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Calibrated Forecasting and Persuasion

Atulya Jain, Vianney Perchet

Abstract

How should an expert send forecasts to maximize her utility subject to passing a calibration test? We consider a dynamic game where an expert sends probabilistic forecasts to a decision maker. The decision maker uses a calibration test based on past outcomes to verify the expert's forecasts. We characterize the optimal forecasting strategy by reducing the dynamic game to a static persuasion problem. A distribution of forecasts is implementable by a calibrated strategy if and only if it is a mean-preserving contraction of the distribution of conditionals (honest forecasts). We characterize the value of information by comparing what an informed and uninformed expert can attain. Moreover, we consider a decision maker who uses regret minimization, instead of the calibration test, to take actions. We show that the expert can achieve the same payoff against a regret minimizer as under the calibration test, and in some instances, she can achieve strictly more.

Calibrated Forecasting and Persuasion

Abstract

How should an expert send forecasts to maximize her utility subject to passing a calibration test? We consider a dynamic game where an expert sends probabilistic forecasts to a decision maker. The decision maker uses a calibration test based on past outcomes to verify the expert's forecasts. We characterize the optimal forecasting strategy by reducing the dynamic game to a static persuasion problem. A distribution of forecasts is implementable by a calibrated strategy if and only if it is a mean-preserving contraction of the distribution of conditionals (honest forecasts). We characterize the value of information by comparing what an informed and uninformed expert can attain. Moreover, we consider a decision maker who uses regret minimization, instead of the calibration test, to take actions. We show that the expert can achieve the same payoff against a regret minimizer as under the calibration test, and in some instances, she can achieve strictly more.
Paper Structure (21 sections, 14 theorems, 52 equations, 4 figures, 1 table)

This paper contains 21 sections, 14 theorems, 52 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model
  3. Calibration Test:
  4. Main results
  5. Persuasion problem
  6. Optimal forecasting strategy
  7. Uninformed expert
  8. Application: Financial Application
  9. Forecasting against regret minimizers
  10. Conclusion
  11. Omitted Results and Proofs
  12. Proposition \ref{['propseqoferror']}
  13. Proof of Proposition \ref{['bayeaffine']}
  14. Proof of Lemma \ref{['mpclemma']}
  15. Proof of Lemma \ref{['stationaryergodic']}
  16. ...and 6 more sections

Key Result

proposition 1

If $Supp(P)$ is affinely independent, then the solution of the persuasion problem $(P,\hat{u}_S)$ is given by where, $Cav \; { \left.\nulldelimiterspace \hat{u}_S \newline \right|_{C} }$ denotes the concave envelope of $\hat{u}_S$ restricted to domain $C= Co (Supp(P)))$ and $\mathcal{B}(P)= \sum_{i=1}^n \lambda_i p_i$ denotes the barycenter (or mean) of the distribution $P$. $Co(A)$ ref

Figures (4)

  • Figure 1: $\hat{u}_S^{sig}(p)$
  • Figure 2: $\hat{u}_S^{rep}(p)$
  • Figure 3: $\hat{u}_S(p)$
  • Figure 4: Financial App

Theorems & Definitions (25)

  • definition 1: Finite
  • Example 1
  • definition 2: Asymptotic
  • definition 3
  • definition 4
  • proposition 1
  • corollary 1
  • Example 1: continued
  • theorem 1
  • lemma 2
  • ...and 15 more