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\subsection{MI-based metric}  According to the previous properties of mutual information we can define a mutual information-based metric between two players' player  trajectories as: \[d_{MI}(X,Y) = 1 - \frac{2MI(X,Y)}{S(X) + S(Y)} \]  Where $X$ and $Y$ are the positions of the two players. 
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\begin{enumerate}  \item $d_{MI}$ is symmetric.  \item $d_{MI}(X,Y) \in [0,1]$.  \item $d_{MI}(X,Y) = 0$ iff $X$ and $Y$ have the same probability law. distribution.  \item in particular $d_{MI}(X,X) = 0$ for any trajectory.  \item $d_{MI}(X,Y) = 1$ iff $X$ and $Y$ are independent.  \end{enumerate}