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.

翻译：我们研究了一个动态博弈，其中专家向决策者发送概率预测。决策者基于历史数据使用校准检验来验证这些预测。专家应如何发送预测以在通过检验的同时最大化其收益？对于平稳遍历过程，我们通过将动态博弈简化为静态说服问题，刻画了最优预测策略。在校准条件下可产生的预测分布恰好是条件分布的均值保持压缩。我们比较了知情专家与不知情专家可获得的收益，为信息价值提供了基准。最后，我们考虑一个后悔最小化的决策者，并证明专家总能至少保证校准基准，有时甚至能严格超过该基准。