Smooth Calibration, Leaky Forecasts, Finite Recall, and Nash Dynamics

TL;DR

This paper introduces a deterministic method for smooth calibration of forecasts that guarantees good performance even with leaked forecasts, finite recall, and finite grid constraints, impacting game dynamics.

Contribution

It presents a novel deterministic procedure for smooth calibration that works with leaked forecasts, finite recall, and finite grids, extending calibration theory.

Findings

Smooth calibration can be guaranteed deterministically.

Leaked forecasts do not prevent smooth calibration.

Smooth calibrated learning leads to approximate Nash equilibria.

Abstract

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).