notes-groupDecisionMaking-votingMethods-mixedApprovalScoreCumulativeMultiwinner

Multiwinner mixed approval, pr-score(2), and stochastic-cumulative (?)

One issue with score voting (range voting) is that under some sets of assumptions (todo cite), it is strategically optimal for each voter to always give either the maximal or the minimal score to each candidate; reducing score voting to approval voting. One issue with approval voting, in some contexts, is that it can favor "safe" candidates/centrists/those who become popular by avoidng controversy or through the support of 'the system' (see http://rangevoting.org/Extremism.html) at the expense of 'oddball' candidates. One issue with multiwinner proportional score voting is that, in some contexts, if no faction wins a majority, this could lead to extended negotiations after the election.

Here i present a 'mixed' method that (a) provides a strategic incentive for voters to make use of a 'middle score', (b) gives "unsafe" candidates more of a chance than approval voting, and (c) provides a bias in favor of the largest faction.

This method is arrived by starting with proportional score voting and then applying the following ideas; (1) in a multiwinner election, multiple election methods can be mixed by assigning some proportion of seats to each method; (2) (non-proportional) approval voting provides a bias towards the largest faction or coalition of factions, but in a way that promotes centrist rather than divisive campaigning; (3) 'unsafe' candidates are those without a broad base of support; so, choose some candidates who are intensely liked by a small group; (4) stochasticity can elect some candidates who are unpopular and who are not good at running campaigns.

The procedure is as follows:

• Each voter gives a score to every candidate. The available scores are 0, 1, or 2. The intended interpretation of these is "0: no comment, 1: good: 2: very good", or more verbosely, "0: i don't like them or i have no opinion; 1: i can live with them; 2: one of my favorites".
• The number of available seats is divided into 3 parts, A, B, and C. These could each be given 1/3 of the seats, or some other distribution could be used (i like giving A and C 1/5 of the seats each, and giving B 3/5 of the seats).
• All candidates who do not meet the following two criteria are eliminated: (1) each candidate must receive at least p votes with score greater than or equal to 1; (2) each candidate must receive at least q votes equal to 2. (where p and q are free parameters; obviously, we should have q <= p) That is to say, each candidate must be approved by at least p voters, and must be a favorite of at least q voters. My preference would be to keep p and q small, similar to thresholds for signature collection/nomination.
• Now we select the A seats. We do this by (non-proportional) approval voting, where a voter who scored a candidate as either 1 or 2 is said to "approve" that candidate. So, for each candidate, sum the number of approvals they received, and the top A candidates win seats.
• Now we select the B seats. We do this using proportional score voting (reweighted range voting) (todo cite). But first, we reweight the ballots (using the reweighted range voting formula) to downweight those ballots who approved of candidates who won a seat in round A.
• Now we select the C seats. The ballots retain the weight given them by proportional score voting (including a downweighting after the last B seat was selected). For this round, we only consider the 'favorite' votes, that is, when a voter scored a candidate as '2' (you can imagine that the 1 votes are now set to zero). We start by penalizing voters who selecting many favorites, by dividing each score on each ballot by the total number of '2's on that ballot. Then for each candidate we assign them a score which is the sum of the (reweighted, and also penalized) scores that they were given. As with reweighted range, we then proceed through each open seat, select a winner, and then reweight all ballots to downweight those who selected that winner as a favorite. But, instead of selecting the winner as the candidate who has the highest score, we instead randomly choose a candidate, with probability corresponding to their score.

Observe that each voter has an incentive not to give too many 2s, suggesting that strategic voters will make use of the '1' score.

Observe that the seats assigned in rounds B are proportional (and i guess maybe in C also?), but the seats assigned in round A are majoritarian. However, since round A is first and the other rounds downweight in response to the winners of round A, too, the overall result is proportional (as long as the plurality faction has more than 1/5 support) -- so the effect of round A may be to encourage the plurality faction candidates to run more centrist campaigns.

Observe that the stochasticity in round C means that if most voters' favorite candidates tend to be those who are bad at mass campaigning, then most of the seats assigned in round C will tend to go to candidates who are bad at mass campaigning, even if no such candidate is a favorite of a large number of voters.

An issue with this method is its complexity. Also, the system used in round C is vulnerable to strategic nomination.