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In electoral systems based on parties, it is easy to publish the results: it is sufficient to publish the number of votes each party received. These numbers contain all the information that is needed to compute the election outcome.

But in single transferable vote, each voter votes by ranking the candidates. The potential number of different votes is the factorial of the number of candidates, which may be enormous. It may be impossible to publish all these results.

Is there an efficient way to publish the results of STV elections, such that the published information can be used by observers to compute the election outcome?

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    In practice, if there are n candidates, at it takes r rounds to determine a winner, only npr categories are relevant, not n!. And I would expect them to just publish the vote totals at each round, not the full ballots. Sep 27, 2023 at 2:23
  • @Acccumulation what is "npr categories"? Sep 27, 2023 at 2:29
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    Read as "n pick r", computed as (n! / (n-r)!). See en.wikipedia.org/wiki/Permutation
    – AAM111
    Sep 27, 2023 at 3:57
  • @ErelSegal-Halevi I did a lot of revisions. I intended to write nPr, but it ended up npr. Sep 28, 2023 at 2:16
  • They usually publish the results for round one (first preference, all candidates), then round 2 (first preference is still in the running, otherwise second preference), and so on, don't they?
    – jcaron
    Sep 28, 2023 at 12:23

3 Answers 3

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Let's consider the classic example: Australia, which has used STV/IRV for more than a century.

For calculations of votes, we need to consider each electorate separately. Calculation of the global outcome (seats won etc.) is trivial.

It should be noted that in any voting system, tallying the final result given "the number of votes each party received" is a technical process and is rarely controversial. The process is more complicated for STV, but it's still pure math. Controversy may arise at the earlier stages: how exactly ballots are counted, what is considered informal (invalid) vote, how ballots are presented, etc. This is where most scrutiny is needed.

Generally, for the public, the results are presented already calculated and ranked. They still present the "number of votes" for each candidate, but this number is calculated in a less trivial manner than just adding up votes. Still, the process of calculation is also presented for those who are interested.

Here is an example from the most recent federal elections for the electorate of Adelaide. (This is the lower house election, which is IRV - a single winner version of STV. It is a bit easier to demonstrate, but the principle is the same. Note that Senate is still elected by candidates, not parties; just there are multiple winners in the last round and the threshold is lower than 50%).

The (abbreviated) "final" table looks rather "normal":

Candidate Votes %
GRANTHAM, Amy 42,948 38.09
GEORGANAS, Steve 69,816 61.91

The catch is that there are only two candidates shown out of 7! This is because other candidates were eliminated in the STV counting process, and their votes were redistributed to these candidates. This is essentially the result of the last round of counting. In Australia, this is often called Two party preferred or Two candidate preferred result.

Next thing presented (and often publicised) is the First preference count for each candidate. This is the exact number of ballots which gave first preference ("1") to this candidate, like if it were "traditional" elections. We can see that for these two candidates, the count is different and much lower:

Candidate Votes %
GRANTHAM, Amy 36,080 32.00
GEORGANAS, Steve 45,086 39.98

In STV, this first preference count has largely psychological/symbolic value. In this example, we can see that a third of votes for the ultimate winner Steve came from other parties, whereas Amy had much higher share of "personal" votes. Of course, it is a well-known feature of STV that the final winner may not be the one who had the most first-preference votes.

Nevertheless, there is an "implementation detail" in Australia that parties/candidates who get enough first-preference votes (a fairly small percentage) get certain monetary compensation for the election expenses. This is to avoid a flood of spurious candidates who might register just "for fun".

The next thing one may wish to examine is the Full distribution of preferences table on the same page. There, each round of counting is presented, starting from first preference and culminating in the two candidate preferred result. It is clearly observable how, for example, the 870 votes from the first-eliminated candidate are distributed to the remaining 6 candidates in the second round, and so on.

Here, by the way, is where Senate election with its multi-winner count gets more complicated. There, if a candidate gets more votes than the quota requires (at any round), the "excess" votes are taken away and redistributed to other candidates with the proportional weight. As a result, each elected candidate gets exactly the same number of votes in the final round, and everyone else gets 0! This is not required for IRV as such process will not change the ultimate winner. The calculation is also published, but it is a multi-page PDF document (example of a smaller one for Tasmania - 30 pages).

For those who want to scrutinise further, AEC publishes the results in machine-readable formats as well.

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    @ErelSegal-Halevi Looks like for the senate they do however publish individual votes. Sep 27, 2023 at 4:59
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    @quarague, well, I was thinking with respect to the first preference votes: final > 1.5*first. But as worded, you are right, I'll fix that. My point though was simply that the "preference share" of the final count is larger for the winner (in this specific case) than for the runner-up; some intuitively feel this is "cheating", but this is a feature, not a bug.
    – Zeus
    Sep 27, 2023 at 7:35
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    @GregoryCurrie That’s a function of how the votes are counted: for the House of Representatives the votes are counted by hand (essentially by sorting them into piles based on the current preference state). The Senate ballot papers are scanned into a computer which computes the preference flow, so they have every single vote available in a digital form.
    – Ben Murphy
    Sep 27, 2023 at 9:38
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    Publishing all the individual votes helps serves an election integrity function, as it means nobody has to trust that the election software carried out the STV process correctly, as anyone can recompute the results themselves based on the individual votes. You, of course, still have to trust that the reported individual votes accurately represent the valid votes cast, but that's the same concern you'd have in any non-STV election, and there are various audit and public inspection procedures used in all types of elections to provide assurance of that. Sep 28, 2023 at 8:24
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    @ZachLipton interestingly, it's not clear that "trust that the election software carried out the STV process correctly" is actually a concern for the AEC. There was a court case a few years back to try and get the code's source made available for scrutiny, but the AEC fought against it hard, and actually got the plaintiff declared a "vexatious applicant". (search: "mjec EasyCount" for details)
    – Jim Cullen
    Sep 28, 2023 at 23:50
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Adding on further context to Zeus's very good answer:

In Australia, where the Senate results are done using STV on a per-state level, voters are allowed to vote for a party as a whole, rather than being required to vote for each candidate individually. This is called "above-the-line" voting. The vast majority of voters choose this path, though the option still exists to number each candidate individually ("below-the-line").

The headline figure seen most often is the number of first preference votes cast above-the-line, or the sum of the party's above-the-line votes and the votes given to each candidate below-the-line. See this page for the Queensland results of the last election for an example.

In reality, to make them more consumable, these raw vote counts/percentages are often converted into "quotas", representing the fractions of a seat that have been won by that party.

Group Candidate Party Votes % Quotas
J Above-the-line Votes Queensland Greens 345,470 11.46 0.8024
ALLMAN-PAYNE, Penny Queensland Greens 21,671 0.72 0.0503
SRI, Anna Queensland Greens 1,718 0.06 0.0040
Group Total Queensland Greens 373,460 12.39 0.8674
S Above-the-line Votes Liberal National Party of Queensland 1,013,227 33.62 2.3533
McGRATH, James Liberal National Party of Queensland 21,411 0.71 0.0497
CANAVAN, Matt Liberal National Party of Queensland 11,822 0.39 0.0275
STOKER, Amanda Liberal National Party of Queensland 11,981 0.40 0.0278
Group Total Liberal National Party of Queensland 1,061,638 35.23 2.4658
Y Above-the-line Votes Australian Labor Party 713,051 23.66 1.6561
WATT, Murray Australian Labor Party 21,847 0.72 0.0507
CHISHOLM, Anthony Australian Labor Party 2,183 0.07 0.0051
Group Total Australian Labor Party 744,212 24.69 1.7285

Note that the "group total" counts are indicative only, and not of any formal value in the ballot count.

From this table you can see that Penny Allman-Payne received 0.85 quotas on first preferences (above the line group J votes give first preference to her), meaning she needs to receive 0.15 more quotas in subsequent round to get elected. James McGrath received 2.4, so he passes 1.4 quotas down to second preferences, at least 1.35 of which will definitely go to Matt Canavan (because above-the-line group S votes give second preference to Canavan). So Canavan receives at least 1.37, and passes at least 0.35 to Amanda Stoker, who therefore is guaranteed at least 0.37 quotas.

For most people, this high-level count is sufficient. Guesses at how many total quotas a candidate will receive can be made by an understanding of party preferences. For example, "Animal Justice Party" above-the-line voters are likely to preference the Greens, while "One Nation" voters probably preference the LNP. So Greens' total quotas can be estimated by adding their own first preference vote to the vote of AJP, as long as you assume AJP candidates will not win a seat, and their voters will not preference a different party which does win a seat first.

For those who want to fully validate the election results, a full list of every formal vote is published. So is a summary of the distributions of preferences at each round of the count. You can see the links to these for the 2022 election here. You can also get counts of things like how many votes were informal (often referred to as "spoilt" ballots, i.e. ballots which were not filled out in a way that allows them to be counted).

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In the case of New York City's latest mayoral primary the data was made available in the form of a collection of CSV files (click on 'Download CVR data'). There's no metadata or readme accompanying it, but Alec Barrett has published an 'unofficial' readme on GitHub. This data was previously used on Politics.SE to calculate the Condorcet winner of that primary.

I suspect that the same method of releasing data is used in all jurisdictions where STV is used.

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    Might be worth looking at Alaska data as they use ranked choice as well if my memory is right.
    – Joe W
    Sep 26, 2023 at 23:15
  • Interesting, thanks! Does it mean that every individual vote is published? Sep 27, 2023 at 2:28
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    @ErelSegal-Halevi yes every single vote is published. Sep 27, 2023 at 3:01

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