ATAR3

ATAR3 voting
===========

ATAR3 is a voting system which is a hybrid of approval voting and Condorcet, which aims to capture some of the advantages of Condorcet over approval voting[1] while being easier to count by hand and easier to understand than Condorcet. ATAR3 stands for Approval Then Automatic Runoff on Three.

Ballots are marked "strongly favor", "favor", "oppose" or "strongly oppose" for each candidate. In the first round, only the favor/oppose distinction matters[2], which reduces to approval voting.

Unmarked candidates are opposed; they should probably be taken to equal "strongly oppose". Since candidates in the runoff must be notorious (many "favor") they will not likely receive many non-votes, making this a minor question.

In ATAR2, the top two candidates are then taken in "automatic runoff", and each ballot is counted based on whether one outranks the other. This allows relative preferences among popular candidates to be counted without counting some votes more than others.

In ATAR3, we extend this principle to a three-candidate runoff. Each candidate's relative preference versus the others is counted; the strength is not (all ballots count equally). So there are three simultaneous races counted from each ballot: A vs B, B vs C, C vs A.

There are two possible outcomes among three candidates excluding natural ties (which are always unavoidable): either we have a clear winner who beats both other candidates, or a rock-paper-scissors cycle, the simplest Condorcet cycle[3]. In the 3-cycle, most Condorcet rules (Kemeny, RP, Schulze, minimax) give the same outcome, which is to drop the weakest of the three paradoxical preferences, i.e. to choose the candidate with the lowest pairwise opposition. This is "regret minimization". Most of the time[4], though, a winner who beats both others is found.

Approval voting is generally considered a solved problem from a counting perspective. ATAR2 is very simple to count. ATAR3 can be counted by either separately counting the three pairwise preferences among the three candidates on each ballot, which has slightly complex accounting, or by sorting ballots into each of thirteen piles corresponding to the thirteen possible rankings of three candidates -- that includes all possible ties.

For four-candidate runoffs, there are six pairwise preferences, 75 possible rankings (Fubini numbers), and complex cycle resolution. So ATAR3 is as close as we get to Condorcet while staying "simple".

---------- Footnotes:

1: In particular, there may be a lot of voters who did not *express* a relative preference between the first-and-second-place candidates in an approval-voting election, but those voters *had* a preference.

2: We contrast with scoring methods, which make people worry that "some votes count more than others"; here we always compare votes on equal footing.

3: A Condorcet cycle occurs when each voter's well-ordered preferences add up to cyclic preferences. So Alice's supporters vote Alice>Bob>Charlie, Bob's vote Bob>Charlie>Alice, and Charlie's vote Charlie>Alice>Bob; now we have a problem. In this perfectly disciplined case, regret minimization gives it to whoever has the most first choice votes. If the voters are not so disciplined, we cancel out all pairs of opposite ballots (Alice>Bob>Charlie cancels Charlie>Bob>Alice), and we either obtain the first situation or we find a well-ordered election; this method is (much) more well-behaved than only looking at first choices when we ask what difference does each ballot make on the outcome.

4: Surveys of ranked and/or scored elections using various systems have found that a Condorcet winner usually exists; cycles are rare because my preference for one candidate over another is linked to my preferences among other candidates by dint of my beliefs.