Difference between revisions of "Ballot"

Jump to navigation Jump to search
5,973 bytes removed ,  1 year ago
[[Pairwise counting]] can be done on all four ballot types, though ranked and rated ballots offer the most information for this purpose.
 
NoteA thatgeneralization ranked,of rated,all andfour Approvalmain ballotsballot cantypes bementioned generalizedabove intois onethe [[rated pairwise preference ballot]]; type:it allowing theallows voters to express theirsubmit rated preferences infor every [[head-to-head matchup]]. Withbetween somethe simplification,candidates thiswhile canfollowing becertain visualized[[transitivity]] as (example for a single voter, with 6 candidates A through F):<blockquote>A>B=C>Drequirements.
{| class="wikitable"
|+Margins-based rated matchups table
!
|1st
|2nd
|3rd
|Last
|-
|1st
| ---
|20%
|60%
|75%
|-
|2nd
|0%
| ---
|50%
|60%
|-
|3rd
|0%
|0%
| ---
|40%
|-
|Last
|0%
|0%
|0%
| ---
|}
</blockquote>So this voter expressed a ranked preference, and also expressed, in the head-to-head matchup table, their strength of preference in every head-to-head matchup between each of the candidates in each rank. "1st" here refers to "1st choice", and "20%" here can be read as "20% of a vote" or "20% support", equivalent to 0.2 votes (or a 2 out of 10 on a rated ballot). This can be read as, for example, "1st>3rd" referring to the voter's support for A>D, and "2nd>last" referring to the voter's support for either B or C over all candidates they prefer less than D. This table captures the margin in strength of preference; it is instead possible to capture the strength of preference in a way that captures both margins and "[[winning votes]]"-relevant information (i.e. the voter's rated preference for both candidates in the matchup) by, instead of writing 20% for the more-preferred candidate and 0% for the less-preferred candidate, writing, say, 80% and 60% respectively, if that's what the voter's actual preference was.
 
Certain minimum requirements for [[transitivity]] are apparent simply from looking at this table; for example, since the voter expressed a 50% difference in support for their 2nd choice>3rd choice, it wouldn't have made sense for them to express less than 50% support for their 1st choice>3rd choice. Another example is that, because they expressed 20% support for 1st>2nd, they must have had at least 20% support for 1st>3rd as well. To put it succinctly, for whatever degree of support a voter indicates in a given pairwise matchup cell, they must indicate at least that much support in all cells above, to the right, or to the upper-right of this cell. Thus, one way of collecting this pairwise information in a digital interface is to ask voters to start out by filling out the pairwise comparison between "Last choice>1st choice" (which is in the very bottom-left), and then accordingly allow the voter to fill out match-ups going up and/or right while imposing the required transitivity constraints. See [[Order theory#Strength of preference]] for further notes on transitivity in this framework.
 
This approach is a generalization of the above 3 ballot types in the sense that if every voter expresses the same margins-based or winning votes-based preference for each candidate in each head-to-head matchup as they would if they were rating them on a scale with all other candidates (i.e. a voter who would give a candidate 80% support on a rated ballot's scale would give that candidate a 30% margin in a head-to-head matchup against a candidate they'd rate a 50% on the same scale), then it reduces to a rated ballot (with the same logic following for an Approval ballot, since an Approval ballot is a restricted form of a rated ballot), and if every voter expresses a maximal preference for their preferred candidate in each matchup, then it reduces to a ranked ballot. See [[Pairwise counting#Cardinal methods]] and [[Order theory#Strength of preference]] for more information on this ballot type.
 
One particular, easier approach to implementing this generalized ballot type is to allow the voters to score the candidates on a scale, and then allow them to check a box indicating whether they have rated or ranked preferences. If using [[pairwise counting]], this can be counted by, for voters who indicate rated preferences, collecting their scores directly, and for those with ranked preferences, doing regular pairwise counting. For example:
{| class="wikitable"
|+
!
!A
!B
!C
|-
|A
|''15''
|3
|4
|-
|B
|2
|''12''
|5
|-
|C
|1
|7
|''13''
|}
This can be interpreted as a regular [[Pairwise comparison matrix]] except that the score/points total for each candidate is recorded in their cell (i.e. A>A shows A's score). Supposing the max score is 5 and min score is 0, in this example 15/5=3 votes would be added to every [[Head-to-head matchup]] in favor of A, 12/5=2.4 votes in favor of B's matchups, and 7/5=1.4 votes in favor of C's matchups. So the final table would be:
{| class="wikitable"
|+
!
!A
!B
!C
|-
|A
| ---
|6
|7
|-
|B
|4.4
| ---
|7.4
|-
|C
|2.4
|8.4
| ---
|}
If using [[:Category:Condorcet-cardinal hybrid methods|Category:Condorcet-cardinal hybrid methods]], it is likely best to store the scores of each voter in one of two separate data values in each candidate's cells i.e. if a voter expressed a rated preference, put their score for a candidate only in the "rated preference" value, but if they expressed a ranked preference, put the score only in the "score for candidate" value. So, for example, a voter expressing a ranked preference who scored candidate A a 5 would be treated as giving A 0 points in the "rated preference" data value but 5 points in the "score for candidate" data value (which could be read as "0, 5" in the A>A cell). This would then be tabulated by giving each candidate as many points as they have in the rated preference data value i.e. a candidate with 51 points in the rated preference value and 37 in the score value would have those values treated such that, supposing a max score of 5, 51/5=10.2 votes would be added to all of their pairwise matchups in favor of them, and 51 points would be added to their score value to find that they have 88 points overall. This actually is easier to count than having to do pairwise counting with only ranked ballots, because for each voter who expresses a rated rather than a ranked preference, their support for a candidate in a head-to-head matchup can be summarized as one data value (the score for the candidate) rather than up to (number of candidates - 1) data values (i.e. the fact that they give that candidate 1 vote in each head-to-head matchup against a lower-ranked candidate).
 
== External links ==
 
Cookies help us deliver our services. By using our services, you agree to our use of cookies.

Navigation menu