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.

💡 Research Summary

Defensive forecasting is a game‑theoretic framework that recasts probabilistic laws as a competition between a Sceptic, who places bets based on the forecaster’s announced probability distribution, and a Forecaster, who must choose forecasts so that the Sceptic’s capital does not grow without bound. Two variants of this framework have been studied. In the continuous version the Sceptic’s betting strategy is required to be (semi‑)continuous in the forecast; this restriction yields deterministic (non‑randomized) forecasts and allows the use of classical topological arguments to guarantee that the Forecaster can keep the Sceptic’s cumulative loss below a pre‑specified bound. In the randomized version no continuity assumption is imposed on the Sceptic’s moves; consequently the Forecaster is allowed to randomize his own forecasts, typically by sampling from a distribution over probability measures. The randomized approach is more general but appears to demand a fundamentally different algorithmic machinery.

The present note bridges this apparent gap. The author shows that any arbitrary (possibly discontinuous) Sceptic strategy can be smeared—that is, convolved with a small, smooth kernel—to produce a new strategy that is continuous (or at least semi‑continuous) while remaining arbitrarily close to the original one. Formally, if (S_t(p)) denotes the Sceptic’s payoff function at round (t) as a function of the forecast (p), the smeared version is defined by
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