Ranked ARAM
A lot of people play this game mode as their main role, i'm not saying to tweak the gameplay too much, but just add a ladder. It could end up being competitive for people who would like to play competitve ARAM.
People will say you can't have ranked with random, but if you manage to win more games on random champions, maybe you should play people who win more games, plus I know a lot of people would probably play this.
I look at it sort of as a free for all, and those have always been competitive in the shooters competitive scenes.
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If you fail to ban on your turn a random champion will be banned(