# 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

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

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.