Voting methods part 2: RCV

Borda count: Example 1

Going back to the example, we’d have a situation that looks like this:

This may seem a bit odd. The three columns at the right denote the three unique rankings of candidates that were cast. There are three more possible orders of the preference ranking, which I’ll consider in a second example. The numbers at the top represent the proportion of voters who cast a ballot with that order.

For now, I’ve just extended the votes so that:

• All of Calvin’s voters prefer Hobbes to Suzie
• All of Hobbes’ voters prefer Suzie to Calvin.
• All of Suzie’ voters prefer Hobbes to Calvin.

We can see this as a split vote on one side, creating a “spoiler effect.”

Just by inspection, we can kind of see that Hobbes is a more broadly popular candidate, right? He’s ranked 1st or 2nd, while Calvin has only 1st and 3rd place votes. Applying the Borda count would be a weighted average of the points (1st = 3 points, 2nd = 2 points, 3rd = 1 point), using the proportion of votes at each rank as the weight. For example, to get Suzie’s borda count, we took a weighted average of the ranks:

\[ BC_{S} = (0.40 \times 1) + (0.35 \times 2) + (0.25 \times 3) \] Applying the same to all of the candidates, we would get the following results:

And so Hobbes would be declared the winner of the election.

Borda count: Example 2

Going to a slightly more complex ballot, let’s say that Calvin and Suzie are the “extreme” candidates, while Hobbes is a more centrist candidate. So Hobbes’ voters second choice are a bit more split, but Suzie’s and Calvin’s would primarily go to Hobbes.

Then applying the Borda count, we would get:

So while Calvin is now a bit more popular than Suzie, Hobbes still wins the election.

Borda count: Example 3

One more example, just because I wrote a function to turn these ballots into a borda count, and I want to get my use out of it, let’s say that Hobbes is actually in last place when considering strictly first-place votes. To put that in real terms: People are voting for the “extreme” candidates of their party (Calvin with his G.R.O.S.S. club, Suzie with her tea parties), but have a more moderate/centrist as a backup choice (Hobbes just wants a tuna sandwich for everyone).

So here, only 10% of voters put Hobbes first, while 90% voted for their preferred “extreme” candidate. But when we tally up the Borda count, we get:

So Hobbes still wins the day, because enough votes from either side thought he was an acceptable backup.

Using a Borda count isn’t the only way to tally ranked ballots, but it’s an example of how to better represent the desires of the voters compared to plurality.

STV: Example

Let’s say that a district needs 5 representatives - so each represents 20% of the vote - and the vote breakdown is shown below.

Clearly Blue and Gold get a representative, right? But then how are the other three representatives allocated? Nobody got 40%, so there’s not really an argument that Blue or Gold should get two representatives (yet). One way of going forward is to eliminate take the party that got the least votes and allocate them to their second choice. In this case, let’s say that Light Blue voters would have otherwise voted for Blue. So the new picture looks like:

Now with 40% of the vote, it seems reasonable to give Blue two of the five representatives. But we still have two more, with Green and Purple being tied for last place. If they both would have opted for Gold otherwise, then we would add them to Gold’s tally, as shown below.

Now it seems rational to give Blue two representatives, and give Gold three representatives, thus proportionally allocating the representatives according to the actual vote of the people.

Now, you might say that if Green and Purple are “close” to each other, that they might instead group together and get a representative that’s more aligned with them rather than someone from Gold. That’s true. What I showed here is just one possible way to go about allocating representatives, and it’s knowingly a bit simplistic just to illustrate the point. There are more sophisticated methods for proportional allocation and STV which can help with this.

It’s also rather unlikely that vote tallies will wind up being so nicely aligned with whatever threshold is needed to claim one of the representatives, so there will be some degree of mismatch between the vote and the allocation of representatives. Generally speaking, the more representatives in a region (e.g., State), the more able the system is to reflect the makeup of the people.