A game-theoretic version of Oakes' example for randomized forecasting

Using the game-theoretic framework for probability, Vovk and Shafer [10] have shown that it is always possible, using randomization, to make sequential probability forecasts that pass any countable set of well-behaved statistical tests. This result generalizes work by other authors, who consider only tests of calbration. We complement this result with a lower bound. We show that Vovk and Shafer’s result is valid only when the forecasts are computed with unrestrictedly increasing degree of accuracy. When some level of discreteness is fixed, we present a game-theoretic generalization of Oakes’ example for randomized forecasting that is a test failing any given method of deferministic forecasting; originally, this example was presented for deterministic calibration.