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Why not have the server choose 3 unique champions at random from each player's owned champions, making a total pool of 9 champions that the team can choose from, in addition to the free week champions. This is similar to Dota 2's single draft mode, except the whole team can choose from a single pool instead each player being limited to only 3. Of course, each player can still only choose from the champions they own (to ensure people know what they're doing to some degree), but it's unlikely (except for low level players) that they would not own some of the same champions their teammates do. What would this do? I think It would force different team comps and they would be more viable because some of the most "op" champions would likely not always be available in the pool - since it's random based on the total pool of all champions the team has available to them. Also, perhaps the enemy team's random available pool could be displayed somehow on champion select as well, so that, while you don't know what they will pick, like blind pick, you do know what they CAN pick, so you can pick champions to counter some of those strengths of that pool. I can see this being a problem for newer players who don't know what most or every champion can do to some degree, but they would be no better off in standard blind pick anyways, at least this way they may have a chance ahead of time to think about what would be good for their team against the enemy pool.
Does anyone think this could be a good idea? If you think it's stupid because it's kindof similar to Dota 2's single draft, thats fine, but I think it could enable more varied team comps - additionally the random selection could weight the selections to try to ensure a balanced pool (so it's not randomly all adc's or supports for example). The amount of this weight would be dependant on the amount and kinds of champions each player on the team had to contribute to the pool.
I'm sure something like this would be time consuming to impliment, but I do think it would largely solve the bruiser heavy team comp problem, particularly if the weighting was done to ensure it's not just a bunch of toplane bruisers in the pool, or adc's, or supports - at least it would try it's best to ensure that, much like how riot tries their best to have a balanced and viable set of free champions each week.
Sorry for such a long post for such a simple idea. Any thoughts?