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D21 (also known as the D21 – Janeček method or Democracy 2.1) is an electoral system applicable for both single-winner and multi-winner voting, which allows voters to cast more votes than there are open seats. It is a cardinal method like [[approval voting]] and [[combined approval voting]]. The method was developed by Czech mathematician [[W:Karel Janeček|Karel Janeček]].<ref>{{cite web |url=https://assets-global.website-files.com/5b046d892298cf84d9a87d56/5ff43cf8b01c725e9df3c415_5.1.2021.pdf |title=D21 – Janeček method |last=Janeček |first=Karel |date=2021-01-05 |access-date=2024-04-29}}</ref>

==Voting==
The basic prerequisite of the D21 method is that the voter always has more votes available than the number of existing winning opportunities. All votes have the same value. The voter can, but does not have to, use all of them. Multiple votes cannot be accumulated; a candidate can receive only one vote from each voter.<ref>{{cite web |url=https://www.ih21.org/en/guidelines |title=Guidelines for using the D21 method |author=<!--Not stated--> |date=2024 |website=Institute H21 |access-date=2024-04-29}}</ref>

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.

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

{{math|big=1|''P''(''W'') ≐ (''2W'' - [''W'' - ''2'']∙''φ'')}}

where {{math|'''φ''' {{=}} '''½ ({{sqrt|5}} - 1) ~ 0.618'''}} and the symbol ≐ denotes rounding to a whole number.

The number of 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.

{{math|big=1|''P'' {{=}} ''min''(''P'', ⌊½∙{{sqrt|(''W'' - ''1'')<sup>''2''</sup> + ''4C''}} + ''W'' - ''1'')⌋)}}

where half brackets indicate the lower whole part, or rounding down. The term "min" instructs you to choose the lower of two values separated by a comma.

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 following table shows the recommended vote allowances for each number of candidates:
{| class="wikitable"
|+ Number of votes allowed by the D21 method
|-
! Number of winners !! Number of candidates !! Number of votes
|-
| rowspan="3" | 1 || 2 || '''1'''
|-
| 3-8 || '''2'''
|-
| 9 or more || '''3'''
|-
| rowspan="3" | 2 || 3-5 || '''2'''
|-
| 6-11 || '''3'''
|-
| 12 or more || '''4'''
|-
| rowspan="3" | 3 || 4-7 || '''3'''
|-
| 8-14 || '''4'''
|-
| 15 or more || '''5'''
|-
| rowspan="4" | 4 || 5-9 || '''4'''
|-
| 10-17 || '''5'''
|-
| 18-27 || '''6'''
|-
| 28 or more || '''7'''
|-
| rowspan="4" | 5 || 6-11 || '''5'''
|-
| 12-20 || '''6'''
|-
| 21-30 || '''7'''
|-
| 32 or more || '''8'''
|}

Note that in an election where the number of candidates is only slightly greater than the number of winners, the D21 effect of more votes than winners is not possible. Instead, these cases use [[bloc voting]].

==Example==
{{Tenn_voting_example}} 

Suppose that voters were able to vote for their two top preferences - their own city and the next closest. The results would be as follows.

*Memphis: 42 total votes
*Nashville: 68 total votes
*Chattanooga: 58 total votes
*Knoxville: 32 total votes

'''Nashville wins.'''

==History & Development==
The D21 method was created in 2013 by Czech mathematician and anti-corruption activist Karel Janeček in response to corruption within the Czech political system.<ref>{{cite web |url=https://www.politico.eu/article/recalculating-democracy-representative-form-government-corruption/ |title=Recalculating democracy |last=Cunningham |first=Benjamin|date=2015-08-15 |website=Politico |access-date=2024-04-29}}</ref> Originally enabling voters to cast a minus vote in addition to the plus vote(s) and designed for two-seat districts, the method aimed to encourage voters to support moderate and consensual candidates across different parties and to penalize candidates associated with corruption. Subsequently, based on research conducted on the method, Janeček adjusted the algorithm to permit additional plus votes, favoring candidates capable of creating a consensual overlap among voters.

In 2016, Janeček founded Institute H21, initially named Institute for Democracy 21, with the objective to promote the D21 method and research it comparatively with other voting methods. The insights gained from research projects conducted by Institute H21 led to further refinements in the D21 voting method. The primary adaptation was maintaining the method as a predominantly plus vote system, with the inclusion of minus votes only in specific cases.

==Theory==
The D21 method is designed to address the limitations of traditional voting systems, such as FPTP or the two-round system, while maintaining simplicity for both voters and electoral administrations. It focuses solely on the strong preferences that voters already hold. The ability to cast multiple votes aims to provide a more accurate reflection of voter preferences and foster societal consensus.

The allocation of multiple plus votes is intended to produce the following effects:
*It favors candidates who are broadly acceptable to a larger segment of the electorate and disadvantages the most polarizing figures. This theoretically leads to the election of candidates who can garner wider support across diverse voter bases.
*It reduces strategic voting as the structure of D21 diminishes the incentives for it. Voters can support their genuine preferences without the fear of wasting their vote, as the impact of vote splitting is mitigated. Consequently, D21 is designed to facilitate the entry of new consensual candidates to the electoral contest, as the penalty for not being an already established candidate is lower.
*It weakens negative campaigning since candidates need to appeal not only to their primary voters but also to voters whose first choice are their opponents. Direct competition between candidates who are close to each other in the political spectrum is not tactical, as it lowers the chance of receiving second and third votes from the supporters of such candidates.

==Strategy==
The D21 method passes a form of the [[monotonicity criterion]], in that voting for a candidate never lowers that candidate's chance of winning. There is never a reason for a voter to tactically vote for a candidate X without voting for the candidate(s) he or she prefers to win over candidate X.
===Compromise===
Due to D21’s limiting of the number of votes available to voters, compromise may occur when a voter chooses to support a less-prefered candidate due to the unviability of their preferred candidate. Effective use of compromise hinges on voters being aware of the main frontrunners in the election. When the voter leaves their preferred candidate(s) out of their selections entirely, to instead support those with a higher chance of winning, decapitation (an extreme form of compromise) occurs. Decapitation is relatively common under FPTP, where some voters opt to cast their vote for a frontrunner to avoid wasting it on an non-viable candidate. By offering more but still a limited number of votes, D21 reduces the effect of compromise compared to FPTP, but trades a full elimination of the compromise strategy for the effect of reduced concerns about uninspiring candidates that can be associated with approval voting.

===Truncation===
[[Truncation]], and at its extreme bullet voting, occurs when voters refrain from voting for each candidate they sincerely support to instead vote for only a single favorite candidate. This strategy relies on voters being informed of the prospects of many candidates but can occur under D21.

==Minus vote==
There exists a variant of the D21 method that includes the minus vote. In such cases, voters can use a minus vote so long as they cast at least two plus votes. Minus votes carry the same absolute value as plus votes. It is not recommended to use minus votes in political elections in societies with major internal divisions among ethnic, religious or linguistic lines. With its effect, the minus vote is designed to enhance the effect of more votes.

Suppose we want to choose W winners out of C ≥ 4 candidates. A voting system follows the method described herein if:
*Each voter is allowed to cast up to P ≥ W (plus) plus votes and up to M (minus) minus votes, where P ≥ 2M (i.e., number of plus votes has to be at least twice as large as the number of minus votes; ideally, M ≐ P/3), and P should not exceed half the total number of candidates C/2.
*Each voter can cast no more than one vote for any candidate.
*Each vote has the same absolute weight (+1 or -1). The W candidates receiving the greatest net sum of all votes win. In case of parity between two candidates, the one with more plus votes wins.
*The voter can, but does not have to, use all available votes.

Hence, if there are two seats per district and a competing party nominates one or two candidates, voters may cast up to four plus votes and (if minus vote is employed) up to one minus vote, which they may distribute across all candidates in the district with only one vote per candidate. The same rules and rationale apply in single-winner elections.

==Case studies and surveys==
Since 2013, there have been numerous representative surveys conducted by Institute H21 as reported on their website,<ref>{{cite web |url=https://www.ih21.org/aktuality/pruzkum-2020-jak-by-dopadly-volby-s-vice-hlasy |title=Průzkum 2020 - Jak by dopadly volby s více hlasy |last=Oreský |first=Jan |date=2020-02-27 |website=Institut H21 |publisher=Institut H21 |access-date=2024-04-29}}</ref><ref>{{cite web |url=https://www.ih21.org/aktuality/simulovane-volby-metodou-d21 |title=Simulované volby metodou D21 |last=Oreský |first=Jan |date=2018-12-31 |website=Institut H21 |publisher=Institut H21 |access-date=2024-04-29}}</ref> researching the real-world effects of D21. Some of the studies compared D21 to other electoral methods in simulated elections, finding the effects of the methods to be similar to approval voting.<ref>{{cite web |url=https://www.ih21.org/aktuality/pavel-by-se-stal-prezidentem-kazdou-volebni-metodou-jak-je-to-s-dalsimi-kandidaty |title=Pavel by se stal prezidentem každou volební metodou. Jak je to s dalšími kandidáty? |last1=Oreský |first1=Jan |last2=Haase Formánková |first2=Zuzana |last3=Líbal |first3=Miroslav |date=2023-02-07 |website=Institut H21 |publisher=Institut H21 |access-date=2024-04-29}}</ref>

In 2018, Institute H21 launched a voting game called Prezident 21, enabling Czech citizens to vote for presidential candidates using the D21 method. This initiative aimed to increase public interest in the presidential election and the candidates' nomination process, as well as to initiate discussions on alternative voting systems including D21. Over 320,000 people participated in the game. 

==References==
<references />