The D21 method

# Guidelines for using the D21 method

How many votes should you use?

## How does the method work?

The D21 method can be used both in single-winner elections (presidential, senatorial) and in multi-winner elections, such as parliamentary or local elections. It can also be used as a voting system in everyday life. For any given election, the number of votes available for voters to use can be determined based on that’s elections number of winners and of candidates, according to the following guidelines:

• A voter always has more votes than the number of options that will win or seats to be elected
• The voter may give any option no more than one vote
• All votes have equal weight
• The voter has the choice of using all or only some of his votes
• The number of plus and minus votes is determined according to the election’s conditions (see recommendations below)

The minus vote can be an additional element of the D21 method. This can serve as a useful tool in certain situations to further weaken controversial and extremist candidates. Minus votes further amplify the effect of multiple votes, yet it should be carefully considered whether their use is well-suited to particular elections. In order to cast a minus vote, a voter must also cast at least two plus votes.

#### Example from politics:

In the current electoral system, populists and extremists have a systematic advantage in societal division. It can be sufficient for them to attract some large group of voters, even if this base is far from constituting a majority of the public. To their benefit, the votes of the majority can easily be split among other consensus candidates.

These elections produce outcomes that satisfy only a minority of society. And they leave out of the picture altogether the candidate who would have been the most acceptable to the majority of the public.

Imagine a presidential election with 7 candidates. In this race, each voter is able to cast a maximum of 2 votes. Even though these voters have different political preferences, the D21 voting method helps them find a candidate they can more mutually agree on.

The outcome of this election will satisfy the majority of society. Even if a voter’s top favourite does not win, a moderate and consensual candidate will. Controversial and extremist candidates are much less likely to succeed.

## Single-winner elections

With the D21 method, the voter has more plus votes than there are winning options. If there were three to eight candidates in a presidential race, voters would have two votes to use. If there were nine candidates, they could cast up to three votes. Among the few basic rules, all of the votes carry equal weight and it is up to each voter to decide how many to use.

The D21 method offers several major advantages over other common methods for majority elections.1 It removes the need for a two-round system, since voters are able to express more preference right away. This eliminates the divisive effect of the second round that brings forward and opposes two irreconcilable halves. At the same time, by creating the possibility to vote for more candidates, the method increases the likelihood of voter satisfaction with the outcome of the election. D21 provides enough votes to express strong preferences, but not so many that it strengthens bland candidates. It allows a more accurate translation of the will of the voters into a result, while also saving time and money.

1) If the D21 method is used in multi-member districts, it is a semi-proportional electoral system.

## Parliamentary Elections

What would happen if people were given just one extra vote, releasing them from having to select only one platform to support? Parties would have to start appealing to their opponents’ voters. Those unable to court anyone outside their core electorate would weaken. At the same time, there are currently parties that are consensually attractive, but have small voter bases. Prospective voters may agree with their platforms yet still not support them in an election, for fear of wasting their votes. By giving people a second vote, that barrier to support falls away, thereby strengthening parties that are closer to others for the prospects of coalitions. The D21 method also works well for voting for individual candidates in an open list system. It ensures that the parliamentary benches are to be occupied by strong consensus figures with a chance to outweigh party cadres.

### General application

Although parliamentary structures and election conditions vary widely from country to country, there are some general features that the D21 election method will introduce in most cases. With the implementation of D21, in general:

• Procedures governing ballot lists, candidate lists, and qualifications for running in the election or getting on the ballot remain unchanged
• Each voter can cast a vote for up to two parties
• If a country has thresholds in place for passing of an election round or the gaining of parliamentary seats, this can be determined as a percentage of votes by total votes cast or adjusted to consider the percentage of votes by participating voters, subject to the country’s discretion
• If voters are able to also vote directly for individual candidates on lists, a maximum number of votes is determined and allowed for each voter to use
• Implementation of the D21 method can include other aspects, selected in consideration of a country’s existing laws and electoral attributes, to best achieve a proportionate result at any relevant levels

### Case study for implementation: the Czech Chamber of Deputies

#### Voting for parties

• Parties generate lists for the country’s 14 regional voting districts - this remains unchanged from before D21 implementation
• Each voter casts a vote for up to two parties by placing one or two ballot papers into the submitted voting envelope.
• An election threshold is set such that parties must achieve at least 5% of the total number of votes cast nationwide to earn any seats - this also remains unchanged from before D21 implementation
• The electoral system has two tiers of representation, so the allocation of seats happens based on both the national and regional results. At the national level, seats are distributed among the parties using the Hare quota.
• The size of the regions is calculated using the Hare quota with the largest remainder method on all valid votes cast.
• Seats are distributed at the regional level. Unallocated seats are further distributed using the largest remainder method - within this process, if a seat is to be allocated in a particular region that is already full, then this seat is transferred to whichever of the remaining unfilled regions has the highest remainder for the party. This process continues until all seats have been allocated.
• After the distribution of seats is completed, specific seats are allocated based on the results of the candidate-level election.

#### Voting for candidates

• Parties nominate candidates for their lists and decide on their order. In the Czech Republic, the maximum number of candidates on a candidate list is determined by law based on the number of voters in a given region - this remains unchanged from before D21 implementation
• Each voter may vote for no more than Φ² (roughly 38.2%) of the maximum number of candidates on the list. The specific value is to be stipulated as an integer in the Annex to the Electoral Act along with the corresponding maximum number.
• The seats allocated to specific candidates are filled by the candidates with the highest number of votes.

## Where might you come across the D21 method?

The D21 method is used in the public sphere to help citizens make decisions about their environment through participatory budgeting. It has been used in many Czech and Slovak municipalities and schools, with even wider usage planned for the future. In 2021, it was also implemented for the annual Czech music award show Český slavík to help select that year’s award winners.

### President 21

President 21 was an electoral game that let Czechs nominate candidates for president and vote for those they would like to see in office. With the D21 method, they had three plus votes and one minus vote, whereas the real election gave them only one vote. Over 320,000 people participated in the game.

## Number of votes in the D21 method

The number of votes allowed in an election by the D21 method depends primarily on the number of winners, although an insufficient number of candidates (or options) may limit the number of votes. The method’s key difference from approval or combined approval voting is its limited number of votes, which gives a higher value to the votes used, i.e. the "rarity" of the votes. D21 evaluates only strong (mostly positive) voter preferences in this way. Thus it combines the effect of more votes with the motivation to think critically. When voters are given an unlimited number of votes, such as with approval voting, they tend to overuse them (especially negative votes) and are not motivated to think as deeply about which candidates or parties are closer or further away from their policy preferences.

For the number of winners W and a large enough number of candidates, we recommend the number of plus votes PL:

PL(W) ≐ [2W - (W-2) × Φ],
where Φ = ½ (√5 – 1) ∿ 0,618.

The number of plus and minus votes can be reduced if the number of candidates participating is not large enough. For example, it does not make sense to give a voter four plus votes for a two-winner election when only four candidates are participating.

When C is the number of candidates. The number of plus votes is then:

The formula above is obtained from the rule where we add the nth additional plus vote to the base number W of plus votes if the number of candidates is greater than or equal to C ≥ (W + n). (n + 1). Thus, we add W + 1 votes to the base number of W plus votes if C ≥ (W + 1) 2, W + 2 if C ≥ (W + 2) 3, etc.

The number of minus votes M is then:

M ≐ P/3 ,

It must always be the case that the number of all votes (plus and minus) is less than the number of candidates, i.e. P+M < C. Should this not be the case, we reduce the number of minus votes until this condition is met.

#### Why doesn't the number of votes continue to increase with an increasing number of candidates?

It makes no fundamental difference whether 21 or 121 candidates run in an election with one winner. In an election with a large number of candidates, even voters with a strong interest in politics will not be familiar with everyone on the ballot, and only a limited number of candidates will have a chance of success. It therefore does not make sense to add more plus votes because the "effective" number of candidates will remain similar. The formula for PL is set to provide an optimal consensus rate for a very large - theoretically infinite - number of candidates. (The value of PL can, of course, be varied for different applications.)

#### When to (not) use the minus votes

We do not recommend the use of minus votes at the beginning of the implementation process of D21 for political elections. We also do not recommend the use of negative votes in political elections in societies with major internal divisions among ethnic, religious or linguistic lines. Further, the minus vote may not be appropriate in apolitical contexts where the choice is between purely positive options (e.g. charitable projects) or when a positive atmosphere is desired (e.g. options are proposed and decided on by children). The minus vote may be best introduced only gradually, after voters have gained experience using multiple plus votes.

Minus votes can be well-suited to the application of participatory budgeting in cities, for example. In this kind of election with mostly positive options and outcomes, the minus vote is able to transparently highlight controversial or unpopular projects and allow voters a more informed and detailed expression of their preferences.

#### In the event of a tie

Given enough voters participating in an election, the occurrence of a tie is highly unlikely. If it does happen that the sum of plus and minus votes is equal, the candidate who has the greater number of plus votes is declared the winner.If two candidates have the same total and the same number of plus votes, the winner will be decided by lot.

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