# 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.