I think that the feeling is there for this idea. I do hate losing my ranked games from a troll and/or toxic player who becomes the ultimate reason as to why our team ends up losing, whether it was a long-drawn battle or an utter stomp. I do think that there are some issues with the method that you posted here. I would compare this to League's current honour system in which the player that is making the votes for the enemy team earns no personal benefit for doing so. Unlike the report system where the beautiful message pops up that says "Thank you for reporting summoner x", there is no real satisfication from doing so. Yes, I do understand that there are some nice summoners on the rift that see the injustice and want to make some good votes but in other circumstances, there is a high chance that problems will arise.
The first problem that will arise is the idea of pleading sympathy from the losing team. In reality, nobody wants to lose LP unless the summoner was intentionally trying to lose the game. In a game that has no leavers, no trolls and no salt, the losing team is most likely going to find an excuse that they tried their hardest and were disadvantaged by a player on their team, even though they lost fair-and-square. The losers would probably try their hardest to win the hearts of the winning team so that they will be voted as good and then marginalize another player's efforts to the point where they seem bad. There would be a lot of self-defence and conflict as the losers try to argue why they deserve sympathy points.
The second problem is whether or not the winning team is actually going to care enough to cast their vote and try to reduce the amount of LP that you lose. Going back to the honour system, there is no icon or banner that Riot hands out for players that consistently honour others because that can be easily abused. I personally think that it takes up time to honour players and I just want to click that 'Play Again' button. This relates back to the point above as the winners do not exactly want to listen to the losers' feud. Also, the voting would have to take place immediately after the game so that the losers would get their sympathy points as soon as possible. With this, a winner would have to essentially make their decision quickly and who's going to remember your sacrifice at the bottom lane that got your adc a double kill, right.
The third and probably most important problem is how players would try and play more selfishly in order to insure themselves in the event that they lose and this was already brought up by Randomonium. Players will try to maintain a positive reputation which can be achieved easily through one method: playing selfishly. Quite similar to how the champion rating looks more at the kda ratio more over teamplay efforts, the same might happen for this system and that doesn't exactly help players win games better.
So yeah these were my comments and it may just look like a wall of roasting but I honestly don't think that this will be effective. Riot needs to find a different way to reduce the lost LP fairly for those trying their hardest.
