My friends and I play rounds of golf at 9 hole courses. In order to make the game more competitive, I've devised a point system for defining a winner other than just the lowest score at the end of the round. However, I'm unsure as to whether or not winning this scoring system is fair, in that is it possible to win the new scoring system without winning the overall score? The scoring system is as follows:
If you win a hole (you get the lowest score in that hole): 1pt, 9 opportunities
If you win a set of 3 (you get the lowest score out of 3 holes): 2pts, 3 opportunities (holes 1-3, 4-6, 7-9)
If you win the overall round (you get the lowest score overall): 5pts, 1 opportunity
Any ties in the categories and the points are zeroed. For example, if 2 players shoot a low score of a 3 on a hole, neither receives the 1 point for that hole.
I've tried to figure out how to code the permutations, but I'm wondering if there's a more efficient way of calculating if it's a fair scoring system.
Below are the scores for two players on each of the nine holes:
Obviously, the numbers are a bit absurd, but Player $2$ can certainly win without the lowest overall shot score.
Edit: I misread the original post: I added $3$ points instead of $2$ for each win of a three hole round. Please refer below to the comments until I have a chance to properly edit my post.