Collective transferable vote
Collective transferable vote (CTV) – a kind of proportional, preferential, transferable electoral system. A voter votes for candidates by indicating (marking) his preferences for them (marked with numbers: 1, 2 etc. - for candidates from the most to less preferred (marking the worst candidates is not necessary)).
Candidates for voting may be persons or other choices (e.g. projects, goals, methods). The voter may indicate any number of candidates on his ballot paper. The result of voting may be the election of one or more candidates (depending on the planned goals of this vote).
Introductory information
The idea, application and features of the collective transferable vote (CTV) system are similar to the Single transferable vote (STV) system (in the case of multi-member constituencies) and to the Alternative Vote (AV) system (in the case of single-member constituencies).
About criteria and features of electoral systems
There are various criteria and features of electoral systems by which these systems can be assessed and compared.
Among the positive (preferred, pro-democratic) features of electoral systems, there is monotonicity, or actually two her kinds:.[1]
- monotonicity in the sense of lack of negative reactivity: i.e. that always, in each vote, improving the ballot papers (in the second version of this vote) in favour of the candidate (previously the winner) cannot cause him to lose; in this monotonicity, it is about non-decreasing of the function that associates a candidate and set of ballots, to the electoral result of the candidate;[2]
- monotonicity due to the size of a compact coalition: i.e. that in each vote, a smaller compact coalition cannot elect more candidates than a larger compact coalition (a "compact coalition" is a group of voters that the first votes for the entire group of "their" candidates, i.e., each such voter votes in the first place for all these "his" candidates, and only in further preferences for some others).
An empirical method of assessing and comparing electoral systems
In order to compare (evaluate) the two electoral systems, it would be necessary to carry out the voting at the same time by both methods in the same group of voters, then to announce the results of both votes in this group and, if these results were different, conduct a third vote on the matter: "Which vote was the better result?" with three possible answers:[lower-alpha 1]
- according to the first method,
- according to the second method,
- abstention (because: "both were the same" or "difficult to evaluate").
Similarities and differences between the collective transferable vote and other transferable vote systems
Common ideas
The common basis for the electoral systems of the transferable vote (collective transferable vote, single transferable vote, alternative vote) are two general ideas:
- voting: the voter on the ballot paper may indicate many candidates by preference numbers for them (No. 1, 2 etc.), i.e. taking into account which candidates he would like to select more and which less;
- calculation: when calculating the election result, in order to consider the next candidate as elected, an appropriate, required number of ballots (with him) is needed, then allocated to that candidate - and this number of ballots is to be excluded from further calculations (i.e. when selecting candidates for subsequent places), and the remaining number of ballots may then be allocated to the next selected candidates.
Differences between STV and CTV
In STV systems, the voter has certain limitations, such as these, that he cannot evaluate two candidates in the same way, i.e. he must not assign the same preference number to different candidates. There are no such restrictions in CTV.
The STV and AV systems are non-monotonic and in the sense reactivity, and in the sense coalition.[1] On the other hand, the CTV system is monotonic in both senses.
Comparison based on actual elections
In 2002, STV elections were held in Dublin (Ireland) and completed ballots were released from the constituencies:[5] North (almost 44 thousand ballots and 4 candidates elected from 12) and West (almost 30 thousand ballots and 3 candidates elected from 9). This makes it possible to calculate the results of these elections also by other methods (for testing and comparison purposes).
According to the calculated result of these elections using the CTV method:[6] In the West constituency, the same candidates were elected (and in the same order), whereas in the North constituency three of the same candidates were elected, and one other (Kennedy, Michael, F.F. instead of Wright, G.V., F.F.). The programme that calculated the result of those votes using the CTV method,[7] used only the first two preferences (i.e. the most important ones - the others were not needed for the calculations (for this vote)), whereas in the STV method (used then, in 2002) all preferences were used in the calculation. When calculating using the STV method, the quota was reduced (i.e. the number of ballots required to elect one candidate), whereas in the CTV method the quota has not been reduced.
When calculating the election results using CTV and STV methods, a fairly significant difference in the two parameters has emerged, i.e. in the amount of preferences used and in the reduction of the quota. This would probably be most likely due to that the CTV method does not lose information about the ballots, while in the basic STV method used in Dublin, in the next stages of calculations, information about ballots is lost, which could result in using too large preference numbers (thus less important for voters) in the calculations, and reducing the "quota", and thus the worse quality of such an algorithm (its calculation results).
Ballot paper
On the ballot paper, the preferences of various candidates are marked by the voter by entering:
- in STV: usually successive preference numbers (in one column), but a multi-column method is also used (similar in appearance to CTV); a voter is not allowed to indicate the same preference for two candidates;
- in CTV: (e.g.) crosses in the columns of selected preferences; candidates' preference numbers may be repeated and omitted.
The number of preferences (which can or should be used by the voter) on ballot papers in STV systems usually equals the number of candidates. However, in the CTV system number of preferences can be any - it seems that a few preferences (e.g. 5 or 9 or somewhat more) should usually be enough for a voter, almost regardless of the number of candidates.
Problems during voting:
- in STV: if the voter evaluates two candidates exactly the same, he must not mark his true preferences for them on the ballot paper, otherwise the voter will be deprived of the right to vote by invalidating his ballot;[lower-alpha 2]
- in CTV: If the voter has marked more than one preference for a candidate, then, for his ballot to be correct and valid, this error should be automatically corrected (when reading the ballot paper) by leaving only one of these preferences selected - the principle of recognising the first of these preferences seems most natural here.[lower-alpha 3]
An example of a completed CTV voting ballot paper:
| No | Candidates | Preferences | ||||
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | ||
| 1 | First Candidate | X | ||||
| 2 | Second Candidate | |||||
| 3 | Third Candidate | X | ||||
| 4 | Fourth Candidate | X | ||||
| 5 | Fifth Candidate | X | ||||
| 6 | Sixth Candidate | |||||
The above ballot paper in the STV record looks like this:
| No | Candidates | Preference |
|---|---|---|
| 1 | First Candidate | 2 |
| 2 | Second Candidate | |
| 3 | Third Candidate | 1 |
| 4 | Fourth Candidate | 1 |
| 5 | Fifth Candidate | 5 |
| 6 | Sixth Candidate |
Method of calculating the voting result
The basic varieties of STV can use a fairly simple calculation method, with the manual transfer of completed ballot papers into stacks corresponding to the candidates. The CTV system is not adapted to this procedure. For CTV only in the case of the single-member version, the 'manual' calculation is simple: to calculate the result, it is enough (e.g. on a piece of paper) to count the ballot papers for each candidate, separately for each preference.
In contrast, advanced STV varieties (e.g. fractional (Meek, Warren)) and multi-member CTV usually requires the use of a computer.
The ideas of calculating the voting result in the STV and CTV systems differ in that once a candidate has been elected (by finding the number of ballots required for this purpose):
- in STV: These ballots (this number) are removed from further calculations (which, however, results in the loss of information about their content at further stages of the calculations (in basic STV varieties)), while the remaining, redundant ballots are transferred to other candidates;
- in CTV: this number of ballots is removed from further calculations (i.e. only this number, but not the ballots themselves - so that at each stage of the calculation the information about the contents of all the ballots is preserved and used); while for the next candidates, the number of ballots for them is calculated from the set of all ballots, but taking into account the earlier removal of their numbers.
When calculating the result of voting in STV in some of its varieties (e.g. basic) only integers are used, and other varieties of STV also use fractional numbers. However, in CTV (in all its varieties), only integers are used in the calculation.
Basic algorithm of CTV for calculating the voting result
In the following algorithms (to simplify them), ties are not resolved and the number of ballots required to select a candidate (the quota) is not reduced.
Definitional note: Standard value of the required number of ballots = 1 + IntegerPart(NumberOfBallots / (NumberOfSeats + 1))
General CTV algorithm
REPEAT
FindTheSmallestPreferenceNumberToWhichAnyCandidateNotElectedReachesRequiredNumberOfBallotsFree
RecognizeElectedForNextSeat(CandidateNotElectedWithTheMostBallotsFreeToThisPreference)
UNTIL AllSeatsAreAlreadyOccupied
Def.1. Number of ballots free to current preference (=NoPref) for a candidate not yet elected (=Cand):
NumberOfBallotsFree(Cand, NoPref) :=
Minimum( { IIF( NumberOfBallotsFromSubsetButWithoutCand(Ssec,Cand,NoPref) ≥
NumberOfBallotsAllocatedToSubset(Ssec),
NumberOfBallotsWithCand(Cand,NoPref),
NumberOfBallotsWithCand(Cand,NoPref) +
NumberOfBallotsFromSubsetButWithoutCand(Ssec,Cand,NoPref) -
NumberOfBallotsAllocatedToSubset(Ssec) )
: Ssec ⊆ SetOfElectedCandidates } )
Def.2. IIF(condition, w1, w2) = w1, if the condition is true; else = w2
Def.3. NumberOfBallotsFromSubsetButWithoutCand(Ssec,Cand,NoPref) := [ Number of ballots in which:
in any preference from No 1 to NoPref any candidate from Ssec is marked,
but there is no Cand on this ballot (in these preferences) ]
Def.4. NumberOfBallotsWithCand(Cand,NoPref) := [ Number of ballots in which:
in any preference from No 1 to NoPref is marked Cand ]
Def.5. NumberOfBallotsAllocatedToSubset(Ssec) := [ The sum of the number of ballots allocated
(i.e.=Required number of ballots (i.e.=quota) (then))
for candidates from the Ssec subset
when considering them as elected ]
The algorithm for electing the first candidate
The idea of calculating the CTV voting result in the case of electing the first candidate: the number of ballots (for each candidate separately) is counted for successive preferences (together, from the first), up to the preference in which any candidate reaches the Required Number of Ballots (meaning: at least such a number). If more than 1 candidate reaches this number, the one with the highest number of ballots is considered elected.
The general algorithm of CTV in the case of electing the first candidate reduces to the algorithm:
FindSmallestPreferenceNumberToWhichAnyCandidateReachesRequiredNumberOfBallots RecognizeElected(CandidateWithTheMostBallotsToThisPreference)
And in another notation (more detailed):
- RequiredNumberOfBallots := 1 + IntegerPart( NumberOfBallots / (NumberOfSeats + 1) ) .
- The current preference is 1.
- For subsequent candidates (separately), ballots with them are counted in preferences from 1st to the current.
- It is checked whether any candidate has reached the RequiredNumberOfBallots:
- if Yes, the candidate with the greatest number of these ballots is considered elected;
- if No, increment the current preference number by 1 and return to item 3; (however, if this was the last preference, the candidate with the most ballots is considered the elected one).
The algorithm for electing the second candidate
The general algorithm of CTV in the case of electing the second candidate is reduced to a similar algorithm as the above one, except that instead of "ballots" one should count "ballots free", so:
FindSmallestPreferenceNumberToWhichAnyCandidateNotElectedReachesRequiredNumberOfBallotsFree RecognizeElected(CandidateNotElectedWithTheMostBallotsFreeToThisPreference)
Def. Number of ballots free for Cand (in preferences from No 1 to current):
IF NumberOfBallotsWithElectedCandidateButWithoutCand ≥ RequiredNumberOfBallots
THEN NumberOfBallotsFreeOfCand = NumberOfBallotsOfCand
ELSE NumberOfBallotsFreeOfCand = NumberOfBallotsOfCand +
NumberOfBallotsOfElectedCandidateButWithoutCand - RequiredNumberOfBallots
Explanation of the idea of calculating in CTV
In the above formula, one of the basic ideas of the CTV is shown. The example below may make it easier to understand.
Situation: candidate Ce has already been elected and it should now be checked whether candidate Cn could also be considered elected, that is, whether it has reached the required number of ballots. The number of ballots for candidates is counted from preference number 1 to some "current" one.
NBCe = number of ballots with Ce, but no Cn NBCn = number of ballots with Cn, but no Ce NBCen = number of ballots with Ce and Cn on them
In this situation, for Cn to reach [required number of ballots free], the Cw must have Required Number of ballots, consisting of NBCe and possibly some part of NBCen. If the remainder of NBCen + NBCn ≥ required number of ballots, then Cn has reached the required number of ballots. More formally, it should be done like this: first check that [the number of ballots free for Cn relative to the set {} (i.e. Ø)] ≥ [required number of ballots], and then check whether [number of Kn ballots free relative to the set {Kw}] ≥ [required number of ballots].
The same applies to the election of the next candidates, but instead of Ce there is a "set of candidates already elected" = Ce = {Ce1,Ce2,...}, and Cs is a subset of Ce.
NBCs = number of ballots with any candidate from Cs, but no Cn NBCn = number of ballots with Cn, but no candidate from Cs NBCsn = number of ballots with any candidate from Cs and with Cn on them
To calculate for Cn [number of ballots free], you first need to calculate [number of ballots free for Cn relative to Cs]. In this case, Cs must have [required number of ballots for Cs] = #Cs[lower-alpha 4] * [required number of ballots], consisting of NBCs and possibly some part of NBCsn. Then [the remaining part of NBCsn] + NBCn = [number of ballots free for Cn relative to Cs].
Then [number of ballots free] for Cn = the smallest value [number of ballots free for Cn relative to Cs], among all Cs ⊆ Ce.
Example of calculating the CTV voting result
Voting example:[8]
3 candidates: A, B, C; 2 seats; 200 ballots: ABC x160 i.e. 160 ballots with candidates A, B, C successively in preferences 1, 2, 3 BAC x10 CBA x30 The required number of ballots to select a candidate = 200 /(2+1) + 1 = 67. Calculation of the voting result: Search for seat 1: Preference 1: Numbers of ballots free for candidates: A: 160 B: 10 C: 30 Candidate A has been selected (number of ballots used in this = 67). Search for place 2: Preference 1: Numbers of ballots free for candidates: B: 10 C: 30 (not enough) Search for place 2: Preference 2 (that is, from 1 to 2): Numbers of ballots free for candidates: B: 133 = 160 (from ABC from pref. 1..2) - 67 (used by A) + 10 (from BAC from pref. 1..2) + 30 (from CBA from pref. 1..2) C: 30 Candidate B has been selected (number of ballots used for this = 67). ( And the remaining number of ballots free in the preference 3 (i.e. 1..3) for candidate C = 66 )
Tie-breaking
When calculating voting results, a tie (conflict) is a situation in which more than 1 candidate reaches the same number of ballots free, greater than or equal to the required number of ballots.
When tie in the CTV algorithm, the sequence of the tie-breaking methods is as follows:[9][10]
- Half[lower-alpha 5] - the purpose of this (initial) method is to optimize both directional methods[lower-alpha 6]; among the various varieties of this method, only the simplified version is independent of the choice of directional method[lower-alpha 7];
- Directional - backward[lower-alpha 8] or progressive[lower-alpha 9];
- List[lower-alpha 5] - e.g. full[lower-alpha 10], simplified[lower-alpha 11] or extended[lower-alpha 12];
- Group - simultaneous election of a group of tied candidates;
- Final[lower-alpha 5] - e.g. by lot or by the greatest number of signatures for a candidate.
Monotonicity of the CTV algorithm
The CTV algorithm is monotonic (in both senses). The rationale for this monotonicity[lower-alpha 13]:
- When electing the first candidate, monotonicity in the sense of lack of negative reactivity, results from the fact that the election of the candidate is determined by the number of ballots counted for him from the first to the next preferences (in total), so improving him any ballot (preference) cannot reduce his value (i.e. election result), and therefore cannot worsen his election result.
- Completion of the inductive justification started above: if a certain number of candidates have already been elected, then the above type of justification would also apply to the election of the next candidate.
- Another argument: when electing a candidate for any seat, the non-decreasing of the CTV algorithm seems to be fulfilled, because each part of the algorithm (each element of the formula (function)) is non-decreasing.[11]
- The CTV algorithm is also monotonic in the sense of a coalition, because each voter would place all the candidates of his coalition already in the first preference, and the total number of their ballots determines the number of seats granted.
Notes
- Actually a series of such votes (in different groups and on different topics) should be held. It would not be advisable to compare more than two electoral systems at the same time, if this could cause a dispute as to the choice of the last voting system (the one evaluating the previous ones) - e.g. CTV or FPTP (they are compatible in the case of only two options)
- although this is a principle somewhat incompatible with the idea of democracy
- that's the most important preference indicated by the voter be used in the calculation of the result of the vote
- card(Cs)
- not necessary
- as if their combination reducing the opposition of their directions
- one-step method - i.e. comparing the number of ballots free (counting them in total from preference 1): in the first preference found, in which the best tied candidate reaches half of the required number of ballots
- comparing the number of ballots free (counting them in total from preference no. 1): from the current preference to ever smaller preference numbers
- comparing the number of ballots free: from preference no. 1 to ever larger numbers of preferences
- comparing the values of successive elements of tied-candidates' lists; it is a non-decreasing list, each element of which corresponds to one of the subsets of the set of elected candidates; the value of a list element is equal to the number of candidate's ballots free in relation to the subset; e.g. the value of the first element of such a list is equal to the number of ballots free of the candidate
- comparing values of only one element from each list: the one concerning the empty set; its value is equal to the number of ballots of the candidate
- compare lists not only to the current preferences (from no. 1), but also to others
- or more precisely: not-decreasing
References
- Paweł Przewłocki. "Single Transferable Vote - Features" (in Polish). Retrieved 2017-12-30.
- "Monotonicity of the CTV algorithm" (in Polish). Retrieved 2018-01-03.
- "Former description of the CTV (ZGP) system (in an already deleted portal)" (in Polish). Retrieved 2018-01-06.
- "Former description of the CTV (ZGP) system" (in Polish). Retrieved 2018-08-14.
- "Dublin County Returning Officer, General Election May 2002 - 8 links to DOC and ZIP files". Archived from the original on 2006-01-03. Retrieved 2018-09-08.
- "Comparison of STV and CTV methods on the example of the actual elections in Dublin in 2002" (in Polish). Retrieved 2018-09-08.
- "File (zip): ZGP program (in MS ACCESS) calculating voting results using the CTV method (8MB)" (in Polish). Retrieved 2018-09-08.
- dr Paweł Przewłocki (2015). Single Transferable Vote (STV). Instytut Spraw Obywatelskich. p. 11. ISBN 978-83-936035-7-2.
- "Rozstrzyganie konfliktów" (in Polish). Retrieved 2017-12-30.
- "Rozstrzyganie remisów" (in Polish). Retrieved 2018-09-08.
- "Monotoniczność - Negatywna Reaktywność - Porównanie wyników obliczanych różnymi metodami" (in Polish). Retrieved 2017-12-30.