Guidelines for using D21 - Janeček method

Rules of D21 - Janeček method

• The basic element of the D21 - Janeček method is that the voter can identify more preferences in a single ballot than the number of seats to be filled.
• Only one vote per candidate/option is allowed.
• All votes have the same absolute value.
• The voter can, but does not have to, use all available votes.
• The voter can cast a minus vote/downvote an option he disagrees with.
• In order to cast a minus vote one must cast at least two plus votes.
• The number of plus and minus votes depends on the specific situation, see the recommendations below.

Example: In the presidential election there are 9 candidates, but only one will become a president. As voters we have 3 plus and 1 minus votes available (see table below).  We can either distribute all these votes or just some (for example just two votes). We can assign only one vote per candidate. If we really disagree with someone we can vote against them with a minus vote. But in order to cast a minus vote, one must cast at least two plus votes. Why? Because elections should be mainly based on positive attitudes.
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The effect of multiple votes, as presented in D21 - Janeček method favours more consensual candidates. (1) Moreover, it also significantly reduces the motivation for strategic voting, hence people do not have to opt for “lesser evil”. The option of minus vote helps to uncover controversial candidates and contributes further to consensus. It therefore strengthens the effect of more votes (2). The minus vote could motivate more voters to the polls. (3)
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1. D21 - Janeček Method focuses on strong preferences. That allows it to find a candidate who has a strong and broad support among voters.
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2. This does not have to be true each time. Theoretically, there can be a candidate who is consensual (receives a lot of additional votes) and polarising at the same time when they get a lot of additional plus votes and also a lot of minus votes.
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3. DOLINSKI, D. and DROGOSZ, M. (2011), Regulatory Fit and Voting1. Journal of Applied Social Psychology, 41: 2673-2688. doi:10.1111/j.1559-1816.2011.00841.x.

Number of votes in D21 - Janeček method

For the number of winners W and for large number of candidates, we recommend the number of plus votes PL:
Number of votes in D21 - Janeček method depends mainly on the number of winners. Although if  the number of candidates (options) is too low, it will limit the number of votes. The key difference between D21 - Janeček method and approval or combined approval voting is the capped number of votes, therefore each vote has higher relative value. This way, D21 - Janeček method focuses only on strong (mainly positive) preferences of voters. It combines the effect of more votes with motivation to think critically. If the number of votes is unlimited, such as  in approval voting, voters often have a tendency to use votes excessively (especially minus votes) as (s)he is not motivated to think about which candidate is closest to his/her opinions.

‍PL(W) = round [2W - (W-2)*Phi],
where Phi = (sqrt(5)-1)/2 ~ 0.618.

The number of plus and minus votes is reduced, when there are less candidates participating. I.e. it would make no sense to allow voters to cast four plus votes in a two-winner's election, if there are only four candidates.

When  C is a number of candidates. Number of plus votes is then:

P = floor [min(PL, ½*(sqrt{(V-1)ˆ2 + 4C} + V-1) ],
where “floor” means always rounding down to the nearest whole number

This formula is extrapolated from the rule:We add nth additional plus vote(s) to the W plus vote(s), if the number of candidates is bigger or  equal than -C>= (W+n)*(n+1). Thus we add W+1st plus vote to the W plus vote(s), if C>=(W+1)*2, W+2th if C>=(W+2)*3, etc.

The number of minus votes M is then

M = round (M=P/3),

The total number of (plus and minus) votes must always be higher than the number of candidates, P+M < C. Should the total number be equal to the number of candidates, the number of minus votes needs to be reduced until the condition of P+M < C is met.

Why is the optimal number of votes limited even if the number of candidates is getting higher?

In one-winner elections, there is no substantial difference between 21 and 121 candidates. If the number of competing candidates is high, not even voters with high interest in politics will know all candidates. Therefore it makes no sense to add any other plus votes, because the “effective” number of candidates will stay relatively stable. The formula for PL provides an optimal level of consensus for a really high (theoretically unlimited) number of candidates. (PL can differ for different applications.)

When (not) to use minus vote(s)?

We do not recommend to use minus vote(s) in political elections with ethnic or religious minorities running for an office. Minus vote(s) is also not convenient in the apolitical elections with only positive oriented options (e.g. charity projects) or where the strictly positive atmosphere is required (e.g. when children propose and vote about projects). The minus vote can be introduced “incrementally” - after people have some experience with using multiple plus votes.

What to do when more candidates get the same number of votes?

The same number of votes on winning places will happen only sporadically in the elections with a lot of voters. When it happens, the winner is the candidate who received more plus votes. If two or more candidates receive the same number of net votes and plus votes, the winner is selected by drawing lots.

Table 1:  Recommended number of votes for a big number of candidates

Table 2: Recommended number of votes for a limited number of candidates
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