User:R.H.

Judge and jury system where the judge gets rated
User:R.H./Judge_and_jury

New(?) single winner method
User:R.H./Condorcet,_IRV,_travel_through_time_and_dimensions

Approval Random
The ballot

Every voter marks one candidate as "favoured" and any number of candidates as "also accepted". Those marks are added together and counted as "approved". If no candidate gets more than 50% approval, the ballots are used in a lottery to determine the winner. Otherwise the one with most approval wins.

Reasoning

Assumption 1: While random elements are a solid way to make elections less vulnerable to cheating, they are too radical for many voters.

It follows that an election method that uses randomness should have that feature not as a first step, but only as a fallback and every voter should have some control over whether that happens.

Assumption 2: If an approval voting election is hold and none of the approved candidates of a voter wins, s/he would prefer to have the election result discarded and a random ballot election instead.

It follows that if no candidate in an approval voting process gets approval by more than half the voters, the majority of the voters is in favour of a random ballot election (it can happen that the result of that is even more disliked, though).

Criteria passed and failed

Approval and Random Ballot pass FBC, so it is no surprise that this combined method does, too. Normal Approval gives people with a preference for a candidate known as weakly supported a reason to approve several candidates and people with a preference for a candidate known as strongly supported a reason to not approve anybody else. That's a bit unfair. This method adds some incentive for people with a preference for a candidate known as weakly supported to not approve anybody else (the lottery chance) and for people who prefer a candidate known as strongly supported a little incentive to also approve somebody else (to prevent the possibility of a lottery win for extremists). Approval Random does not satisfy the Majority Criterion. It is not impossible that a candidate with 60% "favoured" votes loses to a candidate with no first votes but 61% "also approved" votes. It is also not impossible that a candidate with 1% "favoured" votes wins to a candidate with 49% "favoured" votes.

Vote-Trading: The most simple proxy vote system?
A system for non-proportional representation that allows trading of votes. Independently from me, many other people had ideas about "proxy voting" or "delegated voting" which are much more refined. I think mine is the most primitive possible, which is also sort of a quality.

Your ballot looks like this: Vote for 1 of the candidates: A B  C  D  Allow trading with your vote? YES [ ]

Counting

First, we look if a candidate gets more than 50% of the votes, because that would be such a clear support that this should be the final result. Otherwise, some votes can get traded for a defined timeframe. Candidates can get such votes if they make concessions to candidates that give those votes. After time is up, the one with most votes wins.

Given that proxy voting is quite a new idea, this proposal is a rather tame version. First, once traded votes can't get traded again. "Allow trading with your vote?" means two things: It is the option for every voter to allow one's vote given to another candidate. And it is the option for every voter to accept that one's candidate will make concessions to others. A candidate's uncompromising votes can't be traded at all and in case the candidate makes concessions to others his/her uncompromising votes expire. This gives both voters for strong and weak candidates a reason to show some will to compromise.

I think Plurality Proxy Choice would be a good name.

Instant version of approval with a second round
Every voter gives a ranked ballot with an approval cutoff. First, the two with most approval are determined. Between these two the winner is decided by looking at all ballots for who is more often ranked ahead of the other.

D'oh.