Two roads diverged in a yellow wood (part 2)
Retina, 28 bytes

\d
$*
%r`1\G
-
Os`.
+`-1

1+


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Prints 0 for left and 1 for right. Assumes that there are not trailing spaces on any lines.

Explanation

\d
$*


Convert each digit N to a run  of N ones.

%r`1\G
-


One each line (%), match consecutive (\G)  ones from the end (r) and replace each of them with - (i.e. turn the right branch into -s).

Os`.


Sort all characters, so that all -s are directly in front of all 1s.

+`-1



Repeatedly cancel a pair of - and 1.

1+


Try to match at least one 1 (if so, there were more weights in the left path).