Calibrated Mechanism Design

We study mechanism design when a designer repeatedly uses a fixed mechanism to interact with strategic agents who learn from observing their allocations. We introduce a static framework, calibrated mechanism design, requiring mechanisms to remain incentive compatible given the information they reveal about an underlying state through repeated use. In single-agent settings, we prove implementable outcomes correspond to two-stage mechanisms: the designer discloses information about the state, then commits to a state-independent allocation rule. This yields a tractable procedure to characterize calibrated mechanisms, combining information design and mechanism design. In private values environments, full transparency is optimal and correlation-based surplus extraction fails. We provide a microfoundation by showing calibrated mechanisms characterize exactly what is implementable when an infinitely patient agent repeatedly interacts with the same mechanism. Dynamic mechanisms that condition on histories expand implementable outcomes only by weakening incentive constraints, but not by enriching the designer’s ability to obfuscate learning.

💡 Research Summary

This paper investigates a setting in which a designer repeatedly uses a single, fixed mechanism to interact with strategic agents who can learn about the designer’s private information by observing the outcomes (allocations) they receive over time. The authors introduce a static solution concept called calibrated mechanism design. A mechanism is paired with a “calibrated information structure” that describes exactly what each agent learns from the allocation rule; the structure must be calibrated in the sense of Foster and Vohra (1997), i.e., the interim allocation rule observed by the agent must faithfully represent the true probabilities of allocations conditional on the agent’s report and the hidden state.

The key insight is that the more a mechanism conditions on the hidden state, the more information it leaks, tightening the incentive‑compatibility (IC) and individual‑rationality (IR) constraints. In private‑values environments this tension forces the designer toward full transparency: Theorem 1 shows that under the calibration constraint the designer cannot do better than running, in each realized state, the optimal direct mechanism that would be used if the state were common knowledge. Consequently, classic correlation‑based surplus extraction (e.g., Crémer‑McLean) fails when agents can learn the state.

In single‑agent settings the authors prove a powerful equivalence (Theorem 2): any outcome that can be implemented by a calibrated mechanism can also be implemented by a two‑stage mechanism. In a two‑stage mechanism the designer first discloses some signal about the state, thereby shaping the agent’s posterior belief, and then commits to an allocation rule that depends only on the agent’s report and not on the state itself. This representation yields a tractable algorithm: for each possible belief induced by a signal, solve the standard mechanism‑design problem (e.g., Myerson’s optimal auction) and then concavify the resulting value function to choose the optimal belief‑signal pair.

The paper extends the analysis to multiple agents via generalized two‑stage mechanisms. Here each agent receives an individualized belief about the state and a state‑independent interim allocation rule. Because agents only observe their own belief, additional consistency constraints are required to ensure that the collection of interim rules is mutually compatible. Not every generalized two‑stage mechanism is calibrated; the gap reflects the fact that designers may sometimes wish to reveal less information than the calibrated structure would dictate.

Dynamic mechanisms—where the designer can condition each period’s allocation on the entire history of past reports and allocations—are examined in Section 5. Theorem 4 shows that dynamic mechanisms expand the set of implementable outcomes only by weakening the IC and IR constraints; they do not enable the designer to hide additional information beyond what is already captured by calibrated structures. In environments with transferable utility, Rahman (2024) demonstrates that eliminating profitable undetectable deviations is equivalent to standard IC, so dynamic mechanisms achieve exactly the same distribution over allocations, types, and states as static calibrated mechanisms.

A micro‑foundational justification is provided by modeling an infinitely patient agent who repeatedly interacts with the same mechanism while the hidden state remains fixed and the agent’s private type is redrawn each period. Theorem 3 proves that the long‑run expected frequency of allocation‑type‑state triples coincides precisely with those generated by incentive‑compatible two‑stage mechanisms. Thus the static calibrated framework captures exactly what is implementable in the repeated‑interaction limit.

The authors also compare calibrated mechanisms to the Myersonian benchmark. Under certain regularity conditions, the optimal Myersonian mechanism already satisfies state‑by‑state IC, so the calibration constraint does not reduce the designer’s payoff. However, when the ordering of types depends on the state, optimal calibrated mechanisms may either fully reveal the state or withhold it, depending on which yields higher expected surplus.

Overall, the paper contributes a novel theoretical lens for analyzing mechanisms that are fixed yet repeatedly used, highlighting how information leakage fundamentally limits the set of implementable outcomes. By reducing the problem to a combination of information design (choice of signal) and classic mechanism design (choice of allocation rule), the authors provide both a clear conceptual framework and a practical algorithm for optimal design in settings such as online advertising auctions, credit‑scoring policies, and regulatory rule‑making where agents learn over time.