Distributionally Robust Contract Design with Deferred Inspection

TL;DR

This paper develops a distributionally robust contract design framework with deferred inspection, relaxing common-prior assumptions and using moment information to bound worst-case revenue. For a single agent and first-moment ambiguity, it provides a complete characterization: a robustly optimal contract with a concave allocation rule and a linear payment rule, plus a Pareto robustly optimal variant that extracts maximal feasible payments, indicating nonuniqueness in transfers under robustness. When extending to higher moments, robust optimality imposes a degree-$N$ polynomial lower bound on aggregate payments, but these do not pin down a unique mechanism, reflecting the expanded feasibility from inspection and rewards. In symmetrical multi-agent settings, robustness induces dominant-strategy feasibility and leads to complex, often correlated worst-case distributions, with three or more agents potentially collapsing to a Dirac distribution at the mean and necessitating lotteries; overall, robust contract design with deferred inspection yields tractable results in the one-agent case but substantial analytical complexity in multi-agent environments. The work highlights a sharp distinction between bilateral robustness (where simple linear contracts often suffice) and robust multi-agent mechanism design (where complexity and randomness become intrinsic).

Abstract

We study a robust contract design problem with deferred inspection, in which a principal allocates a scarce resource to an agent, observes the agent's realized outcome ex post at negligible cost, and conditions transfers on this information through rewards. The principal faces ambiguity about the agent's value distribution and seeks to maximize worst-case expected revenue subject to incentive compatibility and limited liability. In contrast to existing work on inspection mechanisms, which relies on common-prior assumptions, we adopt a distributionally robust approach based on moment information. Our main contribution is a complete characterization of the robust contract design problem with a single agent. When the ambiguity set is defined by the first moment, we identify a robustly optimal contract with a concave allocation rule and a linear payment rule. We further show that robustness does not uniquely pin down transfers: we construct a Pareto robustly optimal contract that preserves the same allocation while extracting maximal feasible payments from all types, yielding strictly higher expected revenue under non-worst-case distributions. We also derive structural results for multi-agent extensions. For ambiguity sets defined by the first $N$ moments, we show that robust optimality requires aggregate payments to be lower bounded by a multi-dimensional polynomial of degree $N$. However, unlike the single-agent case, robust multi-agent mechanisms are substantially more complex: dominant-strategy incentive compatibility becomes necessary, simple monotone mechanisms are no longer tractable, and worst-case distributions may involve correlated types or degenerate to a Dirac distribution at the mean. These results highlight a sharp contrast between robust contract design and robust multi-agent mechanism design with inspection.

Figures (21)

• Figure 1: Timeline of the Deferred Inspection Mechanism. The net payment is calculated as $p(\nu) = \hat{\nu} - r(\nu)$.
• Figure 2: Expected payoffs of the mechanism from alaei2024optimal and the robustly optimal mechanism with maximal payments.
• Figure 3: Optimal mechanism from alaei2024optimal compared to the optimal Posted Price mechanism when $\nu \sim \mathcal{U}[0,1]$.
• Figure 4: The plots of the nonlinear functions that determine $\mu'$.
• Figure 5: A robustly optimal mechanism with a linear payment rule for different $\mu$ values.

Theorems & Definitions (14)

• Proposition 3.1: alaei2024optimal
• Definition 3.2
• Lemma 3.3: carrasco2018optimal
• Proposition 3.4
• Lemma 3.5
• Theorem 3.6
• Theorem 3.7
• Lemma 4.1
• Proposition 4.2
• Lemma 4.3