Calibrated Forecasting and Persuasion

We study a dynamic game where an expert sends probabilistic forecasts to a decision-maker. The decision-maker verifies these forecasts using a calibration test based on past data. How should the expert send forecasts to maximize her payoff while passing the test? For a stationary ergodic process, we characterize the optimal forecasting strategy by reducing the dynamic game to a static persuasion problem. The distributions of forecasts that can arise under calibration are precisely the mean-preserving contractions of the distribution of conditionals. We compare the payoffs attainable by an informed and uninformed expert, providing a benchmark for the value of information. Finally, we consider a regret-minimizing decision-maker and show that the expert can always guarantee at least the calibration benchmark and sometimes strictly more.

💡 Research Summary

The paper studies a dynamic sender‑receiver game in which an expert (the sender) repeatedly issues probabilistic forecasts about a stochastic state that evolves over time, while a decision‑maker (the receiver) evaluates the expert’s credibility using a calibration test based on past forecasts and realized outcomes. The central question is how the expert should choose forecasts to maximize her long‑run payoff while still passing the calibration test.

Model.
The state space Ω is finite and the state sequence (ωₜ)ₜ≥1 follows a stationary ergodic stochastic process with distribution μ. In each period t the expert knows the conditional distribution pₜ = μ(· | ω^{t‑1}) and can issue any forecast fₜ ∈ Δ(Ω). The receiver has no prior knowledge of μ or the expert’s strategy; instead he computes a calibration error after each period. A sequence of forecasts is ε‑calibrated if, for every forecast value f that appears, the empirical distribution of states observed when f was issued is within ε of f (in Euclidean norm). If the expert passes the test, the receiver treats the forecast at face value and chooses an optimal action aₜ; otherwise the expert suffers a punishment cost.

Calibration and feasible forecast distributions.
The authors show that, under the stationary ergodic assumption, the set of long‑run forecast distributions that satisfy calibration is exactly the set of mean‑preserving contractions of the distribution of the true conditionals (the “truthful” forecasts). Intuitively, the expert may pool low‑probability and high‑probability truthful forecasts with appropriate weights so that the overall average forecast matches the true frequencies, thereby preserving the mean while reducing informational content. Thus calibration imposes an average‑accuracy constraint but allows the expert to be less informative than truthful reporting.

Reduction to a static Bayesian persuasion problem.
The core contribution is an equivalence theorem (Theorem 1): the expert’s optimal payoff in the dynamic forecasting game equals the value of a static Bayesian persuasion problem. In this persuasion formulation, the prior is the distribution of the conditionals, the signal is the forecast, and the receiver’s action depends only on the posterior mean. By constructing a period‑by‑period forecasting rule that implements any optimal signaling policy, the authors demonstrate that the dynamic problem can be solved by solving the static persuasion problem. This provides a micro‑foundation for the commitment assumption in Bayesian persuasion: calibration itself plays the role of a commitment device.

Informed vs. uninformed expert.
When the expert is uninformed about μ, prior literature (Foster & Vohra, 1998) shows that she can still pass calibration for any process. The paper extends this by characterizing the uninformed expert’s attainable payoff via a persuasion problem. Two environments are examined:

1. Adversarial nature: No fixed prior; the payoff is evaluated as a function of the realized empirical distribution of states. The uninformed expert can guarantee at least the minimal payoff in the canonical persuasion problem whose prior’s mean equals that empirical distribution (Theorem 2). This payoff is weakly lower than the informed expert’s truthful‑forecast benchmark.

2. Ergodic Markov chain: The uninformed expert can essentially achieve the “no‑disclosure” payoff of the same persuasion problem that the informed expert faces (Proposition 3). Hence, with enough regularity, lack of information does not severely limit strategic advantage.

Application to a financial platform.
The authors illustrate the theory with a platform that forecasts a Markovian market state. The platform’s revenue depends on forecast precision (reputation) and user engagement (time spent). Using the concavification approach, they derive the optimal forecasting policy and show that, counter‑intuitively, coarse (less informative) forecasts can dominate truthful ones because they better balance the reputation‑engagement trade‑off.

Regret‑minimizing receiver.
The paper also replaces the calibration‑based receiver with a regret‑minimizing learner, a common online‑learning benchmark. They exploit the known link between calibration and external regret (Perchet, 2014). Proposition 4 shows that if the receiver best‑responds to any calibrated strategy, he incurs no regret, justifying calibration as a decision‑making heuristic. Moreover, when facing a mean‑based learner (a natural class of regret‑minimizing algorithms), the informed expert can guarantee at least the calibration benchmark and, for many algorithms, strictly exceed it (Proposition 5, Theorem 3). This demonstrates that calibration not only protects the receiver from exploitation but also leaves room for the expert to extract additional surplus.

Contributions and implications.

1. Dynamic‑to‑static reduction: By mapping calibration to a commitment device, the paper bridges dynamic information transmission with static Bayesian persuasion, offering a novel micro‑foundation for commitment.
2. Characterization of feasible forecasts: The mean‑preserving contraction result precisely delineates the informational limits imposed by calibration.
3. Comparison of informed vs. uninformed experts: The analysis quantifies the value of information under both adversarial and ergodic environments.
4. Regret‑minimization link: The work connects two strands of literature—statistical testing of forecasts and online learning—showing that calibration is robust against regret‑minimizing behavior while still allowing strategic advantage.

Overall, the paper provides a comprehensive theoretical framework for understanding how experts can strategically manipulate forecasts under calibration constraints, how such manipulation relates to classic persuasion models, and how different receiver behaviors (calibration‑based or regret‑minimizing) affect the equilibrium outcomes.