; R(\pi_{ij}, \{Y_{ij} = k\}) = 2\pi_{ijk} - \sum_{\ell = 1}^K \pi_{ij\ell}^2, \quad [-1, 1],\\\\
\mbox{spherical:} &amp; R(\pi_{ij}, \{Y_{ij} = k\}) = \pi_{ijk} \Bigg / \sqrt{\sum_{\ell = 1}^K \pi_{ij\ell}^2}, \quad [0, 1].\\
\end{array}\]</span> The quadratic and spherical scoring rules often result in extreme ranking differences when compared to the logarithmic scoring rule. Furthermore, due to being non-local, the quadratic and spherical scoring rules allow for the possibility of one model receiving the