There is no way you could do a US Presidential election this way.
Maybe. Scale it up in steps. Groups of 12 citizens who are known to each other get into rooms to conduct a vote. One is chosen to take their group's decision to the next level, where 12 group representatives who know each other get together to vote. And so on...you'd only need seven levels of voting to reach the final 12 representatives in the current US voting population.
I'm using a fuzzy interpretation of the Six Degrees of Kevin Bacon game to assume that you could always arrange the representative groups to all know each other. Probably that means that the person in each group with the highest socio-economic status would be the logical representative, since they would be the most likely to have direct ties to other high socio-economic status members of other identical level groups.
I've not fleshed this out logically or mathematically any further than that. I'm also not suggesting that this would work well (or that it would be better than our current system), just proposing that it might be a theoretically possible way to do a US presidential election.