  We consider a general class of forecasting protocols, called "linear
protocols", and discuss several important special cases, including multi-class
forecasting. Forecasting is formalized as a game between three players:
Reality, whose role is to generate observations; Forecaster, whose goal is to
predict the observations; and Skeptic, who tries to make money on any lack of
agreement between Forecaster's predictions and the actual observations. Our
main mathematical result is that for any continuous strategy for Skeptic in a
linear protocol there exists a strategy for Forecaster that does not allow
Skeptic's capital to grow. This result is a meta-theorem that allows one to
transform any continuous law of probability in a linear protocol into a
forecasting strategy whose predictions are guaranteed to satisfy this law. We
apply this meta-theorem to a weak law of large numbers in Hilbert spaces to
obtain a version of the K29 prediction algorithm for linear protocols and show
that this version also satisfies the attractive properties of proper
calibration and resolution under a suitable choice of its kernel parameter,
with no assumptions about the way the data is generated.
