Game Of Prediction

A game of prediction is a triple $(\Omega,\Gamma,\lambda)$ , where $\Omega$ is an outcome space, $\Gamma$ is a decision space, and $\lambda:\Omega\times\Gamma\to[-\infty,\infty]$ is the loss function. Important special cases are those of:

point predictions, where $\Gamma=\Omega$ ;
probability predictions, where $\Gamma$ is the set of all probability measures on $\Omega$ ; if $\Omega$ is finite, the usual loss function is the log loss;
region predictions, where $\Gamma$ is the set of all subsets of $\Omega$ ; the usual loss function is

$\lambda(\omega,\gamma) = \begin{cases} 1 & \text{if $\omega\in\gamma$}\\ 0 & \text{otherwise}. \end{cases}$

The case of region predictions (in particular, interval predictions) is the main object of study in conformal prediction but has never been studied in competitive on-line prediction. This is stated as open problem in the article competitive on-line interval prediction.