Smooth Calibration, Leaky Forecasts, Finite Recall, and Nash Dynamics
This work addresses the challenge of reliable forecasting and game theory dynamics, offering a deterministic solution that is robust to information leaks and has practical implications for multi-agent systems, though it builds incrementally on existing calibration concepts.
The paper tackles the problem of ensuring calibration in forecasting by introducing a smoothed version of the calibration score, which allows deterministic procedures to guarantee it, unlike regular calibration that requires randomization. As a result, this approach enables uncoupled finite-memory dynamics in games where players achieve approximate Nash equilibria in most periods, improving upon prior methods that only yield correlated equilibria on average.
We propose to smooth out the calibration score, which measures how good a forecaster is, by combining nearby forecasts. While regular calibration can be guaranteed only by randomized forecasting procedures, we show that smooth calibration can be guaranteed by deterministic procedures. As a consequence, it does not matter if the forecasts are leaked, i.e., made known in advance: smooth calibration can nevertheless be guaranteed (while regular calibration cannot). Moreover, our procedure has finite recall, is stationary, and all forecasts lie on a finite grid. To construct the procedure, we deal also with the related setups of online linear regression and weak calibration. Finally, we show that smooth calibration yields uncoupled finite-memory dynamics in n-person games "smooth calibrated learning" in which the players play approximate Nash equilibria in almost all periods (by contrast, calibrated learning, which uses regular calibration, yields only that the time-averages of play are approximate correlated equilibria).