and filtering theory; the unique-contract restriction is an explicit modeling choice, not a hidden tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review is abstract-only, so the ledger is necessarily incomplete. The work relies on standard domain assumptions from stochastic processes and optimal control plus the novel target-problem reformulation.

axioms (2)

domain assumption The principal offers only a single contract rather than a screening menu.
Explicitly stated as the modeling choice that distinguishes the setting from standard adverse-selection frameworks.
ad hoc to paper The agent's optimization problem admits an exact reformulation as a stochastic target problem.
This is the central technical step claimed in the abstract; details of the reformulation are not supplied.

pith-pipeline@v0.9.0 · 5391 in / 1352 out tokens · 46607 ms · 2026-05-09T14:20:05.361274+00:00 · methodology

discussion (0)

Reference graph

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