I think if you do well in a game you should lose less LP.
I don't think its really fair that someone should lose 20 LP when they themselves are doing well but other players on their team are unfortunately not.
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I don't think it's fair that at work, I have better stats than the rest of my peers, but they get paid the same I do...
How would you determine how you did in game?
That could easily lead to selfish behaviour instead of teamwork.
If a player has won lane but their allies have lost lane, they may be less likely to help since it would jepordize their KDA.
Splitpushers may refuse to group and teamfight since they would lost CS to risk dying.
Supports and tanks may refuse to peel, engage, or sacrifice themselves when appropriate to protect their KDA, losing teamfights.
People may abandon teamfights when low instead of staying around to help clean up killed, potentially losing won teamfights.
Just a few examples, but many people are already selfish enough in a team game and don't need further incentive.
If you consistently play well then you do lose less LP because you lose less games. Individual games are insignificant.
What do you consider "doing well" if you lost? You can't just look at your score and say "we won or lost because of this". The game isn't about getting high scores, it's about killing the Nexus. The Nexus kill is the only point that ever matters, and how that happens is irrelevant.
A person with 0/9/6 could have actually been split pushing and back doored for the win or just drawing the enemy away, while the 12/2/3 guy was kill stealing and not actually helping in teamfights, but with a personal score system the 12/2/3 guy will get all the credit anyway.
I believe your KDA has impact with your LP. Like if you don't die, you lose less LP or win more LP. LP system is not just based on MMR from both teams, there are plenty of thigns that affect it.