Game-Theoretic Foundations for Probability and Finance

1Borel's Law of Large Numbers

This chapter introduces game‐theoretic probability in a relatively simple and concrete setting, where outcomes are bounded real numbers. We use this setting to prove game‐theoretic generalizations of a theorem that was first published by Émile Borel in 1909 44 and is often called Borel's law of large numbers.

In its simplest form, Borel's theorem says that the frequency of heads in an infinite sequence of tosses of a coin, where the probability of heads is always , converges with probability one to . Later authors generalized the theorem in many directions. In an infinite sequence of independent trials with bounded outcomes and constant expected value, for example, the average outcome converges with probability one to the expected value.

Our game‐theoretic generalization of Borel's theorem begins not with probabilities and expected values but with a sequential game in which one player, whom we call Forecaster, forecasts each outcome and another, whom we call Skeptic, uses each forecast as a price at which he can buy any multiple (positive, negative, or zero) of the difference between the outcome and the forecast. Here Borel's theorem becomes a statement about how Skeptic can profit if the average difference does not converge to zero. Instead of saying ...