ELECTION OF APRIL 2010

On 22 March 2010, the Green National Committee (GNC) opened to Virginia's Greens two seats that had previously been assigned to other states.

In April 2010, the Interim Committee of the Green Party of Virginia (GPVA) held an election to choose two delegates to fill the two vacant seats. A decision was made (not by the committee) to tabulate the votes by the method of Single Transferable Vote (STV), the general form of ranked-choice or preference voting. Of the 14 members, 8 cast ballots. An identification letter is has been assigned to each ballot by the editor for reference. The numbers on the ballot indicate ordinal preference for each candidate: 1 for first choice, 2 for second choice, etc.

BALLOT	CANDIDATE
	Tamar	Kirit	Sheri	Chris	Charlie	"Leave a seat vacant"
A	1	2	3
B	1	2	4	3	5
C	1	3	4	2	5
D	2	1	4	3	5
E	2	3	5	1	4
F	3	2	1	4	5
G	5	1	2	4	3
H		5	1	4	3	2

STV (like most forms of ranked-choice voting) requires that each winning candidate receive a minimum number of votes, a quota or threshold. Several quotas are in common use.

The GPVA Interim Committee's three-fifths quota

The bylaws of GPVA set the threshold for decision at two-thirds of all votes, assuming consensus cannot be reached:

• XX. Interim Committee
  • D. Decision Making
    • 2. The Interim Committee shall strive for consensus in all of its decisions.
    • 4. If consensus is not achievable, then approval by three-fifths of Interim Committee members participating in the vote for the resolution shall affirm a decision.

These tabulations show how a quota of three-fifths of ballots would be employed to fill

• two delegate seats; or
• four seats, including two delegates and two alternates.

This is the resulting tabulation for the two delegate seats:

Fill 2 delegate seats by GPVA-IC quota.

Quota = 8*3/5 = 4.8	Tamar	Kirit	Sheri	Chris	Charlie	"Leave a seat vacant"	Running total
Round 1	3	2	2	1	0	0	8
transfer from "Leave a seat vacant"						-0
transfer from Charlie					-0
Round 2	3	2	2	1	0	0	8
transfer from Chris	+1			-1
Round 3	4	2	2	0	0	0	8
Result	win by default	tied	tied	eliminated	eliminated	eliminated

Within the framework of transferability of unused votes and portions of votes, we see no non-random way to break this tie.

Random methods include tossing a coin, drawing lots, picking a high card from a deck, throwing a die, etc.

As an instant runoff would produce the same tie, some electorates might prefer to eliminate the winning candidate from the runoff, allowing the votes that elected that candidate to be used again. If transferability is extended beyond the single vote, allowing those who voted for a winner to vote a second time, a solution will be produced at the expense of voter equality. Based on recorded preferences, the result would probably resemble this:

Delayed runoff to break tie for 2nd delegate seat by GPVA-IC quota

Quota = 8*3/5 = 4.8	Tamar	Kirit	Sheri	Chris	Charlie	"Leave a seat vacant"	Running total
Round 1		4	2	2	0	0	8
transfer from "Leave a seat vacant"						-0
transfer from Charlie					-0
Round 2		4	2	2	0	0	8
Result	bye: already won	win by default	tied	tied	eliminated	eliminated

This is the tabulation for four seats, including two delegates and two alternates:

Fill all 4 seats by GPVA-IC quota.

Quota = 8*3/5 = 4.8	Tamar	Kirit	Sheri	Chris	Charlie	"Leave a seat vacant"	Running total
Round 1	3	2	2	1	0	0	8
transfer from "Leave a seat vacant"						-0
transfer from Charlie					-0
Round 4	3	2	2	1	0	0	8
Result	win by default	win by default	win by default	win by default	eliminated	eliminated

Although the three-fifths quota is required, it does not appear to have been enforced in this case.

Let's take a look at some standard quotas for STV.

Majority as quota

Using a majority as threshold requires that a candidate receive the votes of more than half the valid ballots (either as first choice or by transfer):

• Quota = 8/2 = 4

Tabulation of this election with a majority quota produces results identical to the previous three-fifths quota, as long as the single transferable vote (STV) concept is observed.

If STV is discarded, allowing all voters to participate in a runoff (as if their votes had not been used previously), that runoff proceeds slightly differently, as shown here:

Delayed runoff to break tie for 2nd delegate seat by majority threshold

8/2=4	Tamar	Kirit	Sheri	Chris	Charlie	"Leave a seat vacant"	Running total
Round 1		4	2	2	0	0	8
transfer from "Leave a seat vacant"						0	0
transfer from Charlie					0		0
Round 2		4	2	2	0	0	8
transfer from Chris		2		-2			0
transfer from Sheri		2	-2				0
Round 3		8	0	0	0	0	8
Result	bye: already won	win	eliminated	eliminated	eliminated	eliminated

Hare quota

The Hare quota is defined as the number of valid ballots divided by the number of seats to be filled. Although it may produce the right number of winning candidates, it often falls short.

Here's how the election would be tabulated using the Hare quota:

Fill 2 delegate seats by Hare quota

Quota = 8/2 = 4	Tamar	Kirit	Sheri	Chris	Charlie	"Leave a seat vacant"	Running total
Round 1	3	2	2	1	0	0	8
transfer from "Leave a seat vacant"						-0
transfer from Charlie					-0
Round 2	3	2	2	1	0	0	8
transfer from Chris	+1			-1
Round 3	4	2	2	0	0	0	8
excess from Tamar	-0	+0
Round 4	4	2	2	0	0	0	8
Result	win	tied	tied	eliminated	eliminated	eliminated

Within the framework of transferability of unused votes and portions of votes, we see no non-random way to break this tie.

If transferability is extended beyond the single vote, allowing those who voted for a winner to vote a second time, a solution will be produced at the expense of voter equality. Based on recorded preferences, the result would probably resemble this:

Inclusive runoff to break tie for 2nd delegate seat

Quota = 8/2 = 4	Tamar	Kirit	Sheri	Chris	Charlie	"Leave a seat vacant"	Running total
Round 1		4	2	2	0	0	8
Result		win

This is the Hare tabulation for four seats, including two delegates and two alternates:

Fill all 4 seats by Hare quota

Quota = 8/4 = 2	Tamar	Kirit	Sheri	Chris	Charlie	"Leave a seat vacant"	Running total
Round 1	3	2	2	1	0	0	8
excess from Tamar	-1			+0.67
excess from Kirit		-0		+0	+0
excess from Sheri			-0	+0		+0
Round 2	2	2	2	1.67	0	0	7.67
transfer from "Leave a seat vacant"						-0
transfer from Charlie					-0
Round 3	2	2	2	1.67	0	0	7.67
Result	win	win	win	win by default	eliminated	eliminated

Droop quota

The Droop quota is a weird system that was probably developed to simplify arithmetic. Unfortunately, it produces erratic numbers. It may be smaller or larger than the Hare quota. (For example, the Droop quota in an election with 24 ballots and 7 seats is 4, compared to a Hare quota of only 3.43.) Because of its unpredictability, we cannot recommend its use when other quotas are available.

Fill 2 delegate seats by Droop quota

Quota = INT(8/(2+1)+1) = 3	Tamar	Kirit	Sheri	Chris	Charlie	"Leave a seat vacant"	Running total
Round 1	3	2	2	1	0	0	8
excess from Tamar	-0			+0
Round 2	3	2	2	1	0	0	8
transfer from "Leave a seat vacant"						-0
transfer from Charlie					-0
Round 3	3	2	2	1	0	0	8
transfer from Chris		+1		-1
Round 3	3	3	2	0	0	0	8
Result	win	win		eliminated	eliminated	eliminated

This is the tabulation for four seats, including two delegates and two alternates:

Fill all 4 seats by Droop quota

Quota = INT(8/(4+1)+1)= 2	Tamar	Kirit	Sheri	Chris	Charlie	"Leave a seat vacant"	Running total
Round 1	3	2	2	1	0	0	8
excess from Tamar	-1			-0.67
excess from Kirit		-0		+0	+0
excess from Sheri			-0	+0		+0
Round 2	2	2	2	1.67	0	0	7.67
transfer from "Leave a seat vacant"						-0
transfer from Charlie					-0
Round 3	2	2	2	1.67	0	0	7.67
Result	win	win	win	win by default	eliminated	eliminated

Hagenbach-Bischoff quota (exclusive)

The Hagenbach-Bischoff quota marks the dividing line between producing just enough winners and the possibility of producing one too many. It is sometimes unwisely used as the threshold a candidate must meet to win, the inclusive limit of eligibility, which is where the extra winner can be found. Tostead, the Hagenbach-Bischoff quota should be the threshold a candidate must surpass to win, the exclusive limit of eligibility. If the preference ballots are filled out and tabulated properly, it will be impossible to have too many winners.

Here, we use the Hagenbach-Bischoff quota as the threshold that must be crossed, not merely reached.

Fill 2 delegate seats by Hagenbach-Bischoff quota

Quota = 8/3 = 2.67	Tamar	Kirit	Sheri	Chris	Charlie	"Leave a seat vacant"	Running total
Round 1	3	2	2	1	0	0	8
excess from Tamar	-0.33	+0.22		+0.11
Round 2	2.67	2.22	2	1.11	0	0	8
transfer from "Leave a seat vacant"						-0
transfer from Charlie					-0
Round 3	2.67	2.22	2	1.11	0	0	8
transfer from Chris	+2			-2
Round 4	2.67	3.33	2	0	0	0	8
Result	win	win		eliminated	eliminated	eliminated

This is the tabulation for four seats, including two delegates and two alternates:

Fill all 4 seats by Hagenbach-Bischoff quota

Quota = 8/5 = 1.6	Tamar	Kirit	Sheri	Chris	Charlie	"Leave a seat vacant"	Running total
Round 1	3	2	2	1	0	0	8
excess from Tamar	-1.4			+1.4
excess from Kirit		-0.4		+0.2	+0.2
excess from Sheri			-0.4	+0.2		+0.2
Round 2	1.6	1.6	1.6	2.8	0.2	0.2	7.8
Result	win	win	win	win

Hagenbach-Bischoff quota (inclusive)

For this data set, no candidate meets the Hagenbach-Bischoff quota exactly. Therefore, the tabulation for inclusive use, as a threshold that must be reached, produces identical results to its use in the previous example, as an exclusive threshold that must be exceeded. Please consult the previous section to see the computation.

Imperiali quota

The Imperial quota, used in Ecuador, often produces too many winners because of its unusually low quota. Here is an example in a two-seat election:

Fill 2 delegate seats by Imperiali quota

Quota = 8/4 = 2	Tamar	Kirit	Sheri	Chris	Charlie	"Leave a seat vacant"	Running total
Round 1	3	2	2	1	0	0	8
Result	win	win	win

Yet the correct number of winners pass the quota in this four-seat election:

Fill all 4 seats by Imperiali quota

Quota = 8/6 = 1.33	Tamar	Kirit	Sheri	Chris	Charlie	"Leave a seat vacant"	Running total
Round 1	3	2	2	1	0	0	8
excess from Tamar	-1.67			+1.11
excess from Kirit		-0.67		+0.33	+0.33
excess from Sheri			-0.67	+0.33		+0.33
Round 3	1.33	1.33	1.33	2.78	0.33	0.33	7.44
Result	win	win	win	win

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Last revised: 26 April 2010
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