Game-theoretic probability and defensive forecasting

In 2001, Vladimir Vovk and I demonstrated how game theory can replace measure theory as a foundation for classical probability theory, discrete and continuous {Probability and Finance: Its Only a Game!, Wiley 2001). In the game-theoretic framework, classical probability theorems are proven by betting strategies that make a player rich without risking bankruptcy if the theorem's prediction fails. These strategies can be specified explicitly, and so the theory has a constructive flavor that lends itself to applications in economics and statistics. Defensive forecasting is one of the most interesting of these applications. It identifies a comprehensive betting strategy, which becomes rich if the probabilities fail in a relevant way (say by being uncalibrated or having poor resolution), and it chooses probabilities to defeat this comprehensive betting strategy. The fact that this is possible gives us new insight into the very meaning of probability.