Game-Theoretic Foundations for Probability and Finance

12Defensive Forecasting

In Part I, we emphasized strategies Skeptic can use to test the agreement between Forecaster and Reality. In this chapter, we consider how Forecaster can turn knowledge of such strategies to his own advantage. As we show, he can defeat any of the strategies for Skeptic that we studied in Part I. He can do this in a very strong sense: he can keep Skeptic's capital from growing at all. When he does this, we say that he is practicing defensive forecasting.

Defensive forecasting enables Forecaster to produce good forecasts when the tests the forecasts need to pass can be represented by a single strategy for Skeptic. This is often the case. In particular, there is often a single strategy for Skeptic that forces a wide range of types of calibration, so that Forecaster can produce reasonably calibrated forecasts by playing against it. If the forecasts take the form of probability measures calibrated with respect to a particular loss function, then they can be used to make decisions that perform well on average with respect to that loss function.

The main message of this chapter is that good probability forecasting and decision making – probability forecasting and decision making that achieve the rate of success we expect when outcomes have known or partly known probabilities – is possible even when probabilities are not known and even if we deny their existence, ...