Game-theoretic Probability and Defensive
Forecasting
Glenn Shafer (Rutgers Business School)
Abstract:
In Probability and Finance: It's Only a Game, published
by Wiley in 2001, Vladimir Vovk and I demonstrated how game theory can replace
measure theory as a foundation for classical probability theory, discrete and
continuous. 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.
