Glenn Shafer, Rutgers Business School
In game-theoretic probability, Forecaster gives probabilities (or upper expectations) on each round of the game, and Skeptic tests these probabilities by betting, while Reality decides the outcomes. Can Forecaster pass Skeptic's tests?
As it turns out, Forecaster can defeat any particular strategy for Skeptic, provided only that each move prescribed by the strategy varies continuously with respect to Forecaster's previous move. Forecaster wants to defeat more than a single strategy for Skeptic; he wants to defeat simultaneously all the strategies Skeptic might use. But as we will see, Forecaster can often amalgamate the strategies he needs to defeat by averaging them, and then he can play against the average. This is called defensive forecasting. Defeating the average may be good enough, because when any one of the strategies rejects Forecaster's validity, the average will reject as well, albeit less strongly.
This result has implications for the meaning of probability. It reveals that the crucial step in placing an evidential question in a probabilistic framework is its placement in a sequence of questions. Once we have chosen the sequence, good sequential probabilities can be given, and the validation of these probabilities by experience signifies less than commonly thought.