Statistical Calibration: a simplification of Foster's Prof

Consider the following problem: at each date in the future, a given event may or may not occur, and you will be asked to forecast, at each date, the probability that the event will occur in the next date. Unless you make degenerate forecasts (zero or one), the fact that the event does or does not occur does not prove your forecast wrong. But, in the long run, if your forecasts are accurate, the conditional relative frequencies of occurrence of the event should approach your forecast. [4] has presented an algorithm that, whatever the sequence of realizations of the event, will meet the long-run accuracy criterion, even though it is completely ignorant about the real probabilities of occurrence of the event, or about the reasons why the event occurs or fails to occur. It is an adaptive algorithm, that reacts to the history of forecasts and occurrences, but does not learn from the history anything about the future: indeed, the past need not say anything about the future realizations of the event. The algorithm only looks at its own past inaccuracies and tries to make up for them in the future. The amazing result is that this (making up for past inaccuracies) can be done with arbitrarily high probability! Alternative arguments for this result have been proposed in the literature, remarkably by [3], where a very simple algorithm has been proved to work, using a classical result in game theory: Blackwell´s approachability result, [1]. Very recently, [2] has especialized Blackwell´s theorem in a way that (under a minor modification of the algorithm) simplifies the argument of [3]. Here I present such modification and argument.