Continuous and randomized defensive forecasting: unified view

Defensive forecasting is a method of transforming laws of probability (stated in game-theoretic terms as strategies for Sceptic) into forecasting algorithms. There are two known varieties of defensive forecasting: “continuous”, in which Sceptic’s moves are assumed to depend on the forecasts in a (semi)continuous manner and which produces deterministic forecasts, and “randomized”, in which the dependence of Sceptic’s moves on the forecasts is arbitrary and Forecaster’s moves are allowed to be randomized. This note shows that the randomized variety can be obtained from the continuous variety by smearing Sceptic’s moves to make them continuous. New as compared to version 1 (17 August 2007) of this report: The assumption of version 1 that the outcome space Ω is finite is relaxed, and now it is only assumed to be compact. In the case where Ω is finite, it is shown that Forecaster can choose his randomized forecasts concentrated on a finite set of cardinality at most |Ω|.