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arxiv: 2605.01080 · v2 · pith:N3ZCCGVGnew · submitted 2026-05-01 · 💰 econ.TH · math.OC

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

Guillermo Alonso Alvarez , Ibrahim Ekren , Liwei Huang This is my paper

Pith reviewed 2026-05-20 23:42 UTC · model grok-4.3

classification 💰 econ.TH math.OC
keywords principal-agent problemadverse selectionstochastic target problemcontract designstochastic controlpartial informationscreening contractsstate constraints
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The pith

The agent's optimization in an adverse selection contract problem is recast as a stochastic target problem whose credible domain turns the principal's task into a control problem with partial information.

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

The paper studies a principal who must design one contract for an agent whose private cost is unknown. Rather than menus that permit self-selection, only a single contract is offered. The agent's best response is reformulated as a stochastic target problem, after which the set of reachable outcomes is characterized as the credible domain. This domain converts the principal's problem into a stochastic optimal control task with state constraints and partial observations. The same domain also supplies the value of screening contracts.

Core claim

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.

What carries the argument

The credible domain of the stochastic target problem, which encodes the set of outcomes the agent can credibly reach and supplies the state constraints and value function for the principal's control problem.

If this is right

  • The principal's contract design reduces to solving a stochastic optimal control problem subject to the credible domain constraints.
  • Partial information about the agent's type is handled directly inside the control formulation.
  • Screening contract values are recovered from the boundary or description of the credible domain.
  • State constraints in the control problem arise precisely from the agent's reachable set.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may extend to repeated or dynamic interactions in which the credible domain evolves with new observations.
  • It could link to mechanism design settings where the principal must maintain credibility of the offered contract over time.
  • Numerical solution of the resulting control problem might be tested on standard cost distributions to compare efficiency with menu-based screening.

Load-bearing premise

The principal is limited to offering a single contract rather than a menu of contracts that would let the agent self-select.

What would settle it

A concrete cost distribution and contract family for which the agent's problem cannot be equivalently stated as a stochastic target problem or for which the derived credible domain fails to bound the principal's achievable payoffs.

Figures

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

Figure 1
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 1
Figure 1. Figure 1: The credible set {(y0, y1) : W(t, Xt) ≤ β1y0 − β0y1 ≤ W(t, Xt)} for general β0, β1 > 0. The constraint on β1y0 − β0y1 defines a strip unbounded in the direction (β0, β1) by additive invariance. The boundaries have slope β0/β1 (here shown for β0 = 1, β1 = 2, giving slope 1/2). Inside the strip, (Z 0 t , Z1 t ) is unconstrained; on the boundaries, the matching conditions via V, V as Z 0 t = β0Z 1 t ±Zt β1 ap… view at source ↗
Figure 2
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 2
Figure 2. Figure 2: A typical trajectory of the gap process Y 0 t − Y 1 t under Theorem 5 and Theorem 1(b), illustrated in the simpler x-independent setting of Section 7 (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. 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 3
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. Figure 4: Individually (or conditionally) rational view at source ↗
Figure 4
Figure 4. Figure 4: Individually (or conditionally) rational [PITH_FULL_IMAGE:figures/full_fig_p033_4.png] view at source ↗
Figure 5
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 5
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. Figure 6: Cross-Sectional Slices of V (t, y0, y1, p) (CR) view at source ↗
Figure 6
Figure 6. Figure 6: Cross-Sectional Slices of V (t, y0, y1, p) (CR) [PITH_FULL_IMAGE:figures/full_fig_p035_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of the principal’s values view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of the principal’s values [PITH_FULL_IMAGE:figures/full_fig_p035_7.png] view at source ↗
Figure 8
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 8
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. Figure 8b reports the corresponding optimal promised utilities assigned to type 0 and type 1. We observe that the principal’s value increases with the initial belief. Although the agents’ cost functions are not ordered in the non-dominated case, … view at source ↗
Figure 9
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 9
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. Figure 10: Comparison of the principal’s values As in view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of the principal’s values As in [PITH_FULL_IMAGE:figures/full_fig_p038_10.png] 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.

Referee Report

2 major / 2 minor

Summary. The paper examines principal-agent problems with adverse selection in which the principal offers a single contract rather than a menu allowing self-selection. The agent's utility maximization is reformulated as a stochastic target problem; the credible domain of this target problem is characterized and then used to impose state constraints on the principal's stochastic optimal control problem with partial information. The same domain characterization is employed to recover the value of screening contracts.

Significance. If the claimed equivalence between the agent's problem and the stochastic target formulation holds with the stated regularity conditions, the approach supplies a technically novel route for embedding adverse-selection screening inside a stochastic control framework with state constraints. This could facilitate the analysis of dynamic or continuous-time screening problems that are difficult to treat with standard mechanism-design tools, particularly when the principal's information is partial and the contract must be unique.

major comments (2)
  1. [§3] §3 (Agent's problem reformulation): the passage from the agent's expected-utility maximization to the stochastic target problem is asserted to be an equivalence, yet the argument appears to require the agent's cost parameter to enter linearly and the noise to be additive; these restrictions should be stated as standing assumptions before the target-problem statement, because they are load-bearing for the subsequent credible-domain construction.
  2. [§5] §5 (Principal's control problem): the partial-information stochastic control problem is formulated with the credible domain as a state constraint, but the paper does not verify that the resulting value function remains continuous up to the boundary of the credible domain; without this verification the optimal contract may not be attainable and the claimed screening values could be overstated.
minor comments (2)
  1. Notation for the credible domain (denoted D or C in different sections) is inconsistent; a single symbol should be fixed throughout.
  2. [Introduction] The abstract states that the principal 'can only offer a unique contract'; this modeling choice is repeated in the introduction but should be contrasted more explicitly with the standard menu-of-contracts benchmark in a dedicated paragraph.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The suggestions help clarify the scope of our reformulation and strengthen the analysis of the principal's problem. We address each major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (Agent's problem reformulation): the passage from the agent's expected-utility maximization to the stochastic target problem is asserted to be an equivalence, yet the argument appears to require the agent's cost parameter to enter linearly and the noise to be additive; these restrictions should be stated as standing assumptions before the target-problem statement, because they are load-bearing for the subsequent credible-domain construction.

    Authors: We agree that the equivalence between the agent's expected-utility maximization and the stochastic target problem relies on the cost parameter entering linearly and the noise being additive. These conditions are essential for the reformulation to hold without additional terms and for the credible-domain construction to proceed as stated. We will revise Section 3 to list these explicitly as standing assumptions immediately before the target-problem formulation. This change will make the load-bearing restrictions transparent and support the subsequent analysis. revision: yes

  2. Referee: [§5] §5 (Principal's control problem): the partial-information stochastic control problem is formulated with the credible domain as a state constraint, but the paper does not verify that the resulting value function remains continuous up to the boundary of the credible domain; without this verification the optimal contract may not be attainable and the claimed screening values could be overstated.

    Authors: The referee correctly identifies that continuity of the value function up to the boundary is needed to guarantee attainability of the optimal contract. Our current development assumes standard regularity (bounded continuous payoffs and compact credible domain) under which continuity holds by standard stochastic-control arguments, but we did not provide an explicit verification. We will add a remark in Section 5 (and a short appendix note) stating the precise conditions that ensure continuity at the boundary, referencing relevant results on constrained stochastic control problems. This addresses the concern without overstating the screening values. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central derivation reformulates the agent's utility maximization under a unique contract as a stochastic target problem, then characterizes the credible domain to impose state constraints on the principal's stochastic control problem with partial information. This equivalence follows directly from the problem setup and is presented as a mathematical reformulation rather than a fitted prediction or self-referential definition. The unique-contract restriction is explicitly identified as the modeling departure from standard screening, with no load-bearing self-citations, ansatzes smuggled via prior work, or renaming of known results. The derivation chain remains self-contained against the stated assumptions and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Paper relies on domain assumptions from principal-agent theory and introduces the credible domain as a new mathematical object; no free parameters or invented entities are identifiable from the abstract.

axioms (1)
  • domain assumption Principal offers only a unique contract rather than a menu allowing self-selection
    Explicitly contrasted with standard screening frameworks in the abstract.

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