Solution:

Swapping wins and losses is a probability-preserving symmetry, so the probability $P$ of more wins than losses equals the probability of more losses than wins, giving $P = \frac{1 - p_0}{2},$ where $p_0$ is the probability of equally many wins and losses.

An outcome with $k$ wins, $k$ losses, and $6 - 2k$ ties can be arranged in $\frac{6!}{k! \, k! \, (6 - 2k)!}$ ways: $1,$ $30,$ $90,$ $20$ for $k = 0, 1, 2, 3,$ totaling $141.$ Each of the $3^6 = 729$ outcome sequences is equally likely, so $p_0 = \frac{141}{729} = \frac{47}{243}.$

Therefore $P = \frac{1}{2}\left(1 - \frac{47}{243}\right) = \frac{98}{243},$ and $m + n = 98 + 243 = 341.$