I think I know how to deal with alt accounts (and everyone is going to HATE it).
Make League cost money to start up. Simply put, WoW did it to stop hurting their toxicity and it (kind of) worked. It's the only thing I can think of that would honestly make people go "Maybe this was a bad idea..." when just trolling and then making alternate accounts. Offset this by giving a set amount of RP when buying the game and done. This will get downvoted to oblivion, I know it will, and there's probably a good amount of reasons why NOT to do this (such as no other big name MOBA does it even though Smite has that 30 dollar every god pack from now till end o' time deal), but hey why not? I don't really know many other options and unless Riot has something better cooking up I think this might be what it'll come to. That or continue letting alt account trolls run wild or LCS players fuck up your promo's
EDIT: So people have been mention WoW's player base steadily declining since. I think that this does lead to a load of questions about whether this is a good idea or not (and judging by that minus 2 up there you guys say not) and as I have mentioned in the comments:
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I should have mentioned in my comparison to WoW's process, these are not the same games. So therefore things tend to be less... Coherent. Things that are considered toxic there might not be here, vice versa, and things that would be a problem in a MOBA (such as dodging que's and feeding) won't be present, at least to the same degree, in League.
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The steady decline of WoW's player base is a concern. However, I think we need to ask ourselves which is more important: A small, tight knit community that has a massively cut player base but a harder nut to crack in terms of trolls, toxic players, and general bad attitudes, or a much larger player base with more money but also heavily increased toxicity and rage. Personally I prefer smaller community with less strength but this is a game of millions, not one
EDIT 2: Ok some people seem to think I mean a pay to play constantly deal. I was thinking more of a pay to start UP than a pay to play constantly.