  We consider the weighted version of the Tron game on graphs where two
players, Alice and Bob, each build their own path by claiming one vertex at a
time, starting with Alice. The vertices carry non-negative weights that sum up
to 1 and either player tries to claim a path with larger total weight than the
opponent. We show that if the graph is a tree then Alice can always ensure to
get at most 1/5 less than Bob, and that there exist trees where Bob can ensure
to get at least 1/5 more than Alice.
