Reasons Behind the Toxicity
Edit: Some people are misunderstanding my points. I am nothing trying to say we should fix these problems. It's the opposite that no one, even Riot, can fix them. People have to accept the system is the core cause, not the playerbase.
I am a decent player who has experience of over 10-year playtime for many games of various styles. LoL is the most frustrating game I have ever played. One might believe the player base of LoL is to blame. However, I believe the system is more likely to be the fundamental problem.
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The Game Forces You to Stay Every time you start a game, you have to stay until the very end, either win or loss. If you meet any toxic players, say an intentional feeder, your next at least 20 minutes will go to waste. You cannot simply leave because the system does not allow you to start another game until the current one ends. Though you understand you are playing with a selfish troll with no chance to win, the system even punishes you more if you decide to leave or AFK. This kind of frustration reminds me of the Sisyphus Greek mythology. In many other games, say Counter-Strike, Maple Story, or FEZ, if you meet any troll, just leave the game and join another. If an intentional KSer keeps following you just to ruin your day, just logout and breath some fresh air. There is always an option to disengage the toxicity in many other games. LoL, however, traps you in the current game no matter how ruined it is.
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Win/Loss is Everything Personal performance has limited influence in LoL. Even if you have godly skilled carrying ability, it is nothing when a pair of duo Q decides to feed because it is fun. No matter how well you perform, you feel awful because trolls can easily ruin it. Unlike other games I mentioned like Counter-Strike and FEZ, you still enjoying killing people. Losing is no big deal because your high score still rewards you a sense of achievement. In LoL's case, you only feel like, "I am 10-0 and has taken two towers in 10 minutes, but the trolls make my incredible outcome worthless.
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Measures Against Toxicity The only counter to trolls in LoL is by reporting. It may be useful in long-term, but hey let's be honest, we want that feeder gets punished IMMEDIATELY. You don't go like "ahh, I have reported and I feel much much better even though I'm not sure if it matters!" You know getting that particular troll banned doesn't improve your game experience in the future, because tons of other trolls are awaiting your following games. This leaves you as the victim helpless with no chance to revenge your anger.
TL;DR: LoL forces players to deal with toxicity with bare hands. Quitting the game seems to be the only option to get rid of it.
PS: Trolls are people trying to ruin others' games intentionally, aka asshole, jerk. Someone asked for the definition.