Game-theoretic probability and its uses, especially defensive forecasting

This expository article reviews the game-theoretic framework for probability and the method of defensive forecasting that derives from it. The game-theoretic framework, introduced by Vladimir Vovk and myself in Probability and Finance: It’s Only a Game! (Wiley 2001 [51]), can replace measure theory as a mathematical framework for classical probability theory, discrete and continuous. Classical theorems are proven by betting strategies that multiply a player’s stake by a large factor if the theorem’s prediction fails. These strategies can be specified explicitly, giving probability a constructive character appropriate for applications. Defensive forecasting, introduced by Vovk, Takemura, and myself in 2005 [77], is one of the most interesting applications. It identifies a betting strategy that succeeds if probabilistic forecasts are inaccurate, and it makes probabilistic forecasts that will defeat this betting strategy. The fact that this is possible provides new insight into the meaning of probability.