User:BetterVotingAdvocacy/Negative vote-counting approach for pairwise counting: Difference between revisions
User:BetterVotingAdvocacy/Negative vote-counting approach for pairwise counting (view source)
Revision as of 23:38, 18 July 2020
, 3 years ago→Formula for counting the required number of marks to be made
Line 281:
!Regular approach
!Negative counting
!Semi-negative counting
|-
|1
|N-1 [0, '''1+''']
|1
|-
|2
|2N-3 [1, '''3+''']
|[1, '''3''']
▲|3
|[1, 2, '''3''']
|-
|3
|3N-6 [3, '''6+''']
|[3, '''6''']
|[2, 4, '''6]'''
|-
|4
|4N-10 [6, '''10+''']
|[6, '''10''']
|[4, 6, 8, 9, '''10''']
|-
|5
|5N-15
|[10, '''15''']
|[6, 9, 11, 13, 14, '''15''']
|}
Notes:
<nowiki>*</nowiki> The way in which last-ranked candidates are counted can change how many marks need to be made; 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 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 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.
▲
** 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).
*** If 3 candidates run, then in negative counting, the 2nd choice is counted with 2 additional marks, so at least ''3'' marks are made; but in semi-negative counting, only 1 additional mark needs to be made (because 2nd choice is ranked above only 1 candidate), so at least '''2''' marks are made.
*** If 4 candidates run, then in both approaches, '''3''' marks are made (2nd choice is ranked above 2 candidates and below 1 candidate).
** Note that regular pairwise counting's formula can be derived as follows: when only 1 candidate is ranked, the formula is N-1. When 2 candidates are ranked, it is the formula for when only 1 candidate is ranked plus another formula i.e. (N-1)+(N-2), which becomes 2N-3. For 3 candidates ranked, it is (2N-3)+(N-3), which is 3N-6, etc.
Note that negative counting is faster when voters rank only a few of all candidates, and potentially slower otherwise.
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