This simulation adds reasonable voter behavior, (explored in SAVE Voters), and looks at how the next focus is chosen in more detail.

The general idea driving the focus selection rules is to converge as rapidly as possible (in terms of rounds) to the best current motion.

First, two definitions:

Approval Vote

Approval voting (AV) is a voting system in which all the motions have a checkbox next to the motion names. Voters may check as many or few as they choose. Checking a box indicates a voter's approval for the checked motion. Leaving a box empty means the associated motion is not approved of by that voter. The normal winner of an AV election is the motion receiving the largest number of votes. There are two AV rounds in the SAVE process. The initial AV round has the sole purpose of selecting the first focus. The final AV round, occurring after the final SAVE winner is determined, is there to explicitly measure the mandates of all the motions. The final mandate AV round does not change the final winner.

Focused Approval Vote

A focused approval vote (FAV) is a variation of normal the normal AV ballot in which one motion, F, is designated the focus motion, and instead of asking voters for voting for the motions they approve of, there are two different questions. For all motions M such that M is not F, the question is whether the voter prefers M over F. If so, the voter is instructed to vote for M. If not, the voter is instructed to leave M's checkbox empty. If the voter considers M and F to be equivalent, it is (as always) up to the voter to check the box for M or not. For the focus motion F, the question is whether the voter wants the SAVE process to stop with F as the final winner, or not. Checking the box for F is a vote for F as the final winner, and leaving the box empty means the voter wants the SAVE process to continue. (Going back to an M considered equivalent to F, I think it is reasonable to vote the same way for F and its equivalent M, but again that is up to each voter.

These are the current rules:

Tie Breaking Rule

Whenever there is a situation in which the other rules are not sufficient to isolate a unique best focus for the next round, such ties are decided in favor of the earliest registered motion in the tied set. This rule is unambiguous and strictly favors motions registered earlier in the process. Later motions already have an advantage due to the availability of the text of all earlier registered motions,

First Focus Rule

The first focus is the winner of the initial approval vote that starts the SAVE process. Normal AV rules apply when there is a clear winner, and any possible ties are broken as above.

Next Focus rules

These are the rules when the ballots for a focused approval vote (FAV) round have been tallied:

Final Winner Rule

If, when the votes for an FAV round are tallied, the current focus motion receives strictly more votes that any other motion, and its vote count is a strict majority of the ballots in the round, the current focus motion becomes the final winner. If either of these conditions is not met, there will be another FAV, with the focus motion for the next FAV determined by the following rules.

Condorcet Winner Rule

If, when the voter for an FAV round are tallied, no non-focus motion M has a vote count that is a strict majority of the ballots cast, the focus motion F is a Condorcet winner and as such will continue as the focus for the next FAV round. This is true whether F is a strong Condorcet winner (no ties) or a weak Condorcet winner (one or more motions M received votes equal to exactly half the ballots cast). If F is neither a strong nor weak Condorcet winner, the focus for the next FAV will be some other motion.

Potential Focus Set

When F is not a Condorcet winner, there is a non-empty set of motions with vote counts equal to or greater than half the number of ballots cast, and for at least one motion, the vote count is strictly greater than a tie. Call this the PF set. The focus motion for the next FAV must be a member of PF.

Single Choice Rule

If the PF set contains exactly \(1\) motion, that motion will be the focus for the next FAV round. If the PF set contains more than one motion, we go to the next rule.

History Period

Since SAVE takes multiple rounds, it has access to not only the current tally of votes, but also the vote tallies from previous rounds. The history period, HP, includes the vote tally from the most recent round, and all consecutive prior rounds such that none of the focus motions are repeated. For example, consider this sequence of rounds and foci: R1 with B, R2 with A, R3 with C, R4 with B, R5 with A, R6 with D, and R7 with D. Ignoring how the foci are selected for the moment, the history periods are: R1 with \({R1}\), R2 with \({R1,R2}\), R3 with \({R1,R2,R3}\), R4 with \({R2,R3,R4}\), R5 with \({R3,R4,R5}\), R6 with \({R3,R4,R5,R6}\), and R7 with \({R7}\).

Best Wins+Ties Count Rule

Using the history period for the round, the Wins+Ties count is calculated for each motion in the PF set. If we look at R4 in the history period description, motion D was proposed at the end or R3 when it was determined that B would be a repeat focus. We can see from the progression that motion D is a Condorcet winner, but we do not know that in R4. All we know is the PF set contains both A and D, and the history period is \({R2,R3,R4}\). During this period, A was the focus in R2, which counts as a Tie, lost to C in R3, and won over B in R4, for a total Wins+Ties count of \(2\). Looking at D, we see its first round was R4, so its Wins+Ties count is just \(1\). Thus A became the focus for R5 under the Best Wins+Ties Count Rule.

Best Vote Deficit Score Rule

The goal of the focus selection rules is to pick a Condorcet winner or the closest motion we have to a Condorcet winner. It does not matter what the margin of victory is for a Condorcet winner in any one-on-one race. All that matters is that the motion does not lose to another motion. If there are two or more motions in the PF set with the same Wins+Ties Count, the next thing to look at is the Vote Deficit Score, which is simply the total of the votes by which a voter missed a tie during the history period. If motion A defeated a focus motion in one round, was the focus in the next round, and only received 40 votes when a tie would have been 50 votes, A's vote deficit is \(-10\). If another motion had the same Wins+Ties Count, but had 45 votes with a vote deficit of \(-5\), that other motion would be selected under the Best Vote Deficit *Score Rule. There is a special case in this rule when one motion has a shorter history than the history period. The vote deficit can only get worse with an increasing history, so it is unfair for a motion with a shorter history be selected over a motion with a longer history under this rule. Thus, a less negative vote deficit score will not cause a motion with a better vote deficit score over a shorter history to knock out a motion with a worse vote deficit over a longer history. On the other hand, if the motion with the shorter history has a worse vote deficit score, it is gone.

Best Wins Count Rule

When the Best Vote Deficit Score Rule is not enough to reduce the PF set to a single motion, the next rule comes into play. This is just the number of wins the motion had against the foci during the history period. Looking at the history period example, the first time D won against a focus, it had a Wins+Ties Score of \(1\), while A had a Wins+Ties Score of \(2\). In the next round, D had a Wins+Ties Score of \(2\) which was matched by C. Motion C also had a Vote Deficit Score of \(-10\), but since D had not been around for the full history period, D's Vote Deficit Score of \(0\) did not help it. However, when the Best Wins Scores were checked, C had only a \(1\), while D had a \(2\), so D became the focus motion for R6.

Moderate Winner Rule

The prior three rules, wins+ties, vote deficit, and wins, all used the data from prior rounds. The Moderate Winner Rule returns to just using the most recent vote tally. The idea behind this rule comes from the observation in the one dimensional single-issue case, in which it can be seen that if there are a number of motions on the issue line that beat the focus, the motion closest to the focus will have the highest vote total, and the motion with the smallest margin (1 vote more than a tie), will be furthest from the focus. The topology dictates that as a motion's position gets closer to the focus, the motion's vote count increases. The goal, however, is not to maximize the margin of victory, but rather to get closer to a Condorcet winner. Accordingly, the Moderate Winner Rule finds the motion that beat the focus by the largest margin, and sets a target vote count equal to a tie vote plus half the maximum margin. It then scores each motion by how close the motion's margin is to the target. For example, if there are four motions in the PF set, with vote counts of \([50,54,59,62]\) over \(100\) ballots, the highest vote count is \(62\) so the target vote count is \(66\). The motions with vote counts of \(50\) and \(62\) are both \(6\) votes from the target. The motion with \(54\) votes is \(2\) votes from the target, and the motion with \(59\) votes is \(3\) votes from the target. Under this rule, the motion with \(54\) votes is closest to the moderate target vote count and becomes the next focus.

Adjustment to the Moderate Winner Rule

The ideal target value for the Moderate Winner Rule is based on the maximum possible margin of victory, but it is often not the case that a motion is positioned to get that maximum possible vote count. So the target is frequently lower than it should be. This is not a bad thing overall, but it does mean the focus series can jump to the other side of a Condorcet winner as the foci converge toward it. As a partial compensation for this, when there are motions equally close to the target t with different vote counts, i.e. \(t-n\) and \(t+n\), the motions with the higher vote counts are preferred.

Finalizing the focus for the next round

As each rule is applied in order, the number of motions in the PF set is possibly reduced. If PF is ever reduced to a single motion, that motion is the next focus. If all the rules are applied and there is still more than one motion in the PF set, it means all the remaining motions beat or tied the focus, that they all have exactly the same number of Wins+Ties as each other, that they all have the same Vote Deficit (or might have different history lengths that make the Vote Deficit rule inapplicable), that they all have the same Win count, the same Moderate Winner score, and the same vote count in this round. With that level of equivalent performance, the Tie Breaker Rule is applied and the earliest registered motion remaining in the PF set becomes the focus for the next FAV round.

• SAVE Home page