Collective transferable vote

Short description: Kind of proportional, preferential, transferable electoral system

Collective transferable vote (CTV) – a kind of proportional, preferential, transferable, monotonic 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:.

• 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;

• 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:

• according to the first method,
• according to the second method,
• abstention (because: "both were the same" or "difficult to evaluate").

History of CTV

The "Collective Transferable Vote" is a relatively new system; the earliest description of it on the Internet is from 2013 - on the Polish Internet: "Zbiorczy Głos Przechodni" (ZGP).

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 elect 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 electing candidates for subsequent places), and the remaining number of ballots may then be allocated to the next elected 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. On the other hand, the CTV system is monotonic in both senses.

Comparison based on actual elections in Dublin 2002

In 2002, STV elections were held in Dublin (Ireland) and completed ballots were released from the constituencies: 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).

Calculation of the voting result using the CTV sequential method

According to the calculated result of these elections using the CTV method: 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 sequential method, 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).

Calculation of the voting result using the CTV global method

In the West constituency, the set of numbers of candidates who have reached the standard quota to preference No. 2, = {2, 4, 5} (i.e. "Burton,Joan,Lab", "Higgins,Joe,SP", "Lenihan,Brian,FF" ). So it is the only group and NOBFC( {2, 4, 5}, 2 ) = 8214. The same group of candidates also won according to the CTV sequential method (with the standard quota = 7498). This number 8214 is also the maximum value of the quota at which these candidates win to the 2nd preference according to the CTV sequential method.

In the North constituency, the set of numbers of candidates who have reached the standard quota to preference No. 2, = {2, 4, 6, 9, 10, 12}. There are therefore 15 groups (sets of 4 elements) of these candidates in the North constituency.

The highest value of NOBFC was achieved by group 11: NOBFC( {4, 6, 9, 10}, 2) = 9170, so it was this group of candidates that won according to the CTV global method. The same group of candidates also won according to the CTV sequential method (with the standard quota = 8789). This number 9170 is also the maximum value of the quota at which these candidates win to the 2nd preference according to the CTV sequential method.

The second next result was achieved by group 14: NOBFC ( {4, 9, 10, 12}, 2 ) = 9105. It was these candidates who won according to the STV method in 2002.

The last, worst result (=7270, below the standard quota) was obtained by 3 groups: group 3 (={2, 4, 6, 12}), group 12 (={4, 6, 9, 12}) and group 13 (={4, 6, 10, 12}).

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. Preliminary observations indicate that in large votings only a small number of preferences is used to calculate the result of the CTV method voting result.

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;
• 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.

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 quota is not reduced.

Definitional note: Standard value of the quota (=required number of ballots) = 1 + IntegerPart(NumberOfBallots / (NumberOfSeats + 1))

General CTV sequential algorithm

REPEAT
    FindTheSmallestPreferenceNumberToWhichAnyCandidateNotElectedReachesRequiredNumberOfBallotsFree
    RecognizeElectedForNextSeat(CandidateNotElectedWithTheMostBallotsFreeToThisPreference)
UNTIL AllSeatsAreAlreadyOccupied 

Def.1. Number of ballots free to current preference (=CurPref) for a candidate not yet elected (=Cand):
       NumberOfBallotsFree(Cand, CurPref) := 
       Minimum( { IIF( NumberOfBallotsFromSubsetButWithoutCand(SsEC, Cand, CurPref) ≥ 
                       NumberOfBallotsAllocatedToSubset(SsEC), 
                                             NumberOfBallotsWithCand(Cand, CurPref), 
                                             NumberOfBallotsWithCand(Cand, CurPref) + 
                       NumberOfBallotsFromSubsetButWithoutCand(SsEC, Cand, CurPref) - 
                                             NumberOfBallotsAllocatedToSubset(SsEC)  ) 
                  : SsEC ⊆ SetOfElectedCandidates } )  

Def.2. IIF(condition, w1, w2) = w1, if the condition is true; else = w2 

Def.3. NumberOfBallotsFromSubsetButWithoutCand(SsEC, Cand, CurPref) := [ Number of ballots in which: 
                            in any preference from No 1 to CurPref any candidate from SsEC is marked, 
                            but there is no Cand on this ballot (in these preferences) ]

Def.4. NumberOfBallotsWithCand(Cand, CurPref) := [ Number of ballots in which:  
                                                 in any preference from No 1 to CurPref 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 ]

More traditional notation of the algorithm

FUNCTION NumberOfBallotsFree(   Cand,  // Candidate number, for which his number of ballots free is to be calculated
                             CurPref   // the Current Preference, to which (from 1.) ballots are to be counted
                            )    

   // NEC == Number of Elected Candidates
   // SEC == Set of Elected Candidates
   // SsEC is a natural number that denotes a subset of set SEC; 
   //      in SsEC (in binary notation) the NEC of the next least significant bits    
   //      stands for NEC of subsequent candidates from SEC (0 == none, 1 == there is in subset)
   //      (note: this is a different numbering than the 'normal' candidate numbers)    

   NoOfBallotsFreeCand = 2^30 // large number

   FOR SsEC = 0 TO 2^NEC - 1
      // Number of ballots in which any candidate from the SsEC is marked, but Cand is not there   
      NoOfBallotsWithSubsetButWithoutCand = CountNoOfBallotsWithSubsetButWithoutCand(SsEC, Cand, CurPref)

      NoOfBallotsWithCand = CountNoOfBallotsWithCand(Cand, CurPref)

      // Sum of No of ballots allocated to candidates from SsEC at the time of their election (=sum of their quotas)
      NoOfBallotsAllocatedToSubset = CountNoOfBallotsAllocatedToSubset(SsEC)

      IF NoOfBallotsWithSubsetButWithoutCand ≥ NoOfBallotsAllocatedToSubset  THEN
         NoOfBallotsFreeOfCandRelativeToSubset = NoOfBallotsWithCand
      ELSE
         NoOfBallotsFreeOfCandRelativeToSubset = NoOfBallotsWithCand +
                                                 NoOfBallotsWithSubsetButWithoutCand - 
                                                 NoOfBallotsAllocatedToSubset
      ENDIF

      NoOfBallotsFreeCand = Min(NoOfBallotsFreeCand, NoOfBallotsFreeOfCandRelativeToSubset) 
   ENDFOR SsEC

   RESULT = NoOfBallotsFreeCand
ENDFUNCTION

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):

1. RequiredNumberOfBallots := 1 + IntegerPart( NumberOfBallots / (NumberOfSeats + 1) ) .
2. The current preference is 1.
3. For subsequent candidates (separately), ballots with them are counted in preferences from 1st to the current.
4. 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 Cn ballots free relative to the set {Ce}] ≥ [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 * [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:

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 quota = required number of ballots to elect 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 elected (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 elected (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 is a situation in which more than 1 candidate reaches the same number of ballots free, greater than or equal to the quota (the required number of ballots).

In the case of a tie in the CTV algorithm, the basic standard method of tie-breaking consists of 4 stages performed in the following order:

1. Basic directional - backward or forward;
2. List simplified directional;
3. Group - simultaneous election of the entire group of tied candidates;
4. Final - by lot.

In directional tie-breaking there are two, as if symmetrical, directions (backward and forward), but only one of them may be used. The forward direction favors an advantage due to initial preferences when calculating the number of ballots free, while the backward direction favors the advantage due to the end preferences used in calculating the number of ballots free.

It is also possible to use an additional 'Simplified Half' stage which is a method somewhat intermediate between the backward directional method and the forward directional method. If it were to be used, it should be at the beginning (before basic directional).

In addition to the above methods, there are also more complicated variations of these methods, but it seems that they would rather not be so necessary or needed.

Algorithms in Tie-Breaking

When a tie occurs, a tie group is formed (there is more than one candidate in it). Each subsequent stage or sub-stage of the tie-breaking is an attempt to reduce this group (i.e. the reduction of the group changed during the previous eventual reduction).

Note: the calculation of the number of ballots for a certain preference described below should be understood as the calculation of the number of ballots from preference 1 to that preference.

Types (stages) of tie-breaking in the CTV:

1. Basic directional tie-breaking

This is the most important tie-breaking. In this type of tie-breaking the number of ballots free of candidates from the tie group should be compared. The subsequent sub-steps of basic directional backward tie-breaking concern the following preference numbers (as in the FOR statement): CurrentPreference - 1 DOWNTO 1, CurrentPreference + 1 TO NumberOfPreferences. Whereas in the forward: 1 TO CurrentPreference - 1, CurrentPreference + 1 TO NumberOfPreferences.

Note: for "CurrentPreference" it is not necessary, because it has already been calculated and checked.

2. List simplified directional tie-breaking

In this type of tie-breaking the number of ballots (ordinary) of candidates from the tie group should be compared. The subsequent sub-steps of list directional backward tie-breaking concern the following preference numbers (as in the FOR statement): CurrentPreference DOWNTO 1, CurrentPreference + 1 TO NumberOfPreferences. Whereas in the forward: 1 TO NumberOfPreferences.

Note: in list full method, the values of successive elements of the lists of tied candidates are compared; 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; the value of the first element of such a list is equal to the number of ballots free of the candidate; in list simplified method 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.

3. Group tie-breaking

Group tie-breaking involves electing the entire Tie Group (TG). Each candidate from TG has the same number of ballots free, counted independently of the other TG candidates. However, if the whole group were to be elected, some other number of ballots free, common in TG, would have to be calculated for each candidate, using the NOBFC(TG, CurPref) function, and this number should be >= quota.

Such election of the whole group is supposed to be and is equivalent to drawing subsequent candidates from TG, but it is a deterministic method, not a random one.

General CTV global algorithm

In addition to the sequential CTV method described above, there is also the global CTV method of calculating voting result. Its algorithm:

FindTheSmallestPreferenceNumberToWhichAny[NumberOfSeats-ElementSubset]OfCandidatesNotElectedReached NOBC ≥ Quota 
RecognizeElectedForAllSeats(CandidatesFromThisSubsetWhichToThisPreferenceHasReachedTheHighestValueOfNOBC)

The NOBC(GC, Pref) value is usually equal to the maximum value of the quota at which all candidates from the GC group would be elected (to the 'Pref' preference). If there were some other variation of this method where some candidates were already selected during the calculation, then NOBFC should be used instead of NOBC.

In the idea of the sequential method is to elect the first candidate reaching the quota: to preferences as small as possible. However, in the idea of global method, it is to elect 'first' candidate (although at the same time with others) reaching the quota: to the maximum preference necessary (its preferences number is the same as in the sequential method). In the sequential method, in the initial stages of calculation, preferences initial are more important (when making decisions about electing) from the others, however, in the global method, all preferences (up to the maximum preference necessary) are always just as important.

Usually the candidates selected in these methods are the same. Both these methods are optimal, but in a slightly different way: the sequential method more prefers earlier preferences, and the global method more prefers a greater common number of ballots. Therefore if the number of seats > 1, then the calculated results in both methods may be different. Example: 2 seats, candidates A, B, C; in pref.1: A x Quota; in pref.2: B x Quota+1 and C x Quota+2, no A; B and C on different ballots, A and C on different ballots - then in the sequential method the first elected candidate is A and the second is C (and NOBC({A,C},2) = Quota), while in the global method elected candidates are B and C with NOBC({B,C},2) = Quota+1.

Tie-breaking

Here tie-breaking is similar to that in the sequential method (but without the 'group'):

1. Basic directional - backward or forward: using the NOBC value (instead of 'number of ballots free' in sequential method) for subsequent preferences.
2. List directional - backward or forward: Same as in the sequential method. The elements of the list (in non-decreasing order by NOBC value) are non-empty subsets of the GC set. If the list was to be simplified, it would be e.g. to single-element sets (because there is no empty set).
3. Final.

NOBFC and NOBC functions

The word "common" in this name means the common, maximum, same number of ballots that each candidate in this group can have (some candidates may have more). This ensured number is the same, common, but the subsets of cards (of this number) for each candidate must be separate (disjointed).

Def.:
NOBFC(GC, Pref) := NumberOfBallotsFreeCommon(GC, Pref) 
                := Minimum( { NumberOfBallotsFree(SGC, Pref) \ #SGC: #SGC > 0 ∧ SGC ⊆ GC } )

Notes:
GC ('Group of Candidates') is a subset of the set of candidates not elected.
The sign "#" means the number of elements.
The sign "\" means integer division.

'Ballots free' for a Group is calculated in the same way as for its element:

Def.1G. Number of ballots free to current preference (=CurPref) for a group (=Group):
      NumberOfBallotsFree(Group, CurPref) := 
      Minimum( { IIF( NumberOfBallotsFromSubsetButWithoutGroup(SsEC, Group, CurPref) ≥ 
                      NumberOfBallotsAllocatedToSubset(SsEC), 
                                                NumberOfBallotsGroup(Group, CurPref), 
                                                NumberOfBallotsGroup(Group, CurPref) + 
                      NumberOfBallotsFromSubsetButWithoutGroup(SsEC, Group, CurPref) - 
                                              NumberOfBallotsAllocatedToSubset(SsEC)  ) 
                 : SsEC ⊆ SetOfElectedCandidates } )  

Def.3G. NumberOfBallotsFromSubsetButWithoutGroup(SsEC, Group, CurPref) := [ Number of ballots in which: 
                        in any preference from No 1 to CurPref any candidate from SsEC is marked, 
                        but there is no candidate from Group on this ballot (in these preferences) ]

Def.4G. NumberOfBallotsGroup(Group, CurPref) := [ Number of ballots in which:  
                               in any preference from No 1 to CurPref is marked any candidate from Group ]

In a situation where there are no elected candidates yet, the NOBC function can be used instead of the NOBFC function, because then NOBC(SGC, Pref) = NOBFC(SGC, Pref), because then NumberOfBallotsFree(SGC, Pref) = NumberOfBallots(SGC, Pref).

Def.:
NOBC(GC, Pref) := NumberOfBallotsCommon(GC, Pref) 
               := Minimum( { NumberOfBallots(SGC, Pref) \ #SGC: #SGC > 0 ∧ SGC ⊆ GC } )

Monotonicity of the CTV algorithm

The CTV algorithm is monotonic (in both senses). The rationale for this monotonicity:

• 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.
• 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.

See also

• Single transferable vote
• Counting single transferable votes

Bibliography

• Zbiorczy Głos Przechodni (on the website already removed)
• Zbiorczy Głos Przechodni (ZGP) (en Collective Transferable Vote, CTV)

External links

• Single Transferable Vote - Preferential voting (pl)
• Calculation of voting results with multiple STV methods (without CTV)
• Website (pl) with the CTV (=ZGP) program (free, in MS ACCESS), calculating voting results using the CTV method, with examples (8MB)