Long live the idiot!
imagine all the cities are in a line.
A-2-B-1-C-1-D-1-E-1-F (letters are cities, numbers are distances(cost).)
if the start point is B, the smart guy will do 1+1+1+1, then go back 1+1+1+1+2 for a total of 10.
The idiot will do 2 then go back 2+1+1+1+1 for a total of 8
EDIT
Assuming it is forbidden to go to the same city two times, I think the genius would always win or tie.
The only way for the idiot to win, would be to trick the genius on a long segment that the idiot would not use. To go to every city, only 1 segment can be ignored and that has to be the long one for the idiot. And the only way to make sure the idiot would not use that long path would be to keep it for last, but since a path connects 2 cities, it is impossible to ignore it until the end.