Indeed, perfect democracy cannot be mathematically achieved, there will always be someone left out somehow. But you can approximate it, see game theory.
Ok, so we are a group of friends and every week we get together to watch a movie, and we have to go through a vote to decide which movie to see but we waste too much time on it. I want to see if it is possible to design a voting procedure in which there always is a definitive result in the first round of voting, whilst keeping everyone's voting rights equal.
>3 people have to be present for the vote to happen. So far I keep hitting a brick wall and I think it is mathematically impossible for such a thing to be designed, proving so or otherwise wil; be equally redeeming. Any ideas?
The subject of interest to you seems to be Social choice theory. One of the most basic results in this area is that an anonymous, neutral choice function exists iff there does not exist such that , where n is the number of voters and m the number candidates. This basically means that in many cases, you won't even be able to find a 'fair' algorithm that will guarantee someone will be chosen, let alone in one round!
In any case, it does seem impossible that you can have a fair algorithm that always terminates in one round - taking voters for one candidate, voters for another candidate and at most one voter for a third candidate will always leave us at a tie.
If you want to read more about this, I would recommend Moulin's The Strategy of Social Choice.