The Principal of Red Barn Elementary School is allowing the students to select three community pets for the school grounds. There are a total of 10 pets the students may choose from:

Buddy

Penny

Cotton

Misty

Rocky

Scout

Ivory

Splash

Pumpkin

Thumper

The students fell into 7 voting groups:

• The Cat Voters only wanted cats.
• The Dog Voters only wanted dogs.
• The LongEar Voters only wanted pets with long ears.
• The BigEye Voters only wanted pets with large eyes.
• The Stripe Voters only wanted pets with stripes.
• The WhiteFur Voters only wanted pets with white fur.
• The BlueNose Voters only wanted pets with blue noses.

All of the students were indifferent after their top 3 choices. They filled out their ballots:

11 Cat Voters

14 Dog Voters

15 LongEar Voters

12 BigEye Voters

10 Stripe Voters

8 WhiteFur Voters

9 BlueNose Voters

The school received a total of \(V_{TOTAL}=79\) student ballots. Since the school is selecting \(S=3\) pets, the election quota is:

So any pet reaching 20 votes is guaranteed to be selected. All 79 ballots begin with an initial voting strength of 1:

We then calculate the initial first-place votes:

No pet initially reached the 20-vote quota needed to be selected. Misty, Cotton, and Penny received no first-place votes and were eliminated before the first transfer. Ivory then had the fewest remaining first-place votes, so Ivory was eliminated, and all 8 WhiteFur ballots transferred from Ivory to Scout:

Still no pet had reached the 20-vote quota, so Rocky, with the fewest votes, was eliminated next, and all 10 Stripe ballots transferred from Rocky to Pumpkin.

Pumpkin reached 21 votes and crossed the quota, becoming the first selected pet. Since Pumpkin only needed 20 votes to be selected, each of those 21 ballots was reduced proportionally so the excess voting strength could continue transferring onward. The reduced ballot strengths were calculated as:

The 11 Cat ballots had no remaining choices, since Ivory and Misty had already been removed, so those ballots became exhausted and no longer participated in the election.

The 10 Stripe ballots transferred to Thumper with a combined voting strength of:

So the totals after those ballots transfer become:

No pet reached the quota, so Splash was eliminated next. All 12 LargeEyes ballots transferred from Splash to Buddy.

Buddy increased to 26 votes and became the second selected pet. Since Buddy only needed 20 votes to be selected, each of those 26 ballots was reduced proportionally so the excess voting strength could continue transferring onward. The reduced ballot strengths were calculated as:

The 12 BigEye ballots had no remaining choices, since Misty had already been removed, so those ballots became exhausted and no longer participated in the election.

The 14 Dog ballots transferred to Scout with a combined voting strength of:

So the totals after those ballots transfer become:

Scout increased from 17 votes to 21.2 votes, crossing the quota and becoming the third selected pet. So the pets chosen are:

Pumpkin

Buddy

Scout

Even though Thumper received the most initial first-place votes, Thumper was not selected. Scout began with only 9 first-place votes, but later transfers from other ballots allowed Scout to eventually reach quota and become the final selected pet.

This example also shows how lower-ranked choices can continue influencing the outcome after earlier choices have either been selected or eliminated. Some ballots eventually became exhausted because the students did not rank additional remaining pets beyond their first 3 choices. The next example shows how these same mechanics can affect representation inside a much larger system.

We return to the Wilderlands, which will be represented by a 20-member assembly. The 4 major factions are:

As we discussed previously, the Wilderlands is divided into 20 regions:

• The Foresters have the largest total population across all regions.
• The Felines have the second-largest population, but much of their population is concentrated in the 11 northern regions.
• The Avians are smaller in population, and their population is spread evenly across all regions.
• The Mariners are even smaller in population, and are spread across the southern coastal regions.

How the factions vote in each region are described in the following chart.

There were several broad assumptions made about these voters:

• The first-choice candidate for each voter in each region and faction is shown in the chart.
• Voters then continue ranking additional candidates from their own faction highly.
• Voters then continue ranking candidates from 2 additional factions according to the coalition assumptions above.

This example intentionally assumes unusually consistent coalition voting behavior in order to make the transfer process easier to follow. Real elections are generally far less organized, and voters within the same coalition often rank candidates very differently. Later articles will discuss some of the ways large STV elections can be simplified through ballot design, regional organization, coalition coordination, and other approaches that can both simplify ballots for voters and improve transparency for election observers.

For this election with \(V_{TOTAL}=2000\) and \(S=20\) elected members:

Each candidate will need 96 votes to claim a seat in the assembly. The election progress is summarized through the following transfer markers:

Marker 1: 7 candidates (the Lion, Bear, Wolf, Fox, Eagle, Falcon, and Dolphin) immediately begin above the 96-vote threshold and secure seats. Their remaining excess votes begin transferring to allied candidates, particularly the Tiger and several additional Forester candidates.

Marker 2: The Tiger rises above quota with 111 votes and secures another seat. The Tiger’s remaining excess votes then transfer onward to additional Feline candidates.

Marker 3: As smaller candidates are gradually eliminated, transfers consolidate support into the remaining Feline candidates, including the Leopard, Panther, Cheetah, Jaguar, and Lynx.

Marker 4: The Panther, Cheetah, Jaguar, and Lynx all eventually cross the 96-vote threshold. Their remaining excess votes then transfer onward to the Cougar.

Marker 5: The Cougar secures a seat with 140 votes. Because all remaining Feline, Avian, and Mariner candidates had either already secured seats or been eliminated, the remaining 44 Cougar ballot strength became exhausted instead of transferring onward.

Marker 6: As additional smaller candidates are eliminated, Forester support gradually consolidates into the Badger, Boar, Moose, and Beaver.

Marker 7: The Badger, Boar, Moose, and Beaver all cross the 96-vote threshold and secure seats. Their remaining excess votes then transfer onward to the Elk and Raccoon.

Marker 8: The Elk rises above the 96-vote threshold and transfers additional votes to the Raccoon.

Marker 9: The Raccoon eventually crosses the 96-vote threshold. Because every remaining candidate had either already secured a seat or been eliminated, additional remaining ballot strength became exhausted.

After the transfers are complete, the 20-member assembly becomes:

The Foresters had the largest total population across the Wilderlands, but with only 45% of the population they could only claim 9 of the 20 seats. The Felines held majorities in 11 of the 20 regions, but with only 40% of the total population they could only claim 8 of the 20 seats. The Avians and Mariners, with respective 10% and 5% shares of the population, were still able to secure representation within the assembly even though their populations were spread out across the regions.

Unlike territorial systems, no single coalition was able to convert regional advantages into overwhelming assembly control. The regional boundaries could be redrawn, but because votes transfer across regions, representation in this system can remain broadly proportional if voters continue transferring support within their coalitions. This example also demonstrates how STV can reward transferable coalition support rather than simply rewarding the largest initial voting bloc.

Our final example illustrates some of the important strategic incentives that should be understood for STV elections:

The Wilderlands has decided to create a 5-member executive council. Seven candidates have emerged to compete for those seats:

• The Foresters support the Bear, Wolf, and Fox, and hope to secure 3 of the 5 council seats. They support the Eagle as a next choice, as they do not want the Felines to control the council.
• The Felines support the Lion, Tiger, and Cheetah, and also hope to secure 3 of the 5 council seats. They also support the Eagle next, as they do not want the Foresters to control the council.
• The Avians support the Eagle, and primarily want the Eagle to become the deciding voice on the council. If they cannot secure that final seat, they prefer the Felines over the Foresters.

A total of 975 Foresters, 875 Felines, and 150 Avians fill out their ballots as follows:

340 Voters

320 Voters

315 Voters

325 Voters

290 Voters

260 Voters

150 Voters

The election quota is:

The initial first-place totals are:

The Bear immediately crosses the 334 vote quota and claims the first council seat. The Bear’s 6 excess votes transfer to the Wolf:

After the transfer, no other candidate reaches quota. The Eagle, having the fewest votes, is eliminated, and the 150 Avian votes transfer to the Lion:

The 150 votes from the Avians put the Lion at 475 votes, so the Lion claims the second council seat. The Lion’s 141 excess votes transfer to the Tiger:

The Lion’s transferred 141 votes put the Tiger at 431 votes, so the Tiger claims the third council seat and transfers 97 excess votes to the Cheetah:

With 357 votes, the Cheetah claims the 4th council seat, giving the Felines majority control of the council. Because no Foresters appeared on any of the ballots that elected the Cheetah, the remaining 31 votes become exhausted. With one seat remaining and 2 candidates left, neither candidate has over 334 votes, so the Fox is eliminated and his 310 votes are transferred to the Wolf:

The Wolf is then elected to the final seat with 630 votes, which is 296 votes over the threshold. The final 5-member executive council becomes:

Bear

Wolf

Lion

Tiger

Cheetah

Giving the Felines a majority of the Executive Council. Both the Foresters and the Avians now have a legitimate gripe. The Eagle started with 150 votes, and the Foresters finished with 296 votes over the threshold, which combined would have been more than enough to elect the Eagle and create the best achievable outcome for both groups. But the Eagle was eliminated early, giving the Felines majority control of the council.

This creates an important strategic incentive in STV elections. In some situations, coalitions may need to think not only about their own candidates, but also about helping candidates in allied coalitions secure transfers to achieve their best possible outcome. This can become very difficult for coalitions to manage, and very complicated for voters to navigate.