Firstly this is a theory that I'm pretty certain must be true at least under certain conditions, but I don't know how to explain it mathematically. I understand there are many variables I'm not accounting for.
FYI: In a battle royale game (BR) players drop in to a specified area that closes over time, fighting till the last man standing, with no respawning. Outside of game-specific mechanics like fall-deaths, the total kill-death ratio (kdr) is a fraction below 1, because the last player doesn't die.
Here's my logic now:
From the stats I've seen, as well as experience, the median kd should be lower than the mean, I'd guess around 0.7-0.85 for most BR games. It's quite obvious why that is. The top players can sometimes kill 1/4 of the lobby. But I wanted to explain it more mathematically.
If you convert the battle royale into a 1v1 single elimination tournament, the distribution is obvious. If there are 8 players, 4 get no kills, 2 get 1, 1 gets 2, and 1 gets 3. From my understanding you can't calculate the mean kd from this because the winner doesn't die, but the median is 0.5. To get the mean I'd add a second game, where all the players who didn't get a kill get on the scoreboard. The KDs would now be 4 with 0.5, 2 with 1 and 2 with 3 (the mean being 1.5 and the median being 0.75). So even with the skill levels being completely flattened (the placing reversed after the first game), the median is still 50% of the mean. If you added skill, and assumed the same players would place highly most of the time, the median would be even lower.
Obvious counters to this as a relevant model: yes, players don't fight each other one at a time, and some players will camp for placement. IMO the former would actually accentuate the skewedness of the distribution, as good players are more able to take advantage of third parties. The latter would have no/little effect on kd distribution, as camping till the end to win won't help you if you have to fight good players, and doesn't necessarily improve your kd or harm a high-kd player's anyway.
So here's an alternative model: players kill each other in a circle, p1 kills p2, p8 kills p1... and p3 kills p4 to win. Playing twice assuming equal skill you would end up with a distribution like {0, 0.5, 1, 1, 1, 1, 2, 3} the mean being 1.1875 and the median being 1}. Even though I think this is less similar to how BRs play out, it still would give a lower mean than median.
I was wondering how I could improve upon the logic and make it more useful. Is it a convincing argument? Is there a way I could combine the first and second models? And also if there are any general theorems that would apply.