Calibrated Forecasting and Persuasion

1. Problem Statement

The paper investigates a dynamic game between an informed sender (expert) and an uninformed receiver (decision-maker).

• Setting: The state of the world evolves according to a stochastic process. In each period, the sender issues a probabilistic forecast regarding the next state.
• Receiver's Behavior: The receiver does not hold a prior belief about the process or the sender's strategy. Instead, they evaluate the sender's credibility using a calibration test based on the history of forecasts and realized outcomes.
  • If the sender passes the test, the receiver treats the forecast as the true probability distribution and plays a best response.
  • If the sender fails, they incur a punishment cost, and the receiver reverts to a default action.
• Sender's Objective: The sender aims to maximize their long-run average payoff by strategically choosing forecasts. They face a trade-off: truthful forecasts guarantee passing the test but may yield lower payoffs than strategic, biased forecasts that still pass the test.
• Core Question: How should an expert forecast to maximize payoff while satisfying the calibration constraint? What is the value of information (informed vs. uninformed sender)?

2. Methodology

The authors employ a combination of game theory, stochastic processes, and online learning techniques.

• Dynamic to Static Reduction: The primary methodological innovation is reducing the infinite-horizon dynamic forecasting game to a static Bayesian persuasion problem.
  • They define the "state" of the persuasion problem as the conditional probability of the next state given the history ($p_t$).
  • The "prior" is the long-run distribution of these conditionals ($C_\mu$).
  • The "signals" are the forecasts sent by the expert.
• Calibration as a Constraint: They characterize the set of feasible long-run forecast distributions under the calibration constraint. They prove that for stationary ergodic processes, any calibrated forecast distribution must be a mean-preserving contraction (MPC) of the distribution of conditionals.
  • Intuitively, the expert can "garble" or pool truthful forecasts (conditionals) into coarser forecasts, provided the weighted average of the pooled forecasts equals the original conditional probabilities.
• Regret Minimization: In the second part of the paper, they replace the calibration test with a receiver who uses regret minimization (specifically mean-based learning algorithms) to make decisions, analyzing how this changes the sender's attainable payoffs.

3. Key Contributions and Results

A. The Informed Sender (Stationary Ergodic Processes)

• Main Theorem (Theorem 1): For a stationary ergodic process, the sender's maximum payoff in the dynamic game is exactly equal to the value of a static Bayesian persuasion problem where:
  • The prior is the distribution of conditionals ($C_\mu$).
  • The sender's utility depends on the receiver's best response to the forecast.
• Mechanism: The optimal strategy involves the sender committing to a signaling policy (in the static sense) that maps the current conditional probability $p_t$ to a forecast $f_t$. This policy is implemented period-by-period.
• Implication: Calibration provides a micro-foundation for the commitment assumption in Bayesian persuasion. Even without explicit commitment power, the requirement to pass a calibration test forces the sender to act as if they have committed to an optimal signaling policy.
• Feasible Forecasts: The set of attainable forecast distributions is precisely the set of mean-preserving contractions of $C_\mu$. The sender can improve upon truthful forecasting (which yields $E[\hat{u}_S(p)]$) by "concavifying" their utility function over the support of $C_\mu$.

B. The Uninformed Sender

The authors analyze senders who do not know the data-generating process.

• Adversarial Setting: If nature adapts to minimize the sender's payoff, the sender can only guarantee the closed convex hull of their indirect utility function, evaluated at the realized empirical distribution of states. This is generally lower than the informed sender's payoff.
• Ergodic Markov Chain: If the unknown process is an ergodic Markov chain, the uninformed sender can approximate the payoff of the informed sender (specifically, the payoff corresponding to the "no-disclosure" policy in the persuasion problem with prior $C_\mu$).
• Value of Information: The difference between the informed sender's payoff (Theorem 1) and the uninformed sender's adversarial guarantee provides a rigorous benchmark for the value of information in strategic forecasting.

C. Application: Financial Platform

The authors apply the model to a financial platform that balances reputation (favored by precise forecasts) against user engagement (favored by coarse, less extreme forecasts).

• Result: The optimal strategy is to issue coarse forecasts (e.g., 15% and 85%) rather than truthful conditionals (e.g., 5% and 95%), even though the platform knows the true probabilities. This maximizes the concave envelope of the utility function, demonstrating that strategic "garbling" is optimal under calibration.

D. Regret Minimization vs. Calibration

• Proposition 4: A receiver who best responds to calibrated forecasts incurs zero regret. This justifies calibration as a rational decision-making heuristic.
• Theorem 3: When facing a mean-based regret-minimizing receiver (who learns based on cumulative payoffs), the sender can sometimes achieve a payoff strictly higher than the calibration benchmark (the persuasion value).
  • Mechanism: The sender exploits the lag in the receiver's learning. By swapping forecasts in a specific pattern, the sender manipulates the receiver's cumulative payoff history, inducing them to play actions that are not best responses to the current belief but are optimal given the receiver's historical data.

4. Significance and Impact

1. Bridging Literature: The paper connects three major fields: Strategic Information Transmission (Bayesian Persuasion), Expert Testing (Calibration), and Online Learning (Regret Minimization).
2. Micro-foundation for Persuasion: It resolves a key theoretical gap by showing that the "commitment" required in Bayesian persuasion can emerge endogenously from the requirement to pass a statistical calibration test in a repeated game.
3. Strategic Value of Calibration: It demonstrates that calibration does not force experts to be truthful; rather, it constrains them to a specific class of "garbled" signals (MPCs), allowing for strategic manipulation that can increase payoffs.
4. Robustness of Decision Heuristics: It validates calibration as a robust decision rule (zero regret) while also highlighting its vulnerability to strategic exploitation by sophisticated senders facing learning agents.

5. Conclusion

The paper establishes that for stationary ergodic processes, the dynamic problem of strategic forecasting under calibration is equivalent to a static persuasion problem. The optimal strategy involves mapping true conditional probabilities to coarser forecasts via a mean-preserving contraction. While uninformed senders can pass calibration tests, they generally achieve lower payoffs than informed senders unless the process has specific structure (like a Markov chain). Finally, the paper shows that while calibration ensures zero regret for the receiver, it does not prevent the sender from extracting surplus, and in some cases, a sender can outperform the calibration benchmark against learning receivers.