On calibration error of randomized forecasting algorithms

It has been recently shown that calibration with an error less than Δ>0 is almost surely guaranteed with a randomized forecasting algorithm, where forecasts are obtained by random rounding the deterministic forecasts up to Δ. We show that this error cannot be improved for a vast majority of sequences: we prove that, using a probabilistic algorithm, we can effectively generate with probability close to one a sequence “resistant” to any randomized rounding forecasting with an error much smaller than Δ. We also reformulate this result by means of a probabilistic game.