Game-theoretic versions of Kolmogorov’s strong law of large numbers

We prove two variants of Kolmogorov’s strong law of large numbers in a completely worst-case framework, eschewing any probabilistic assumptions. The first variant is an assertion about a game involving the Bookmaker predicting the values of unprobabilized random variables; in an intuitive sense it is much stronger than the usual strong law of large numbers for martingales. The second variant is an assertion about a security market. 1 Predictive strong law of large numbers In this section we consider the following perfect-information game between 3 players, the Bookmaker, the Statistician, and the Nature. The game proceeds in trials. At each trial i, i = 1, 2, . . ., the Bookmaker tries to predict the real number Xi the Nature is to produce at the end of the trial. His prediction consists of two numbers, Ei and Di ≥ 0; roughly, Ei is the Bookmaker’s expectation of Xi, and Di is his expectation of the accuracy Li := (Xi−Ei) (measured by the Brier scoring rule, see Dawid [1]) of the prediction Ei. ∗The research described in this publication was made possible in part by Grant No. MRS000 from the International Science Foundation.