The thing is this in no way fixes the problem and can be abused. There is a bullet list at the bottom if you don't feel like reading explanations. There is also an idea that you may like that I put at the bottom. It may help alleviate your issue.
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Please create a feature that if 4 out of 5 players in a queue vote to dodge due to a player trolling them in the champion select lobby they can dodge that game. It's really necessary especially in Bronze with all the trolls. What do you guys think? It's stupid to have to go a full game with the troll and most likely end up losing it and lose LP for it too only to report the guy at the end of the game.
Edit: Wow this got a bit of attention. What I hope they think about implementing is something similar to what Postal Badger wrote about making it work like ARAM's rerolls, you are limited to 1-2 and have to play ranked to regain the privilege.
This fixes nothing. All this does, if you are serious about all the trolls is delay the inevitable. You get to dodge 1 or 2 games and then play your game with a troll. Nothing solved. This also only happens if all 4 of the people who are voting to dodge even have all their votes. If 1 of the 4 has used their 2 votes, guess what, you are playing or dodging the old way. Also, to dodge 1 ranked game a day is - 3LP and 5 minutes. The second dodge is -10LP and 15 minutes. This is less than 20 LP in 20 minutes. You would probably lose an equivalent amount just playing through a game and losing except with the extra benefit that your MMR is also affected.
Trolls are already empowered enough because somebody on that team will most likely take the 3LP hit and dodge for them. But with this system you are giving people a free dodge, something that trolls will jump at. Then what if somebody regular dodges? Just use this old -3LP the first dodge system and we are back where we started except trolls also get the option of a get out of jail free card?
You also cannot change the trolls LP. Riot has a policy that they do not go into people's accounts and change their LP. They don't do this for Tribunal cases, they won't do this because 4 people decide they don't like one person just because he "picks off meta" or some other thing that the team decides they don't like.
This can also be used for people not trolling. "Holy crap their team comp will stomp ours, let's dodge guys. What 4 of us have votes?" or "I want to get so I'm going to leave them open. Let's dodge if the enemy takes them." I mean it promotes teamwork but I don't think the type of teamwork Riot wants promoted.
------------if you skipped past the above explanation here is your tl;dr--------------------
I hope this helps a little bit. What you have proposed is not a good solution as it :
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Does not address the root problem. It just attempts to circumvent it. Much like how the current system does not actually address trolls pre-game.
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It actually empowers trolls by giving them more tools to exploit.
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It also opens up avenues of abuse for players who think the game is not in their favor (without a negative side effect).
One thing I would like to see, if I am so damn smart about why your idea doesn't work :
If somebody dodges a game, have a pop-up that allows each player in the game to explain why the game was dodged (and who caused the dodge). Basically an option to report somebody for pre-game abuses that were big enough to cause a dodge.
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Just like tribunal, this would not go on a game by game basis but more of a "This person has been reported repeatedly, what is going on here?" basis.
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This does contain the old system (somebody will lose LP) but also contains the chance that trolls will receive punishment for repeated behavior. This actually puts a real consequence on trolls while imposing a minimal consequence on the dodger.
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This will be slow. However, it will allow a build up of evidence. So that when action is taken against an account, Riot has plenty of proof for why they are receiving actions such as suspensions, bans, etc...
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As a bonus, this uses systems that Riot already has. They have a report system. You have to update the look/categories for pre-game but the foundation of the report system is there. Riot already has a tribunal so they already store these reports somewhere. Just add in this category (either separately or combined).
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This empowers the non-trolls by allowing players to take action against trolls before having to sit through an entire games worth of abuse.
I hope this helps you with you with critical analysis of your proposed features in the future!! :}