I am trying to find a good descriptive statistic for a game I'm playing. I would like to compare my kills in a game to a person on the enemy team, and have some sort of normalization for each game. So I'm looking for some f(x,y) where x is my kills and the y is the enemy's kills (I might do the same thing with other stats such as KDA, gold earned, etc.) with the following properties:
1. if x=y, then f(x,y)=0 positive if I do better, negative if they do better
2. if x>y, then f(x,y)>0
3. if x<y, then f(x,y)<0
4. f(x,y)= -f(y,x) statistic is the same size but opposite sign if our kills are reversed
5. some sort of normalization, so if I have a match where I go 2-1, and then a match where I go 20-10, my outputted stat for each game is comparable, something like f(cx,xy)=f(x,y)
6. Is defined for all x>=0, y>=0. Exception: defining f(0,0)=0 is okay
7. The hardest one, an intuitive sense of scale, like, if I do twice as well as my opponent, then the number I get should give that impression somehow.
I might also include other variables in the function if they made sense like the total kills on each team, but I think it would skew how I did versus that one other player.
f(x,y)=x-y gives the first 4 properties, but doesn't normalize.
f(x,y)=(x-y)/(x+y) gives 1-6, but not 7, say that x=2y, then f(x,y)=1/3=33%, which I feel doesn't represent the performance.
f(x,y)=(x-y)/x doesn't give property 4
Everything else that I've thought of has either the same problems as these or worse. Thanks for the help.