Robust Learning with Private Information

Firms increasingly delegate decisions to learning algorithms in platform markets. Standard algorithms perform well when platform policies are stationary, but firms often face ambiguity about whether policies are stationary or adapt strategically to their behavior. When policies adapt, efficient learning under stationarity may backfire: it may reveal a firm’s persistent private information, allowing the platform to personalize terms and extract information rents. We study a repeated screening problem in which an agent with a fixed private type commits ex ante to a learning algorithm, facing ambiguity about the principal’s policy. We show that a broad class of standard algorithms, including all no-external-regret algorithms, can be manipulated by adaptive principals and permit asymptotic full surplus extraction. We then construct a misspecification-robust learning algorithm that treats stationarity as a testable hypothesis. It achieves the optimal payoff under stationarity at the minimax-optimal rate, while preventing dynamic rent extraction: against any adaptive principal, each type’s long-run utility is at least its utility under the menu that maximizes revenue under the principal’s prior.

💡 Research Summary

The paper investigates a repeated screening interaction between a privately informed agent and a platform (principal) who may either follow a stationary policy or adapt strategically to the agent’s behavior. The agent has a fixed type θ drawn once from a known finite set Θ and must commit ex‑ante to an online learning algorithm ℒ that dictates his actions over T periods. In each period the agent chooses an action aₜ∈A, the principal selects a mechanism (allocation rule xₜ, payment rule pₜ), and payoffs are uₜ=θ·xₜ(aₜ)−pₜ(aₜ) for the agent and vₜ=pₜ(aₜ) for the principal. The principal’s policy belongs to one of two classes: (i) Stationarity, where mechanisms are i.i.d. draws from an unknown distribution; (ii) Adaptation, where the principal observes the agent’s algorithm and past actions and then chooses mechanisms to maximize long‑run revenue given a prior over types.

The first major contribution is a negative result: any algorithm satisfying the standard no‑external‑regret property (including popular bandit methods such as EXP3, Hedge, Follow‑the‑Regularized‑Leader) is vulnerable to an adaptive principal. Because no‑external‑regret learners adjust their action probabilities based on observed payoffs, their behavior becomes type‑dependent: a higher‑valuation type will shift probability toward a profitable action faster than a lower‑valuation type. An adaptive principal can exploit this by running a short probing phase with a price that yields opposite signs of payoff for the two types, thereby inferring the agent’s type with high probability. After the type is identified, the principal switches to a personalized mechanism (e.g., a reserve price just below the inferred valuation) and extracts essentially the entire surplus for the remainder of the horizon. Consequently, the agent’s average surplus converges to an arbitrarily small ε, while the principal’s average revenue approaches the agent’s full valuation. This violates “weak extraction robustness” and shows that all no‑external‑regret algorithms permit asymptotic full surplus extraction in this setting.

The second major contribution is a constructive positive result: a new algorithm ℒ* that is robust to the principal’s possible adaptation while retaining optimal learning performance under stationarity. ℒ* consists of three components:

1. Type‑independent exploration – a brief initial phase where the agent randomizes uniformly over actions, generating a baseline performance that does not reveal his type.
2. Statistical test for stationarity – during the subsequent learning phase the algorithm monitors the observed allocation‑payment pairs. If the empirical distribution deviates significantly from what would be expected under any stationary mechanism, the test flags “non‑stationarity.”
3. Opt‑out threat – upon detection of non‑stationarity, the algorithm permanently switches to a pre‑specified opt‑out action a₀ that yields zero payoff for both parties. This credible threat forces an adaptive principal to behave as if his policy were stationary if he wishes to sustain revenue.

When the principal’s policy is truly stationary, ℒ* behaves like a standard no‑external‑regret learner and achieves the minimax‑optimal regret rate O(√(T log|A|)), guaranteeing that the agent’s time‑average payoff approaches that of the best fixed action. When the principal adapts, the opt‑out threat ensures that the agent’s long‑run utility is at least the utility he would obtain under the static, revenue‑maximizing menu that the principal could design given his prior over types. In other words, no dynamic rent extraction occurs: the adaptive principal cannot extract more than the static optimal revenue benchmark.

The paper’s findings have two important implications. Theoretically, they demonstrate that regret minimization alone is insufficient to protect persistent private information in strategic environments; robustness must incorporate detection of policy deviation and credible commitment devices. Practically, firms that simply deploy off‑the‑shelf bandit algorithms in platform markets risk having their private valuations fully harvested by adaptive platforms. Incorporating a statistical stationarity test and an opt‑out mechanism provides a practical way to safeguard against such exploitation while still enjoying fast learning when the environment is benign.

The work connects to three strands of literature: (i) exploitation of no‑regret learners, where prior studies considered either normal‑form games with known payoffs or Bayesian games with independently redrawn types each period; this paper extends the analysis to a setting with a fixed, economically valuable private type, showing a stronger impossibility result; (ii) design of learning rules under opponent ambiguity, building on recent work that treats the learner’s algorithm as a commitment device; and (iii) repeated games with incomplete information, differing by modeling a behaviorally constrained learner rather than a fully rational best‑responding player. Overall, the paper provides a rigorous characterization of the vulnerability of standard learning algorithms to adaptive platforms and offers a novel, implementable solution that balances learning efficiency with protection of private information.