According to the previous properties of mutual information we can define a mutual information-based metric between two 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.
Indeed it comes from the previous section that:
\(d_{MI}\) is symmetric.
\(d_{MI}(X,Y) \in [0,1]\).
\(d_{MI}(X,Y) = 0\) iff \(X\) and \(Y\) have the same probability distribution.
in particular \(d_{MI}(X,X) = 0\) for any trajectory.
\(d_{MI}(X,Y) = 1\) iff \(X\) and \(Y\) are independent.
This quantity may be interpreted as to the proportion of uncertainty that is not shared between \(X\) and \(Y\).