Cumulative voting and minority rights

[J. J.]

On 2/13/2024 at 6:47 PM, Dan Honemann said:

I believe these two things to be true:

1. If 120 members vote in an election to elect 4 members to a committee, a candidate must receive at least 61 votes in order to be elected.

2. The minimum number of members that can cast at least 61 votes for a candidate is 16.  This, then, is the number of members protected by the rule. 

Do you (Mr. Gerber, that is) disagree with either or both of these statements?

 

1.  Yes.

2.  I am going with no.  The minority protected is the number of members that, by this rule, is the number that can always elect at  least one member.  16 members may not be able to elect in all circumstances, even if they have a majority.  Even in straight majority voting for multiple positions, a majority of the votes cast does not necessarily elect. 

Edited February 14, 2024 at 02:30 PM by J. J.

[J. J.]

If I understand correctly, this is what Mr. Gerber is suggesting:

In this example, there are five candidates, G, H, J, K, & M.  120 voters each cast 4 votes for the seats, using cumulative voting.

Ballots 1-24 has all 4 votes for G (96 vote)

Ballot 25 has one vote for G and 1 vote each for H, J, & K.

Ballot 26-120 has one vote each for H, J, K, & M. (Each gets 95 votes)

The total are:

G gets 97 votes

H gets 96 votes

J gets 96 votes

K gets 96 votes

M gets 95 votes

G, H, J and K are elected because they have the highest votes (though all have more than more than a majority).

G will always get enough to be in the top 4 matter how you distribute the votes among M, H, J, & K.

If J, for example, gets 63 votes and the other 3 get them evenly, the result is:  G 97, H 111, K 111, M 110 and J 63.  G is still elected.  Even if H got all of those votes the result would be G 97, H 129, K 96, M 95 & J 63.  Even if J got just 3 vote and they were distributed evenly, the result would be G 97, H 127, K 127, M 126 & J 3. 

There is no circumstance where any minority greater than 1/5 cannot elect G in this case. 

I think this what Mr. Gerber is suggesting. 

 

[Dan Honemann]

On 2/14/2024 at 10:23 AM, J. J. said:

If I understand correctly, this is what Mr. Gerber is suggesting:

In this example, there are five candidates, G, H, J, K, & M.  120 voters each cast 4 votes for the seats, using cumulative voting.

Ballots 1-24 has all 4 votes for G (96 vote)

Ballot 25 has one vote for G and 1 vote each for H, J, & K.

Ballot 26-120 has one vote each for H, J, K, & M. (Each gets 95 votes)

The total are:

G gets 97 votes

H gets 96 votes

J gets 96 votes

K gets 96 votes

M gets 95 votes

G, H, J and K are elected because they have the highest votes (though all have more than more than a majority).

G will always get enough to be in the top 4 matter how you distribute the votes among M, H, J, & K.

If J, for example, gets 63 votes and the other 3 get them evenly, the result is:  G 97, H 111, K 111, M 110 and J 63.  G is still elected.  Even if H got all of those votes the result would be G 97, H 129, K 96, M 95 & J 63.  Even if J got just 3 vote and they were distributed evenly, the result would be G 97, H 127, K 127, M 126 & J 3. 

There is no circumstance where any minority greater than 1/5 cannot elect G in this case. 

I think this what Mr. Gerber is suggesting. 

 

Why do each of the voters cast only 4 votes instead of 5?

[Shmuel Gerber]

On 2/14/2024 at 11:06 AM, Dan Honemann said:

Why do each of the voters cast only 4 votes instead of 5?

Five just happens to be the number of candidates in JJ's example; there could have been more. There are still only four seats being elected. 

[Dan Honemann]

On 2/14/2024 at 11:14 AM, Shmuel Gerber said:

Five just happens to be the number of candidates in JJ's example; there could have been more. There are still only four seats being elected. 

Got it.

[J. J.]

On 2/14/2024 at 11:14 AM, Shmuel Gerber said:

Five just happens to be the number of candidates in JJ's example; there could have been more. There are still only four seats being elected. 

Is my example an accurate description of what you are suggesting?  The minority protected would be any number greater than 24 voters?

[J. J.]

On 2/14/2024 at 11:15 AM, Dan Honemann said:

Got it.

I do not believe that the number of candidates would change the minority protected, if I understand Mr. Gerber's premise. 

[Dan Honemann]

On 2/14/2024 at 12:59 PM, J. J. said:

Is my example an accurate description of what you are suggesting?  The minority protected would be any number greater than 24 voters?

In other words, in this case the fraction of members protected by the rule is 5/24. Is this correct?

[J. J.]

On 2/14/2024 at 1:07 PM, Dan Honemann said:

In other words, in this case the fraction of members protected by the rule is 5/24. Is this correct?

No, technically.  The fraction election is any amount above 1/5 of the members voting (for at least one candidate).  The minority protected is any about greater than the total number of members voting (for at least one candidate) divided by the sum of the number of positions to be elected plus one.

Minority to be protected = M

Total number of voters voting for at least one candidate = T

Total number of positions to be elected = P

M > T/(P+1)

M is the minimal number that would be capable of electing one person without any additional votes.

[Note that I have not used the term ballot, because the vote would secret and not subject to suspension.  If this were non secret ballots, i.e. signed ballots, T would be the total number of ballots with at least one vote on it.]

Edited February 14, 2024 at 06:30 PM by J. J.

[J. J.]

On 2/14/2024 at 1:07 PM, Dan Honemann said:

In other words, in this case the fraction of members protected by the rule is 5/24. Is this correct?

Just to follow up on your example, if 100 people voted, the rule could not be suspended in the face of any number above 20. 

[Dan Honemann]

On 2/14/2024 at 1:25 PM, J. J. said:

Just to follow up on your example, if 100 people voted, the rule could not be suspended in the face of any number above 20. 

But I would say that this should be stated in the form of a fraction so that it can be applied to whatever number of members vote on the motion to suspend the rules.  I gather that you are saying that this fraction is 5/24. 

[Shmuel Gerber]

On 2/13/2024 at 6:47 PM, Dan Honemann said:

1. If 120 members vote in an election to elect 4 members to a committee, a candidate must receive at least 61 votes in order to be elected.

As I said in the topic I referred to earlier, I don't really cotton to the notion that there is something special about a majority of the ballots cast in an election being done by cumulative voting. It does not represent a majority of the voters or of the votes cast. It is just some strange artifact of the rules relating to the way a ballot vote is conducted when a majority of the ballots cast represents a majority of the voters.

However, simply for the purpose of this discussion, I'm willing to assume that in order to be among the winners, a candidate must not only be in the top 4 vote-getters (where there are 4 positions) but also must attain a number of "votes" (in the cumulative system, i.e., where a voter can assign multiple votes to a single candidate) that is a majority of the number of ballots cast. As I believe J.J. pointed out earlier, this means more than 60 votes, which then translates to 61 when dealing with whole numbers.

On 2/13/2024 at 6:47 PM, Dan Honemann said:

2. The minimum number of members that can cast at least 61 votes for a candidate is 16.  This, then, is the number of members protected by the rule. 

I'm beginning to see where you're getting these numbers from — i.e., each voter can place 4 votes, so for a candidate to attain more than 1/2 the total ballots, the minimum number of voters who could yield this majority is more than 1/8 of the voters, all giving their total support (⅛ × 4 = ½). I don't think it's helpful to convert this into a particular number of votes, as it stems from a fraction that always applies, regardless of how it may translate into whole numbers under specific numbers of voters; but, yes, in the case of 120 members voting, 16 voters (all giving all of their votes to the same candidate) would be needed for a majority.

But I don't agree that this is the number (or fraction) that is protected by the rules. As J.J. has noted, such a majority does not necessarily win an election even when cumulative voting is in effect. In fact, if a certain set of 4 candidates is supported by more than 4/5 of the voters, and a different candidate is supported by a minority of more than 1/8 (but not 1/5 — which cannot exist in this scenario, because the total minority remaining is not that large), then the minority candidate is sure to lose (assuming all the votes are well coordinated).

I would think that if anything, where a majority vote is required with cumulative voting, the rules protect not the minority of more than 1/8, but rather the majority of 7/8, by ensuring that 1/8 or less cannot elect their choice (without additional support). But even that protection is not really necessary, because 4/5 (which is even less than 7/8) can get what they want regardless of the "majority" threshold.

I still say that the minority that is protected by the rules is a minority of more than 1/5, which can always (with proper coordination) elect one candidate of its choice in a 4-seat election.

Edited February 14, 2024 at 08:32 PM by Shmuel Gerber
inserted the underlined words

[Shmuel Gerber]

On 2/14/2024 at 12:59 PM, J. J. said:

Is my example an accurate description of what you are suggesting?  The minority protected would be any number greater than 24 voters?

 

On 2/14/2024 at 1:07 PM, Dan Honemann said:

In other words, in this case the fraction of members protected by the rule is 5/24. Is this correct?

 

On 2/14/2024 at 1:54 PM, Dan Honemann said:

But I would say that this should be stated in the form of a fraction so that it can be applied to whatever number of members vote on the motion to suspend the rules.  I gather that you are saying that this fraction is 5/24. 

Yes, I think J.J. has illustrated my point. The minority protected is any greater than 1/5. This is equivalent to 24/120. So with 120 voters in total, a minority of more than 24, by coordinating their votes in a cumulative-voting election for 4 seats, can be sure to elect one candidate.

[Dan Honemann]

On 2/14/2024 at 3:29 PM, Shmuel Gerber said:

 

 

Yes, I think J.J. has illustrated my point. The minority protected is any greater than 1/5. This is equivalent to 24/120. So with 120 voters in total, a minority of more than 24, by coordinating their votes in a cumulative-voting election for 4 seats, can be sure to elect one candidate.

So then, would you say that, for purposes of 25:2(7), a rule mandating cumulative voting protects a minority of 5/24, and that this will be so regardless of how many members vote in the election and how many positions are to be filled? 

[J. J.]

Just for the record, if there are more than one position to be chosen, and the vote is by straight majority voting, it is entirely possible for someone to get a majority and not be elected.  That is why I am seeing something comparable. 

[J. J.]

On 2/14/2024 at 4:05 PM, Dan Honemann said:

So then, would you say that, for purposes of 25:2(7), a rule mandating cumulative voting protects a minority of 5/24, and that this will be so regardless of how many members vote in the election and how many positions are to be filled? 

I would not.

Would you say that for the purposes of 44:1 a majority is 61/120 and that this will be so regardless of how many members vote in the election and how many positions are to be filled?  (I hope not.)

[Shmuel Gerber]

On 2/14/2024 at 4:05 PM, Dan Honemann said:

So then, would you say that, for purposes of 25:2(7), a rule mandating cumulative voting protects a minority of 5/24, and that this will be so regardless of how many members vote in the election and how many positions are to be filled? 

I don't know where the number 5/24 is coming from. My formula to find the minority protected by a rule allowing cumulative voting is that it's the size of the minority that is guaranteed the ability, by coordinating their votes, to elect one candidate. This number is anything more than 1/(N+1), where N is the number of seats being elected. For 2 seats, it would be a minority greater than 1/3; for 3 seats, it would be greater than 1/4; and for 4 seats, greater than 1/5. (And for the trivial case of 1 seat, it's more than 1/2, which is not a minority.)

[Dan Honemann]

On 2/14/2024 at 4:15 PM, Shmuel Gerber said:

I don't know where the number 5/24 is coming from. My formula to find the minority protected by a rule allowing cumulative voting is that it's the size of the minority that is guaranteed the ability, by coordinating their votes, to elect one candidate. This number is anything more than 1/(N+1), where N is the number of seats being elected. For 2 seats, it would be a minority greater than 1/3; for 3 seats, it would be greater than 1/4; and for 4 seats, greater than 1/5. (And for the trivial case of 1 seat, it's more than 1/2, which is not a minority.)

Okay, so then the fraction of members protected by a rule mandating cumulative voting will vary depending upon the number of seats to be filled.  The question is whether this is the sort of "minority of a particular size" contemplated by 25:2(7).

[Shmuel Gerber]

On 2/14/2024 at 5:46 PM, Dan Honemann said:

Okay, so then the fraction of members protected by a rule mandating cumulative voting will vary depending upon the number of seats to be filled.  The question is whether this is the sort of "minority of a particular size" contemplated by 25:2(7).

In my opinion it's not helpful to be too particular about the meaning of particular. 🙂

I would note that in RONR, the rule describing cumulative voting itself depends on the number of seats to be filled: "In this form of voting, each member is entitled to cast one vote for each position, so that if, for example, three directors are to be elected, each member may cast three votes..." (46:43)

[Dan Honemann]

On 2/14/2024 at 6:34 PM, Shmuel Gerber said:

In my opinion it's not helpful to be too particular about the meaning of particular. 🙂

I would note that in RONR, the rule describing cumulative voting itself depends on the number of seats to be filled: "In this form of voting, each member is entitled to cast one vote for each position, so that if, for example, three directors are to be elected, each member may cast three votes..." (46:43)

Yes, I know what 46:43 says, and I gather your answer to my question is yes.

[Shmuel Gerber]

On 2/14/2024 at 7:42 PM, Dan Honemann said:

Yes, I know what 46:43 says, and I gather your answer to my question is yes.

I won't purport to read the mind of  25:2(7), but my answer is yes. 

[J. J.]

I would be interested in hearing Dr. Kapur's premise that the rule does not protect a minority of, in this case, less than 1/3.

I will note that the fraction protected by the rule is always the same fraction, based on the number of positions, under Mr. Gerber's premise.  For example, if there were all 200 members voting, the minority protected would be any number greater than 1/5 of the voters*.  Likewise, if only 20 people voted the minority protected would be any number greater than 1/5 of the voters. 

 

 

*A voter here is any member who casts at least one vote for the position.  The vote is not by secret ballot. 

[Dan Honemann]

On 2/15/2024 at 9:23 AM, J. J. said:

  The vote is not by secret ballot. 

I assume this is referring to the election, and so I don't see where this makes any difference.  As I understand it, the bylaws do not require a ballot vote, but I would think that a motion could have been adopted ordering one.  If so, this should have no effect on the responses to the questions asked.

[J. J.]

On 2/15/2024 at 10:40 AM, Dan Honemann said:

I assume this is referring to the election, and so I don't see where this makes any difference.  As I understand it, the bylaws do not require a ballot vote, but I would think that a motion could have been adopted ordering one.  If so, this should have no effect on the responses to the questions asked.

It could, indirectly, reveal voter preference.  I am factoring that issue out, however. 

[Dan Honemann]

On 2/15/2024 at 9:23 AM, J. J. said:

I would be interested in hearing Dr. Kapur's premise that the rule does not protect a minority of, in this case, less than 1/3.

I will note that the fraction protected by the rule is always the same fraction, based on the number of positions, under Mr. Gerber's premise.  For example, if there were all 200 members voting, the minority protected would be any number greater than 1/5 of the voters*.  Likewise, if only 20 people voted the minority protected would be any number greater than 1/5 of the voters. 

 

While you are waiting for a response from Dr. Kapur, I can't help but feel it a bit strange to refer to "any number greater than 1/5" as being the number of members protected by a rule. I suppose lots of numbers will be greater than 1/5. Do you mean the smallest one of these numbers?