User:BetterVotingAdvocacy/Negative vote-counting approach for pairwise counting: Difference between revisions

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****If this ballot also ranked a 2nd choice, the negative approach requires indicating that the 2nd choice is also "preferred to themselves", and preferred less than the 1st choice, while the regular approach requires indicating that the 2nd choice is preferred over each of 8 candidates i.e. all of the 10 candidates that aren't the 1st choice or 2nd choice.
****If this ballot also ranked a 2nd choice, the negative approach requires indicating that the 2nd choice is also "preferred to themselves", and preferred less than the 1st choice, while the regular approach requires indicating that the 2nd choice is preferred over each of 8 candidates i.e. all of the 10 candidates that aren't the 1st choice or 2nd choice.


Here are some examples for the first numbers in each series:
Here are some examples for the first numbers in each series (with the upper bound bolded):
{| class="wikitable"
{| class="wikitable"
|+Number of marks required in each vote-counting approach ''when equal-ranking isn't allowed'' ("N" refers to total number of non-write-in candidates in election)
|+Number of marks required in each vote-counting approach ''when equal-ranking isn't allowed'' ("N" refers to total number of non-write-in candidates in election)
!Number of candidates ranked
!Number of candidates ranked ("R")
!Regular approach
!Regular approach
!Negative counting
!Negative counting
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|3N-6 [3, '''6+''']
|3N-6 [3, '''6+''']
|[3, '''6''']
|[3, '''6''']
|[2, 4, '''6]'''
|[2, 4, 5, '''6]'''
|-
|-
|4
|4
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Notes:
Notes:


* For regular counting, a lower bound has been provided (for when the number of candidates ranked is equal to the number of candidates running in the election), and since there is no upper bound, instead the number of marks required when the number of candidates running is one higher than the number of candidates ranked by the voter is provided.
* For regular counting, a lower bound has been provided (when N=R), and since there is no upper bound, instead the number of marks required when N is one higher than R is provided.
* For both negative and semi-negative counting, the numbers provided are a series that include both lower and upper bounds on the number of marks that have to be made, depending on how many candidates are running (starting from the number of candidates ranked by the voter, and sequentially increasing by one, up to twice that), and how last-ranked candidates are counted.
* For both negative and semi-negative counting, the numbers provided are a series that include both lower and upper bounds on the number of marks that have to be made, depending on N (which starts at R, and sequentially increases by one, up to twice that i.e. 2R), and how last-ranked candidates are counted.
** For counting last-ranked candidates, see [[#Dealing with last-place candidates]]. If no marks are made for them, then a ballot that ranks all candidates requires the same number of marks as a ballot that ranks all candidates except the last-ranked candidate(s). For example, a ballot that ranks 5 candidates when there are 5 candidates total can be thought of as ranking the top 4 candidates, and leaving the 5th candidate unranked.
** For counting last-ranked candidates, see [[#Dealing with last-place candidates]]. If no marks are made for them, then a ballot that ranks all candidates requires the same number of marks as a ballot that ranks at least one less candidate i.e. all candidates except the last-ranked candidate(s). For example, a ballot that ranks 5 candidates when there are 5 candidates total can be thought of as ranking the top 4 candidates, and leaving the 5th candidate unranked.
** Semi-negative counting's performance is always better than or equal to negative counting's performance when the same number of candidates run and the same number of candidates are ranked in both approaches. As an example for a voter who ranks 2 candidates:
** Semi-negative counting's performance is always better than or equal to negative counting's performance when the same number of candidates run and the same number of candidates are ranked in both approaches. As an example for a voter who ranks 2 candidates:
*** When only those 2 candidates run, then in both approaches, at least '''1''' mark needs to be made (to count the 1st choice; the 2nd choice can be skipped because they are ranked last).
*** When only those 2 candidates run, then in both approaches, at least '''1''' mark needs to be made (to count the 1st choice; the 2nd choice can be skipped because they are ranked last).
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* But negative counting only requires 3 marks: 1 each for A and B to indicate they are preferred in every matchup, and 1 to indicate that this isn't the case for B>A.
* But negative counting only requires 3 marks: 1 each for A and B to indicate they are preferred in every matchup, and 1 to indicate that this isn't the case for B>A.


==== Regular pairwise counting but done by counting first choices separately ====
Note that regular pairwise counting can have its required number of marks reduced without using any negative numbers by counting 1st choices separately from all other ranks; see the section above [[Pairwise counting#Uses for first choice information]]. The regular approach requires [(number of candidates)-2] less marks if using this modification, though when voters equally rank multiple candidates 1st, then whether or not the modification is applied to count all of those 1st choices separately or not can make a significant difference in overall number of marks.
Note that regular pairwise counting can have its required number of marks reduced, without using any negative numbers, by counting 1st choices separately from all other ranks; see the section above [[Pairwise counting#Uses for first choice information]]. The regular approach requires [(number of candidates)-2] less marks if using this modification i.e. a voter who ranks 2 candidates sequentially when there are 10 candidates only requires 1+8=9 marks rather than 9+8=17 marks to have their ballot counted, an [(10)-2=] 8-mark difference.

However, when voters equally rank multiple candidates 1st, then whether or not the modification is still applied can make a significant difference in overall number of marks.


=== Election example comparisons ===
=== Election example comparisons ===
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